AN IEEE PRESS CLASSIC REISSUE
ELECTROMAGNETIC
THEORY
BY
JULIUS ADAMS STRATTON
Profeccor of Physics
Mu rsuchusetis Institute of Technology
IEEE Anfennas and Propagation Society, Sponsor
IEEE Press Series on Electromagnetic Wave Theory
Donald G. Dudley, Series Editor
IEEE
IEEE PRESS
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CI*IL11*I*L
" I c I x I I * * I a L
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FOREWORD TO THE REISSUED EDITION
The purpose of the IEEE Press Series on Electromagnetic Wave Theory is to publish books of long-term archival significance in electromagnetics. Included are new
titles as well as reissues and revisions of recognized classics. The book Electromagnetic Theory, by J. A. Stratton is one such classic.
Originally published in 194I , Stratton’s book has formed an integral part of the
electromagnetic education of both physics and electrical engineering graduate students for over sixty years. In addition, virtually every electromagnetic researcher I
know has a copy in hidher library. Unfortunately, the book is out of print. It is our
purpose to rectify this situation.
When I consult my copy of this classic book, I never cease to be amazed at its
timeliness. The equivalence principle is strongly rooted in the Stratton-Chu inversion of the vector wave equations, contained on pages 464-470. The Hansen vectors
are introduced and exploited in Chapter VII. The Sommerfeld problem is discussed
in Section 9.28. Natural modes along circular cylinders embedded in a lossy medium are developed in Section 9.15. I could go on and on. Suffice it to say that the development of the theory and the inclusion of examples remain of current interest today.
Professor Stratton had an amazing career, culminating with his succession to the
presidency of the Massachusetts Institute of Technology (MIT). Another former
president, Professor Paul E. Gray, has kindly agreed to write an introduction to this
reissued edition, highlighting Stratton’s achievements. I am indebted to Professor
Stratton’s surviving spouse, Kay Stratton, who kindly agreed to our reissue plans
and then assisted in making the project possible. In addition, I should like to thank
Carol Fleishauer, Associate Director for Collection Services of the MIT Libraries
who made available from their archives a pristine copy of the book for our use in
the reissue.
Over the past twelve years, I have received many requests worldwide to reissue
this book. It is with great pleasure that I welcome it to the IEEE Press Series on
Electromagnetic Wave Theory.
DONALD
G . DUDLEY
University of Arizona
V
Professor Julius Adams Stratton. Courtesy MIT Museum.
INTRODUCTION
Julius Adams Stratton served the Massachusetts Institute of Technology, The Radiation Laboratory at MIT, the Federal Government, the National Academies of Science and Engineering and the Ford Foundation during his long and productive life.
His work at MIT, both as a member of the faculty, and as provost and subsequently
president, was important in the development of both research and education during
intervals of rapid growth and change in the Institute.
Stratton was born on May 18, 1901 in Seattle, Washington. His father, Julius A.
Stratton, was an attorney who founded a law firm well known and respected
throughout the Northwest. He also served as a judge. His mother, Laura Adams
Stratton, was an accomplished pianist. Following his father’s retirement, the family
moved to Germany in 1906 where Stratton attended school through age nine and
developed fluency in the German language. The family returned to Seattle in 1910
where he completed his public school education.
Stratton came to MIT, with which he was associated for 74 years, as the result of
an accident at sea and the advice of a student friend. He had an early interest in finding out how things worked and in building things, particularly those that involved
electricity. In high school he became very interested in radio in those early days of
spark-gap transmitters and galena crystal detectors. These interests, coupled with
the shutdown of amateur radio operations during World War I and the desire to
serve the nation, led him to study and qualify as a commercial radio operator-second grade-and to sign on during summer vacations as a shipboard radio operator.
Stratton had been admitted to Stanford for matriculation in September 1919, and
signed on for that summer as radio operator on the SS Western Glen out of Seattle
for a trip to Japan and Manchuria. The ship encountered a typhoon near Kobe,
Japan and participated in the rescue of another American ship in distress. Also, it
experienced an engine failure at the start of the return voyage that required a return
to port in Japan. These accidents made him late in returning to the US-too late to
enroll at Stanford that year. As an alternative, he succeeded in enrolling late at the
University of Washington in Seattle. During that year, in which he pursued his interests in electricity and mathematics, a conversation with a UW classmate persuaded him to apply for transfer to MIT, where he was admitted in 1920. He traveled to
the Institute as a radio operator-this time first grade--on the SS Eastern Pilot via
Balboa, Panama and New York City, arriving in August 1920, a week before the
start of classes.
At MIT Stratton enrolled in the Electrical Communications; Telegraph, Telephone and Radio option of the Department of Electrical Engineering, which he described in a letter home as “. . . far more interesting than that of ordinary dynamoelectric machinery. Line telegraphy and telephony involve some of the most
complex mathematics known.” He received his SB Degree in June 1923 with a thesis entitled “The Absolute Calibration of Wavemeters.” The equipment he developed generated harmonics up to 30 megahertz from a one kilohertz tuning fork.
During his senior year, Stratton determined to continue his studies in Europe. He
vii
...
Vlll
INTRODliC‘TlON
traveled to Paris via Cherbourg (this time as a passenger), with the dual goals of
continuing his engineering studies and becoming fluent in the French language.
During the ’23-‘24 year he traveled to and studied in Nancy, Grenoble, Toulouse
and Italy returning to the US in August 1924.
From September 1924 through June 1926 he was, as a research assistant in communications, enrolled at MIT in a master’s degree program, graduating with a thesis
entitled “A High Frequency Bridge.”
Upon completion of his graduate studies at MIT Stratton received a traveling fellowship that enabled him to again return to Europe, where many universities
seethed with excitement about quantum theory and atomic structure. He enrolled
for a doctor of science degree at the Eidgenossische Technische Hochschule (ETH)
in Zurich, Switzerland where he studied with Debye, graduating in March 1928
with a thesis entitled “Streuungskoeffzient von Wasserstoft nach der Wellenmechanic” (The Scattering Coefficient of Hydrogen According to Wave Mechanics).
He was invited to return to MIT as an assistant professor in electrical engineeringa modern physicist embedded in the engineering department. This appointment
marked the beginning of 38 years of continuous active association with the Institute.
Stratton’s desire to see the world was obviously very strong. During the summers he traveled to Africa, the Yukon and to Ecuador. His interest in other cultures
and other nations was deep.
On June 14, 1935, in Saint Paul’s Chapel at Ivy Depot, Virginia, Julius Adams
Stratton and Catherine Nelson Coffman were married. From this fortunate marriage
came three daughters: Catherine, Cary, and Laura. Their mother, known to all as
Kay, is active in the MIT community, as a founding member of the Council for the
Arts and as the guiding force behind two annual panel discussions: one each fall in
a topic of current interest; and one each spring on some aspect of ageing gracefully.
His experiences at the ETH changed Stratton’s career interests and further developed his passion for mathematics and physics. As he put it “In the years 1923-1924
I was thinking of a doctorate in literature or philosophy. This was to be the subject:
The Influence of Science on 19th Century French Literature. I decided to go into
pure physics. The years 1925 through 1928 changed my mind.” In 1930 his appointment was moved to the Department of Physics.
Karl Taylor Compton, the newly arrived president, set out to strengthen the sciences at MIT generally, and to give greater emphasis to modern physics. Stratton
became part of that transformation. He was promoted to professor in 1941, the same
year this book, Electromugnetic Theory, was published by McGraw Hill. His book,
although long out of print, is still widely used and referenced by 21st century writers. The decision of the IEEE to republish it as part of their series of electrical engineering classics will be much appreciated by engineers
Much of his research in the 1930s was done at the Round Hill Experiment Station in South Dartmouth, Massachusetts. Stratton’s research there involved the
propagation of very short radio waves and light through rain and fog. He also studied the possibility of using intense electromagnetic radiation to disperse fog, and
made measurements of the field of an antenna over the open sea, employing the
INTRODUC’TION
ix
Mqjlower, a dirigible loaned to him by the Goodyear Zeppelin Company. He prepared and published, through the National Academy of Sciences, tables of spheroidal functions-solutions of differential equations that arose in his studies of antennas. Between 1927 and 1942 Stratton published eleven technical papers in
refereed journals.
The German invasion of Poland in 1939 changed everything for Stratton. Prior
to the start of the war, British scientists employed high-frequency radio waves (ca
100 megahertz) as an early form of radar by exploiting reflections from aircraft.
They realized that higher frequencies would both enable much smaller antennas and
yield greater precision in target location, but were unable to generate radiation of
sufficient intensity at frequencies in the gigahertz range. Their invention of the microwave magnetron, in the months before the war started, enabled the creation of
radar systems that would be of much greater effectiveness in the war. Unsure that
they would be able to pursue the essential development in Britain under wartime
conditions, they sent the Tizard Mission with the magnetron and its developers to
the US, where American engineers and scientists could develop militarily useful
microwave radar systems quickly.
The Federal government established the Radiation Laboratory at MIT in October
1940 and Stratton was one of many who took on the tasks of making microwave
radar useful to the military on land and on sea as well as in the air. He was a natural
for this work given his understanding of electromagnetic radiation and the applications of Maxwell’s equations. He was appointed in November 1940 as a volunteer
consultant to the Microwave Section of Division D of the National Defense Research Council, and seconded by MIT to the Rad Lab (as it was universally called).
In August 1942 Stratton was appointed Expert Consultant to the Secretary of
War, Henry G. Stimson. He served in that capacity until December 1945. In this
role he made frequent visits to the theaters of war. In October 1942, as a member of
a committee investigating communication problems in the North Atlantic he traveled by air to England with extended stops in Presque Isle, Labrador, Greenland and
Iceland. In 1943, soon after the Allied invasion of North Africa, he traveled to Algiers, Tunis, Italy, and London to assess radar utility and communications effectiveness.
Robert Buderi wrote in The Invention that Changed the World (Touchstone,
1997): “The Atomic Bomb only ended the war. Radar won it.” Stratton was a very
significant part of that critically important development.
In August 1945 the Office of Scientific Research and Development, that had
overseen all of the laboratories created to aid the war effort, was shut down, and the
Radiation Laboratory was told to wind up its affairs. On January 1, 1946 Stratton
took over administration of the Division of Basic Research of the Lab, which had
been created in August 1945 following the surrender of the Japanese government.
At the suggestion of John Slater, head of the physics department, he named the Division the MIT Research Laboratory of Electronics (RLE).
Research support in the early days of RLE came from the Department of Defense through a multi-services contract, much of it in the form of a block grant at
the level of $600,000 per year. RLE was “responsible for extending the useful range
of the electromagnetic spectnim . . . to shorter wavelengths, approaching ultimately
that of infra-red.”
Title to the temporary buildings in the heart of the MIT campus that housed the
Rad Lab and to all the equipment contained therein was transferred to RLE in July
1946. The largest of those “temporary buildings” became “Building 20.” Until its
demise in 1998 at age 55, is was cherished prime research space at the Institute.
Stratton, as the founding director, led RLE during its formative years as the nation’s first university-based interdepartmental research laboratory. Its history, now
spanning more than sixty years of scientific and engineering accomplishments, has
its origins in Stratton’s vision and leadership.
In 1949 James Rhyne Killian, Jr. succeeded Karl Taylor Compton as president of
the Institute, and Killian appointed Stratton as the Institute’s first Provost.
The fifties and sixties were years of extraordinary growth at MIT. Vannevar
Bush’s landmark report, Science, the Endless Frontier led to the creation of the National Science Foundation and the National Institutes of Health. Cold War tensions,
much increased in 1957 by the Soviet’s launch of Sputnik, caused enrollments in
engineering and science to grow rapidly throughout this period. The Federal Government greatly expanded financial support for research and for students in the
fields of science and engineering. Charles Stark Draper’s Instrumentation Laboratory (later to become the independent Draper Laboratory) expanded greatly to add the
Apollo mission to its development of inertial navigation systems for the military
services. The compound annual growth rate of sponsored research at the Institute
was in double digits until 1969.
Stratton was primarily responsible for management of the physical and intellectual growth of MIT in these decades and for the thoughtful development and implementation of necessary structures, policies and procedures that became the foundations of, and the models for, the modem MIT. During his time as provost, two new
schools were created, the School of Humanities and Social Sciences in 1950 and the
Sloan School of Management in 1952.
In 1957 Killian was called to Washington to serve as President Eisenhower’s
Science Advisor, and Stratton, who had been appointed Chancellor in 1956, became
acting president and an ex oficio member of the governing board. In 1958 he was
elected eleventh president of the Institute. His presidency was a time of physical expansion for MIT. New buildings for Chemistry, Earth Sciences, Biology and the
Center for Materials Science and Engineering filled former parking lots. McCormick Hall, the first dormitory for undergraduate women, was completed in
1965. This was the crucial first step in increasing the number of women at MIT,
who now comprise 45% of the undergraduate student body. Stratton was liked and
respected by MIT students, who suggested that the new student center, completed
in 1966 be named for him.
As founding director of RLE, as provost, and as chancellor and president, Jay
Stratton deserves, along with Compton and Killian, a large share of credit for the
transformation of MIT from the premier school of engineering to a modern research
university.
Stratton was elected a member of the National Academy of Sciences (NAS) in
1950. Although there was a section of the NAS for distinguished engineers, there
INTRODUL‘T/ON
xi
were very few such members. The national engineering societies affiliated with the
Engineers’ Joint Council suggested in 1963 that a new organization to be called the
National Academy of Engineering (NAE) be created. Stratton, who was at the time
a member of the NAS Council, chaired a committee that worked through the complex issues of the idea and that led to the creation in 1964 of thc NAE as an affiliate
of the NAS. The two academies, joined some years later by the Institute of Medicine, comprise the premier organization for responding to questions or requests for
studies that come from the Federal government. Stratton’s clarity of vision and persuasiveness were very important in shaping the expansion of the enterprise, which
is now commonly referred to simply as “The National Academies.”
Following his retirement from MIT in 1966, Stratton accepted appointment as
Chairman of the Board of the Ford Foundation in New York City. At that time the
Ford Foundation was the nation’s largest grant-making charitable foundation. Stratton’s interest in MIT affairs continucd during his time in New York. Hc remained a
member of the Institute’s governing board and served on several of its committees.
During his years at the Ford Foundation Stratton accepted presidential appointment in 1967 as Chairman of the Committee on Marine Sciences, Engineering and
Resources (COMSER). The charge to the committee was, in part, to recommend
“National Policy to develop, encourage and maintain a coordinated, comprchensive, and long range program in marine science for the benefit of mankind . . . expanding scientific knowledge of the marine environment and of developing an
ocean engineering capability to accelerate exploration and development of marine
resources. . . .”
COMSER’s report: Our Nation and the Seu. A Plan for Nutional Action was presented to a different president and published two years later in January 1969. An
outcome of the study was the creation of NOAA, the National Oceanic and Atmospheric Agency.
Stratton returned to MIT in 1971 full time when his term as chairman of the Ford
Foundation ended. His affection and concern for the university that was integral to
his professional life for more than 50 years was undiminished by his years in New
York, and his renewed engagement in the life of MIT was immediately evident.
Jay Stratton died on June 22, 1994. During the last twenty years of his life, he
had the opportunity to see up-close the transformation from premier engineering
school to important world-class research university that he, together with Compton
and Killian, had commenced sixty years earlier.
In the preface to Electromugnetic Theory, Stratton noted that his wife Catherine
had assisted him in its preparation by proofreading the galleys with him. In the
preparation of this edition she has worked enthusiastically and closely with the
IEEE to enable the book’s republication.
PAULE. GRAY
Professor of Electrical Engineering
President Emeritus
MIT
October 2006
PREFACE
The pattern set nearly 70 years ago by Maxwell’s Treatise on Electricity and Magnetism has had a dominant influence on almost every subsequent English and American text, persisting to the present day. The
Treatise was undertaken with the intention of presenting a connected
account of the entire known body of electric and magnetic phenomena
from the single point of view of Faraday. Thus it contained little or
no mention of the hypotheses put forward on the Continent in earlier
years by Riemann, Weber, Kirchhoff, Helmholtz, and others. It is
by no means clear that the complete abandonment of these older theories
was fortunate for the later development of physics. So far as the
purpose of the Treatise was to disseminate the ideas of Faraday, i t was
undoubtedly fulfilled; as an exposition of the author’s own contributions,
i t proved less successful. By and large, the theories and doctrines
peculiar t o Maxwell-the concept of displacement current, the identity
there in scarcely
of light and electromagnetic vibrations-appeared
greater completeness and perhaps in a less attractive form than in the
original memoirs. We find that all of the first volume and a large part
of the second deal with the stationary state. In fact only a dozen pages
are devoted to the general equations of the electromagnetic field, 18 to
the propagation of plane waves and the electromagnetic theory of light,
and a score more t o magnetooptics, all out of a total of 1,000. The
mathematical completeness of potential theory and the practical utility of
circuit theory have influenced English and American writers in very
nearly the same proportion since that day. Only the original and
solitary genius of Heaviside succeeded in breaking away from this course.
For an exploration of the fundamental content of Maxwell’s equations
one must turn again to the Continent. There the work of Hertz, Poincar6, Lorentz, Abraham, and Sommerfeld, together with their associates
and successors, has led t o a vastly deeper understanding of physical
phenomena and to industrial developments of tremendous proportions.
The present volume attempts a more adequate treatment of variable
electromagnetic fields and the theory of wave propagation. Some attention is given to the stationary state, but for the purpose of introducing
fundamental concepts under simple conditions, and always with a view
to later application in the general case. The reader must possess a
general knowledge of electricity and magnetism such as may be acquired
from an elementary course based on the experimental laws of Coulomb,
...
Xlll
xiv
PREFACE
AmpBre, and Faraday, followed by an intermediate course dealing with
the more general properties of circuits, with thermionic and electronic
devices, and with the elements of electromagnetic machinery, terminating in a formulation of Maxwell’s equations. This book takes up
a t that point. The first chapter contains a general statement of the
equations governing fields and potentials, a review of the theory of units,
reference material on curvilinear coordinate systems and the elements of
tensor analysis, concluding with a formulation of the field equations in
a space-time continuum. The second chapter is also general in character, and much of i t may be omitted on a first reading. Here one will
find a discnssion of fundamental field properties that may be deduced
without reference to particular coordinate systems. A dimensional
analysis of Maxwell’s equations leads to basic definitions of the vectors
E and B, and an investigation of the energy relations results in expressions for the mechanical force exerted on elements of charge, current, and
neutral matter. I n this way a direct connection is established between
observable forces and the vectors employed t o describe the structure of a
field.
In Chaps. 111 and IV stationary fields are treated as particular cases
of the dynamic field equations. The subject of wave propagation is
taken up first in Chap. V, which deals with homogeneous plane waves.
Particular attention is given to the methods of harmonic analysis, and
the problem of dispersion is considered in some detail. Chapters VI and
VII treat the propagation of cylindrical and spherical waves in unbounded
spaces. A necessary amount of auxiliary material on Bessel functions
and spherical harmonics is provided, and consideration is given to vector
solutions of the wave equation. The relation of the field to its source,
the general theory of radiation, and the outlines of the Kirchhoff-Huygens
diffraction theory are discussed in Chap. VIII.
Finally, in Chap. IX, we investigate the effect of plane, cylindrical,
and spherical surfaces on the propagation of electromagnetic fields.
This chapter illustrates, in fact, the application of the general theory
established earlier to problems of practical interest. The reader will
find here the more important laws of physical optics, the basic theory
governing the propagation of waves along cylindrical conductors, a
discussion of cavity oscillations, and an outline of the theory of wave
propagation over the earth’s surface.
It is regrettable that numerical solutions of special examples could
not be given more frequently and in greater detail. Unfortunately the
demands on space in a book covering such a broad field made this impractical. The primary objective of the book is a sound exposition of
electromagnetic theory, and examples have been chosen with a view t o
illustrating its principles. No pretense is made of an exhaustive treat-
I’REFA CL
xv
ment of antenna design, transmission-line characteristics, or similar
topics of engineering importance. It is the author’s hopc that, the
present volume will provide the fundamental background necessary for
a critical appreciation of original contributions in special fields and satisfy
the needs of those who are unwilling t o accept engineering formulas
without knowledge of their origin and limitations.
Each chapter, with the exception of the first two, is followed by a
set of problems. There is only one satisfactory way t o study a theory,
and that is by application t o specific examples. The problems have been
chosen with this in mind, but they cover also many topics which it was
necessary t o eliminate from the text. This is particularly true of the
later chapters. Answers or references are provided in most cases.
This book deals solely with large-scale phenomena. It is a sore
temptation t o extend the discussion to that fruitful field which Frenkel
terms the “quasi-microscopic state,” and to deal with the many beautiful
results of the classical electron theory of matter. I n the light of contemporary developments, anyone attempting such a program must soon
be overcome with misgivings. Although many laws of classical electrodynamics apply directly t o submicroscopic domains, one has no basis
of selection. The author is firmly convinced that the transition must be
made from quantum electrodynamics toward classical theory, rather
than in the reverse direction. Whatever form the equations of q u a n t u n
electrodynamics ultimately assume, their statistical average over large
numbers of atoms must lead t o Maxwell’s equations.
The m.k.s. system of units has been employed exclusively. There
is still the feeling among many physicists t h a t this system is being forced
upon them by a subversive group of engincers. Perhaps i t is, although
i t was Maxwell himself who first had the idea. At all events, i t is a good
system, easily learned, and one that avoids endless confusion in practical
applications. At the moment there appears t o be no doubt of its universal adoption in the near future. Help for the tories among us who
hold to the Gaussian system is offered on page 241.
I n contrast t o the stand taken on the m.k.s. system, the author
has no very strong convictions on the matter of rationalized units.
Rationalized units have been employed because Maxwell’s equations are
taken as the starting point rather than Coulomb’s law, and it seems
reasonable to make the point of departure as simple as possible. As a
result of this choice all equations dealing with energy or wave propagation
are free from the factor 47F. Such relations are becoming of far greater
practical importance than those expressing the potentials and field
vectors in terms of their sources.
The use of the time factor eciUL instead of e+wt is another point of
mild controversy. This has been done because the time factor is invar-
xvi
PREFACE
iably discarded, and it is somewhat more convenient t o retain the positive
exponent e+ikRfor a positive traveling wave. To reconcile any formula
with its engineering counterpart, one need only replace -i by +j.
The author has drawn upon many sources for his material and is
indebted to his colleagues in both the departments of physics and of
electrical engineering at the Massachusetts Institute of Technology.
Thanks are expressed particularly t o Professor M. F. Gardner whose
advice on the practical aspects of Laplace transform theory proved
invaluable, and to Dr. S. Silver who read with great care a part of the
manuscript. In conclusion the author takes this occasion to express his
sincere gratitude t o Catherine N. Stratton for her constant encouragement during the preparation of the manuscript and untiring aid in the
revision of proof.
JULIUSADAMSSTRATTON.
CAMBRIDQE,
MASS.,
January, 1941.
CONTENTS
PAGE
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
PREFACE
...
Xlll
CHAPTER I
T H E FIELD EQUATIONS
E ~ U A T K .I N. S. . . . . . . . . . . . . . . . . .
MAXWELL’S
1.1 The Field Vectors . . . . . . . . . . . . . . . . . . .
1.2 Charge and Current . . . . . . . . . . . . . . . . . . .
1.3 Divergence of the Field Vectors . . . . . . . . . . . .
1.4 Integral Form of the Field Equations . . . . . . . . . .
PROPERTIES
MACROSCOPIC
. .
OF MATTER
1.5 The Inductive Capacities t and p .
1.6 Electric and Magnetic Polarization
1.7 Conducting Media . . . . . . .
. . . . .
. . . . .
. . . . .
1
i
. . . . . .
2
6
. . . . . .
6
. . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . .
10
10
. . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . .
13
DIMENSIONS. . . . . . . . . . . . . . . . . . . . . . . . .
1.8 M.K.S. or Giorgi System . . . . . . . . . . . . . . . . . . . . .
UNITS AND
. . . . . . . . . . . . . . . .
THE ELECTROMAGNETIC
POTENTIALS
1.9 Vector and Scalar Potentials . . . . . . . . . . . . . . . . .
1.10 Conducting Media . . . . . . . . . . . . . . . . . . . . .
1.11 Hertz Vectors, or Polarization Potentials . . . . . . . . . .
1.12 Complex Field Vectors and Potentials . . . . . . . . . . .
. . . .
. . .
. . .
11
16
16
. . . .
23
23
26
28
32
BOUNDARY
CONDITIONS. . . . . . . . . . . . . . . . . . . . . . . . .
1.13 Discontinuities in the Field Vectors . . . . . . . . . . . . . . . . .
34
34
COORDINATE SYSTEMS . . . . . . . . . . . . . . . . . . .
1.14 Unitary and Reciprocal Vectors . . . . . . . . . . .
1.15 Differential Operators . . . . . . . . . . . . . . . . .
1.16 Orthogonal Systems . . . . . . . . . . . . . . . . . .
1.17 Field Equations in General Orthogonal Coordinates . . .
1.18 Properties of Some Elementary Systems . . . . . . .
. . . .
. . . . . . . 38
38
44
47
. . . . . .
. . . . . . . 50
. . . . . . . 51
. . . . . . .
. . . . . .
TENSORS
. . . . . . . . . . . . . . . . . . . . . . . . . . .
THEFIELD
1.19 Orthogonal Transformations and Their Invariants . . . . . . . . . .
1.20 Elements of Tensor Analysis . . . . . . . . . . . . . . . . . . . .
1.21 Space-time Symmetry of the Field Equations . . . . . . . . . . . .
1.22 The Lorentz Transformation . . . . . . . . . . . . . . . . . . . .
1.23 Transformation of the Field Vectors to Moving Systems . . . . . . . .
xvii
59
59
64
69
74
78
xviii
CON TENTS
PAon
CHAPTER I1
STRESS AND ENERGY
STRESS AND
ELASTICMEDIA . . . . . . . . . . . . . . . . . . .
83
2.1 Elastic Stress Tensor . . . . . . . . . . . . . . . . . . . . . .
2.2 Analysis of Strain . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Elastic Energy and the Relations of Stress to Strain . . . . . . . . .
83
87
93
STRAIN I N
FORCES
ON CHARGES A N D CURRENTS
ELECTROMAGNETIC
2.4 Definition of the Vectors E and B . . . . . . .
2.5 Electromagnetic Stress Tensor in Free Space . .
2.6 Electromagnetic Momentum . . . . . . . . .
96
96
. . . . . . . . . . . 97
. . . . . . . . . . 103
. . . . . . . . . .
. . . . . . . . . . .
ELECTROSTATIC
ENERGY. . . . . . . . . . . . . . . . . . . . . . . . .
2.7 Electrostatic Energy as a Function of Charge Density . . . . . . . . .
2.8 Electrostatic Energy as a Function of Field Intensity . . . . . . . . .
2 3 A Theorem on Vector Fields . . . . . . . . . . . . . . . . . . . .
2.10 Energy of a Dielectric Body in an Electrostatic Field . . . . . . . . .
2.11 Thomson’s Theorem . . . . . . . . . . . . . . . . . . . . . . .
2.12 Earnshaw’s Theorem . . . . . . . . . . . . . . . . . . . . . . .
2.13 Theorem on the Energy of Uncharged Conductors . . . . . . . . . .
MAGNETOSTATIC
ENERGY
. . . . . . . . . . . . . . . . .
2.14 Magnetic Energy of Stationary Currents . . . . . .
2.15 Magnetic Energy as a Function of Field Intensity . .
2.16 Ferromagnetic Materials . . . . . . . . . . . . . . .
2.17 Energy of a Magnetic Body in a Magnetostatic Field .
2.18 Potential Energy of a Permanent Magnet . . . . . .
104
104
107
111
112
114
116
117
. . . . . . . . 118
. . . . . . . . 118
. . . . . . . . 123
. . . . . . .
125
. . . . . . . . 126
. . . . . . . . 129
ENERGYFLOW
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
2.19 Poynting’s Theorem . . . . . . . . . . . . . . . . . . . . . . .
131
135
2.20 Complex Poynting Vector . . . . . . . . . . . . . . . . . . . . .
FIELD
. .
FORCESON A DIELECTRIC
IN A N ELECTROSTATIC
2.21 Body Forces in Fluids . . . . . . . . . . . . .
2.22 Body Forces in Solids . . . . . . . . . . . . .
2.23 The Stress Tensor . . . . . . . . . . . . . .
2.24 Surfaces of Discontinuity . . . . . . . . . . .
2.25 Electrostriction . . . . . . . . . . . . . . . .
2.26 Force on a Body Immersed in a Fluid . . . . . .
FORCES
IN THE MAGNETOSTATIC
FIELD
. . . . . . . .
. . . .
2.27 Nonferromagnetic Materials . . . . .
. . . .
2.28 Ferromagnetic Materials . . . . . . .
. . . .
FORCESIN THE ELECTROMAGNETIC
FIELD
2.29 Force on a Body Immersed in a Fluid . . . . . .
. . . . . . . . . . 137
. . . . . . . . . . 137
. . . . . . . . . 140
. . . . . . . . . .
146
. . . . . . . . . . 147
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
149
151
153
153
155
156
156
CHAPTER I11
T H E ELECTROSTATIC FIELD
PROPERTIES
OF AN ELECTROSTATIC
FIELD. . . . . . . . . . . . . 160
GENERAL
3.1 Equations of Field and Potential . . . . . . . . . . . . . . . . . . 160
3.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 163
( ‘ONTENTS
XIX
PAQE
CALCULATION
OF THE FIELD
FROM THE CHARGE
DISTRIBUTION
. .
3.3 Green’s Theorem . . . . . . . . . . . . . . . . . . . .
3.4 Integration of Poisson’s Equation . . . . . . . . . . . .
3.5 Behavior a t Infinity . . . . . . . . . . . . . . . . . . .
3.6 Coulomb Field . . . . . . . . . . . . . . . . . . . . .
3.7 Convergence of Integrals . . . . . . . . . . . . . . . .
. . . . . . 165
. . . . .
165
. . . . . . 166
. . . . .
167
. . . . .
169
. . . . .
170
OF THE POTENTIAL
IN SPHERICAL
HARMONICS
. . . . . . . . . . .
EXPANSION
3.8 Axial Distributions of Charge . . . . . . . . . . . . . . . . . . .
3.9 TheDipole . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.10 Axial Multipoles . . . . . . . . . . . . . . . . . . . . . . . . .
3.11 Arbitrary Distributions of Charge . . . . . . . . . . . . . . . . .
3.12 General Theory of Multipoles . . . . . . . . . . . . . . . . . . .
172
172
175
176
178
179
183
DIELECTRIC
POLARIZATION. . . . . . . . . . . . . . . . . . . . . . . .
3.13 Interpretation of the Vectors P and n . . . . . . . . . . . . . . . 183
OF INTEGRALS
OCCURRING
IN POTENTIAL
THEORY
. .
DISCONTINUITIES
3.14 Volume Distributions of Charge and Dipole Moment . . . . .
3.15 Single-layer Charge Distributions . . . . . . . . . . . . . .
3.16 Double-layer Distributions . . . . . . . . . . . . . . . . . .
3.17 Interpretation of Green’s Theorem . . . . . . . . . . . . .
3.18 Images . . . . . . . . . . . . . . . . . . . . . . . . . .
BOUNDARY-VALUE
PROBLEMS
. . . . . . . .
3.19 Formulation of Electrostatic Problems
3.20 Uniqueness of Solution . . . . . . .
3.21 Solution of Laplace’s Equation . . . .
. . . . 185
. . . . 185
. . . . 187
. . .
188
. . . . 192
. . .
193
. . . . . . . . . . . . . . . 194
. . . . . . . . . . . . . . . 194
. . . . . . . . . . . . . . .
. . . . . . . . . . . . . . .
PROBLEM
OF THE SPHERE
. . . . . . . . . . . . . . . . .
3.22 Conducting Sphere in Field of a Point Charge . . .
3.23 Dielectric Sphere in Field of a Point Charge . . . .
3.24 Sphere in a Parallel Field . . . . . . . . . . . . .
.
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.
.
.
.
.
196
197
.
201
. . 201
. . 204
.
205
PROBLEM
OF THE ELLIPSOID. . . . . . . . . . . . . . . . . . . . . . . 207
3.25 Free Charge on a Conducting E:llipsoid . . . . . . . . . . . . . . . 207
3.26 Conducting Ellipsoid in a Parallel Field . . . . . . . . . . .
3.27 Dielectric Ellipsoid in a Parallel Field . . . . . . . . . . .
3.28 Cavity Definitims of E and D . . . . . . . . . . . . . . . . .
3.29 Torque Exerted on an Ellipsoid . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . .
PROBLEMS
. . . . 209
. . . . 211
. . .
213
. . . . 215
. . . . 217
CHAPTER IV
T H E MAGNETOSTATIC FIELD
GENERAL
PROPERTIES
OF A MAGNETOSTATIC
FIELD. . . . . . . . . . . . . 225
4.1 Field Equations and the Vector Potential . . . . . . . . . . . . . . 225
226
4.2 Scalar Potential . . . . . . . . . . . . . . . . . . . . . . . . .
228
4.3 Poisson’s Analysis . . . . . . . . . . . . . . . . . . . . . . . .
CALCULATION
OF THE FIELD
OF A CURRENT
DISTRIBUTION
. . . . . . . . . . 230
4.4 Biot-Savart Law . . . . . . . . . . . . . . . . . . . . . . . . . 230
4.5 Expansion of the Vector Potential . . . . . . . . . . . . . . . 233
xx
CONTENTS
4.6 The Magnetic Dipole .
4.7 Magnetic Shells . . .
PAQ~
. . . . . . . . . . . . . . . . . . . . . .
236
. . . . . . . . . . . . . . . . . . . . . . 237
A DIGRESSION
O N UNITS AND DIMENSIONS
. . . . . . . . . . . . . . . . . 238
4.8 Fundamental Systems . . . . . . . . . . . . . . . . . . . . . . . 238
. . . . . . . . . . . . . . . .
POLARIZATION
. . . . . . . . . . . . . . . . . . . . . . . . .
MAGNETIC
4.9
Coulomb's Law for Magnetic Matter
241
242
4.10 Equivalent Current Distributions . . .
4.11 Field of Magnetized Rods and Spheres
. . . . . . . . . . . . . . . 242
. . . . . . . . . . . . . . . 243
DISCONTINUITIES
OF THE VECTORS
A AND B . . . . . . . . . . . . . . . . . 245
4.12 Surface Distributions of Current . . . . . . . . . . . . . . . . . . 245
4.13 Surface Distributions of Magnetic Moment . . . . . . . . . . . . . 247
INTEGRATION
OF THE EQUATION
v X v X A = pJ . . . . . . . . . . . . . 250
4.14 Vector Analogue of Green's Theorem . . . . . . . . . . . . . . . . 250
4.15 Application to the Vector Potential . . . . . . . . . . . . . . .' . . 250
BOUNDARY-VALUE
PROBLEMS
. . . . . . . . . . . . . . . . . . . . . . .
4.16 Formulation of the Magnetostatic Problem . . . . . . . . . . .
4.17 Uniqueness of Solution . . . . . . . . . . . . . . . . . . . . . .
. .
254
254
256
ELLIPSOID. . . . . . . . . . . . . . . . . . . . . . .
257
4.18 Field of a Uniformly Magnetized Ellipsoid . . . . . . . . . . . . . . 257
4.19 Magnetic Ellipsoid in a Parallel Field . . . . . . . . . . . . . . . . 258
PROBLEM O F THE
CYLINDER
IN A PARALLEL
FIELD. . . . . . . . . . . . . . . . . . . . .
4.20 Calculation of the Field . . . . . . . . . . . . . . . . . . . . . .
4.21 Force Exerted on the Cylinder . . . . . . . . . . . . . . . . . . .
PROBLEMS
. .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
258
258
261
262
CHAPTER V
PLANE WAVES I N UNBOUNDED. ISOTROPIC MEDIA
PROPAQATION
OF PLANEWAVES
. . . . . . . . . . . .
5.1 Equations of a One-dimensional Field . . . . .
5.2 Plane Waves Harmonic in Time . . . . . . .
5.3 Plane Waves Harmonic in Space . . . . . . .
5.4 Polarization .
5.5 Energy Flow .
5.6 Impedance . .
. . . . . . . . . .
268
. . . . . . . . . . . 268
. . . . . . . . . . . 273
. . . . . . . . . . . 278
. . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . .
279
281
282
GENERAL
SOLUTIONS
OF THE ONE-DIMENSIONAL
WAVEEQUATION
. . . . . . . 284
5.7 Elements of Fourier Analysis . . . . . . . . . . . . . . . . . . . .
285
5.8 General Solution of the One-dimensional Wave Equation in a Nondissips,
tive Medium . . . . . . . . . . . . . . . . . . . . . . . . . . .
292
5.9 Dissipative Medium; Prescribed Distribution in Time . . . . . . . . . 297
5.10 Dissipative Medium; Prescribed Distribution in Space . . . . . . . . . 301
5.11 Discussion of a Numerical Example . . . . . . . . . . . . . . . . . 304
5.12 Elementary Theory of the Laplace Transformation . . . . . . . . . . 309
5.13 Application of the Laplace Transformation to Maxwell's Equations . . . 318
xxi
PREFACE
PAGE!
DISPERSION. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.14 Dispersion in Dielectrics . . . . . . . . . . . . . . . . . . . . . .
5.15 Dispersion in Metals . . . . . . . . . . . . . . . . . . . . . . .
5.16 Propagation in an Ionized Atmosphcre . . . . . . . . . . . . . .
VELOCITIESOF PROPAGATION
. . . . . . . . . . . . . . . . . . . . . . .
.
321
321
325
327
330
5.17 Group Velocity . . . . . . . . . . . . . . .
5.18 Wave-front and Signal Velocities . . . . . .
. . . . . . . . . . .
330
. . . . . . . . . . . . 333
PROBLEMS
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
340
CHAPTER VI
CYLJ ND RIC AL WAVES
EQUATIONS
OF A CYLINDRICAL
FIELD
. . . . . . . . . . . . . . . . . . . .
349
6.1 Representation by Hertz Vectors . . . . . . . . . . . . . . . . . . 349
351
6.2 Scalar and Vector Potentials . . . . . . . . . . . . . . . . . . . .
6.3 Impedances of Harmonic Cylindrical Fields . . . . . . . . . . . . . 354
OF THE CIRCULAR
CYLINDER. . . . . . . . . . . . . . . 355
WAVEFUNCTIONS
6.4 Elementary Waves . . . . . . . . . . . . . . . . . . . . . . . . 355
6.5 Properties of the Functions Z n ( p ) . . . . . . . . . . . . . . . . . . 357
6.6 The Field of Circularly Cylindrical Wave Functions . . . . . . . . . . 360
INTEGRAL
REPRESENTATIONS
O F WAPEFUNCTIONS
. . . . . . . . . . . .
6.7 Construction from Plane Wave Solutions . . . . . . . . . . . . . .
6.8 Integral Representations of the Functions Z, ( p ) . . . . . . . . . .
6.9 Fourier-Bessel Integrals . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . .
6.10 Representation of a Plane Wavt
6.11 The Addition Theorem for Circularly Cylindrical Waves . . . . . . .
. 361
. 361
. 364
369
. 371
. 372
WAVE FUNCTION5 O F THE ELLIPTIC
CYLINDER . . . . . . . . . . . . . . . . 375
375
6.12 Elementary Waves . . . . . . . . . . . . . . . . . . . . . . . .
6.13 Integral Representations . . . . . . . . . . . . . . . . . . . . . .
6.14 Expansion of Plane and Circular Waves . . . . . . . . . . . . .
380
. . 384
PROBLEMS
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387
CHAPTER VII
SPHERICAL WAVES
THEVECTORWAVEEQUATION. . . . . . . . . . . . . . . . . . . . . . 392
7.1 A Fundamental Set of Solutions . . . . . . . . . . . . . . . . . . 392
7.2 Application to Cylindrical Coordinates . . . . . . . . . . . . . . . 395
THESCALAR
WAVEEQUATION
IN SPHERICAL
COORDINATTS
. . . . . . . . . . 399
7.3
7.4
7.5
7.6
7.7
7.8
7.9
7.10
Elementary Spherical Waves . . . . . . . . . . . . . . .
Properties of the Radial Functions . . . . . . . . . . .
Addition Theorem for the Legendre Polynomials . . . . .
Expansion of Plane Waves . . . . . . . . . . . . . . . .
Integral Representations . . . . . . . . . . . . . . . . .
A Fourier-Bessel Integral . . . . . . . . . . . . . . .
Expansion of a Cylindrical Wave Function . . . . . . . .
Addition Theorem for zo(kR). . . . . . . . . . . . . . .
. . . . . 399
. . . . 404
. . . . 406
. . .
408
. . .
409
. .
. .
. .
. .
. .
. .
. .
. . . . 411
. . . . 412
. . .
413
CON7FIV 73
XXll
PAQE
THEVECTOR
7.11
7.12
7.13
7.14
WAVE
EQUATION
I N SPHERICAL COORDINATES . . . . . . . . . . 414
Spherical Vector Wave Functions . . . . . . . . . . . . . . . . .
Integral Representations . . . . . . . . . . . . . . . . . . . . . .
Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . .
Expansion of a Vector Plane Wave . . . . . . . . . . . . . . . . .
PROBLEMS
. .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
414
416
417
418
420
CHAPTER VIII
RADIATION
THEINHOMOGENEOUS
SCALAR
WAVEEQUATION
. . . . . . . . . . . . . . . 424
8.1 Kirchhoff Method of Integration . . . . . . . . . . . . . . . . . . 424
428
8.2 Retarded Potentials . . . . . . . . . . . . . . . . . . . . . . . .
430
8.3 Retarded Hertz Vector . . . . . . . . . . . . . . . . . . . . . .
A MULTIPOLE
EXPANSION
. . . . . .
8.4 Definition of the Moments . . .
8.5 Electric Dipole . . . . . . . .
8.6 Magnetic Dipole . . . . . . .
. . . . . . . . . . . . . . . . . .
431
431
. . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 434
. . . . . . . . . . . . . . . . . .
RADIATION
THEORY
OF LINEAR
ANTENNA
SYSTEMS. . . .
8.7 Radiation Field of a Single Linear Oscillator . . . .
8.8 Radiation Due to Traveling Waves . . . . . . . .
8.9 Suppression of Alternate Phases . . . . . . . . .
8.10 Directional Arrays . . . . . . . . . . . . . . . .
8.11 Exact Calculation of the Field of a Linear Oscillator
8.12 Radiation Resistance by the E.M.F. Method . . . .
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. .
437
438
438
445
446
448
454
457
.
.
.
.
. . . . . . .
460
THEKIRCHHOFF-HUYGENS
PRINCIPLE
8.13 Scalar Wave Functions . . . . . . . . . . . . . . . . . . . . . .
460
8.14 Direct Integration of the Field Equations . . . . . . . . . . . . . . 464
8.15 Discontinuous Surface Distributions . . . . . . . . . . . . . . . . 468
470
O F THE RADIATION
PROBLEM . . . .
8.16 Integration of the Wave Equation . . . . . . . . . . . . . . . . . 470
8.17 Field of a Moving Point Charge . . . . . . . . . . . . . . . . . . 473
477
PROBLEMS
. . . . . . . . . . . . . . . . . . . . . . . . . . .
FOUR-DIMENSIONAL
FORMUL.4TION
CHAPTER I X
BOUNDARY-VALUE PROBLEMS
THEOREMS
. . . . . . . . .
GENERAL
9.1 Boundary Conditions . . . . .
9.2 Uniqueness of Solution . . . .
9.3 Electrodynamic Similitude . . .
. . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . .
REFLECTION
AND REFRACTION
AT A PLANE
SURFACE
. .
9.4 Snell’s Laws . . . . . . . . . . . . . . . . .
9.5 Fresnel’s Equations . . . . . . . . . . . . . .
9.6 Dielectric Media . . . . . . . . . . . . . .
9.7 Total Reflection . . . . . . . . . . . . . .
9.8 Refraction in a Conducting Medium . . . . .
9.9 Reflection a t a Conducting Surface . . . . . .
.
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483
483
486
488
. . . . . . . . 490
. . . . . . .
490
. . . . . . .
492
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. 494
. 497
. 500
. 505
Cob TEiL TS
XXlll
PAW
PLANESHEETS. . . . . . . . . . . . . . .
9.10 Reflection and Transmission Coefficients .
9.11 Application to Dielectric Media . . . .
9.12 Absorbing Layers . . . . . . . . . . .
. . . . . . . . . . . . .
511
. . . . . . . . . . . . . . 511
. . . . . . . . . . . . . . 513
. . . . . . . . . . . . . .515
SURFACE
WAVES. . . . . . . . .
9.13 Complex Angles of Incidence .
9.14 Skin Effect . . . . . . . .
516
518
520
PROPAGATION
A L O N G A CIRCULAR
CYLINDER
. . .
9.15 Natural Modes . . . . . . . . . . . .
9.16 Conductor Embeddcd in a Dielectric . .
9.17 Further Discussion of the Principal Wave .
9.18 Waves in Hollow Pipes . . . . . . . . .
COAXIAL
LINES. . . . . . . . . . . . .
9.19 Propagation Constant . . . . . . .
9.20 Infinite Conductivity . . . . . . .
9.21 Finite Conductivity . . . . . . . .
. . . . . . . . . . . . 524
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. 524
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. 527
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. 531
. . . . . . . . . . . .
537
. . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . .
545
545
548
551
.
.
.
.
.
.
.
. . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . .
OF A SPHERE
. . . . . . . . . . . . .
OSCILLATIONS
9.22 Natural Modes . . . . . . . . . . . . . . .
9.23 Oscillations of a Conducting Sphere . . . . .
9.24 Oscillations in a Spherical Cavity . . . . . .
. . . . . . . . . . .
554
. . . . . . . . . . . 554
. . . . . . . . . . . . 558
. . . . . . . . . . . . 560
DIFFRACTION
O F A P L A N E WAVE BY A SPHERE . . .
9.25 Expansion of the Diffracted Field . . . . . .
9.26 Total Radiation . . . . . . . . . . . . . .
9.27 Limiting Cases . . . . . . . . . . . . . . .
. . . . . . . . . . . . 563
. . . . . . . . . . . . 563
. . . . . . . . . . . 568
. . . . . . . . . . .
570
EFFECT
OF TRC EARTH
ON THE PROPAGATION
OF RADIO
WAVES. . . . . . . . 573
9.28 Sommerfeld Solution . . . . . . . . . . . . . . . . . . . . . . .
573
9.29 Weyl Solution . . . . . . . . . . . . . . . . . . . . . . . . . .
577
9.30 van der Pol Solution . . . . . . . . . . . . . . . . . . . . . . .
582
9.31 Approximation of the Integrals . . . . . . . . . . . . . . . . . . . 583
PROBLEMS
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
588
APPENDIX I
A . NUMERICAL
VALUESOF FUNDAMENTAL
CONSTANTS
. . . . . . . . . . . . 601
B. DIMENSIONS
OF ELECTROMAQNETIC
QUANTITIES
. . . . . . . . . . . . . 601
C . CONVERSIONTABLES
. . . . . . . . . . . . . . . . . . . . . . . . .
602
F O R M U L A S FROM VECTOR
APPENDIX 11
ANALYSIS . . . . . . . .
APPENDIX I11
CONDUCTIVITY O F V A R I O U S h’f ATERIALS . . . . . .
SPECIFIC INDUCTWE
CAPACITY
O F DIELECTRICS
. .
. . . . . . . . . . . .
604
. . . . . . . . . . . . . 605
. . . . . . . . . . . . . 606
APPENDIX IV
ASSOCIATEDLEGENDRE
FUNCTIONS
. . . . . . . . . . . . . . . . . . . .
608
INDEX
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
609
ELECTROMAGNETIC THEORY
CHAPTER I
THE FIELD EQUATIONS
A vast wealth of experimental evidence accumulated over the past
century leads one to believe that large-scale electromagnetic phenomena
are governed by Maxwell’s equations. Coulomb’s determination of the
law of force between charges, the researches of Ampere on the interaction
of current elements, and the observations of Faraday on variable fields
can be woven into a plausible argument to support this view. The
historical approach is recommended to the beginner, for it is the simplest
and will afford him the most immediate satisfaction. In the present
volume, however, we shall suppose the reader to have completed such a
preliminary survey and shall credit him with a general knowledge of the
experimental facts and their theoretical intcrpretation. Electromagnetic
theory, according to the standpoint adopted in this book, is the theory of
Maxwell’s equations. Consequently, we shall postulate these equations
at the outset and proceed to deduce the structure and properties of the
field together with its relation to the source. No single experiment
constitutes proof of a theory. The true test of our initial assumptione
will appear in the persistent, uniform correspondence of deduction with
observation.
In this first chapter we shall be occupied with the rather dry business
of formulating equations and preparing the way for our investigation.
MAXWELL’S EQUATIONS
1.1. The Field Vectors.-By
an electromagnetic field let us understand the domain of the four vectors E and B, D and H. These vectors
are assumed to be finite throughout the entire field, and at all ordinary
points to be continuous functions of position and time, with continuous
derivatives. Discontinuities in the field vectors or their derivatives
may occur, however, on surfaces which mark an abrupt change in the
physical properties of the medium. According to the traditional usage,
E and H are known as the intensities respectively of the electric and
magnetic field, D is called the electric displacement and B, the magnetic
induction. Eventually the field vectors must be defined in terms of the
experiments by which they can be measured. Until these experiments
1
2
THE FIELD EQUATIONS
[CHAP.
I
are formulated, there is no reason to consider one vector more fundamental than another, and we shall apply the word intensity t o mean
indiscriminately the strength or magnitude of any of the four vectors
at a point in space and time.
The source of an electromagnetic field is a distribution of electric
charge and current. Since we are concerned only with its macroscopic
effects, it may be assumed that this distribution is continuous rather
than discrete, and specified as a function of space and time by t.he density of charge p , and by tJhevector current density J.
We shall now postulate that a t every ordinary point in space the field
vectors are subject to the Maxwell equations:
aB
V X E + - =atO ,
f3D
VXH--=J.
dt
By an ordinary point we shall mean one in whose neighborhood the
physical properties of the medium are continuous. It has been noted that
the transition of the field vectors and their derivatives across a surface
bounding a material body may be discontinuous; such surfaces must,
therefore, be excluded until the nature of these discontinuities can be
investigated.
1.2. Charge and Current.-Although the corpuscular nature of electricity is well established, the size of the elementary quantum of charge
is too minute to be taken into account as a distinct entity in a strictly
macroscopic theory. Obviously the frontier that marks off the domain
of large-scale phenomena from those which are microscopic is an arbitrary one. To be sure, a macroscopic element of volume must contain
an enormous number of atoms; but that condition alone is an insufficient
criterion, for many crystals, including the metals, exhibit frequently a
microscopic “grain” or “mosaic” structure which will be excluded from
our investigation. We are probably well on the safe side in imposing
a limit of one-tenth of a millimeter as the smallest admissible element
of length. There are many experiments, such as the scattering of light
by particles no larger than lop3 mm. in diameter, which indicate that
the macroscopic theory may be pushed well beyond the limit suggested.
Nonetheless, we are encroaching here on the proper domain of quantum
theory, and it is the quantum theory which must eventually determine
the validity of our assumptions in microscopic regions.
Let us suppose that the charge contained within a volume element Av
is Aq. The charge density a t any point within Av will be defined by the
relation
(3)
Aq = p Av.
SEC.
1.21
CHARGE A N D CURRENT
3
Thus by the charge density a t a point we mean the average chaage per
unit volume in the neighborhood of t h a t point. I n a strict sense (3)
does not define a continuous function of position, for Au cannot approach
zero without limit. Nonetheless we shall assume t h a t p can be represented by a function of the coordinates and the time which a t ordinary
points is continuous and has continuous derivatives. The value of the
total charge obtained by integrating that function over a large-scale
volume will then differ from the true charge contained therein by a
microscopic quantity a t most.
Any ordered motion of charge constitutes a current. A current distribution is characterized by a vector field which specifies at each point
not only the intensity of the flow but also its direction. As in the study
of fluid motion, it is convenient to imagine streamlines traced through
the distribution and everywhere tangent to the direction of flow. Consider a surface which is orthogonal t o a system of streamlines. The
current density at any point on this surface is then defined as a vector J
directed along the streamline through the point and equal in magnitude
t o the charge which in unit time crosses unit area of the surface in the
vicinity of the point. On the other hand the current I across any surface
S is equal t o the rate at which charge crosses that surface. If n is the
positive unit normal t o an element A a of X,we have
(4)
AI
=
-
J n Au.
Since Au is a macroscopic element of area, Eq. (4)docs not define the
current density with mathematical rigor as a continuous function of
position, but again one may represent the distribution by such a function
without incurring an appreciable error. The total current through S is,
therefore,
(5)
I
=
J-nda.
Since electrical charge may be either positive or negative, a convention
must be adopted as t o what constitutes a positive current. If the flow
through an element of area consists of positive charges whose velocity
vectors form an angle of less than 90 deg. with the positive normal n,
the current is said t o be positive. If the angle is greater than 90 deg., the
current is negative. Likewise if the angle is less than 90 deg. but the
charges are negative, the current through the element is negative. I n
the case of metallic conductors the carriers of electricity are presumably
negative electrons, and the direction of the current density vector is
therefore opposed t o the direction of electron motion.
Let us suppose now that the surface S of Eq. ( 5 ) is closed. We shall
adhere t o the customary convention that the positive normal to a closed
4
[CHAP.I
THE FIELD EQUATIONS
surface i s drawn outward. I n virtue of the definition of current as the
flow of charge across a surface, it follows that the surface integral of the
normal component of J over S must measure the loss of charge from the
region within. There is no experimental evidence to indicate that under
ordinary conditions charge may be either created or destroyed in macroscopic amounts. One may therefore write
where V is the volume enclosed by S , as a relation expressing the conservation of charge. The flow of charge across the surface can originate
in two ways. The surface S may be fixed in space and the density p
be some function of the time as well as of the coordinates; or the charge
density may be invariable with time, while the surface moves in some
prescribed manner. I n this latter event the right-hand integral of ( 6 )
is a function of time in virtue of variable limits. If, however, the surface
is fixed and the integral convergent, one may replace d / d t by a partial
derivative under the sign of integration.
We shall have frequent occasion to make use of the divergence theorem
of vector analysis. Let A(z, y, Z) be any vector function of position
which together with its first derivatives is continuous throughout a
volume V and over the bounding surface S. The surface S is regular
but otherwise arbitrary.l Then it can be shown that
A an da =
sv
V A dv.
As a matter of fact, this relation may be advantageously used as a
definition of the divergence. To obtain the value of V A at a point P
within V , we allow the surface S to shrink about P. When the volume V
has become sufficiently small, the integral on the right may be replaced
by VV A, and we obtain
-
(9)
‘ A regular element of arc is represented in parametric form by the equations
z = z(t), y = y ( t ) , z = z ( t ) such that in the interval a S t 5 b z,y, z are continuous,
single-valued functions of t with continuous derivatives of all orders unless otherwise
restricted. A regular curve is constructed of a finite number of such arcs joined end
t o end but such that the curve does not cross itself. Thus a regular curve has no
double points and is piecewise differentiable. A regular surface element is a portion
of surface whose projection on a properly oriented plane is the interior of a regular
closed curve. Hence it does not intersect itself. Cf.Kellogg, “Foundations of Potential Theory,” p. 97, Springer, 1929.
SEC.
1.21
CHARGE A N D CURRENT
5
The divergence of a vector at a point is, therefore, to be interpreted as the
integral of its normal component over an infinitesimally small surface
enclosing that point, divided by the enclosed volume. The flux of a
vector through a closed surface is a measure of the sources within; hence
the divergence determines their strength a t a point. Since S has been
shrunk close about P , the value of A a t every point on the surface may
be expressed analytically in terms of the values of A and its derivatives
at P, and consequently the integral in (9) may be evaluated, leading in
the case of rectangular coordinates to
On applying this theorem to (7) the surface integral is transformed
to the volume integral
Now the integrand of (11) is a continuous function of the coordinates
and hence there must exist small regions within which the integrand does
not change sign. If the integral is to vanish for arbitrary volumes V , it
is necessary that the integrand be identically zero. The differential
equation
V - J + -a=P O
at
expresses the conservation of charge in the neighborhood of a point.
By analogy with an equivalent relation in hydrodynamics, (12) is freGently referred to as the equation of continuity.
If at every point within a specified region the charge density is constant, the current passing into the region through the bounding surface
must at all times equal the current passing outward. Over the bounding
surface S we have
$s J - n d a = 0,
and at every interior point
(14)
V -J = 0.
Any motion characterized by vector or scalar quantities which are
independent of the time is said to be steady, or stationary. A steadystate flow of electricity is thus defined by a vector J which at every point
within the region is constant in direction and magnitude. In virtue of
the divergenceless character of such a current distribution, it follows
6
THE FIELD EQUATIONS
[CHAP.I
that in the steady state all streamlines, or current filaments, close upon
themselves. The field of the vector J is solenoidal.
1.3. Divergence of the Field Vectors.-Two further conditions satisfied by the vectors B and D may be deduced directly from Maxwell’s
equations by noting that the divergence of the curl of any vector vanishes
identically. We take the divergerxe of Eq. (1) and obtain
The commutation of the operators V and a / d t is admissible, for at an
ordinary point B and all its derivatives are assumed to be continuous.
It follows from (15) that at every point in the field the divergence of B
is constant. If ever in its past history the field has vanished, this constant must be zero and, since one may reasonably suppose that the
initial generation of the field was at a time not infinitely remote, we
conclude that
(16)
V * B = 0,
and the field of B is therefore solenoidal.
Likewise the divergence of Eq. (2) leads to
(17)
d
at
V .J + - V . D
=O,
or, in virtue of (12), to
(18)
d
( V - D- p ) = 0.
at
-
If again we admit that at some time in its past or future history the field
may vanish, it is necessary that
(19)
V - D = p.
The charges distributed with a density p constitute the sources of the
vector D.
The divergence equations (16) and (19) are frequently included as
part of Maxwell’ssystem. I t must be noted, however, that if one assumes
the conservation of charge, these are not independent relations.
1.4. Integral Form of the Field Equations.-The
properties of an
electromagnetic field which have been specified by the differential equations (l),( Z ) , (16), and (19) may also be expressed by an equivalent
system of integral relations. To obtain this equivalent system, we apply
a second fundamental theorem of vector analysis.
According to Stokes’ theorem the line integral of a vector taken
about a closed contour can be transformed into a surface integral extended
SEC.1.41
INTEGRAL FORM OF T H E FIELD EQUATIONS
7
over a surface bounded by the contour. The contour C must either be
regular or be resolvable into a finite number of regular arcs, and it is
assumed that the otherwise arbitrary surface S bounded by C is twosided and may be resolved into a finite numbcr of regular elements. The
positive side of the surface S is related to the positive direction of circulation on the contour by the usual convention that an observer, moving
in a positive sense along C, will have the positive side of S on his left.
Then if A(x, y, z ) is any vector function of position, which together with
its first derivatives is continuous a t all points of S and C, it may be shown
that
(20)
LA.ds=L(VxA)-nda,
where ds is an element of length along C and n is a unit vector normal t o
the positive side of the element of area da. This transformation can
also be looked upon as an equation defining the curl. T o determine the
value of V x A a t a point P on S , we allow the contour t o shrink about P
until the enclosed area S is reduced t o an infinitesimal element of a plane
whose normal is in the direction specified by n. The integral on the
right is then equal t o (V x A) nS, plus infinitesimals of higher order. The
projection of the vector V x A in the direction of the normal is, therefore,
-
The curl of a vector at a point is to be interpreted as the line integral of
that vector about an infinitesimal path on a surface containing the point,
per unit of enclosed area. Since A has been assumed analytic in the
neighborhood of P , its value a t any point on C may be expressed in
terms of the values of A and its derivatives a t P , so t h a t the evaluation of
the line integral in (21) about the infinitesimal path can actually be
carried out. I n particular, if the element X is oriented parallel t o the
yz-coordinate plane, one finds for the x-component of the curl
Proceeding likewise for the y- and z-components we obtain
8
T H E FIELD EQUATIONS
[CHAP.
I
Let us now integrate the normal component of the vector d B / d t over
any regular surface S bounded by a closed contour C. From (1) and
(20) it follows that
If the contour is fixed, the operator a / a t may be brought out from under
the sign of integration.
(25)
By definition, the quantity
(26)
a
=
LB-ncia
is the magnetic flux, or more specifically the flux of the vector B through
the surface. According to (25) the line integral of the vector E about any
closed, regular curve in the field is equal
to the time rate of decrease of the magnetic
flux through any surface spanning that
curve. The relation between the direction
of circulation about a contour and the positive normal to a surface bounded by it is
C
illustrated in Fig. 1. A positive direction
FIG. 1.-Convention
relating
direction of the positive normal about c is chosen arbitrarily and the flux a
n t o the direction of circulation is then positive or negative according to
about a contour C .
the direction of the lines of B with respect
to the normal. The time rate of change of Q, is in turn positive or n e g a
tive as the positive flux is increasing or decreasing.
We recall that the application of Stokes’ theorem t o Eq. (1) is valid
only if the vector E and its derivatives are continuous a t all points of S
and C. Since discontinuities in both E and B occur across surfaces
marking sudden changes in the physical properties of the medium, the
question may be raised as to what extent (25) represents a general law
of the electromagnetic field. One might suppose, for example, that the
contour linked or pierced a closed iron transformer core. To obviate
this difficulty it may be imagined that at the surface of every material
body in the field the physical properties vary rapidly but continuously
within a thin boundary layer from their values just inside to their values
just outside the surface. I n this manner all discontinuities are eliminated
from the field and (25) may be applied to every closed contour.
The experiments of Faraday indicated that the relation (25) holds
whatever the cause of flux variation. The partial derivative implies a,
&
SEC.1.41
INTEGRAL FORM OF T H E FIELD EQUATIONS
9
variable flux density threading a fixed contour, but the total flux can
likewise be changed by a deformation of the contour. To take this into
account the Faraday law is written generally in the form
It can be shown that (27) is in fact a consequence of the differential
field equations, but the proof must be based on the electrodynamics of
moving bodies which will be touched upon in Sec. 1.22.
In like fashion Eq. (2) may be replaced by an equivalent integral
relation,
where I is the total current linking the contour as defined in (5). I n the
steady state, the integral on the right is zero and the conduction current I
through any regular surface is equal to the line integral of the vector H
about its contour. If, however, the field is variable, the vector dD/dt
has associated with it a field H exactly equal to that which would be
produced by a current distribution of density
J‘ = dD
-.
dt
To this quantity Maxwell gave the name (‘displacement current,” a term
which we shall occasionally employ without committing ourselves as
yet to any particular interpretation of the vector D.
The two remaining field equations (16) and (19) can be expressed in
an equivalent integral form with the help of the divergence theorem.
One obtains
(30)
SsB.nda
=
0,
stating that the total flux of the vector B crossing any closed, regular
surface is zero, and
according to which the flux of the vector D through a closed surface is
equal to the total charge q contained within. The circle through the
sign of integration is frequently employed to emphasize the fact that a
contour or surface is closed.
10
THE FIELD EQUATIONS
[CHAP.I
MACROSCOPIC PROPERTIES OF MATTER
1.5. The Inductive Capacities E and p.-No other assumptions have
been made thus far than that an electromagnetic field may be characterized by four vectors E, B, D, and H, which a t ordinary points satisfy
Maxwell’s equations, and that the distribution of current which gives
rise to this field is such as to ensure the conscrvation of charge. Between
the five vectors E, B, D, H, J t,hcre are but two independent relations, the
equations (I) and (2) of the preceding scction, and we are therefore obliged
t o impose further conditions if the system is t o be made determinate.
Let us begin with the assumption that a t any given point in the field,
whether in free space or within matter, the vector D may be represented
as a function of E and the vector H as a function of B.
(1)
D
=
D(E),
H
=
H(B).
The nature of these functional relations is t o be determined solely by the
physical properties of the medium in the immediate neighborhood of the
specified point. Certain simple relations are of most common occurrence.
1. I n free space, D differs from E only by a constant factor, as does H
from B. Following the traditional usage, we shall write
(2)
D = QE,
H = -1B e
PO
The values and the dimensions of the constants €0 and po will depend
upon the system of units adopted. In only one of many wholly arbitrary
systems does D reduce to E and H t o B in empty space.
2. If the physical properties of a body in the neighborhood of some
interior point are the same in all directions, the body is said t o be isotropic. At every point in an isotropic medium D is parallel t o E and H
is parallel t o B. The relations between the vectors, moreover, are linear
in almost all the soluble problems of electromagnetic theory. For the
isotropic, linear case we put then
(3)
D
=
eE,
H
=
1
-B.
P
The factors cand p will be called the inductive capacities of the medium.
The dimensionless ratios
(4)
are independent of the choice of units and will be referred t o as the
specific inductive capacities. The properties of a homogeneous medium
are constant from point t o point and in this case it is customary t o refer
SEC.1.61
ELECTRIC A N D MAGNETIC POLARIZATION
11
t o K~ as the dielectric constant and to K , as the permeability. I n general,
however, one must look upon the inductive capacities as scalar functions
of position which characterize the electromagnetic properties of matter
in the large.
3. The properties of anisotropic matter vary in a different manner
along different directions about a point. I n this case the vectors D and
E, H and B are parallel only along certain preferred axes. If i t may be
assumed that the relations are still linear, as is usually the case, one
may express each rectangular component of D as a linear function of the
three components of E.
(5)
The coefficients ejk of this linear transformation are the components of a
symmetric tensor. An analogous relation may be set up between the
vectors H and B, but the occurrence of such a linear anisotropy in what
may properly be called macroscopic problems is rare.
The distinction between the microscopic and macroscopic viewpoints
is nowhere sharper than in the interpretation of these parameters e and p,
or their tensor equivalents. A microscopic theory must deduce the
physical properties of matter from its atomic structure. It must enable
one t o calculate not only the average field that prevails within a body but
also its local value in the neighborhood of a specific atom. It must tell
us how the atom will be deformed under the influence of that local field,
and how the aggregate effect of these atomic deformations may be
represented in the large by such parameters as e and p.
We, on the other hand, are from the present standpoint sheer behaviorists. Our knowledge of matter is, t o use a large word, purely phenomenological. Each substance is to be characterized electromagnetically
i n terms of a minimum number of parameters. The dependence of the
parameters E and p on such physical variables as density, temperature,
and frequency will be established by experiment. Information given by
such measurements sheds much light on the internal structure of matter,
but the internal structure is not our present concern.
1.6. Electric and Magnetic Polarization.-To describe the electromagnetic state of a sample of matter, it will prove convenient t o introduce two additional vectors. We shall define the electric and magnetic
polarization vectors b y the equations
The polarization vectors are thus definitely associated with matter and
12
[CHAP.I
THE FIELD EQUATIONS
vanish in free space. By means of these relations let us now eliminate
D and H from the field equations. There results the system
(7)
1
V*E=-(p-V*P),
V.B=O,
€0
which we are free to interpret as follows: the presence of rigid material
bodies in a n electromagnetic field m a y be completely accounted for b y a n
equivalent distribution of charge of density -V P, and an equivalent
dP
distribution of current of density - V x M.
dt
-
+
In isotropic media the polarization vectors are parallel to the corresponding field vectors, and are found experimentally to be proportional
to them if ferromagnetic materials are excluded. The electric and
magnetic susceptibilities xe and xmare defined by the relations
(8)
P
=
x~OE,
M = xmH.
Logically the magnetic polarization M should be placed proportional to B.
Long usage, however, has associated it with H and to avoid confusion
on a matter which is really of no great importance we adhere to this
convention. The susceptibilities xe and x,,, defined by (8) are dimensionless ratios whose values are independent of the system of units employed.
In due course it will be shown that E and B are force vectors and in this
sense are fundamental. D and H are derived vectors associated with
the state of matter. The polarization vector P has the dimensions of D,
not E, while M and H are dimensionally alike. From (3), ( 6 ) , and (8) it
follows at once that the susceptibilities are related t o the specific inductive capacities by the equations
(9)
Xe =
Ke
- 1,
Xm =
Km
- 1.
In anisotropic media the susceptibilitirs are represented by the components of a tensor.
It will be a part of our task in later chapters t o formulate experiments
by means of which the susceptibility of a substance may be accurately
measured. Such measurements show that the electric susceptibility is
always positive. I n gases it is of the order of 0.0006 (air), but in liquids
and solids it may attain values as large as 80 (water). An inherent
difference in the nature of the vectors P and M is indicated by the fact
that the magnetic susceptibility X n may be either positive or negative.
Substances characterized by a positive susceptibility are said t o be
SEC.1.71
CONDUCTING M E D I A
13
paramagnetic, whereas those whose susceptibility is negative are called
diamagnetic. The metals of the ferromagnetic group, including iron,
nickel, cobalt, and their alloys, constitute a particular class of substances
of enormous positive susceptibility, the value of which may be of the
order of many thousands. In view of the nonlinear relation of M to H
peculiar to these materials, the susceptibility xmmust now be interpreted
as the slope of a tangent to the M-H curve at a point corresponding to a
particular value of H. To include such cases the definition of susceptibility is generalized to
The susceptibilities of all nonferromagnetic materials, whether paramagnetic or diamagnetic, are so small as to be negligible for most practical
purposes.
Thus far it has been assumed that a functional relation exists
between the vector P or M and the applied field, and for this reason
they may properly be called the induced polarizations. Under certain
conditions, however, a magnetic field may be associated with a ferromagnetic body in the absence of any external excitation. The body is
then said to be in a state of permanent magnetization. We shall maintain our initial assumption that the field both inside and outside the
magnet is completely defined by the vectors B and H. But now the
difference of these two vectors a t an interior point is a fixed vector Mo,
which may be called the intensity of magnetization and which bears no
functional relationship to H. On the contrary the magnetization M,
must be interpreted as the source of the field. If an external field is
superposed on the field of a permanent rnagnct, the intensity of magnetization will be augmented by the induced polarization M. At any interior
point we have, therefore,
+ +
B = po(H M Mo).
(11)
Of this induced polarization we can only say for the present that it is a
function of the resultant H prevailing at the same point. The relation
of the resultant field within the body to the intensity of an applied field
generated by external sources depends not only on the magnetization
Mo but also upon the shape of the body. There will be occasion to
examine this matter more carefully in Chap. IV.
1.7. Conducting Media.-To Maxwell's equations there must now
be added a third and last empirical relation between the current density
and the field. We shall assume that at any point within a liquid or
solid the current density is a function of the field E.
14
T H E F I E L D EQUATIONS
[CHAP.I
The distribution of current in a n ionized, gaseous medium may depend
also on the intensity of the magnetic field, but since electromagnetic
phenomena in gaseous discharges are in general governed b y a multitude
of factors other than those taken into account in the present theory, we
shall exclude such cases from further consideration.
Throughout a remarkably wide range of conditions, in both solids
and weakly ionized solutions, the relation (12) proves to be linear.
J = uE.
(13)
The factor u is called the conductivity of the medium. The distinction
between good and poor conductors, or insulators, is relative and arbitrary.
All substances exhibit conductivity to some degree but the range of
observed values of u is tremendous. The conductivity of copper, for
example, is some lo7 times as great as that of such a “good” conductor
as sea water, and 1019 times that of ordinary glass. I n Appendix 111
will be found a n abbreviated table of the conductivities of representative
materials.
Equation (13) is simply Ohm’s law. Let us imagine, for example, a
stationary distribution of current throughout the volume of any conducting medium. I n virtue of the divergenceless character of the flow
this distribution may be represented by closed streamlines. If a and b
are two points on a particular streamline and d s is an element of its
length, we have
(14)
JbE.ds
=
J ba J .nd s .
A bundle of adjacent streamlines constitutes a current filament or tube.
Since the flow is solenoidal, the current I through every cross section of
the filament is the same. Let S be the cross-sectional area of the filament
on a plane drawn normal to the direction of flow. X need not be infinitesimal, but is assumed t o be so small that over its area the current
density is uniform. Then SJ ds = I d s , and
-
The factor,
(16)
R
=
.I
b l
ads,
1 It is true t h a t to a very slight degree the current distribution in a liquid or solid
conductor may be modified by an impressed magnetic field, but the magnitude of this
so-called Hall effect is so small that it may be ignorcd without incurring an appreciable
error.
CONDUCTING M E D I A
SEC.1.71
15
is equal to the resistance of the filament bstweeil the points a and b.
The resistance of a linear section of homogeneous conductor of uniform
cross section S and length 1 is
a formula which is strictly valid only in the case of stationary currents.
W i t h i n a region of nonvanishing conductivity there can be no permanent
distribution of f r e e charge. This fundamentally important theorem cag
be easily demonstrated when the medium is homogeneous and such that
the relations between D and E and J and E are linear. By the equation
of continuity,
V.J
+ -aatP = V . a E + --dtdP = 0.
On the other hand in a homogeneous rnedium
(19)
V .E
1
= - p,
E
which combined with (18) leads to
The density of charge at any instant is, therefore,
the constant of integration po being equal to the density a t the time t = 0.
The initial charge distribution throughout the conductor decays exponentially with the time a t every point and in a manner wholly independent of the applied field. If the charge density is initially zero, it
remains zero a t all times thereafter.
The time
required for the charge at any point to dacay to l / e of its original value
is called the relaxation time. I n all but the poorest conductors r is
exceedingly small. Thus in sea water the relaxation time is about
2X
sec. ; even in such a poor conductor as distilled water it is not
great,er than
sec. I n the best insulators, such as fused quartz, it
may nevertheless assume values exceeding lo6 sec., an instance of the
extraordinary range in the possible values of the parameter u.
16
T H E FIELD EQUATIONS
[CHAP.I
Let us suppose that at f = 0 a charge is concentrated within a small
spherical region located somewhere in a conducting body. At every
other point of the conductor the charge density is zero. The charge
within the sphere now begins to fade away exponentially, but according
to (21) no charge can reappear anywhere within the conductor. What
becomes of i t ? Since the charge is conserved, the decay of charge
within the spherical surface must be accompanied by an outward flow,
or current. No charge can accumulate a t any other interior point; hence
the flow must be divergenceless. It will be arrested, however, on the
outer surface of the conductor and it is here that we shall rediscover the
charge that has been lost from the central sphere. This surface charge
makes its appearance at the exact instant that the interior charge begins
to decay, for the total charge is constant.
UNITS AND DIMENSIONS
1.8. The M.K.S. or Giorgi System.-An electromagnetic fleld thus
far is no more than a complex of vectors subject to a postulated system of
differential equations. To proceed further we must establish the physical
dimensions of these vectors and agree on the units in which they are
to be measured.
In the customary sense, an “absolute” system of units is one in which
every quantity may be measured or expressed in terms of the three
fundamental quantities mass, length, and time. Now in electromagnetic
theory there is an essential arbitrariness in the matter of dimensions
which is introduced with the factors € 0 and po connecting D and E, H
and B respectively in free space. No experiment has yet been imagined
by means of which dimensions may be attributed to either € 0 or PO as
a n independent physical entity. On the other hand, it is a direct consequence of the field equations that the quantity
c =
(1)
1
d
G
0
shall have the dimensions of a velocity, and every arbitrary choice of BO
and po is subject to this restriction. The magnitude of this velocity
cannot be calculated a priori, but by suitable experiment it may be
measured. The value obtained by the method of Rosa and Dorsey of
the Bureau of Standards and corrected by Curtis1 in 1929 is
(2)
c =
1
-= 2.99790 X los
dGZJ
meters/sec.,
ROSAand DORSEY,
A New Determination of the Ratio of the Electrostatic Unit
of Electricity, Bur. Standards, Bull. 3, p. 433, 1907. CURTIS, Bur. Standards J .
Research, 3, 63, 1929.
SEC.1.81
THE M.K.S. OR GIORGI SYSTEM
17
or for all practical purposes
(3)
c =3
x
108
meters/sec.
Throughout the early history of electromagnetic theory the absolute
electromagnetic system of units was employed for all scientific investigations. I n this system the centimeter was adopted as the unit of length,
the gram as the unit of mass, the second as the unit of time, and as a
fourth unit the factor p o was placed arbitrarily equal t o unity and considered dimensionless. The dimensions of c0 were then uniquely determined by (1) and it could be shown that the units and dimensions of
every other quantity entering into the theory might be expressed in
terms of centimeters, grams, seconds, and po. Unfortunately, this absolute system failed t o meet the needs of practice. The units of resistance
and of electromotive force were, for example, far too small. To remedy
this defect a practical system was adopted. Each unit of the practical
system had the dimensions of the corresponding electromagnetic unit and
differed from it in magnitude by a power of ten which, in the case of
voltage and resistance a t least, was wholly arbitrary. The practical
units have the great advantage of convenient size and they are now
universally employed for technical measurements and computations.
Since they have been defined as arbitrary multiples of absolute units, they
do not, however, constitute an absolutc system. Now the quantities
mass, length, and time are fundamental solely because the physicist has
found it expedient t o raise them t o that rank. That there are other
fundamental quantities is obvious from the fact that all electromagnetic
quantities cannot be expressed in terms of these three alone. The
restriction of the term “absolute” to systems based on mass, length, and
time is, therefore, wholly unwarranted; one should ask only that such a
system be self-consistent and that every quantity be defined in terms of
a minimum number of basic, independent units. The antipathy of
physicists in the past to the practical system of electrical units has been
based not on any firm belief in the sanctity of mass, length, and time,
but rather on the lack of self-consistency within that system.
Fortunately a most satisfactory solution has been found for this
difficulty. I n 1901 Giorgi,’ pursuing an idea originally due to Maxwell,
called attention t o the fact that the practical system could be converted
into an absolute system by an appropriate choice of fundamental units.
It is indeed only necessary to choose for the unit of length the interGIORGI:Units Razionali di Elettromagnetismo, Atti deZ2’ A.E.I., 1901. An
historical review of the development of the practical system, including a report of the
action taken a t the 1935 meeting of the International Electrotechnical Commission
and an extensive bibliography is given by Kennelly, J. Inst. Elec. Engrs., 78, 235The M.K.S. System of Electrical Units, J . Inst.
245, 1936. See also GLAZEBROOK,
Elec. Engrs., 78, pp. 245-247.
I8
T H E FIELD EQUAT IO NS
[CHAP.I
national meter, for the unit of mass the kilogram, for the unit of time the
second, and as a fourth unit any electrical quantity belonging to the
practical system such as the coulomb, the ampere, or the ohm. From
the field equations it is then possible to deduce the units and dimensions
of every electromagnetic quantity in terms of these four fundamental
units. Moreover the derived quantities will be related to each other
exactly as in the practical system and may, therefore, be expressed in
practical units. I n particular it is found that the parameter po must
have the value 47r X lop7, whence from (1) the value of co may be calculated. Inversely one might equally well assume this value of po as a
fourth basic unit and then deduce the practical series from the field
equations.
At a plenary session in June, 1935, the International Electrotechnical
Commission adopted unanimously the m.k.s. system of Giorgi. Certain
questions, however, still remain to be settled. No official agreement
has as yet been reached as to the fourth fundamental unit. Giorgi himself recommended that the ohm, a material standard defined as the
resistance of a specified column of mercury under specified conditions
of pressure and temperature, be introduced as a basic quantity. If
pa = 47r x 10-7 be chosen as the fourth unit and assumed dimensionless,
all derived quantities may be expressed in terms of mass, length, and
time alone, the dimensions of each being identical with those of the corresponding quantity in the absolute electromagnetic system and differing
from them only in the size of the units. This assumption leads, however,
to fractional exponents in the dimensions of many quantities, a direct
consequence of our arbitrariness in clinging to mass, length, and time
as the sole fundamental entities. I n the absolute electromagnetic system, for example, the dimensions of charge are grams? . centimeters*, an
irrationality which can hardly be physically significant. These fractional
exponents are entirely eliminated if we choose as a fourth unit the
coulomb; for this reason, charge has been advocated at various times as a
fundamental quantity quite apart from the question of its magnitude.’
I n the present volume we shall adhere exclusively to the meter-kilogramsecond-coulomb system. A subsequent choice by the I.E.C. of some
other electrical quantity as basic will in nowise affect the size of our units
01. the form of the equations.2
See the discussion by WALLOT:Elektrotechnische Zeilschrift, Nos. 44-46, 1922.
Also SOMMERFELD:
“Ueber die Electromagnetischen Einheiten,” pp. 157-165, Zeeman
Verhandelingen, Martinus Nijhoff, The Hague, 1935; Physik. 2.36, 814-820, 1935.
No ruling has been made as yet on the question of rationalization and opinion
seems equally divided in favor and against. If one bases the theory on Maxwell’s
equations, it seems definitely advantageous t o drop the factors 47r which in unrationalired systems stand before the charge and current densities. A rationalized system
will be employed in this hook.
19
T H E M.K.S. OR GIORGI S Y S T E M
SEC.1.81
To demonstrate that the proposed units do constitute a self-consistent
system let us proceed as follows. The unit of current in the m.k.s.
system is t o be the absolute ampere and the unit of resistance is t o be
the absolute ohm. These quantities are to be such that the work
expended per second by a current of 1 amp. passing through a resistance
of 1 ohm is 1 joule (absolute). If R is the resistance of a section of
conductor carrying a constant current of I amp., the work dissipated in
heat in t sec. is
W
(4)
=
IzRt
joules.
By means of a calorimeter the heat generated mr,y be measured and thus
one determines the relation of the unit of electrical energy to the unit
quantity of heat. It is desired that the joule defined by (4) be identical
with the joule defined as a unit of mechanical work, so that in the electrical
as well as in the mechanical case
1 joule
(5)
=
0.2389
gram-calorie (mean).
Now we shall dejine the ampere on the basis of the equation of continuity
(6), page 4,as the current which transports across any surface 1 coulomb
in 1 sec. Then the ohm is a derived unit whose magnitude and dimensions
are determined by (4) :
watt lohm=l-ampere2
kilogram . meter2
coulomb2 second'
1
since 1 watt is equal to 1 joule/sec. The resistivity of a medium is
defined as the resistance measured between two parallel faces of a unit
cube. The reciprocal of this quantity is the conductivity. The dimensions of u follow from Eq. (17), page 15.
(7)
1 unit of conductivity
=
1
ohm . meter
coulomb2 * second
hlogram . meter3.
I n the United States the reciprocal ohm is usually called the mho,
although the name siemens has been adopted officially by the I.E.C.
The unit of conductivity is therefore 1 siemens/meter.
The volt will be defined simply as 1 wattlamp., or
1 volt = 1
watt ampere
~
kilogram . meter2
-.
coulomb . second2
Since the unit of current density is 1 amp./meter2, we deduce from the
relation J = aE that
(9)
1 unit of E
=
watt
volt 1 ampere . meter -I----meter
kilogram meter
coulomb . second2'
20
[CHAP.I
T H E FIELD EQUATIONS
The power expended per unit volume by a current of density J is therefore E J watts/meter3. It will be noted furthermore that the product
of charge and electric field intensity E has the dimensions of force. Let a
charge of 1 coulomb be placed in an electric field whose intensity is
1 volt/meter.
-
(10)
joule
kilogram * meter
volt 1 coulomb X 1 - 1 --1
meter
meter
second2
The unit of force in the m.k.s. system is called the newton, and is equivalent to 1 joulc/meter, or lo5 dynes.
The flux of the vector B shall be measured in webers,
@=JB.Ilda
(11)
webers,
and the intensity of the field B, or flux density, may therefore be expressed
in webers per square meter. According to ( 2 5 ) , page 8,
L E - d s = -d@
dt
The line integral
webers
s e d '
f E - ds is measured in volts and is usually called the
electromotive force (abbreviated e.m.f.) between t,he points a and b;
although its value in a nonstationary field depends on the path of integration. The induced e.m.f. around any closed contour C is, therefore,
equal to the rate of decrease of flux threading that contour, so that
between the units there exists the relation
1 volt
=
weber
1 ---1
second
or
1 weber = 1
joule
ampere
=
1
kilogram . meter2
-.
coulomb . second
It is important to note that the product of current and magnetic flux
is an energy. Note also that the product of B and a velocity is measured
in volts per meter, and is therefore a quantity of the same kind as E.
1 unit of B
=
weber
1A =
meter2
kilogram
coulomb . second'
The units which have been deduced thus far constitute an absolute
system in the sense that each has been expressed in terms of the four
21
THE M.K.S. OR GIORGI S Y S T E M
SEC.1.81
basic quantities, mass, length, time, and charge. That this system is
identical with the practical series may be verified by the substitutions
(17)
1 kilogram =
lo3 grams,
lo2 centimeters,
1 coulomb = & abcoulomb.
1 meter
=
The numerical factors which now appear in each relation are observed
t o be those that relate the practical units to the absolute electromagnetic
units. For example, from (6),
(18)
1 ohm
=
1 kilogram. meter2 coulomb2 . second
lo3 grams . lo4 centimeters2
abcoulomb2 ‘ seconds
= lo9 abohms;
and again from (8),
(19)
1 volt = 1
kilogram. meter2 - lo3 grams . l o 4 centimeters2
10-I abcoulomb . second2
coulomb . second2
= lo8 abvolts.
The series must be completed by a determination of the units and
1
dimensions of the vectors D and H. Since D = EE,H = - B , it is
P
necessary and sufficient that € 0 and PObe determined such as to satisfy
Eq. (2) and such that the proper ratio of practical t o absoiute units be
maintained. We shall represent mass, length, time, and charge by the
letters M , L , T, and &, respectively, and employ t,he customary symbol [ A ]
as meaning “the dimensions of A.” Then from Eq. (31), page 9,
JD.nda
(20)
=
q
coulombs
and, hence,
[Dl
=
Q
coulombs _ -L2’
meter2
coulombs - Q2T2
--.
The farad, a derived unit of capacity, is defined as the capacity of ft
conducting body whose potential will be raised 1 volt by a charge of
1 coulomb. It is equal, in other words, to 1 coulomb/volt. The
parameter € 0 in the m.k.s. system has dimensions, and may be measured
in farads per meter.
f
H ds taken
along a specified path is commonly called the magnetomotive force
By analogy with the electrical case, the line integral
22
THE FIELD EQUATIONS
(abbreviated m.m.f.).
In a stationary magnetic field
LH-ds=I
(23)
[CHAP.I
amperes,
where I is the current determined by the flow of charge through any
surface spanning the closed contour C. If t h field is variable, I must
include the displacement current as in (28), page 9. According to (23)
a magnetomotive force has the dimensions of current. I n practice,
however, the current is frequently carried by the turns of a coil or winding
which is linked by the contour C. If there are n such turns carrying a
current I , the total current threading C is nI ampere-turns and it is
customary to express magnetomotive force in these terms, although
dimensionally n is a numeric.
(24)
[m.m.f.]
=
ampere-turns,
whence
=
-.
ampere-turns - Q
meter
LT
It will be observed that the dimensions of D and those of H divided by a
velocity are identical. For the parameter po we find
-
ML
volt second - -.
As in the case of c,, it is convenient to express po in terms of a derived
unit, in this case the henry, defined as 1 volt-second/amp. (The henry
is that inductance in which an induced e.m.f. of 1 volt is generated when
the inducing current is varying a t the rate of 1amp./sec.) The parameter
po may) therefore, be measured in henrys per meter.
From (22) and (26) it follows now that
and hence that our system is indeed dimensionally consistent with
Eq. (2). Since it is known that in the rationalized, absolute c.g.s.
electromagnetic system pa is equal in magnitude to 47r, Eq. (26) fixes also
its magnitude in the m.k.s. system.
(28)
po =
4~
gram . centimeters
= 47r
abcoulombs2
kilogram lop2meter
lo2 coulombs2
or
(29)
=
4a
x
10-7
kilogram . meters
coulombs2
=
1.257 X
henry
meter
23
VECTOR A N D SCALAR POTENTIALS
SEC.1.91
The appropriate value of
c=--
(2)
€0
is then determined from
dG0
=
2.998
x
lO8meters
second
to be
(30)
=
8.854 X
coulomb2 . seconds2
= 8.854 X
kilogram meter3
lo-''-.
farad
meter
It is frequently convenient to know the reciprocal values of these factors>
-
(31)
PO
- _- 0.1129 x 1012 meters
Ae0
farad '
meters
henry '
- 0.7958 X lo6-
and the quantities
dg
(32)
=
376.6 ohms,
=
2.655 X
mho,
recur constantly throughout the investigation of wave propagation.
In Appendix I there will be found a summary of the units and dimensions of electromagnetic quantities in terms of mass, length, time, and
charge.
THE ELECTROMAGNETIC POTENTIALS
1.9. Vector and Scalar Potentials.-The analysis of an electromagnetic
field is often facilitated by the use of auxiliary functions known as potentials. A t every ordinary point of space, the field vectors satisfy the
system
+ f3B = 0,
(111) V * B = 0,
f3D
(11) V X H - - = J,
(IV) v * D= p.
(I) V X E
at
According to (111) the field of the vector B is always solenoidal.
quently B can be represented as the curl of another vector Ao.
B
(1)
=
Conse-
V x Ao.
However A. is not uniquely defined by (1); for B is equal also to the curl
of some vector A,
B=VXA,
(2)
where
(3)
and
A
= Ao
- V#J,
9 is any arbitrary scalar function of position.
24
T H E FIELD EQUATIONS
[CHAP.1
If now B is replaced in (I) by either (1) or (2), we obtain, respectively,
(4)
Thus the fields of the vectors E
dA
are irrotational and
+ aAo and E + dt
equal to the gradients of two scalar functions 40and 4.
The functions #I and
+o
are obviously related by
4
(7)
=
a+
+ Z'
$0
The functions A are vector potentials of the field, and the 4 are scalar
potentials. A0 and 40 designate one specific pair of potentials from which
the field can be derived through (1) and (5). An infinite number of
potentials leading to the same field can then be constructed from (3)
and (7).
Let us suppose that the medium is homogeneous and isotropic, and
that e and p are independent of field intensity.
D = eE,
(8)
B
= pH.
I n terms of the potentials
(9)
which upon substitution into (11) and (IV) give
(10)
V
x
V X A
+~
a4
+
d2A
V d t ' L E=
XPJ,
dA
V24+V'dt=
1
--p.
All particular solutions of (10) and (11) lead to the same electromagnetic
field when subjected to identical boundary conditions. They differ
among themselves by the arbitrary function +. Let us impose now upon
A and 4 the supplementary condition
V*A+~
(12)
To do this it is only necessary that
Ea4
-=O.
at
+ shall satisfy
SEC.
25
VECTOR A N D SCALAR POTENTIALS
1.91
where +,, and A. are particular solutions of (10) and (1 1). The potentials
$ and A are now uniquely defined and are solutions of the equations
(14)
V x V x A - VV-A
d2A
+/J
E
= pJ,
T
Equation (14) reduces to the same form as (15) when use is made of
the vector identity
(16)
V X
V x A=VV-A-V*VA.
The last term of (16) can be interpreted as the Laplacian operating on
the rectangular components of A. I n this case
The expansion of the operator V VA in curvilinear systems will be
discussed in Sec. 1.16, page 50.
? The relations (2) and (6) for the vectors B and E are by no means
general, To them may be added any particular solution of the homogeneous equations
(Ia) V X E
+ aB
(IIa) V x H -
aD
at
-
1
0,
(IIIu) V * B
=
0,
=
0,
(IVU)V - D
=
0.
From the symmetry of this system it is at once evident that it can be
satisfied identically by
(1%
D
=
-V X A*,
dA *
H = -V+* - - 7
at
from which we construct
1
E=--vxA*,
B = -
The new potentials are subject only to the conditions
26
T H E FIELD EQUATIONS
[CHAP.
1
A general solution of the inhomogeneous system (I) to ( I V ) is, therefore,
B
=
aA
V X A - p __ - pV+',
2%
at
provided 1.1 and e are constant.
The functions +* and A* are potentials of a source distribution which
is entirely external to the region considered. Usually +* and A* are put
equal to zero and the potentials of all charges, both distant and local,
are represented by 4 and A.
At any point where the charge and current densities are zero a
possible field is 40 = 0, A, = 0. The function I) is now any solution
of the homogeneous equation
@I)
- pr 2
= 0.
at
V'I)
+
Since at the same point the scalar potential satisfies the same equation,
$ may be chosen such that
vanishes. I n this case the field can be
expressed in terms of a vector potential alone.
+
B=VxA,
(24)
aA
E = --
at '
V*A=O.
(25)
Concerning the units and dimensions of these new quantities, we
note first that E is measured in volts/meter and that the scalar potential
4 is therefore to be measured in volts. I f q is a charge measured in
coulombs, it follows that the product q+ represents an energy expressed
in joules. From the relation B = v x A it is clear that the vector
potential A may be expressed in webers/meter, but equally well in either
volt-seconds/meter or in joules/ampere. The product of current and
vector potential is therefore an energy. The dimensions of A* are found
t o be coulombs/meter, while +* will be measured in ampere-turns.
1.10. Homogeneous Conducting Media.-In
view of the extreme
brevity of the relaxation time it may be assumed that the density of
free charge is always zero in the interior of a conductor. The field
equations for a homogeneous, isotropic medium then reduce to
(Ib) V X E
+ aB
at
=
0,
aD
(IIb) V x H - - - UE = 0,
at
( I I I b ) V * B = 0,
( I V b ) V D = 0.
SEC. 1.101
HOMOGENEOUS CONDUCTING M E D I A
27
We are now free to express either B or D in terms of a vector potential.
In the first alternative we have
aA
B=VXA,
(26)
E=-V+--*
at
If the vector and scalar potentials are subjected to the relation
a possible electromagnetic field may be constructed from any pair of
solutions of the equations
a2A
aA
V2A - p~ 7- p~ - = 0 ,
at
at
a2(b
a4 = 0.
v2+ - p c - puat2
at
As in the preceding paragraph one will note that the field vectors are
invariant to changes in the potentials satisfying the relations
4
(30)
=
40
+ a+Z J
A
=
Ao - v$,
+
where $ 0 , A0 are the potentials of a possible field and is an arbitrary
scalar function. In order that A and (b satisfy (27) it is only necessary
that be subjected to the additional condition
+
To a particular solution of (31) one is free to add any solution of the
homogeneous equation
+
Frequently it is convenient to choose such that the scalar potential
vanishes. The field within the conductor is then determined by a single
vector A.
aA
B=VxA,
E=---,
(33)
at
a2A
aA
V2A-p€T-p~X=0,
V*A=0.
(34)
at
The field may also be defined in terms of potentials
(35)
I> = -v
x
A*,
H
=
-v4*
+* and A * by
- aA* - !?A*.
at
28
T H E FZELD EQUATIONS
[CHAP.I
If 4 * and A * are to satisfy (28) and (29)' it is necessary that they be
related by
The field defined by (35) is invariant to all transformations of the potentials of the type
(37)
+* = 4;
a+* c
+ at
+ ;+*,
A*
=
At
-
V+*,
where as above 4: and A t are the potentials of any possible electromagnetic field. To ensure the relation (36) it is only nccessary that +*
be chosen such as to satisfy
Finally, by a proper choice of
to vanish.
+* the scalar
V X A*,
potential 4* may be made
a A*,
1.11. The Hertz Vectors, or Polarization Potentials.-We have seen
that the integration of Maxwell's equations may be reduced t o the
determination of a vector and a scalar potential, which in homogeneous
media satisfy one and the same differential equation. It was shown by
Hertz' that it is possible under ordinary conditions to define an electromagnetic field in terms of a single vector function.
Let 11s confine ourselves for the present to regions of an isotropic,
homogeneous medium within which there are neither conduction currents
nor free charges. The field equations then reduce to the homogencous
system (1a)-(IVa). We assume, for reasons which will become apparent,
that the vector potential A is proportional to the time derivative of a
vector n.
HERTZ,Ann. Physik, 36, 1, 1888. The general solution is due to Righi: Bologna
Mem., (5) 0, 1, 1901, and Zl Nuovo Cimento, (5) 2, 2, 1901.
SEC.1.111 THE HERTZ VECTORS, OR POLARIZATION POTENTIALS
29
and, when in turn this expression for E is introduced into ( I I a ) , it is
found that
(43)
5at (v
x v xn
+ vc#J+ p €
We recall that at points where there is no charge, the scalar function 4
is wholly arbitrary so long as it satisfies an equation such as ( 2 3 ) . In
the present instance it will be chosen such that
(44)
=
-v-n.
Then upon integrating (43)with respect to the time, we obtain
(45)
- + pe
V x V X II - V V Tz
a2n
__ =
at2
constant.
The particular value of the constant does not affect the determination of
the field and we are therefore free to place it equal to zero. Equation
(IVa)is also satisfied, for the divergence of the curl of any vector vanishes
identically. Then we may state that every solution of the vector equation
(46)
v x v x n - v v . n + p € =a m
=
0
determines an electromagnetic jield through
(47)
The condition that 4 shall satisfy (23) is fulfilled in virtue of (46). One
may replace (46) by
provided V 2is understood to operate on the rectangular components of m.
Since the vector D as well as B is solenoidal in a charge-free region,
an alternative solution can be constructed of the form
*
A* = p c a
-,n
at
where n*is any solution of (46)or (48).
From these results we conclude that the electromagnetic field within
a region throughout which e and p are constant, p and J equal to zero,
may be resolved into two partial fields, the one derived from the vector
30
[CHAP.I
THE FIELD EQUATIONS
and the other from the vector n*. The origin of these fields lies exterior
to the region. To determine the physical significance of the Hertz
vectors it is now necessary to relate them to their sources; in other words,
we must find the inhomogeneous equations from which (48) is derived.
Let us express the vector D in terms of E and the electric polarization P. According to (6), page 11, D = QE P. Then in place of
(IIa) and (IVa), we must now write
+
V
X
H - €0-aE
at
aP
=-
at’
1
V * E = --V-P.
€0
It may be verified without difficulty that these two equations, as well
as (Ia) and (IIIa), are still identically satisfied by (47), provided only
that E be replaced by c0 and that ][I be now any solution of
T h e source of the vector n and the electromagnetic jield derived f r o m it i s a
distribution of electric polarization P. In due course we shall interpret
the vector P as the electric dipole moment per unit volume of the medium.
Since n is associated with a distribution of electric dipoles, the partial
field which it defines is sometimes said to be of electric type, and n itself
may be called the electric polarization potential.
In like manner it can be shown that the field associated with ][I* is
set up by a distribution of magnetic polarization. According to (6),
page 11, the vector B is related to H by B = po(H M), which when
introduced into (Ia) and (IIIa) gives
+
Then these equations, as well as (IIa) and (IVa), are satisfied identically
by (50) if we replace there p by PO and prescribe that n* shall be a
solution of
(54)
We shall show later that the polarization M may be interpreted as the
density of a distribution of magnetic moment. The partial field derived
from n*may be imagined to have its origin in magnetic dipoles and is
said to be a field of magnetic type.
The electric polarization P may be induced in the dielectric by the
field E, but it may also contain a part whose magnitude iscontrolled
by wholly external factors. In the practical application of the theory
one is interested usually only in this independent part Po,which will be
Snc. 1.111 THE HERTZ VECTORS, OR POLARIZATION POTENTIALS
31
shown to represent the electric moment of dipole oscillators activated by
external power sources. The same is true for the magnetic polarization.
To represent these conditions we shall write ( 6 ) , page 11, in the modified
form
D = EE Po,
H = -1 B - Mo,
(55)
+
P
in which Po and Mo are prescribed and independent of E and H,and
where the induced polarizations of the medium have again been absorbed
into the parameters e and p. Then the electromagnetic field due to these
distributions of Poand Mo is determined by
when n and
n*are solutions of
1
a2n
V 2 n - p~ - = --Po,
(58)
at2
a m*
V211* - p~ at2 = -Mo.
In virtue of the second of Eqs. (58) and of the identity (16) we may aIso
write (57) as
H=eV X
(59)
an + V
dt
X
V X III*-Mo.
Since B = V x A, it is evident from this last relation that the vector
potential A may be derived from the Hertzian vectors by putting
A
(60)
where
with
an + pV
= pe-
at
X 11*
+ is an arbitrary scalar function.
- V+,
The associated potential
+ is
+ subject only to the condition that it satisfy
The extension of these equations to a homogeneous conducting
medium follows without difficulty. The reader will verify by direct
substitution that the system (1b)-(IVb), in a medium which is free Gf
32
[CHAP.I
THE FIELD EQUATIONS
fixed polarization Po and MO, is satisfied by
(63)
an *
at ’
+ v x v x n*,
E = V X V XII-pV X
H
=
vx
( c % + u ~ )
-
1.12. Complex Field Vectors and Potentials.-It has been shown by
Silberstein, Bateman, and others that the equations satisfied by the
fields and potentials may be reduced to a particularly compact form by
the construction of a complex vector whose real and imaginary parts are
formed from the vectors defining the magnetic and electric fields.’ The
procedure has no apparent physical significance but frequently facilitates
analysis.
Consider again a homogeneous, isotropic medium in which D = EE,
B = pH. If now we define Q as a complex field vector by
(66)
Q=B+~&E,
the Maxwell equations (I)-(IV) reduce to
(67)
V X Q + i G dQ
X
= pJ,
V.Q
=i$p.
The vector operation V X Q may be eliminated from (67) by the
simple expedient of taking the curl of both members. By the identity
(16) we obtain
which, on replacing the curl and divergence of Q by their values from
(67), reduces to
When this last equation is resolved into its real and imaginary com‘SILBERSTEIN,
Ann. phys., 22, 24, 1907. Also Phil. Mag. (6) 23, 790, 1912.
BATEMAN,
“Electrical and Optical Wave Motion,” Chap. I, Cambridge University
Press.
SEC.1.121
COMPLEX FIELD VECTORS A N D POTENTIALS
33
ponents, one obtains the equations satisfied individually by the vectors
E and H.
82H
V'H-epS
(70)
=
-V x J,
Next, let us define Q in terms of complex vector and scalar potentials
L and CP by the equation
(72)
Q
=
vxL
subject to the condition
-
aL
at
v L + r p -a+
at
=
*
(73)
- iz/GV+,
-idep-
0.
It will be verified without difficulty that (72) is an integral of (67) provided the complex potentials satisfy the equations
PL
(74)
V2L - '
1
.
= 1
-pJ,
7
dt
(75)
v2CP- €
8%
p i =
at
1
--
p.
E
If the real and imaginary parts of these potentials are written in the
form
and substituted into (72), one finds again after separation of reals and
imaginaries the general expressions for the field vectors deduced in
Eqs. (21) and (22).
If the free currents and charges are everywhere zero in the region
under consideration, Eq. (67) reduces to
(77)
V x Q
+i
d -~aQp =
x 0,
V - Q = 0.
The electromagnetic field may now be expressed in terms of a single
complex Hertzian vector r.
(78)
Q
=
~ E XV
a r fifiev
X V X
where r is any solution of
(79)
v2r
a2r
- Ep-
at2
=
0.
r,
34
T H E FIELD EQUATIONS
If, finally,
[CHAP.
1
r is defined as
r = n - i$n*
(80)
and substituted into (781, one finds again after separation into real and
imaginary parts exactly the expressions (47) and (50) for the electric
and magnetic field vectors.
When the mcdium is conducting, the field equations are no longer
symmetrical and the method fails. The difficulty may be overcome
if the field varies harmonically. The time then enters explicitly as a
factor such as ekiwt. After differentiating with respect to time, the
system (16)-(IVb) may be made symmetrical by introducing a complex
inductive capacity
E’
=
u
E
C i-.
w
BOUNDARY CONDITIONS
1.13. Discontinuities in the Field Vectors.-The
validity of the field
equations has been postulated only for ordinary points of space; that is
t o say, for points in whose neighborhood the physical properties of the
medium vary continuously. However, across any surface which bounds
one body or medium from another there occur sharp changes in the
parameters E , p , and u. On a macroscopic scale these changes may
usually be considered discontinuous and hence the field vectors themselves
may be expected to exhibit corresponding discontinuities.
Let us imagine at the start that the surface S which bounds medium
(1) from medium ( 2 ) has been replaced by a very thin transition layer
within which the parameters e, p, u vary rapidly but continuously from
their values near S in (1) to their values near S in (2). Within this
layer, as within the media (1) and ( Z ) , the field vectors and their first
derivatives are continuous, bounded functions of position and time.
Through the layer we now draw a small right cylinder, as indicated in
Fig. 2a. The elcments of the cylinder are normal to S and its ends lie
in the surfaces of the layer so that they are separated by just the layer
thickness Al. Fixing our attention first on the field of the vector B, we
have
(1)
$B-nda=O,
when integrated over the walls and ends of the cylinder. If the base,
whose area is Aa, is made sufficiently small, i t may be assumed that B
has a constant value over each end. Neglecting differentials of higher
order we may approximate (1) by
(2)
- + B - n2)Aa + contributions of the walls
(B nl
=
0.
SEC.1.131
D I S C O N T I N U I T I E S I N T H E FIELD VECTORS
35
The contribution of the walls to the surface integral is directly proportional t o AZ. Now let the transition layer shrink into the surface S.
I n the limit, as A1 --to, the cnds of the cylinder lie just on either side
of S and the contribution from the walls becomes vanishingly small.
The value of B a t a point on X in medium (1) will be denoted by B1,while
FIG.2a.--For
the normal boundary condition.
the corresponding value of B just across the surface in (2) will be denoted
b y Bz. We shall also indicate the positive normal t o S by a unit vector
n drawn from (1) into (2). According t o this convention medium (1)
lies on the negative side of S , medium (2) on the positive side, and
nl = -n. Then as A1 + 0, Aa 3 0,
-
(Bz- B,) n
(3)
= 0;
the transition of the normal component of B across a n y surface of discontinuity in the medium i s continuous. Equation (3) is a direct consequence
of the condition V B = 0, and is sometimes called the surface divergence.
S
(1) El>,u,.O1
FIG.2b.--For tho tangential boundary condition.
The vector D may be treated in the same manner, but in this case
the surface integral of the normal component over a closed surface is
equal to the total charge contained within it.
(4)
$ D - n da = (I.
The charge is distributed throughout the transition layer with a density p. As the ends of the cylinder shrink together, the total charge q
remains constant, for i t c.annot be destroyed, and
(5)
q
= p A1 Aa.
I n the limit as A1 --+ 0, the volume density p becomes infinite. It. is then
convenient to replace the product p A1 by a surface density W, defined as
36
T H E FIELD EQUATIONS
[CHAP.
I
the charge per unit area. The transition of the normal component of the
vector D across any surface X is now given by
-
(Dz - D J n = W .
(6)
The presence of a layer of charge on S results in a n abrupt change i n the
normal component of D, the amount of the discontinuity being equal to the
surface density measured in coulombs per square meter.
Turning now to the behavior of the tangential components we replace
the cylinder of Fig. 2a b y a rectangular path drawn as in Fig. 2b. The
sides of the rectangle of length As lie in either face of the transition layer
and the ends which penetrate the layer are equal in length to its thickness Al. This rectangle constitutes a contour COabout which
(7)
where So is the area of the rectangle and no its positive normal. The
direction of this positive normal is determined, as in Fig. 1, page 8, b y
the direction of circulation about CO. Let T I and - 2 be unit vectors in
the direction of circulation along the lower and upper sides of the rectangle as shown. Neglecting differentials of higher order, one rnay
approximate (7) by
(8)
(E
- +Er1
T,)
As
+ contributions from ends = --aB
- noAs Al.
at
As the layer contracts to the surface S, the contributions from th e segments a t the ends, which are proportional to All become vanishingly
small. If n is again the positive normal to S drawn from (1) into (2),
we may define the unit tangent vector T by
(9)
T
=
no x n.
Since
(10)
no x n - E = n o . n x E ,
we have in the limit as A1 ---f 0, A s + 0,
The orientation of the rectangle - and hence also of no - is entirely
arbitrary, from which i t follows th at the bracket in (11) must equal
zero, or
(12)
n x (Ez- El)
=
- lim
AZ+0
dB
-
at
AZ.
SEC.1.131
DISCONTINTJITIES I N THE FIELD VECTORS
37
The field vectors and their derivatives have been assumed to be bounded ;
consequently the right-hand side of (12) vanishes with A l .
(13)
n X (Ez - El)
= 0.
T h e transition of the tangential components of the vector E through a surface
of discontinuity i s continuous.
The behavior of H a t the boundary may be deduced immediately
from (12) and the field equation
We have
n X (Hz
- H1)
=
lim
AZ+O
The first term on the right of (15) vanishes as A1 -+ 0 because D and its
derivatives are bounded. If the current density J is finite, the second
term vanishes as well. It may happen, however, that the current
I = J noA s A1 through the rectangle is squeezed into a n infinitesimal
layer on the surface S as the sides are brought together. It is convenient to represent this surface currant by a surface density K defined
as the limit of the product J A 1 as A1 + 0 and J --+ a. Then
(16)
n x (Hz - H1) = K.
When the conductivities of the contiguous media are finite, there can be
no surface current, for E is bounded and hence the product aE A1 vanishes with Al. I n this case, which is the usual one,
(17)
n x (Hz - HI) = 0,
(finite conductivity).
Not infrequently, however, it is necessary to assume the conductivity
of a body to be infinite in order to simplify the analysis of its field. One
must then apply (16) as a boundary condition rat>herthan (17).
Summarizing, we are now able t o supplement the field equations by
four relations which determine the transition of a n electromagnetic
field from one medium to another separated by a surface of discontinuity.
(18)
n . (Bz- B1) = 0,
n x (El - El) = 0,
n x (Hz - HI) = K,
n (D2 - D1) = W .
-
From them follow immediately the conditions for the transition of the
normal components of E and H.
38
T H E FIELD EQUATIONS
[CHAP.
1
Likewise the tangential components of D and B must satisfy
COORDINATE SYSTEMS
1.14. Unitary and Reciprocal Vectors.-It is one of the principal
advantages of vector calculus that the equations defining properties
common to all electromagnetic fields may be formulated without reference
to any particular system of coordinates. To determine the peculiarities
that distinguish a given field from all other possible fields, it becomes
necessary, unfortunately, to resolve each vector equation into an equivalent scalar system in appropriate coordinates.
In a given region let
(l)
y, z>,
u1 =
u2 = f2(2~
y,
u3 = f3(Z1
Yt z>,
be three independent, continuous, single-valued functions of the rectangular coordinates x. y, z. These equations may be solved with respect
to x, y, z, and give
(2) x
=
Cpl(U1,
u2, u3),
y =
CpZ(U1,
u2, u3),
z = cp3(u1, u2, u3),
three functions which are also independent and continuous, and which
are single-valued within certain limits. In general the functions cpI as
well as the functions fi are continuously differentiable, but at certain
singular points this property may fail and due care must be exercised in
the application of general formulas.
y, z) in the region there is associated by means of
With each point P(x,
(1) a triplet of values ul, u2,u3;inversely (within limits depending on the
boundaries of the region) thcre corresponds to each triplet ul, u2, u3 a
definite point. The functions ul, u2,u3 are called general or curvilinear
coordinates. Through each point P there pass three surfaces
(3)
u1 =
constant,
u2 = constant,
u3 = constant,
called the coordinate surfaces. On each coordinate surface one coordinate is constant and t w o are variablc. A surface will be designated by
the coordinate which is constant. Two surfaces intersect in a curve,
called a coordinate curve, along which two coordinates are constant and
one is variable. A coordinate curve will be designated by the variable
coordinate.
Let r denote the vector from an arbitrary origin to a variable point
P(z, y, 2). The point, and consequently also its position vector r, may
be considered functions of the curvilinear coordinates u ] , u2,u3.
(4)
r = r(ul, u2,u3).
Next Page
SEC.1.141
39
U N I T A R Y A N D RECIPROCAL VECTORS
A differential change in r due to small displacements along the coordinate
curves is expressed b y
(5)
dr
dr
= - du'
aul
+ -au,dr
du2
dr
+ au3
du3.
~-
Now if one moves unit distance along the ul-curve, the change in r is
directed tangentially to this curve and is equal to dr/dul.
The vectors
are known as the unitary vectors associated with the point P.
constitute a base system of reference for
all other vectors associated with that
particular point.
+
They
+
(7) dr = al du'
a2du2 a3du3.
It must be carefully noted that the
unitary vectors are not necessarily of unit
length, and their dimensions will depend
on the nature of the general coordinates.
The three base vectors al, a,, a3 define a parallelepiped whose volume is
(8) V = al * (a,
x a3) = a2* (aa x al)
= a3 (al x ad.
-
FIG.
S.-Base vectors for a curvilinear
coordinate system.
The three vectors of a new triplet defined by
are respectively perpendicular to the plancs determined by the pairs
(az, a3), (a3, al), (al, a2). Upon forming all possible scalar products of
the form a i . ai, i t is easy to see that they satisfy the condition
(10)
ai. a1. =
6ii,
where 6ii is a commonly used symbol denoting unity when i = j , and
zero when i # j . The unitary vectors can be expressed in terms of the
system a', a2, a3by relations identical in form.
Any two sets of noneoplanar vectors related b y the Eqs. (8) to (11) are
said to constitute reciprocal systems. The triplets al, a2, a3 are called
reciprocal unitary vectors and they may serve as a base system quite
as well as the unitary vectors themselves.
Previous Page
40
[CHAP.I
T H E FIELD EQUATIONS
If the reciprocal unitary vectors are employed as a base system, the
differential dr will be written
dr = a' d u l
(12)
+ a2duz + a3du3.
The differentials dul, duz, du3 are evidently components of dr in the directions defined by the new base vectors. The quantities UI, uz, u3 are
functions of the coordinates ul, u2, u3,but the differentials d u l , duz, du3
are not necessarily perfect. On th e contrary they are related to the
differentials of the coordinates by a set of linear equations which in
general are nonintegrable. Thus equating (7) and (12), we have
3
3
dr
(13)
aidui =
=
i=l
2 aidui.
j=1
Upon scalar multiplication of (13) b y ai and by ai in turn, we find, thanks
to (10) :
3
(14)
dui
3
-
ai a; dui,
=
dui
i=l
-
ai ai dui.
=
j=1
It is customary to represent the scalar products of the unitary vectors
and those of the reciprocal unitary vectors by the symbols
The components of dr in the unitary and in the reciprocal base systems
are then related by
A fixed vector F a t the point P may be resolved into components
either with respect to the base syst,em all azl a3, or with respect t o th e
reciprocal system all a2, a3.
3
F
=
xfiai
i=l
3
=
2jiai.
j=1
The components of F in the unitary system are evidently related t o those
in its reciprocal system by
3
(19)
jl
=
2 giifi,
i=l
c
3
fi = j=1 Siifi,
and in virtue of the orthogonality of the base vectors aj with respect to
SEC.1.141
41
U N I T A R Y A N D RECIPROCAL VECTORS
the reciprocal set ai as expressed by (lo), we may also write
fi = F . ai,
(20)
F . a+
fi =
It follows from this that (18) is equivalent to
The quantities? are said t o he the contravariant components of the
vector F, while the components fi are called covariant. A small letter
has been used to designate these components to avoid confusion with the
components Fl, F2, F3 of F with respect t o a base syst,em coinciding with
the ai but of unit length. It has been noted previously that the length
and dimensions of the unitary vectors dcpend on the nature of the
curvilinear coordinates. An appropriate set of unit vectors which, like
the unitary set ai, are tangent to the ui-curves, is defined by
(22)
il
1
a1
= ____
- -G
l
i2 =
filal'
1
- ~-
a2,
6;
1
13
= __-
1/933
a3,
and, hence,
F
(23)
= Flil
+ Fzi2+ Fa&,
with
(24)
Fi
=
6 f i .
The F i are of the same dimensions as the vector F itself.
The vector dr represents a n infinitesimal displacement from the point
P(u', u2, u3) to a neighboring point whose coordinates are u' du',
u2 du2, u3 du3. The magnitude of this displacement, which constitutes a line element, we shall denote by ds. Then
+
+
+
or, in the notation of (15) and (16),
3
3
The gij and g i i appear here as coefficients of two differential quadratic
forms expressing the length of a line element in the space of the general
42
T H E FIELD EQUATIONS
[CHAP.
1
coordinates ui or of its reciprocal set ui. They are commonly called the
metrical coeficients.
It is now a relatively simple matter to obtain expressions for elements
of arc, surface, and volume in a system of curvilinear coordinates. Let
dsl be an infinitesimal displacement a t P(u', u2, u3)along the ul-curve.
dsl = al dul,
(27)
dsl = (dsll =
du1.
Similarly, for rlcmcnts of length along the u2- and uhcurves, we have
dsz
(28)
=
6du2,
ds3 =
dz
du3.
Consider next a n infinitesimal parallelogram in the ul-surface bounded
b y intersecting u2- and u3-curves as
indicated in Fig. 4. The area of such
an element is equal in magnitude to
(29) dual
= Ids2 X d~31
= laz X a3/du2du3
-
= d ( a 2x as) (az x as) du2 du3.
By a well-known vector identity
-
=
FIG.4.-Elernent
(31)
of area in the u*-surface.
-
-
(a x b) (c x d)
(a c)(b d) - ( a . d)(b c),
(30)
-
-
where a, b, c, d are any four vectors,
and hcnce
-
-
-
(az X a,) (a2 X as) = (az az) (as .a3) - (a2 a3)(a3 az)
=
922933
- 9223.
For the area of an clement in the u'-surface we have, therefore,
dai = d g z z g s - g&
(32)
d U 2 du3,
and similarly for elements in thc u2- and u3-surfaces,
daz
da3
(33)
d U 3 dul,
= d g i i g 2 z - giz du' dU2.
=
dq33g11'-
g';
Finally, a volume element bounded by coordinate surfaces is written as
(34)
dv
=
If now in (21) we let F
(35) az x a3
=
-
-
dsl dsz X ds3 = al a2 X a3du' du2 du3.
=
a2 x a3, we have
(a' az x a3)al
+ (a2- a2 x a3)az+ (a3 az x a$)a3,
SEC.1.141
U N I T A R Y A N D RECIPROCAL VECTORS
43
or, on replacing the ai by their values from (8) and (9))
(36)
-
a1 a2 X a3
-
-
a1
[(az x a3 az x a3)al
ale az x a3
(a3 x a1 az x a3)az (al x az az x a3>a3].
= _____
-
+
+
The quantities within parentheses can be expanded by (30) and the terms
arranged in the form
(37)
-
-
-
-
-
.
(a1 a2 x a3)z= a1 al[(a2 az>(a3a3) - (az a3)(a3 aZ)l
a1 az[(az a3) (a3 * al) - (az al)(a3 ad]
a1 * ad(a2 ad (a3 az) - (az az)(aa all].
+ -
-
-
-
+
-
-
-
.
If finally the scalar products in ( 3 7 ) are replaced by their gii, we obtain
as an expression for a volume element
dv
(38)
=
~du1du2du3,
in which
(39)
A corresponding set of expressions for the elements of arc, area, and
volume in the reciprocal base system may be obtained by replacing the
g.i by the gii, but they will not be needed in what follows.
Clearly the coefficients g l i are sufficient t o characterize completely
the geometrical propcrties of space with respect to any curvilinear system
of coordinates; i t is therefore essential that we know how these coefficients
may be determined. To unify our notation we shall represent the
rectangular coordinates z, y, z of a point P by the letters xl, xz,x 3 respectively. Then
(40)
dsz
=
+
( d ~ l ) ~(dx')'
+( d ~ ~ ) ~ ,
I n this most elementary of all systems the metrical coefficients are
(41)
gil
= 6tjl
(&
=
1,
6,j
=
0 when i # j).
From the orthogonality of the coordinate planes and the definition (9))
it is evident that the unitary and the reciprocal unitary vcctors are
identical, are of unit length, and are the base vectors customarily represented by the letters i, j, k.
Suppose now that the rectangular coordinates are related functionally
t o a set of general coordinates as in (2) by the equations
(42) z1= z l ( u l ,u2,u31,
zz =
z2(u1,u2,u 3 ) ,
x3
=
x3(u1,u2,u3).
44
T H E FIELD EQUATIONS
[CHAP.I
The differentials of the rectangular coordinates are linear functions of the
differentials of the general coordinates, as we see upon differentiating
Eqs. (42).
(43)
According to (26) and (40)
3
(44)
ds2 =
3
3
22
gii
d d dui =
2 (dxk)2,
k=l
i = l j=1
whence on squaring the differentials in (43) and equating coefficients
of like terms we obtain
(45)
1.16. The Differential Operators.-The gradient of a scalar function
+(ul,u2, u3)is a fixed vector defined in direction and magnitude as the
with respect to the cordinates. The
maximum rate of change of
variation in + incurred during a displacement dr is, therefore,
+
Now the dui are the contravariant components of the displacement vector
dr, and hence by (20),
(47)
-
dui = ai dr.
This value for dui introduced into (46) leads to
and, since the displacement dr is arbitrary, we find for the gradient of a
scalar function in any system of ciirvilinear coordinates :
(49)
In this expression the reciprocal unitary vectors constitute the base
SEC.1.151
45
T H E DIFFERENTIAL OPERATORS
system, but these may be replaced by the unitary vectors through the
transformation
3
a t u2, the right at u2
(55)
+ du2.
The area
l
0
J
46
THE FIELD EQUATIONS
[CHAP.
1
The curl of the vector F is found in the same manner b y calculating
the line int>cgral of F around an infinitesimal closed path. According
to Eq. (21), page 7, the component of the curl in a direction defined
by a unit normal n is
-
(V x F) n
=
lim
c-io
-
s
(F a3d
0
4
~
-
~ - )(F ~a3d ~~ ~ +) , ~~;
from th e bottom and top parallel t o u2-
a?\,
(57)
This quantity must now be divided by the area of the rectangle, or
d ( a z x as) (a2 x a3) du2du3. As for the unit normal n, we note th a t
the reciprocal vector a', not the unitary vector al, is always normal to the
ul-surface. Its magnitude must be unity; hence
-
These values introduced into (56) now lead to
(59)
a1
(v x F) - dG*
-
1
By (9) and f37)
(60)
-
a2 x a3 = [al (az x a3>]al= d i a l ;
~
~
47
0RT HOGON A L S YST E M S
SEC.1.161
hence (59) reduces t o
The two remaining components of V X F are obtained from (61) by
permutation of indices. Then by (21)
3
vx
F
(v x
=
i=l
-
F a“>;.
Remembering that F * a; is the covariant component fi, we have for the
curl of a vector with respect t o a set of general coordinates
Finally, we consider the operation V24, by which we must understand
We need only let F = V + in ( 5 5 ) . The contravariant components of the gradient are
v - v+.
3
3
f ” = ~ . a i = C a i . .a84
J-==gii~.
(64)
dUi
j=1
j=l
dUj
Then
1.16. Orthogonal Systems.-Thus far no restriction has been imposed
on the base vectors other than that they shall be noncoplanar. Now it,
happens that in almost all cases only the orthogonal systems can be
usefully applied, and these allow a considerable simplification of the:
formulas derived above. Oblique systems might well be of the greatest
practical importancc; but they lead, unfortunately, t o partial differential
equations which cannot be mastered by present-day analysis.
The unitary vectors al, az, a3of a n orthogonal system are by definition
mutually perpendicular, whence i t follows that ai is parallel t o ai and
is its reciprocal in magnitude.
1
1
a$ = ___
ai = - a(.
ai ai
gii
-
(66)
Furthermore
(67)
-
al a,
= dz
- a3 = a 3 . al = 0;
48
[CHAP.I
THE FIELD EQUATIONS
hence gij = 0, when i # j. It is customary in this orthogonal case t c
introduce the abbreviations
The hi may be calculated from the formula
although their value is usually obvious from the geometry of the system.
The elementary cell bounded by coordinate surfaces is now a rectangular
box whose edgcs are
(71)
dsl
=
hi du',
dsz
=
hz dU2,
d ~ 3
=
h3 du3,
and whose volume is
(72)
dv
=
hlhZh3 du' dU2du3.
All off-diagonal terms of the determinant for g vanish and hence
(73)
& = hihzh3.
The distinction between the contravariant and covariant components
of a vector with respect to a unitary or reciproca1,unitary base system is
essential to an understanding of the invariant properties of the differential
operators and of scalar and vector products. However, in a fixed reference system this distinction may usually be ignored. It is then convenient to express the vector F in terms of its components, or projections,
F1,FZ,P3 on an orthogonal base system of unit vectors il, i2, i3. By (22)
and (66)
.
1.
a., - h.i.
aE = - 1 ; .
(74)
h,
%,
In terms of the components Fi the contravariant and covariant components are
1
fi = -Fi,
fz = h z ; .
(75)
h,
Also
The gradient, divergence, curl, and Laplacian in an orthogonal system
of curvilinear coordinates can now be written down directly from the
results of the previous section.
SEC.1.161
49
ORTHOGONAL SYSTEMS
From (49) we have for the gradient
According to (55) the divergence of a vector F is
For the curl of F we have by (63)
It may be remarked that (80) is t’he expansion of the determinant
Finally, the Laplacian of an invariant scalar 4 is
By an invariant scalar is meant a quantity such as temperature or
energy which is invariant to a rotation of the coordinate system. The
components, or measure numbers, Fi of a vector F are scalars, but they
transform with a transformation of the base vectors. Now in the
analysis of the field we encounter frequently the operation
(83)
V X V
X
F
=
V V - F- V - V F .
-
No meaning has been attributed as yet to V VF. In a rectangular,
Cartesian system of coordinates xl, x2,z3,it is clear that this operation is
equivalent to
L e . , the Laplacian acting on the rectangular components of F.
I n gener-
50
T H E FIELD EQUATIONS
[CHAP.I
alized coordinates V x V x F is represented by the determinant
la
(85) V X V X F = s
I
-
The vector V VF may now be obtained by subtraction of (85) from the
expansion of V V F, and the result differs from that which follows
a direct application of the Laplacian operator to the curvilinear components of F.
1.17. The Field Equations in General Orthogonal Coordinates.-In
any orthogonal system of curvilinear coordinates characterized by the
coefficients hl, hz, h3, the Maxwell equations can be resolved into a set of
eight partial differential equations relating the scalar components of the
field vectors.
-
SEC.1.181 PROPERTIES OF SOME ELEMENTARY SYSTEMS
51
It is not feasible to solve this system simultaneously in such a manner
as to separate the components of the field vectors and to obtain equations
satisfied by each individually. .In any given problem one must make thc
most of whatever advantages and peculiaritics the various coordinate
systems have to offer.
1.18. Properties of Some Elementary Systems.-An
orthogonal
coordinate system has been shown to be completely characterized by the
three metrical coefficients, h l , hz, h3. These parameters will now be
determined for certain elementary systems and in a few cases the diffcrential operators set down for convenient
23
reference.
1. Cylindrical Coordinates.--Let P' be
the projection of a point P ( z , y, z ) on the
z-plane and r, 0 be the polar coordinates of P' in this plane (Fig. 7 ) . The
variables
(87)
x
=
r cos 8,
y
=
r sin 8,
z
=
z.
The coordinate surfaces are coaxial cylinders of circular cross section
intersected orthogonally by the planes B = constant and z = constant.
The infinitesimal line element is
ds2 = dr2
(88)
+ r2de2 + dz2,
whence it is apparent that the metrical coefficients are
(89)
If
hl
=
1,
ha = r,
hS = 1.
+ is any scalar and F a vector function we find:
VG
=
a+ it 4- -189.
- 1.
a+ i,.
+-
52
THE FIELD EQUATIONS
2. Spherical Coordinates.-The
(91)
u1
=
variables
u2
l-1
[CHAP.1
=
0,
u3
=
41
related to the rectangular coordinates by the transformation
(92)
= r sin
x
e cos 4,
y
=
r sin 0 sin 4,
z =
r cos 0,
are called the spherical coordinates of the point P. The coordinate
surfaces, r = constant, are concentric spheres intersected by meridian
planes, 4 = constant, and a family of cones, 0 = constant. The unit
vectors i,, i, i, are drawn in the direction of increasing r, 8, and 4 such as to
constitute a right-hand base system, as
indicated in Fig. 8. The line element
is
(93)
2
Js2 = dr2
+ r2de2
+ r2 sin2 0 d@,
whence for the metrical coefficients we
obtain
(94)
hl
=
h2 = r,
1,
h3 = r sin
FIG.8.-Spherical
coordinates.
These values lead to
a*
la*.
1 a+.
~#=-i~+--i~+-dr
r a0
r sin e a+ la*
V
F
i a
= -r 2 ar (r2F1)
8.
--
i a
1 8FS
+ -r sin e a e (sin OFz) + r sin e a+'
3. Elliptic Coordinates.-Let two fixed points PI and PZbe located a t
x = -c on the x-axis and let rl and r2 be the distances of a
variable point P in the x-plane from P1 and P2. Then the variables
1
= c and
(96)
defined by equations
(97)
u1 =
E,
u2
= 7,
u3
=
2,
SEC.1.181
PROPERTIES OF SOME ELEMENTARY S Y S T E M S
From these relations i t is evident that
are called elliptic coordinates.
5 I 1,
(98)
53
- 1 5 9 5 1.
The coordinate surface, 4 = constant, is a cylinder of elliptic cross
section, whose foci are P1 and 1’2. The semimajor and semiminor axes
of an ellipse E are given by
(99)
a = c&
b
=
c d F i ,
and the eccentricity is
The surfaces, q = constant, represent a family of confocal hyperbolic
X
u= T
FIG.g.-Coordinates
of the elliptic cylinder. Ambiguity of sign is avoided by placing
E = cosh u , 7 = COB a.
cylinders of two sheets as illust,rated in Fig. 9.
two confocal systems are
Thc equations of these
from which we deduce the transformation
(102)
z = c[q,
y =c
d(p- 1)(1
- +),
2
=
2.
The variable 9 corresponds to the cosine of an angle measured from the
x-axis and the unit vectors il, iz of a right-hand base system are therefore
54
[CHAP.I
THE FIELD EQUATIONS
drawn as indicated in Fig. 9, with i3 normal to the page and directed
from the reader.
The metrical coefficients are calculated from (102) and (70)) giving
4. Parabolic Coordinates.-If r , 0 are polar coordinates of a variable
point in the z-plane, one may define two mutually orthogonal families
of parabolas by the equations
5
=
4%sin -,2e
The surfaces, t = constant and q
q =
=
constant, are intersecting parabolic
I
5=-5
4%cos -.2e
FIG.10.-Parabolic
7/=.-5
coordinates.
cylinders whose elements are parallel to the z-axis as shown in Fig. 10.
The parameters
u' = {,
(105)
u2
= 7,
u3= - 2
are called parabolic coordinates. Upon replacing r and 0 in (104) by
rectangular coordinates we find
(106)
p
= 1/22
+ - 2)
y2
12
= 1/22
+ +
y2
2)
SEC.1.181 PROPERTIES OF SOME ELEMENTARY SYSTEMS
55
whence for the transformation from rectangular to parabolic coordinates
we have
(107)
z=
g.112
- p),
y = 5.11,
2
=
-2.
The unit vectors il and i, are directed as shown in Fig. 10, with i, normal
to the page and away from the reader. The calculation of the metrical
coefficients from (107) and (70) leads to
(108)
hl = hz = dp
+
72,
h3
=
1.
5. Bipolar Coordinates.-Let, Pl and P z be two fixed points in any
z-plane with the coordinates (a, 0), ( - u , 0) respectively. If 5 is a
parameter, the equation
(109)
(X -
u coth [)'
+ y2 = u2 CsCh2 4,
describes two families of circles whose centers lie on the z-axis. These
two families are symmetrical with respcct to the y-axis as shown in
I
FIG.11.--Bipolar coordinates.
+
Fig. 11. The point P1 at (u, 0) corresponds to 5 = QI , whereas its
image P , at (-a, 0 ) is approached when 5 = - to. The locus of (log),
when $, = 0, coincides with the y-axis. The orthogonal set is likewise
a family of circles whose centers all lie on the y-axis and all of which pass
through the fixed points P1and Pa. They are defined by the equation,
(110)
5 2
+ (y - a cot
r])2
= a 2 csc2 7,
56
[CHAP.I
THE FIELD EQUATIONS
wherein the parameter q is confined t o the range 0 5 q S ZT. I n order
that the coordinates of a point P in a given quadrant shall be singlevalued, each circle of this family is separated into two segments by the
points P I and Pz. A value less than ?r is assigned to the arc above the
x-axis, while the lower arc is denoted by a value of q equal to T plus
the value of q assigned to the upper segment of the same circle.
The variables
(111)
?J1 = f ,
u2 = q,
u3
= 2,
are called bipolar coordinates. From (109) and (110) the transformation
to rectangular coordinates is found t o be
(112)
5 =
a sinh [
cod1 - cos q ’
=
a sin q
cash c; - cos q’
= ”
The unit vectors il and i2 are in the direction of increasing f and q as
indicated in Fig. 11, while is is directed away from the reader along the
z-axis. The calculation of the metrical coefficients yields
6. Spheroidal Coordinates.-The coordinates of the elliptic cylinder
were generated by translating a system of confocal ellipses along the
z-axis. The spheroidal coordinates are obtained by rotation of the
ellipses about an axis of symmetry. Two cases are to be distinguished,
according to whether the rotation takes place about the major or about
the minor axis. In Fig. 9 the major axes are oriented along the z-axis of
a rectangular system. If the figure is rotated about this axis, a set of
confocal prolate spheroids is generated whose orthogonal surfaces are
hyperboloids of two sheets. If 4 measures the angle of rotation from
the y-axis in the x-plane and r the perpendicular distance of a point
from the x-axis, so that
(114)
y
=
r cos 4,
z
=
r sin 4,
then the variables
(115)
u1
= f,
u2 = q,
263
=
4,
defined by (97) and (114) are called prolate spheroidal coordinates. In
place of (101) we have for the equations of the two confocal systems
SEC.1.181
57
PROPERTIES OF SOME ELEMENTARY S Y S T E M S
from which we deduce
(117)
2 =
c.&,
d r p - 1)(1 - q z ) cos 4 ,
y =c
z = c d ( ” - 1)(1 - 72) sin
t L 1,
(118)
-1STJS1,
4,
OSCpS27r.
A calculation of the metrical coefficients gives
(119)
hl
=
c
JF, Jq, d(P
hz
2 - 1
=c
ha
1-
- 1)(1 - 7”.
=c
When the ellipses of Fig. 9 are rotated about the y-axis, the spheroids
are oblate and the focal points P I , P , describe a circle in the plane y = 0.
Let r, 4, y, be cylindrical coordinates about the y-axis,
r eos 6,
z =
(120)
z =
T
sin 4.
If by P1 and P , we now understand the points where the focal ring of
radius c intercepts the plane 4
constant, t h e variables 4 and TJ are
still defined by (97) ; but for the equations of the coordinate surfaces we
have
from which we deduce the transformation from oblatc spheroidal
coordinates
(122)
u1
t,
u2
y =c
d(t2
- 1)(1 - $),
=
= q,
u3
=
4,
to rectangular coordinates
(123)
5 = c.$q
sin 4,
z
= ~ $ cos
7
+.
The surfaces, E = constant, are oblate spheroids, whereas the orthogonal
family, TJ = constant, are hyperboloids of one sheet. The metrical
coefficients are
The practical utility of spheroidal coordinatrs may be surmised from
the fact that as the eccentricity approaches unity the prolate spheroids
berome rod-shaped, whereas the oblate spheroids degenerate into flat,
elliptic disks. I n the limit, as the focal distance 2c and the eccentricity
approach zero, the spheroidal coordinates go over into spherical coordinates, with 5 --+ r , 7 + cos 0.
7. Paraboloidal Coordinates.--Another set of rotational coordinates
may be obtained b y rotating the parabolas of Fig. 10 about their axis
58
[CHAP.I
THE Y I E L D EQUATIONS
of symmetry.
The variables
are called paraboloidal coordinates. The surfaces, 5 = constant,
q = constant, are paraboloids of revolution about an axis of symmetry
which in this casc has been taken coincident with the z-axis. The plane,
y = 0, is cut by these surfaces along the curves
which are evidently parabolas whose foci are located at the origin and
whose parameters are t2and q 2 . The metrical coefficients are
hl
=
h,
=
d m ,
8. Ellipsoidal Coordinates.-The
2 2
y2
h3
= 511.
equation
22
-+y+-=l,
a2 b
c2
(a
> b > c),
is that of a n ellipsoid whose semiprincipal axes are of length a, b, c.
Then
are the equations respectively of an ellipsoid, a hyperboloid of one sheet,
and a hyperboloid of two sheets, all confocal with the ellipsoid (129).
Through cach point of space there will pass just one surface of each kind,
and to each point there will correspond a unique set of values for f , 7,{.
The variables
(131)
u' = '$,
uz = 7,
u3
=
(,
are called ellipsoidal coordinates. The surface, 5 = constant, is a
hyperboloid of one shect and 9 = constant, a hyperboloid of two sheets.
The transformation to rectangular coordinates is obtained b y solving
SEC.1.191
59
ORTHOGONAL T R A N S F O R M A T I O N S
(130) simultaneously for 2, y, z.
This gives
a 2 )(17
+ 4 (.r + a?
u”(c2
- a2)
The mutual orthogonality of the three families of surfaces may be verified
by calculating the c,oefficients g7. from (132) by means of (45). They are
zero when i # j ; for the diagonal terms we find
It is convenient to introduce the abbreviation
46’7U ’ ) ( S + b 2 ) ( s + c2),
For the Laplacian of a scalar + we then have
(134)
R,
=
(s =
t, 17, r).
THE FIELD TENSORS
1.19. Orthogonal Transformations and Their Invariants.-In
the
theory of relativity one undertakes the formulation of the laws of physics,
and in particular the equations of the clectromagnetic field, such t h a t
they are invariant t o transformations of the system of referenw.
Although in the present volume we shall have no occasion to examine the
foundations of the relativity theory, it will nevertheless prove occasionally advantageous t o employ the symmetrical, four-dimensional notation
introduced b y Minkowski and Sommerfeld and t o deduce the Lorcntz
transformation with respect to which the field equations are invariant.
To discover quantities which are invariant to a transformation from one
system of general curvilinear coordinates t o another, it is essential t h a t
60
[CHAP.I
THE FIELD EQUATIONS
one distinguish between the covariant and contravariant components
of vectors and between unitary and reciprocal unitary base systems.
For our present purposes i t will be sufficient, however, t,o confine t h e
discussion to systems of rectangular, Cartesiai coordinates in which, as
we have seen, covariant and contravariant components are identica1.l
Let il, iz,i3be three orthogonal, unit base vectors defining a rectangular coordinate system X whose origin is located a t the fixed point 0, and
let r be the position vector of any point P with respect to 0.
(1)
r
=
zlil
+ xziz+ z&,
and since
ii ik
(2)
= 8jk,
the coordinates of P in the system X are
-
xk = r ik.
(3)
Suppose now that i:, i;, i: are the base vectors of a second rectangular
system X' whosc origin coincides with 0 and which, therefore, differs
from X only by a rotation of the coordinate axes. Since
r
(4)
=
x;ii
+ xkil + zLih,
the coordinates of P with respect to X' are
(5)
z! = r . i!
=
.
.
xlil . ii. + xziz ii. + z3i3 il;
each coordinate of P in X' is a linear function of its coordinates in X ,
whereby the coefficients
(6)
ajk
=
il
.
ik
of the linear form are clearly the direction cosines of the coordinate axes
of X' with respect t o the axes of X. A rotation of a rectangular coordinate system effects a changc in the coordinates of a point which may be
represented by the linear transformation
(7)
The coefficients a j k arc subject to certain conditions which are a
consequence of the fact that the distance from 0 to P , t h a t is to say, the
1 This section is based essentially on the following papers: MINKOWSKI,
Ann.
Physik, 47, 927, 1915; SOMMEREELD,
Ann. Physik, 32, 749, 1910 and 33, 649, 1910;
MIE, Ann. Physik, 37, 511, 1912; PAULI,
Relativitatstheorie, in the Encyklopadie der
mathematischen Wissenschaften, Vol. V, part 2 , p. 539, 1920.
ORTHOGONAL T R A N S F O R M A T I O N S
Ssc. 1.191
61
magnitude of r, is independent of the orientation of the coordinate system.
whence it follows that
when i = k ,
when i f k.
Equation (10) expresses in fa& the relations which must prevail among
the cosines of the angles b c t m c n coordinate axes in order that they be
rectangular and which are, therefore, known as conditions of orthogonality. The system (7), when subject to (lo), is likewise called an
orthogonal transformation. As a direct consequence of (lo), i t may be
shown that the square of the determinant jajkl is equal to unity and hence
la,k[ = A1.
Any set of coefficients a j k which satisfy (10) define a n
orthogonal transformation in the sense that the relation (8) is preserved.
Geometrically the transformat ion (7) represents a rotation only when
the determinant Iajkl = +l. The orthogonal transformation whose
determinant is equal to -1 corresponds t o a n inversion followed by a
rotation.
Since the determinant of an orthogonal transformation docs not
ianish, the x k may be expressed as linear functions of the x:. These
relations are obtained most simply by writing as in ( 5 ) :
(11)
xk
=
-
-
-
r ie = xiii ik + zkii ik + xkii ie,
(L = 1, 2, q,
whence i t follows from (8) that,
Let A be any fixed vector in space, so that
3
A
(14)
=
2 .qkik
k-1
=
3
ALiL.
k=l
The component A; of this vector with respect t o the system X’ is given by
(15)
A!
=
.
A i!
3
3
A k i k .if.
=
k=l
2 aikAk;
:
k= 1
62
[CHAP.I
T H E FIELD EQUATIONS
thus the rectangular components of a fixed vector upon rotation of the
coordinate system transform like the coordinates of a point. Now
while evcry vector has in gencral three scalar components, it does not
follow that any three scalar quantities constitute the components of a
vector. I n order that three scalars A1, Az, AS m a y be interpreted as the
components of a vector, i t i s necessary that they transform like the coordinates
of a point.
Among scalar quantitics onc must distinguish the variant from the
invariant. Quantities such as temperature, pressure, work, and the like
are independent of the orientation of the coordinate system and are,
thcrefore, called invariant scalars. On the other hand the coordinates
of a point, and the measure numbers, or components, of a vector have
only magnitude, but they transform with the coordinate system itself.
We know that the product A B of two vectors A and B is a scalar, but a
scalar of what kind? In virtue of (12) and (13) we have
the scalar product of two vectors i s invariant to a n orthogonal transformation
of the coordinate system.
Let cp be an invariant, scalar and consider the triplet of quantities
(i = 1) 2) 3).
Now by (12),
(18)
and hence
the Bi transform like the components of a vector and therefore the gradient of 4,
calculated at a point P , i s a jixed vector associated with that point.
Let Ai be a rectangular component of a vector A, and
(21)
Then by (18) and (15),
SEC.1.191
whence, from
63
ORTHOGONAL TRANSFORMATIONS
(lo), i t follows th at
the divergence of a vector i s invariant to a n orthogonal transformation of the
coordinate system.
Lastly, since the gradient of an invariant scalar is a vector and since
the divergence of a vector is invariant, it folIows that the Laplacian
(24)
is invariant to an orthogonal tra.nsformation.
The transformation properties of vectors may be extended t o manifolds of more than three dimensions. Let xl, x2,x 3 , 5 4 be the rectangular
coordinates of a point P with respect to a reference system X in a fourdimensional continuum. The location of P with respect to a fixed origin 0
is determined by the vector
2 xjij, . . . = sik.
4
(25)
r
=
ii i k
j=1
The linear transformation
A
will be called orthogonal if the coefficients satisfy the conditions
c
4
ajzajk = 6a.
j=1
The characteristic property of an orthogonal transformation is that it
leaves the sum of the squares of the coordinates invariant:
2
4
2 (xi)Z.
4
=
JZ.(
j= 1
j=1
The square of the determinant formed from the a j k is readily shown to be
positive and equal to unity, and hence the determinant, itself may equal
1. However, if (26) is to includc the identical transformation
(29)
2'. = 2.
1
31
(j
=
1, 2, 3, 41,
it is obvious that the determinant must be positive. Henceforth we
shall confine ourselves to the subgroup of orthogonal transformations
64
T H E FIELD EQUATIONS
[CHAP.I
characterized by (27) and the condition
(30)
lUlkl =
+l.
The transformation then corresponds geometrically to a rotation of the
coordinate axes.
A four-vector is now defined as any set of four variant scalars
Ai (i = 1, 2, 3, 4) which transform with a rotation of the coordinate
system like the coordinates of a point.
4
4
It is then easy to show, as above, that the scalar product of two fourvectors and the four-dimensional divergence of a four-vector are invariant
to a rotation of the coordinate system.
4
k=l
4
~j=1
(33)
Furthermore the derivatives of a scalar,
(34)
(i = 1, 2, 3) 4)
transform like the components of a four-vector and hence the fourdimensional Laplacian of an invariant scalar,
(35)
is also invariant to an orthogonal transformation.
1.20. Elements of Tensor Analysis.-Although most physical quantities may be classified either as scalars, having only magnitude, or as
vectors, characterized by magnitude and direction, there are certain
entities which cannot be properly represented by either of thesc terms.
The displacement of the center of gravity of a metal rod, for example,
may be defined by a vector; but the rod may also be stretched along the
axis by application of a tension at the two ends without displacing the
center a t all. The quantity employed to represent this stretching must
thus indicate a double direction. The inadequacy of the vector concept
becomes all the more apparent when one attempts the description of a
volume deformation, taking into account the lateral contraction of the
SEC.
1.201
ELEMENTS OF TENSOR ANALYSIS
65
rod. I n the present section we shall deal only with the simpler aspects
of tensor calculus, which is the appropriate tool for the treatment of such
problems.
I n a three-dimensional continuum let each rectangular component
of a vector B be a linear function of the components of a vector A.
In order t ha t this association of the components of B with the components
of A in the system X be preserved as the coordinates are rotated, it is
necessary that the coefficients T2ktransform in a specific manner. The
T j k are therefore variant scalars.
A tensor-or more properly, a tensor of
rank two-will now be defined as a linear transformation of the components of a vector A int,o the components of a vector B which is invariant
to rotations of the coordinate system. The nine coefficients Tik of the
linear transformation are called the tensor components.
To determine the manner in which a tensor component must transform we write first (36) in the abbreviated form
(37)
If (37) is to be invariant to the transformation defined by
then the Tjk must transform to Til such that
3
(i = 1, 2, 3).
(39)
Multiply (37) by a;j and sum over the index j.
3
66
T H E FIELD EQUATIONS
[CHAP.
I
The components of a tensor of rank two transform according to the law
3
(43)
Ti', =
2
3
UijaikTjk,
(i, z
=
1, 2, 3);
j=1 k=l
inversely, any set of nine quantities which transform according to (43)
constitutes a tensor.
By an analogous procedure one can show that the reciprocal transformation is
(44)
If the order of the indices in all the components of a tensor may be
changed with no resulting change in the tensor itself, so that Tjk = Tki,
the tensor is said to be completely symmetric. A tensor is completely
antisymmetric if an interchange of the indices in each component results
in a change in sign of the tensor. The diagonal terms Tij of an antisymmetric tensor evidently vanish, while for the off-diagonal terms,
= Tkj, then also Ti', = Tii.
Tjk = - Tkj. It is clear from (43) that if
Likewise if Ti, = -Tkj, it follows that Ti, = -Tii. The symmetric or
antisymmetric character of a tensor is invariant to a rotation of the
coordinate system.
The sum or difference of two tensors is constructed from the sums or
differences of their corresponding components. If 2Ris the sum of the
tensors 2S and 2T,1
its components are by definition
I n virtue of the linear character of (43) the quantities Rjk transform like
the S j k and Tjk and, therefore, constitute the components of a tensor 2R.
From this rule it follows that any asymmetric tensor may be represented
as the sum of a symmetric and an antisymmetric tensor. Assuming 2R
to be the given asymmetric tensor, we construct a symmetric tensor 2S
from the components
and an antisymmetric tensor
2Tfrom the components
Then by (45) the sum of 2S and 2Tso constructed is equal to 2R.
In a three-dimensional manifold an antisymmetric tensor reduces to
three independent components and in this sense resembles a vector. The
Tensors of second rank will be indicated by a superscript as shown.
SEC. 1.201
67
ELEMENTS OF TENSOR ANALYSIS
tensor (36), for example, reduces in this case to
(48)
B1
=
Bz
B3
=
=
+
0 - T ~ I A Z T13A3,
TziA1 0 - T32A3,
-Ti3Ai
T32A2 0.
+
+
+
These, however, are the components of a vector,
B=TxA,
(49)
wherein the vector T has the components
Now it will be recalled that in vector analysis it is customary to distinguish polar vectors, such as are employed to represent translations and
mechanical forces, from axial vectors with which there are associated
directions of rotation. Geometrically, a polar vector is represented by a
displacement or line, whereas an axial vector corresponds to an area. A
typical axial vector is that which results from the vector or cross product
of two polar vectors, and we must conclude from the above that an axial
vector is in fact an antisymmetric tensor and its components should
properly be denoted by two indices rather than one. Thus for the components of T = A X B we write
(51)
Tik = AiBk - AkBi
=
-Tk. 11
( j , k = 1, 2, 3).
If the coordinate system is rotated, the components of A and B are
transformed according t o
3
3
Upon introducing these values into (51) we find
3
(53)
AiBk - AkBi
=
3
2 2 aljaik(AiBi - A:B’,),
2=1 i=l
a relation which is identical with (44) and which demonstrates that the
components of a “vector product” of two vectors transform like the
components of a tensor. The essential differences in the properties of
polar vectors and the properties of those axial vectors by means of which
one represents angular velocities, moments, and the like, are now clear:
axial vectors are vectors only in their manner of composition, not in
their law of transformation. It is important to add that an antisymmetric tensor can be represented by an axial or pseudo-vector only in a
three-dimensional space, and then only in rectangular components.
68
[CHAP.I
T H E FIELD EQUATIONS
Since the cross product of two vectors is in fact an antisymmetric
tensor, one should anticipate that the same is true of the curl of a vector.
That the quantity aAj/axk, wherc Ai is a component of a vector A, is
the component of a tensor is at once evident from Eq. (22).
(54)
The components of
vx
A,
(55)
therefore, transform like the components of an antisymmetric tensor.
The divergence of a tensor is defincd as the operation
3
( j = 1, 2, 3).
k=l
The quantities Bi are easily shown t o transform like the components of
a vector.
(57)
or, on summing over 1 and applying the conditions of orthogonality,
The divergence of a tensor of second rank is a vector, or tensor of first rank.
The divergence of a vector i s a n invariant scalar, or tensor of zero rank.
These are examples of a process known in tensor analysis as contraction.
As in the case of vectors, the tensor concept may he extended t o manifolds of four dimensions. Any set of 16 quantities which transform
according to the law,
(59)
Ti, =
52
j=l
k=l
4
4
i=l
1=1
aijarkTjk,
(i, I
=
1, 2, 3, 4),
aijazkT,!l,
(j,k
=
1, 2, 3, 41,
or its reciprocal,
(60)
Tik
=
]c 2
will be called a tensor of second rank in a four-dimensional manifold.
As in the three-dimensional case, the tensor is said to be completely
SEC. 1.211
69
THE SPACE-TIME SYMMETRY
symmetric if Tjk = Tkj, and completely antisymmetric if Tik = - T k j ,
with Tji = 0. I n virtue of its definition i t is evident that an antisymmetric four-tensor contains only six independent components.
Upon expanding (59) and replacing Tkj by - Tik, then re-collecting terms,
we obtain as the transformation formula of an antisymmetric tensor the
relation
Any six quantities that transform according to this rule constitute a n
antisymmetric four-tensor or, as it is frequently called, a six-vector.
I n three-space the vector product is represented geometrically by the
area of a parallelogram whose sides are defined by two vectors drawn
from a common origin. The components of this product are then the
projections of the area on the three coordinate planes. By analogy, the
vector product in four-space is defned as the “area” of a parallelogram
formed by two four-vectors, A and B , drawn from a common origin.
The components of this extended product are now the projections of the
parallelogram on the six coordinate planes, whose areas are
(62)
Tjk
=
A,Bk - AkBj
=
-Tkj,
(j,k
=
1, 2, 3, 4).
The vector product of two four-vectors is therefore a n antisymmetric
four-tensor, or six-vector.
If again A is a four-vector, the quantities
can be shown as in (54) to transform like the components of a n antisymmetric tensor. The l l l k may be interpreted as the components of
the curl of a four-vector.
As in the three-dimensional case the divergence of a four-tensor is
defined by
a set of quantities which are evidently the components of a four-vector.
1.21. The Space-time Symmetry of the Field Equations.-A remarkable symmetry of form is apparent in the equations of the electromagnetic
field when one introduces as independent variables the four lengths
(65)
x1 = 2,
22
== y,
5 3
= 2,
where c is the velocity of light in free space.
2 4
=
ict,
When expanded in rec-
70
T H E FIELD EQUATIONS
[CHAP.I
tangular coordinates, the equations
aD
(11) V x H - - = J,
(IV) V * D
at
= p,
are represented by the system
0
aHz
. aD1 =
+ aHo
__ - __ - ZCax2
ax3
ax4
J1,
We shall treat the right-hand members of this system as the components
of a “four-current ” density,
(67)
J1
=
J,,
Jp
=
J,,
Js
= Jz,
i ~ p ,
J q =
and introduce in the left-hand members a set of dependent variables
defined by
Gi2
=
613
Gzz
=
Gz3 =
-Hz
Hi
GI4
-Ha
Hz
iCD1
G3z
=
=
0
icDa
G3q
GI1 = 0
Gzi
(68) Gal
G4l
=
=
=
G42
Ha
0
= -Hi
= i~Dz
G33
G43
=
G24
Gqq
= -icDl
= -$Dp
=
=
-wD~
0.
Then in the reference system X , Eqs. (11) and (IV) reduce to
(j
=
1, 2, 3, 4).
Only six of the G 3 k are independent, and the resemblance of this set
of quantities to the components of an antisymmetric four-tensor is
obvious. Since the divergence of a four-tensor is a four-vector, it follows
from (69) that if the G l k constitute a tensor, thcn the J k are the components of a four-vector; inversely, if we can show that J is indeed a
four-vector, we may then infer the tensor character of 2G. However,
we have as yet offered no evidence to justify such an assumption. I n
the preceding sections it was shown that the vector or tensor properties
of sets of scalar quantities are determined b y the manner in which they
transform on passing from one reference system to another. Evidently
an orthogonal transformation of th e coordinates 2 3 corresponds to a
simultaneous change in both the space coordinates x, y, z and the time t ,
and only recourse t o experiment will tell us how the field intensities may
71
T H E SPACE-TIME S Y M M E T R Y
SEC. 1.211
be expected t o transform under such circumstances. I n Sec. 1.22 we
shall set forth briefly the experimental facts which lead one to conclude
that the J k are components of a four-vector, and the Gjk the components
of a field tensor; in the interim we regard (69) and the deductions that
follow below merely as concise and symmetrical expressions of the field
equations in a fixed system of coordinates.
The two homogeneous equations
aB
(I) V x E + 27
=
(111) V * B
0,
=
0,
are represented by the system
After division of the first three of these equations by ic, an antisymmetrical array of Components is defiiicd as follows:
Fzi
=
-Ba
F,, = 0
F61
=
BZ
Fa2
Bi
F23
=
F33
=0
F24
a
= --
C
(71)
i
F ~ =I - El
C
F42
= -B1
=
i
-
C
i
Ez
F43
= C
F34
i
= --
C
E3
F44
=
Ez
E3
0.
Then all the equations of (70) are contained in the system
where i, j , k are any three of the four numbers 1, 2, 3, 4.
The arrays (68) and (71) are congruent in the sense th a t in each the
real components pertain to the magnetic field, while the imaginary components are associated with the electric field. To indicate this partition
it is convenient t o represent the sets of components by the symbols
(73)
,
'G = (H,
-icD).
72
T H E FIELD EQUATIONS
[CHAP.
1
Now the field equations may be defined equally well in terms of the
“ dual” systems :
(74)
2F* =
(-fE, B),
2G* = (-icD, H),
or
FYI = 0
Ffl =
i E3
FFl
=
-- E z
F4:
=
-B1
(75)
C
i
C
i
Fz
= --
F&
=
C
F&
E3
=
5
Ez
-- E 1
p,”z = 2.C El
F&
0
F&
F23 = -B3
-Bz
=
=
B1
FF4
=
B,
FF4
=
B3
i
Ff. =
0
F&
C
F4“4 = 0,
and a corresponding system for the components of 2G*. Upon introducing these values into (I) and (111),and (11) and (IV), respectively,
we obtain
4
k=l
(77)
It has been pointed out by various writers that this last representation
is artificial, in that (74) implies th at E is an axial vector in three-space
and B a polar vector, whereas the contrary is known to be true. The
representation
9%
= Ji,
k=l
must in this sense be considered the I‘ natural” form of the field equations.
To these we add the equation of continuity,
v-J+
=
0,
which in four-dimensional notation becomes
(78)
9%
=
k=
“
1
axk
0.
If the components F j k are defined in terms of the components of a
four-potential ” Q, by
(79)
73
THE SPACE-TIME SYMMETRY
SEC. 1.211
one may readily verify that Eq. (72) is satisfied identically. Now in
three-space the vectors E and B are derived from a vcctor and scalar
potential.
aA
E=-V+--,
B=vxA;
(80)
at
or, in component form,
where the indices i, j , k are l3obe taken in cyclical order. Clearly all
these equations are comprised in the system (79) if we define the components of the four-potential by
=
+I
(82)
A,,
=
A,,
ips =
A,,
a* = -i 4.
C
As in three dimensions, the four-potential is useful only if we can
i
E).
determine from it the field 2G(H, -ZcD) as well as the field 2F(B,
-c
Some supplementary condition must, therefore, be imposed upon @ in
order that it satisfy (69) as well as (72); thus i t is necessary that the
components G i k be related functionally to the F j k . We shall confine the
discussion here to the usual case of a homogeneous, isotropic medium and
assume the relations to be linear. To preserve symmetry of notation
i t will be convenient to write the proportionality factor which characterizes the medium as y j k , so that
(83)
Gik = YikFjk;
but i t is clear from (68) and (71) that'
(84)
yjk
1
= - when j ,
P
k
=
1, 2, 3,
yjk
= ccz
when j or k
=
4.
These coefficients are in fact components of a symmetrical tensor, and
with a view t o subsequent necds the diagonal terms are given the values
Equation (69) is now to be replaced by
( j = 1, 2, 3, 4).
1 A medium which is anisotropic in either its electrical properties or its magnetic
properties may be represented as in (83) provided the coordinate system is chosen to
coincide with the principal axes. This also is the case if its principal axes of electrical anisotropy coincide with those of magnetic anisotropy.
74
[CHAP.I
T H E FIELD EQUATIONS
Upon introducing (79) we find that (86) is satisfied, provided
solution of
is a
(87)
subject to the condition
(j= 1, 2. 3, 4).
k=l
This last relation is evidently equivalent to
(89)
and (87) comprises the two equations
In free space ~ o e o= c-', Y j k = PO-', for all values of the indices. Equations (87) and (88) then reduce to the simple form:
3%
k=l
4
a@k
= -p0Jj,
= 0, ( j = 1, 2, 3, 4).
k=l
1.22. The Lorentz Transformation.-The
physical significance of
these results is of vastly greater importance than their purely formal
elegance. A series of experiments, the most decisive being the celebrated
investigation of Michelson and Morley,' have led to the establishment of
two fundamental postulates as highly probable, if not absolutely certain.
According to the first of these, called the relativity postulate, it is impossible
to detect by means of physical measurements made within a reference
system X a uniform translation relative to a second system X'. That
the earth is moving in an orbit about the sun we know from observations
on distant stars; but if the earth were enveloped in clouds, no measurement on its surface would disclose a uniform translational motion in
space. The course of natural phenomena must therefore be unaffected
by a nonaccelerated motion of the coordinate systems to which they are
referred, and all reference systems moving linearly and uniformly relative
to each other are equivalent. For our present needs we shall state the
relativity postulate as follows : When properly formulated, the laws of
1 MICHELSON
and MORLEY,Am. J . Sci., 3, 34, 1887.
SEC.
1.221
T H E LORENTZ TRANSFORMATION
75
physics are invariant to a transformation f r o m one reference system to
another moving with a linear, u n i f o r m relative velocity. A direct consequence of this postulate is that the components of all vectors or tensors
entering into an equation must transform in the same way, or covariantly.
The existence of such a principle restricted to uniform translations was
established for classical mechanics by Newton, but we are indebted t o
Einstein for its extension to electrodynamics.
The second postulate of Einstein is more remarkable: The velocity
of propagation of a n electromagnetic disturbance in free space i s a universal
constant c which i s independent of the reference system. This proposition
is evidently quite contrary to our experience with mechanical or acoustical
waves in a material medium, where the velocity is known t o depend
on the relative motion of medium and observer. Many attempts have
been made to interpret the experimental evidence without recourse to
this radical assumption, the most noteworthy being the electrodynamic
theory of Rita.l Thc results of all these labors indicate that although a
constant velocity of light is not necessary to account for the negative
results of the Michelson-Morley experiment, this postulate alone is consistent with that experiment and other optical phenomena.2
Let us suppose, then, that a source of light is fixed a t the origin 0
of a system of coordinates X(z,y , z). At the instant t = 0, a spherical
wave is emitted from this source. An observer located at the point
x, y , z in X will first note the passage of the wave a t the instant ct, and the
equation of a point on the wave front is therefore
(92)
52
+ y2 +
22
- c2t2 = 0.
The observer, however, is free to measure position and time with respect
to a second reference frame X’(x’,
y’, z’) which is moving along a straight
line with a uniform velocity relative t o 0. For simplicity we shall assume
the origin 0‘ to coincide with 0 a t the instant t = 0. According to the
second postulate the light wave is propagated in X’ with the same
velocity as in X , and the equation of the wave front in X’ is
(93)
+ / + z’z - c2tJZ= 0.
By t’ we must understand the time as measured by an observer in X’
with instruments located in that system. Here, then, is the key to the
transformation that connects the coordinates x, y , z, t of an observation
or event in X with the coordinates x’, y’, z’,t’ of the same event in X’: it
must be linear and must leave the quadratic form (92) invariant. The
linearity follows from the requirement that a uniform, linear motion of a
particle in X should also be linear in X‘.
2
RITZ,Ann. chim. p h y s . , 13, 145, 1908.
An account of these investigations will be found in Pauli’s article, 20c. cit., p. 549.
76
[CIIAP.I
T H E FIELD EQUATIONS
Let
(94)
5 1
= 5,
5 2
= y,
5 3
= 2,
54
=
ict,
be the components of a vector R in a four-dimensional manifold
X(51, 5 2 , 5 3 , 5 4 ) .
R 2 = X!
(95)
+ X: -+ +
X!
5".
The postulate on the constancy of the velocity c will be satisfied by the
group of transformations which leaves this length invariant. But in
Sec. 1.19 i t was shown that (05) is invariant to the group of rotations in
four-space and we conclude, therefore, th at the transformations which
take one from the coordinates of an event in X to the coordinates of that
event in X' are of the form
(26)
XI =
5
(j = 1, 2, 3, 41,
ajkxk,
k=l
where
4
2
(27)
ajiajk
=
(i, k
6ikl
= 1, 2, 3,
4),
j= 1
the determinant lajk[ being equal t,o unity.
We have now to find t,hese coefficients. T h e calculation will be
simplified if we assume th at the rotation involves only the axes x 3 and 2 4 ,
and the resultant lack of generality is inconsequential. We take, therefore, xi = 51, xk = 2 2 , and write down the coefficient matrix as follows:
The conditions of orthogonality rcduce to
(97)
a23
+ a& = 1,
4 4
+
a"4
= 1,
ckL3a34
+
a43a44
= 0.
If we put a33 = a, a34= iaD, we find from (97) th a t a44 = +a,
~4~ = f i a p , a dl - p 2 = & 1. Only the upper sign is consistent with
the requirement that the determinant of the coefficients be positive unity,
and this in turn is the necessary condition th a t the group shall contain
the identical transformation. I n terms of the single parameter /3 the
coefficients are
1
iP
a33= a44=
a 3 4 = -a43 =
(98)
dFj9'
di+
SEC.
77
T H E LORENTZ T R A N S F O R M A T I O N
1.221
for the transformation itself, we have
.:=
(99)
21,
z: = z.2,
Reverting t o the original space-time manifold this is equivalent to
z’ = 2,
(100)
z‘ =
Y’ = Y,
1
diq5
(2 -
t’ =
pet),
d~1
(1
-
z).
The parameter p may be det,ermined by considering z’, y’, z’ to be the
coordinates of a fixed point in X’. The coordinates of this point with
respect to X are z, y, z. Since dz’ = 0, it follows th a t
_ -- v
dt
dz
= pc,
p
= ?,
C
and hence the rotation defined in (96) and (97) is equivalent to a translation of the system X’ along the z-axis with the constant velocity v relative
to the unprimed system X .
The transformation
obtained from (100) by substitution of the value for p, or its inversion,
has been named for Lorentz, who was the first t o show that Maxwell’s
equations are invariant with respect to the change of variables defined
by (l02), but not invariant under the “Galilean transformation:”
(104)
2’ = z -
vt,
t’
=
t.
All known electromagnetic phenomena may be properly accounted for
if the position and t i m e coordinates of an event in a moving system X’
be related t o the coordinates of that event in an arbitrarily fixed system
X by a Lorentz transformation. The Galilean transformation of classical
mechanics represents the limit approached by (102) when v << c, and
may be interpreted as the relativity principle appropriate t o a world in
which electromagnetic forces are propagated with infinite velocity.
78
T H E FIELD EQUATIONS
[CHAP.I
1.23. Transformation of the Field Vectors to Moving Systems.-We
shall not dwell upon the manifold ronscquences of the Lorentz transformation; the Fitzgerald-Lorentz contraction, the modified concept of simultaneity, the variation in apparent mass, the upper limit c which is imposed
upon the velocity of matter, belong properly to the theory of relativity.
The application of the principles of relativity to the equations of the
electromagnetic field is essential, however, to a n understanding of the
four-dimensional formulation of Sec. 1.21.
The Lorentz transformation has been deduced from the postulate on
the constancy of th e velocity of light and has been shown to be equivalent
to a rotation in a space 21, x2, 2 3 , 2 4 = ict. Now according t o the relativity postulate, th e laws of physics, when properly stated, must have
the same form in all systems moving with a relative, uniform motion;
otherwise, it would obviously be possible to detect such a motion. I n
Secs. 1.19 and 1.20 it was shown th at the curl, divergence, and Laplacian
of vectors and tensors in a four-dimensional manifold are invariant to a
rotation of the coordinate system. Therefore, to ensure the invariance
of the field equations under a Lorentz transformation it is only necessary
to assunie that the four-current J and the four-potential a do indeed
transform like vectors, and that the quantities 2F, 2G transform like
tensors. I n other words, we base the vector and tensor character of these
four-dimensional quantities directly on the two postulates.
The four-current J satisfies the equation
(78)
CI. J
za
4
=
k=l
aJ k
= 0.
xk
Under a rotation of the coordinate system the components transform as
A
or, upon introducing the values for
aik
from (98),
with its inverse transformation
We shall assume henceforth that the reference system X' is fixed within a
material body which moves with the constant velocity v relative to the
TRANSFORMATION OF THE FIELD VECTORS
SEC.1.231
79
system X . This latter may usually be assumed a t rest with respect to
the earth. If the velocity v is very much less than the speed of light,
Eq. (107) is approximately equal to
J,
(108)
=
JL
+
VP’,
p = p‘.
An observer on the moving body measures a charge density p‘ and a
current density J : ; his colleague a t rest in X finds the current JL augmented by the convection current up’.
I n like manner the relations between the electric and magnetic
vectors defining a given field in a fixed and in a moving system are
obtained directly from the rule (61) for the transformation of the components of an antisymmetric tensor. Upon substitution of the appropriate values for the coefficients ark, one obtains for the components of 2F:
F& = a11azzF12 = Flz,
and, hence,
The restriction to translations along the z-axis may be discarded by
writing v as a vector representing the translational velocity of X’ (the
moving body) in any direction with respect to a fixed system X. Since in
(110) the orientation of the z-axis was arbitrary, we have in general
80
[CHAP.I
T H E FIELD EQUATIONS
where 11 denotes components parallel, I components perpendicular to
the axis of translation. Dropping terms in c - ~ , as is justifiable whenever
the body is moving with a velocity v << 3 X lo8 meters/second, we
obtain the approximate formulas
(112)
E‘II = Ell,
B’l! = Bl!,
+
E’I = E l
(V X B)I.
BI,
The implication of these results is striking indeed: the electric and
magnetic fields E and B have no independent existence as separate
i
entities. The fundamental complex is the field tensor 2F = (B, -;E) ;
B’I
=
the resolution into electric and magnetic components is wholly relative
to the motion of the observer. When at rest with respect to permanent
magnets or stationary currenk, one measures a purely magnetic field B.
An observer within a moving body or system X’, on the other hand,
notes approximately the same magnetic field, but in addition an electrostatic field of intensity E’ = v x B. Or, inversely, the moving body
may carry a fixed charge. To an observer on the body, moving with the
charge, the field is purely electrostatic, whereas his colleague aground
finds a magnetic field in company with the electric, identifying quite
rightly the moving charge with a current.
From the tensor 2G = (H, -icD) are calculated in like fashion the
transformations of the vectors H and D from a fixed to a moving system.
(D
+ 21 v X H)L.
The invariancc of Maxwell’s equations to uniform translations amounts
to this: if the vector functions E, B, H, D define an electromagnetic field
in a system X , the equations
V’ X E’
(114)
v’ x
aB’
+7
= 0,
at
H’ -
aD’
~
dt’
= J,
V’ B’ = 0,
v’. D’
= p’,
are satisfied in a system X’ which moves with the constant velocity v
relative to X, the operator v’ implying that differentiation is to be effected
with respect to the variables d,y‘, 2’. An observer at rest in X’ interprets the vectors E’, B’, H’, D’ as the intensities of an electromagnetic
field satisfying Maxwell’s equations. Clearly the ratios of D to E and
H to B are not preserved in both systems. The macroscopic parameters
SEC. 1.231
81
TRANSFORMATION OF T H E FIELD VECTORS
E , p, a are also subject to transformation, which may be ascribed to an
actual change in the structure of matter in motion. I n practice one is
interested usually in the mechanical and electromotive forces, measured in
the fixed system X , which act o n moving matter, rather than in the transformed field intensities E', B', H', D'. The determination of these
forces and of the differential equations which they satisfy within the
framework of the relativity theory was accomplished by Minkowski in
the course of his investigation on the electrodynamics of moving bodies.
The vector character of the four-potential is demonstrated by Eq.
(79) which expresses the field tensor 2Fas the curl of @. Under a Lorentz
transformation
4
a;. =
(115)
2
(j
Ujk%
=
1, 2, 3, 41,
k= 1
or, in terms of vector and scalar potentials,
As in the case of thc field vectors, the resolution into vector and scalar
potentials in three-space is determined by the relative motion of the
observer.
I n conclusion it may be remarked that a rotation of the coordinate
system leaves invariant the scalar product of any two vectors. It was
in fact from thc required invariance of the quantity
+ +
R * R = R 2 = x 2 IJ' x 2 - C 2 t 2 ,
(117)
that we deduced the Lorentz transformation. Since the current density
J and the potential @ have been shown to be four-vectors, it follows th a t
the quantities
J 2 = J:
J i Jf - c2p2,
(118)
*
+ +
@
= A: + A ; + A2, - 2,
*
J
=
A,J,
+ A,J, + AzJz -
4p,
are true scalar invariants in a space-time continuum. There are, moreover, certain other scalar invariants of fundamental importance to the
general theory of the electromagnetic field. From the transformation
formula Eq. (59) the reader will verify that if *Sand 2Tare two tensors
of sccond rank, the sums
S,kTkj
j=1
k=l
j=1
k=l
=
invariant,
82
T H E FIELD EQUATIONS
[CHAP.I
are invariant to a rotation of the coordinate axes. These quantities
may be interpreted as scalar products of the t,wo tensors. Let us form
first the scalar product of 2F with itself. According to (71) and (119)
we find
$3 bz
(120)
Ffk = 2
-
j=1 k = l
Next we construct the scalar product of 2F with its dual 2F*defined in
From the tensor *G and its dual zG* may be constructed the invariants
4
4
25
GjkGS
(123)
j=1
=
-4icH
*
D
=
invariant.
k=l
Proceeding in the same fashion, we obtain
4
4
j=1
k=l
4
4
j=1 k = l
= invariant,
= invariant.
The invariance of these quantities in configuration space is trivial; they
are set apart from other scalar products by the fact that they preserve the
same value in every system moving with a uniform relative velocity.
CHAPTER I1
STRESS AND ENERGY
To translate the mathematical structure developed in the preceding
pages into experiments which can be conducted in the laboratory, we
must calculate the mechanical forces excrted in the field upon elements of
charge and current or upon bodies of neutral matter. I n the present
chapter it will be shown how by an appropriate definition of the vectors
E and B these forces may be deduced directly from the Maxwell equations.
In the course of this investigation we shall have to take account of the
elastic properties of material media. A brief digression on the analysis of
elastic stress and strain will provide an adequate basis for the treatment
of the body and surface forces exerted by electric or magnetic fields.
za
STRESS AND STRAIN IN ELASTIC MEDIA
2.1. The Elastic Stress Tensor.-Let us suppose that a given solid or
fluid body of matter is in static equilibrium under the action of a specified
system of applied forces. Within
this body we isolate a finite volume
V by means of a closed surface S, as
indicated in Fig. 12.
Since equilibrium has been assumed for the body and all its parts,
the resultant force F exerted on the
matter within S must be zero. Contributing to this resultant are volume
Y
or body forces, such as gravity, and
surface forces exerted by elements of
matter just outside the enclosed
region on contiguous elements
within. Throughout
therefore, FIG.12.-A region V hounded by a surface
S in an elastic medium under stress.
we suppose force to be distributed
with a density f per unit volume, while the force exerted by matter outside
S on a unit area of S will be represented by the vector t. The components
of t are evidently normal pressures or tensions and tangential shears. The
condition of translational equilibrium is expressed by the equation
v,
JV
f dv
+ $s t da = 0.
83
84
STRESS A N D ENERGY
[CHAP.I1
T o ensure rotational equilibrium it is necessary also that the resultant
torque be zero, or
(2)
j v r x f dv
+J r
x tda
=
0,
where r is Lhe radius vector from an arbitrary origin 0 to a n element of
volume or surface.
The transition from the integral relations (1) and (2) to a n equivalent
differential or local system expressing these same conditions in the
immediate neighborhood of an arbitrary point P(x, y, z ) may be accomplished by a method employed in Chap. I. The surface X is shrunk
about P and the Components of t are expanded in Taylor series. The
integral can then be evaluated over the surface and on passing to the
limit the conditions of equilibrium are obtained in terms of the derivatives o f t a t P. This labor may be avoided by applying theorems already
a t our disposal. Let n be the unit, outward normal to a n element of S.
There are then three vectors X, Y, Z which satisfy the equations
(3)
t, = X - n ,
t, = Y - n ,
t, =
2.n.
The quantity X , is clearly the x-component of force acting outward on a
unit element of area whose orientation is fixed by the normal n. The
expansion of the scalar products in the form
(4)
+ XUnu+ X z n z ,
+ Y,n, + Yzn,,
Zznz + Z,n, + Zzns
t, = X,n,
t, = Y,n,
t,
=
may also be interpreted as a linear transformation of the components of
n into the components of t, the components n,,n,, n, being the direction
cosines of n with respect to the coordinate axes. The equilibrium of the
x-components of forces acting on matter within X is now expressed by
(5)
sVfz+
dv
X - n da
=
0,
which in virtue of the divergence theorem and the arbitrariness of V is
equivalent t o the condition that a t all points within S
+
f,
v . x = 0.
(6)
For the y- and z-components we have, likewise,
(7)
f,+V.Y=O,
f,+v.Z=0.
The rotational equilibrium expressed by (2) imposes further conditions upon the nine components of the three vectors X, Y, Z. The
x-component of this equation is, for example,
$v (Zlf2 - zf,) dv +
( y t z - zt,) da = 0.
SEC.
85
THE ELASTIC STRESS TENSOR
2.11
Introduction of the values of 1, and t, defined in (3) leads t o
Jr (yj,
(9)
- zj,) dv
+
(YZ- zY)* n da
= 0,
which, again thanks to the divergence theorem and the arbitrariness of
V , is equivalent to
+v
yf, - zfv
(10)
*
(yZ - ZY) = 0.
But
v * ($3)
(11)
=I
:,T
- z + 7 - Qy = y o z + 2,.
Equation (10) reduces to
ycf2
(12)
+ v - Z)
-
+v
x(f,
*
Y)
+ 2, - Y , = 0;
or, on taking account of ( 7 ) ,
2, = Y,.
(13)
I n like manner there may be derived from the y- and z-component,s of
(2) the symmetry relations
x, = Y,,
(14)
x,= 2,.
The nine components X,, X,, . . . Z, representing forces exerted on
unit elements of area, are called stresses. The diagonal terms X,, Y,, 2,
act in a direction normal to the surface element and are, therefore,
pressures or tensions. The remaining six components are shearing
stresses acting in the plane of the element. These nine quantities constitute the components of a symmetrical tensor, as is evident from (4)
and the fact that t and n are true vectors (cf. See. 1.20). For t,he sake of
a condensed notation we shall henceforth represent the components of the
stress tensor by Tik, where
(15)
X,
=
TII, X ,
=
Tn,
...
Y,
=
T21,
...
2,
=
T33.
I n order tlhat a fluid or solid medium under stress shall be in static
equilibrium it is necessary that a t every point
(16)
f
+ div 2T= 0,
T j k = Tkj.
Imagine a n infinitesimal plane element of area containing a point
y, z ) in a stressed medium. The stresses acting across this surface
element will, in general, be both normal and tangential, but there are
three distinct orientations with respect to which the stress is purely
normal. Now if the resultant force t acting on unit area of a plane element is in the direction of the positive normal n, one may put
(5,
(17)
t = Xn,
86
STRESS A N D ENERGY
[CHAP.I1
where h is an unknown scalar function of the coordinates x, y, z, of the
element. When this condition is imposed upon Eqs. (4)one obtains the
homogeneous system
3
Xni =
2
( j = 1, 2, 3).
Tjknk,
k=l
In order that these homogeneous equations be self-consistent, it is
necessary that their determinant shall vanish; whence it follows that the
scalar function X can be det,ermined from
provided, of course, that the stress components T i k with respect to some
arbitrary coordinate system are known. The secular equation (19) has
three roots, A,, Xb, A,, and these fix, through Eq. (18)) three orientations of
the surface element which we shall designate respectively as n@),n@),n@).
If the roots are distinct, the preferred directions defined by the unit
vectors n(a), n(b),ncC)--called the principal axes of stress-are mutually
perpendicular. Let us consider, for example, nca) and db).According
to (W,
3
3
Multiply the first of these equations by nib),the second by nf"),subtract
the second from the first and sum over j .
3
3
The right-hand sum vanishes, leaving
(h, - Xb)n(5) n(b) = 0.
(22)
-
If X, # k b , the vector n(") must be orthogonal to n(b)as stated.
The physical significance of the principal axes may be made clear in
another manner. At a point P in a stressed medium the symmetrical
tensor 2T associates with a unit vector n the resultant force t acting on a
unit surface element normal to n. Let the origin of a rectangular coordinate system be located at P. The scale of length in this new system is
arbitrary and we may suppose that the components of n, drawn from P ,
are 5, q, 5. Now the scalar product of the vectors t and n, which we shall
call is a quadratic in 5, q,
+,
(23) a([, 7, p)
r.
=
t
o
n
=
Td2
+ Tzzq2+ Tad2 + 2T& + 2-bT m2TaitE.
t
SEC. 2.21
ANALYSIS OF STRAIN
87
The surface @ = constant is called the stress quadric. By a rotation of
the coordinate axes (23) can be reduced to the square terms alone, and
it is clear that the principal axes n@),n(*),nee) must coincide with the
principal axes of the quadric. One will observe, furthermore, that
(24)
t
=
+v*.
This property enables one to find the stress across any surface element at
P by graphical construction. We shall suppose the stress to be such that
the quadric is an ellipsoid and draw about P the surface @ = 1. The
radius vector from P to any point P' on this surface is n. According to
(24) the resultant force acting on an element at P whose normal is n is in
= 1 a t P'. The magnitude of t
the direction of V+, or normal to
according to (23) is equal to the reciprocal of the projection NP' indicated
Fro. 13.-Grrtphical
determination of stress on an element of area a t the point P from the
stress quadric.
in Fig. 13. Along the principal axes, a, 6, c, the normal to the surface
@ = 1 coincides with the vector n and the stress on the element a t P is
purely normal. If these principal stresses are known, the quadric can
be constructed and the stress on an arbitrarily oriented element determined graphically.
2.2. Analysis of Strain.-If
surface and volume forces are applied to
a perfectly rigid body, the resultant motion may be described in terms of
a translation of the body as a whole and a rotation about its center of
mass. If, however, the body is deformable, its parts suffer also relative
displacements which increase until equilibrium is reestablished by internal
forces evoked in the deformation. The changes in the relative positions
of the parts of a body under stress are called strains.
In an unstressed, continuous medium a point P is located with respect
to an arbitrary, fixed origin by the radius vector r. Under the influence
of applied forces the medium undergoes a deformation, in the course of
s.
which the matter located initially at P moves to a point P' a t r' = r
The displacement s corresponding to a given deformation depends on
+
88
STRESS AND ENERGY
[CHAP.I1
the coordinates of the initial point P ; we assume s to be a continuous
function of r. Consider now any neighboring point P1 located by the
radius vector rl. The initial position of P1 with respect to P is fixed by
the vector 6r, where
6r
(25)
=
rl - r.
The displacement of P I during the deformation is sl, a function of rl.
The relative displacement of two neighboring points P and P1 occasioned
by the deformation is therefore
(26)
6s = s1 - s = s(r
+
+ 6r) - s(r).
Since s(r
6r) is continuous, it may be expanded in a Taylor series;
for points in a region sufficiently small about P we need retain only the
linear term
(27)
6s = (6r. V)S;
or, in component form,
(j
=
1, 2, 3 ) .
(The indices 1, 2, 3 again replace the subscripts x , y, z for convenience in
summing.)
The nine derivatives dsi/dxk are the components of a tensor (cf.
Eq. (54), Sec. 1.20) which is in general asymmetrical. However, it can
be resolved after the manner of Eqs. (46) and (47), page 66, into a
symmetric part,
and a n antisymmetric part,
(j, k
=
1, 2, 3).
The components of this antisymmetric tensor are evidently identical with
the components of an axial vector b, defined by
b = gV X S,
(31)
(32)
bi = b 3 2 ,
bz = b 1 3 ,
bt = b x .
The relative displacement 6s can likewise be split up into a part 6s’
associated with the symmetric tensor a j k ,and a part 6 ~ associated
’ ~
with
the antisymmetric tensor b j k . Then, by (28) and (30),
(33)
6s” = b x 6r = ;(v x s) x 6r.
Physically this implies that the matter contained within a n infinitesimal
volume about P is subjected to a rotation as a rigid solid; the rotation
ANALYSIS OF STRAIN
SEC. 2.21
89
takes place about a n instantaneous axis through P in the direction of
s, and in circular measure is equal to +iV X sI.
This local rotation
is brought about by stresses whose curl does not vanish.
Whereas the relative positions of points in the immediate vicinity
of P are preserved b y the local rotation (33), thc symmetric tensor (29)
defines a local deformation-an actual stretching and twisting. We ask
first whether there are any vcwtors 6r drawn from P whose direction is
unchanged after the deformation; that is t o say, arc there any points
P I , Fig. 14, which in the course of the
deformation will move along the line
defined by 6r? The necessary condition is that
vx
(34)
6s‘ = X 6r,
where is a n unknown scalar function
of t h e coordinates of P ; or
3
(35)X 6x3
=
a3k6xk,
( j = 1, 2, 3).
k=l
0
T h e parameter ), is found from the FIG. 14.-Vectors characterizing the deformation af a contmuous medium.
condition that the determinant of this
homogeneous system shall vanish and, as in thc analogous case of the
stress tensor discussed in the preceding paragraph, it is easy to show that
t h e three roots fix three principal axes which in general arc mutually
orthogonal. Along these principal axes of strain, and along them only,
the deformation consists of a pure stretching. The nature of the deformation along any other axes can be visualized with the aid of a strain quadric
such as (23). Let the point PI, Fig. 14, have the coordinates tl, E2, E3
with respect t o P and construct the surface
(36)
-
\E = 6r 6s’ =
3
3
2C
alktltk
= constant.
j=l k=l
If all lines issuing from P are extendcd, or if all are contracted, the quadric
is a n ellipsoid; if some lines are extended and others contracted, the surface is an hyperboloid. The relative displacement of P1 with respect to
P is given by
(37)
6s’ =
&v*.
*
The radius vector from P to P1 on the surface
= 1 is 6r. I n Fig. 15
it has been assumed that this surface is an ellipsoid. The direction of
the relative displacement 6s’ due to the deformation is that of the normal
erected a t P 1 and its magnitude is equal t o the reciprocal of the projection
90
STRESS A N D ENERGY
[CHAP.
I1
NPl of the radius vector 6r on the normal. Obviously the vectors 6s’
and 6r are parallel only along the principal axes of the ellipsoid. These
axes represent also the directions of maximum and minimum contraction
or extension.
There is an interesting physical interpretation to be given to the
tensor components aik. If before a deformation the relative position
of two points P and PI, Fig. 14, is fixed by 6r, their relative position
after the deformation is 6r’ = 6r
6s’, the prime over 6s‘ indicating
that local rotations are excluded. Suppose now that 6r is directed along
+
FIG.lS.-Graphical
determination of strain a t the point
P from the strain quadric.
the xl-axis, so that 6x2 = 6x3 = 0. Then 6r’ is a vector whose components are
(38)
62: = (1
+ all) 6x1,
62: =
UZI 621,
62; = a316xl.
The absolute value of the distancc between P and PI after deformation is
(39)
16r’I = 1 / ( 1
+ a d 2 + ail + aPI 6x1,
and the relative change in length is
obtained by expanding (39) and discarding all terms of higher order
than the first. Similar expressions can be found for deformations of line
elements lying initially along the 5 2 - and x3-axes,whence we conclude that
the diagonal components aii = dsi/axi represent extensions of linear elements which in the unstrained state are parallel to the coordinate axes.
Again let 6a be a linear element which in the unstrained state is
parallel to the xl-axis and 6b be a linear element initially parallel to the
x2-axis. A local deformation transforms the vector 6a = il 6xl into an
element
(41)
6a’ = il(l
+
all)
6x1
+ i2a216xl + i3a3,6xl,
ANALYSIS OF STRAIN
SEC. 2.21
91
whilc a t the same time 6b = i2 6x2 is transformed into
(42)
6b’
=
Lals 6x2
+ i2(l + a,,)
6x2
+
ha32
6x2.
I n the initial state 6a and 6b are mutually orthogonal, but after the
deformation, they form an anglc which differs but little from 7r/2 and
which we shall denote by
(43)
But
(44)
6a’
-
2
6b’
=
cos
812.
16a’l 16b’Jcos
-
6
- BIZ).
e12) = sin e12= e12,
and, hence,
(45)
e12= a d 1
+ all> + azl(l + a2J + a31a32;
or, on neglecting terms of higher order than the first,
012 = alz
+ aZ1
=
asl
ax,
+ --asz
axl
Thus, in general, the coeficient a i k , with j # k, measures one-half the
cosine of the angle made by two linear elements after a deformation, which
in the unstrained state were directed along a pair of orthogonal coordinate
axes.
Conforming to t,his interpretation, i t is customary to write the components of the deformation tensor in a slightly altered manner. The
coefficients defined by
(47)
e22 =
as2
-
e23 =
883
c31
ax2
c~~ = -
ax3
=
as3
+ax,
as1
-+ ax, ax3
as2
-
ax3
as3
=
e32
=
el3
are called the components of strain, and the components of the relative
displacement of any two points P and P1 due to a local deformation
are therefore
6s: = e l l 6x1 +el2 6x2
!re13 8x3,
(48)
6s: = +e21 6x1 eZ26x2
ge23 6x3,
6s: = !re31 6x1 +e3, 6x2
e33 6x3.
+
+
+
+
+
+
If the origin of the arbitrary coordinate system XI,2 2 , 2 3 is located a t the
point P and the axes then rotated into coincidence with the principal
axes of the strain ellipsoid associated with that point, all components of
92
STRESS A N D ENERGY
[CHAP.I1
strain are reduced to zero with the exception of those lying on the main
diagonal. We shall measure distances along the principal axes of strain
by the coordinates u1,uz,us. Then the components of any local deformation with respect to P are
(49)
6sk1 = el 6u1,
6s:?
=
6sLJ = e3 6u3,
e2 6u2,
where
(50)
6V
(51)
= (6a
x 6b). 6c
=
6al
6bl
6c1
1
(53)
6V’
=
(6a’
x 6b’)
*
6c’ =
+ all
a12
6az
6a3
6b2
6cz
6b3.
6c3
a21
1
a13
(54)
6V’ _____
6V
6V - a11
a31
+ a22
+ a Z 2+ a33
1
a23
=
+
a32
6x1 6x2 6x3.
a33
-
V s
for the change of volume per unit volume-a quantity called the cubical
dilatation.
I n summary, the analysis has shown that the most general displacement of particles in the neighborhood of a point may be resolved into a
translation and a local rotation, upon which there is superposed a deformation characterized by thc strain components e j h and accompanied by a
change in volume. The rotational component of the total strain can
be induced only by rotational stresses-force functions, that is t o say,
whose curl is not everywhere zero.
SEC.2.31
93
RELATIONS OF STRESS TO STRAIN
2.3. Elastic Energy and the Relations of Stress to Strain.-Under the
influence of external forces the particles composing an elastic fluid or
solid suffer relative displacements. These strains, in turn, evoke internal
forces which, in the case of static equilibrium, eventually compensate
the applied stresses. The work done by the applied volume and surface
forces in displacing every point of the medium an amount 6s is
SW
(55)
=
$ f - 6s dv + $ t - 6s da.
According to Eq. (3)
-
t 6s = 6s,X. n
(56)
+ 6s,Y
+ 6s,Z
n
n,
and, hence,
(57)
$ t - 6s da = $ V
+ Y As, + Z 6sz) dv.
(X 6s,
Now
v * (X SS,)
(58)
and by (6) f,
(59)
=
-V
6W
=
- X.
- x + x - v Fs,,
= 6S,V
Thus (55) reduces to
J- (X - V
6s,
+Y
*
+Z
V 6~,
*
V 6 ~ , ) dv.
The components of the integrand may, however, be written
which in the notation of Eqs. (47) leads to
+ T31 6 e d dv
for the work done by the applied stresses against the elastic restoring
forces in the course of an infinitesimal change in the state of strain. If it
be assumed that this change takes place so slowly that the variation in
kinetic energy may be neglected, and that no heat is added or lost in thc
process, then 6W must be equal to the increase in the potential energy U
stored up in the elastically deformed medium. The elastic energy
stored in unit volume will be denoted by u,whence
(62)
SW
=
6U
=S6UdV.
The energy density u at any point must depend on the local state of
strain and it may: therefore, be assumed that
(63)
u =
u(e11, e12,
- -
4,
94
[CHAP.I1
STRESS A N D ENERGY
or
The strain components e j k are arbitrary; hence it follows that the components of applied stress (which in static equilibrium are equal and
opposite to the induced elastic stresses) can be derived from a scalar
potential function,
The existence of an elastic potential, which has bcen demonstrated
here for the case of quasi-stationary, adiabatic deformations, may be
shown for isothermal changes as well.
When the deformations are not excessivc, the components of elastic
stress may be expressed as linear functions of the components of strain.
TII = cllell
(66)
+
612622
+
c13e33
+
+
c14e12
+
+
c15e23
+
+
c ~ e 3 1 ,
...............................................
~
1
2
=
c4lell+
c42e22
+
c43e33
c44e12
c45e23
c46e31,
...............................................
The coefficients c,, are called the elastic constants of the medium. The
elastic potential u is in this case a homogeneous quadratic function of the
strain components. The existence of a scalar function u ( e i k ) satisfying
(65) and (66) imposes on the elastic constants the conditions
(67)
Cnln
(n, m = 1, 2,
= cnm
-
, 6),
whereby the number of pararnetcrs necessary to specify the relation of
stress to strain in an anisotropic medium reduces from 36 to 21. Any
further reduction is accomplished by taking advantagc of possible symmetries in the structure of the medium. If, in particular, the substance
is elastically isotropic, the quadratic form u must be invariant to orthogonal transformations of the coordinate system, and the number of independent constants then reduces to two. The elastic potential assumes
the form
(68)
= &(ell
+ +
e22
e3J2
+ X d e 5 + + +4e'L + +e& + +e"31>,
4 2
4 3
whence. by (65)
(69)
T11
=
Xl(e11
T22
=
Xl(e11
2'33
=
Xl(el1
+ +
+ +
e22
e33)
e22
e33)
e22
e33)
+ 2Xzelt
+
7'12
212e22,
T23 =
212633,
7'31
+ + +
= X2elr,
=
Xze~,
X2e31.
SEC. 2.31
RELATIONS OF STRESS TO STRAIN
95
Now the conditions of static equilibrium are expressed by
f
(16)
+V
2T = 0.
Upon introducing the components (69) into the expanded divergence
and expressing the strain components e , k in terms of the deformation s,
we find
(70)
f
+ ( A , + X2)VV ’ s + xzv2s = 0
as the equation determining the deformation of an isotropic body subjected to a volume force f. The constants of integration are evaluated
in terms of data specifying either the displacement of points on the
bounding surface, or the distribution of stress over that surface.
The parameters X 1 and Xz are positive quantities. 1, is called the
rigidity, or shear modulus, for it measures the strains induced by tangential, or shearing, stresses. No simple physical meaning can be
attached t o X I ; however, it may be defined in terms of the better known
constants E and u, where Young’s modulus E is the ratio of a simple
longitudinal tension to the elongation per unit length which it produces,
and where the Poisson ratio u is the ratio of lateral contraction to longitudinal extension of a bar under tension. Then
Eu
x -- (1 + u)(l
x -- 2(1 E+
- 24’
r J j
An ideal fluid supports no shearing stress, and hence Xz = 0. The
stress acting on any element of a closed surface within the fluid is, therefore, a normal pressure,
Ti1 = T2Z = T33 = -P,
T12 = T23 = T3l = 0,
(72)
the negative sign indicating that the stress is directed inward.
components of strain e l l , e22, e 3 3 are all equal and
+ +
p = --Xl(ell
e22 e 3 3 ) = --XlV
(73)
while the equation of equilibrium reduces to
The
s,
f - v p = 0.
(74)
Let V o be the initial volume of an element of fluid, VI its volume at a
pressure p l , and V 2 its volume a t a pressure p,. From the definition
(54) of thc cubical dilatation V s we have
-
(75)
pl =
v,
vo ’
v1-
--x1---
pz
vz - vo
= XI
vo ’
and for sufficiently small changes,
(76)
P2 - p1
= -x1
v2
- V?
vo
96
STRESS A N D ENERGY
[CHAP.
I1
For a n infinitesimal change in pressure
(77)
The reciprocal of XI is in this case
where r is the density of the fluid.
called the compressibility.
ELECTROMAGNETIC FORCES ON CHARGES AND CURRENTS
2.4. Definition of the Vectors E and B.-An
electromagnetic field is
defined according to our initial hypotheses by four vectors E, B, D and H
satisfying Maxwell’s equations. The physical nature of these vectors
will be expressed in terms of experiments by means of which they may be
measured. Now it is easy to show that pE and J x B are quantities
whose dimensions are those of force pcr unit volume. For by See. 1.8
(1)
(2)
[PEI
”
coulombs
volts kilograms
x -meter
meter3
second2. meter2
kilograms
amperes webers x -meter2
meter2
second2. meter2’
=-
B1 =
We are, therefore, free to define the vectors E and B as forces exerted
by the field on unit elements respectively of charge and current. More
precisely, we shall suppose that charge is distributed throughout a volume
V with a macroscopically continuous density p. Then E is defined such
that the net mechanical force acting on the charge is
Fe = Jv pEdv,
(3)
distributed with a volume density
f,
(4)
=
pE.
If V is sufficiently small, the field within this region at any instant may
be assumed homogeneous, so that in the limit
(5)
F, = ESVpdv = qE,
(V-0).
I n like manner the association of the magnetic vector B with the
force exerted on a unit volume element of current through the equation
(6)
leads to
(7)
F,
J
f,
=
=
SvJ
X B,
X Bdu
as the net force exerted on a volume distribution of current. If the
current is confined to a linear conductor of sufficiently small cross section,
The definitions of E and B based on Eqs.
(3) and (7) have been shown by purely
dimensional considerations to be compatihle with Maxwell’s equations. In the
(9)
EO(VX E) X E
+ -PO1 (V X B) X B
=
,,d Fm
J
X B
+
a
€0
(E x B).
In a rectangular system of coordinates the first term of (9) may be
represented by the determinant
The x-component of this vector is
(11)
[(V
x E) x E] i
=
E,-
dEZ
dX
aE,
dE,
+ E uaE,
- + Eza?/
az - E,- d X
= ~ ( R i - i 1~ 2 ) + ~a ( E , ~ , ) +
a ~ ( E z-E E, ,)V . E .
ax
transform like the components of a tensor [Eq. (43), page 661 and the
first three terms on the right of (11) constitute therefore the x-component
of the divergence of a tensor 2S(e). The remaining components arc
calculated from thc y- and z-components of EO(V x E) x E, such that
we are led to the identity
eO(V x E)
(13)
the components of
2S(e)
=
L Y : ~O F
-1
1
- (v x
PO
B) X B
THE
TENSOR
IN
FREESPACE
3
2
The transformation of (V x B)
(14)
-
div 2S(e)- EOEV E,
being tabulated below.
T A R L E I.-COMPONENTS
1
xE
x B is effected similarly, giving
= div 2S(m)-
1
-
PO
-
B V B,
where the components of 2S(m)are as represented in Table 11.
T H E ELECTROMAGNETIC STRESS TENSOR
SEC.2.51
r r A B L E II.-cOMPONENTS
I
1
PO
8:;) OF
THE
99
TENSOR
2S(m)I N FREES P A C E
BsBz
Upon replacing the first, two terms of (9) by (13) and (14) and taking
account of (111) and (IV), we obtain an identit,y of the form
(15)
div 2S = EP 4-J X B
+
a (E x B),
€0 -
dt
wherein the components of the tensor 2S are
Equation (15) is a relation through which the forces exerted on elements
of charge and current a t any point in otherwise empty space may be
expressed in terms of the vectors E and B alone.
Let us intcgratc this identity over a volume V . Now thc integral
of the divergence of a tensor throughout V is equal t o the integral of a
vector over the surface bounding V . To demonstrate this tensor analogue of the vector divergence thcorem, Ict n be the outward unit normal
a t a point on the bounding surface and consider S11to be the x-component
of a vector whose 1 ~ -and z-components are zero. The component of this
vector in the direction of the normal n is Slln,, whence by the divergence
theorem
Likewise it is apparent that
100
STRESS A N D ENERGY
[CHAP.IT
Upon adding these three identities, we obtain
The integrand on the right is the x-component of div 2 S , and consequently
t,
(21)
=
Slln,
+ Slzn, + &an,
is the x-component of a vector t which according to (20) is to be integrated
over the surface bounding V . Proceeding similarly for the other components, we have the general theorem:
$ t da = $ div 2Sdv,
(22)
where the components of t are
(23)
or, in abbreviated notation,
(24)
t
=
%-n.
Applied to Eq. (15), the divergence theorem (22) leads to
J
(25)
-
2S n da
=
F,
+ F, +
c0
at
SJEXBdV
with F, and F, representing, as in Eqs. (3) and (7), the resultant forces
acting respectively on the charge and the current contained within V .
Consider first stationary distributions of charge and current. The
fields are then independent of time and the third term on the right of (25)
is zero. Equation (25) now states that the force exerted on stationary
charges or currents can be expressed as the integral of a vector over any
regular surface enclosing thcse charges and currents. It does not state
that the volume forces F, and F, are maintained in equilibrium by the
force 2s n distributed over the surface. The equilibrium must be
established with mechanical forces of some other type, and in fact i t will
be shown shortly that a charge distribution cannot possibly be maintained
in static equilibrium under the action of electrical forces alone. To be
more specific, let us imagine a stationary charge distribution to be divided
into two parts by an arbitrary closed surface I;. The force exerted by
the external charges on those within I; must in some manner be transmitted across this surface. The net force on the interior charges may,
according to (25), be correctly calculated on the assumption that the
force transmitted across an element of area da is 2S(e) n da, and the
components S,!;) are hence effective stresses in the electrostatic field.
-
(26)
E ~ ( E-$ &E2)- X
EDE~E~
coE,Ez
- +E2) - X
eoEyEz
eoEyE,
EO(E~
coEZEz
~oE3,
EO(E: - +E2) -
Ei
=
0.
x
+ E; + E; = E2,
Eq. (26) proves equivalent to
8x3 + 4E"),2 - 2E4x
(27)
The roots of (26) are, therefore,
- E6
= 0.
from which i t is apparent that the stress quadric has a n axis of symmetry.
Let n(") be a unit vector fixing the direction of the principal axis
According to (18) page 86, the components of
associated with A,.
n(5)with respect to a n arbitrary reference system must satisfy
( E i - E2) n(")
, E,E,ng)
E,E,np) = 0,
(29)
EyE,np)
(Ei - E2)nF) E,E,nY) = 0,
E,E,n?)
EzEyng) (E: - E2)np)= 0.
From the theory of homogeneous equations it is known that the ratios of
thc unknowns np), ng), np) art: as the ratios of the minors of the detcrminant of the system, whence one may easily verify from (29) th a t
(30)
n2):nF:nP)= Ex:E,:Ez.
+
+
+
+
+
+
The major axis of the electric strcss quadric at a n y point in thejield is directed
along the vector E at that point. The stress transmitted across a n element
of surface whose normal is oriented in this direction is a simple tension,
t(a)
(31)
=
5 E2 n(4.
2
The stress across any element of surface containing th e vector E i . e . ,
an element whose normal is at right angles to the lines of force-is also
normal but negative, and corresponds therefore to a compression
(32)
t(bf
=
-3 E2 n(b)
2
t(c)
=
-2 E2 n(cf.
2
102
STRESS AND ENERGY
[CHAP.
11
Suppose, finally, that the normal of a surface element in the field is
oriented in an arbitrary direction specified by n. Let the z-axis of a
coordinate system located a t the point in question be drawn parallel to E
and choose the x-axis perpendicular to the plane through E and n. The
angle made by n with the direction of E will be called 0. Then
E,
n,
=
= 0,
E,
=
n,
=
0,
IEI = E,,
sin 0,
n2 = cos 8,
whence according to Eq. (23) the stress components are
(33)
t, = 0,
t, =
-;sin e,
~2
tz =
€0
~2
cos e.
The absolute value of the stress transmitted across any surface element,
whatever its orientation, is therefore
(34)
It] =
9
E2.
2
Furthermore t lies in the plane of E and n in a direction such that E
bisects the angle between n and t as illustrated in Fig. 17.
I n the light of this representation it is
easy to comprehend the efforts of Faraday
and Maxwell to reduce the problem of electric and magnetic fields of force to that of
an elastic continuum. To both, the concept of a force propagated from one point in
space to another without the intervention
of a supporting medium appeared wholly
untenable, and in the absence of anything
FIG. 17.-Relation of tension t
transmitted across an element of more tangible an all-pervading “ether”
surface in an electrostatic field t o was evcntually postulated to fill that role.
the field intensity E.
There was then attributed to the stress
components of the field, even in space free of matter, a physical
reality, and a valiant attempt was made to associate with the &her
properties analogous to the strains of elastic media. These efforts did
not bear fruit. Subsequent research has shown that electromagnetic
phenomena may be formulated without employment of any fixed reference system, and that there are no grounds for the assumption that force
can be propagated only by actual contact of contiguous elements of matter or ether. On the modern view, the representation of an electrostatic
field in terms of the stress componcnts 8;;) has no essential physical
reality. All that can be said-and all that it is necessary t o say-is
that the mutual forces between elements of charge can be correctly
ELECTROMAGNETIC MOMENTUM
SEC.2.61
3 03
calculated on the assumption that there exists throughout the field a
fictitious state of stress as described.
and 2S(e)are of identical form, the discussion of
Since the tensors 2S(m)
the foregoing paragraphs applies in its entirety to the field of a stationary
distribution of current. The force exerted by external sources on the
current within a n arbitrary closed surface is obtained by integrating over
this surface a stress whose absolute value is
jtl
(35)
1
= - B2,
2PO
and whose direction with respect to the orientation of a surface element
and the direction of the field B is as in Fig. 17.
If in Eq. (23) the components Xii, are introduced from Tables I and 11,
it is apparent that the stress transmitted across an element of area whose
positive normal is n may be written vectorially as
(36)
(37)
=
t(e)
t(m) =
.n = to(E- n)E 22 E2n,
1 (B .n)B ZS(4 .n = PO
2S(c)
-
Over any volume bounded by :t regular, closed surface Z we have therefore for stationary distributions of charge and current:
.I[
z
(39)
[h
-
tO(E n)E - 2
2 E2n] da =
1
(B .n)B - - BZn da
2P0
=
Iv
pE dv,
sv
J
x B dv.
2.6. Electromagnetic Momentum.-In a stationary field the net force
transmitted across a closed surface 2 bounding a region containing neither
charge nor current is zero. If, howevcr, the field is variable, it is clear
from Eq. (25) that this is not the case. How, then, are we to interpret
the apparent action of a force on volume elements of empty space?
Since the dimensions of toE = D are QL-2, and those of B = paH are
MQ-IT-’, i t is evident that the quantity
is dimensionally a momentum per unit volume. The dimension Q drops
out of the product D x B and this conclusion, therefore, is not simply a
consequence of the particular system of units employed. The identity
104
[CHAP.I1
STRESS A N D ENERGY
can be interpreted on the hypothesis that there is associated with an
electromagnetic field a momentum distributed with a density g. The
total momentum of the field contained within V is
G
=
J-ga,
kg.-meters/sec.,
and (41) now states that the force transmitted across L: is accounted for
by an increase in the momentum of the field within 2. The vector
2 s n measures the inward flow of momentum per unit area through 2 ,
while the quantity -Sjh may be interpreted as the momentum which in
unit time crosses in the j-direction a unit element of surface whose normal i s
oriented along the k-axis.
A direct consequence of this hypothesis is the conclusion that Newton's third law and the principle of conservation of momentum are
strictly valid only when the momentum of a n electromagnetic field is
taken into account along with that of the matter which produces it. Let
us suppose that within the closed surface 2 there are charges distributed
with a density p, and that the motion of these charges may be specified
by a current density J. The force exerted on the charged matter within
2 is then
d
(43)
F, F, =
(PE J X B) dv = Gmeeh,
-
+
+
where Gmeohis the total linear momentum of the moving, ponderable
charges. The conservation of momentum theorem for a system composed of charges and field within a bounded region is, therefore, expressed
according to Eq. (25) by
(44)
If the surface Z is extended t o enclose the entire field, t h e right-hand
side of (44) must vanish, and in this case
(45)
Gmeoh
+
Gelectro-
= constant.
There appears t o be associated with a n electromagnetic field an inertia
property similar t o that of ponderable matter.
ELECTROSTATIC ENERGY
2.7. Electrostatic Energy as a Function of Charge Density.-A finite
charge concentrated in a rrgion so small as to be of negligible extent
relative to other macroscopic dimensions will be referred to as a point
charge. Now the force excrt,ed on such a point charge q in the field of a
stationary charge distribution is qE, and the work done in a displacement
SEC.2.71
CHARGE D E N S I T Y
of q from a point r = rl t o a second point r
=
105
r2 is
Since the curl of E vanishes a t every point in an electrostatic field, the
vector E is equal to the negative gradient of a scalar potential 4, and we
have
(2)
E - d s == -V+. ds
-d+,
1
where d+ is the change in potential along an clement ds of the path of
integration. From this it follows clearly Ihal thc work done in a displacement of q from rl to r2 is independent of the choice of path and is a
function solely of the initial and terminal valucs of the potential.
(3)
w = -q J: ci+
=
q[+(rl) - +(rz)~.
I n particular, the work done in the course of a displacement about a
closed path is zcro. A field of force is said to bc conservative if the work
done in a displacement of a system of particles from one configuration
t o another depends only on the initial and final configurations and is
independent of the sequence of infinitesimal changes by which the finite
displacement is effected. The conservative nature of the electrostatic
field is established b y (3), or more precisely by the condition V x E = 0.
Displacements and variations in a static field must be understood to occur
so slowly as to be equivalent to a sequence of stationary states. Such
changes are said in thermodynamics t o be reverszble.
I n Chap. I11 i t will be shown that if all the sources of an electrostatic
field are located at finite distances from some arbitrary origin the potential and field intensities become vanishingly small a t points which are
sufficiently remote. The work done by a charge q as i t recedes from an
initial point r = rl t o rz = 00 is, therefore,
(4)
w = n+(r>.
Obviously the scalar potential itself may be interpreted as the work done
against the forces of the field in bringing a unit charge from infinity t o
the point r, or as the work returned by the system as a unit element of
charge recedes to infinity.
We shall use the term energy of a n electrostatic system somewhat
loosely t o mean the work done on the system in carrying its elements of
charge from infinity t o the specified distribution by a sequence of reversi-
106
STRESS A N D ENERGY
[CHAP.I1
ble steps. It will be assumed that the temperature of all dielectric or
magnetic matter in the ficld is held constant.'
The energy of a point charge q 2 in the field of a single point source q1 is
u=
(6)
q2421,
where 4 2 1 is the potential a t q2 due t o ql. Now the work done in bringing
q z from infinity to a terminal point in the field of q1 would be returned
were q1 allowed next t o recede t o infinity, and therefore
(7)
IT=
p1412.
The mutual energy of thc two charges may consequently be expressed by
the symmetrical relation
u=
(8)
m 4 1 2
+
q2421).
If first q2 and then q3 be introduced into the field of ql, the energy is
u=
(9)
q2421
+
q3(431
+
4321,
which in virtue of the reciprocal relations betwecn pairs is equivalent t o
(10)
u=
+(412
+
413)41
+
&A421
+
423142
+
i(431
+
432)q3.
By induction it follows that the energy of a closed system of n point
charges is
where ~ $ i is the potential a t qi due t o the remaining n - 1 charges of the
system.
Note that (11) is valid only if t h e system is complete, or closed. If
on the contrary the n charges are situated in an external field of potential
+o, a term appears which does not involve the factor
I n this case
+.
Let us consider now a region of space V within which there are to be
found fixed conductors and diclcct ric matter. Within the dielectrics
charge is distributed with a volume drnsity p, while over the surfaces of
these dielectrics and on the conductors there may be thin layers of charge
of surface density w . At a point (z, y, x ) within V the potential of the
distribution plus that of possible sources situated outside V is 4. The
work required t o increase the charge a t ( 5 , y, z ) b y an infinitesimal
1 The
energy in question is in fart the free energy in a thermodynamic sense.
107
FIELD I N T E N S I T Y
SEC.2.81
+
amount 6q is 6q, and the increase in energy resulting from an increase
in the volume density of charge by an amount 6p, or the surface density
by a n amount 6w, a t every point in V is
sUl
(13)
=
$ 4
6p
dv
+ $9
6w da.
The second integral is to be extended over all surfaces within I
' bearing
charge. On the other hand, the addition of an element of charge 6q at
(z, y, z ) increases the potential a t all points both inside and outside V
by an amount 64,with a resultant increase of the cnergy of the charges
already existing within V of
6Uz =
(14)
$
p
64 dv
+
o 64 da.
The work 6U1 done on the system in building up the charge density is
e'qual to the potential cnergy 6U2 stored in the field provided the system
is closed; provided, th at is, that the region V of integration is extended
to include all cha-rges contributing t o the field. I n that case
su = su1=
6U2,
which upon integration leads to
as the electrostatic energy of a charge system referred to the zero state
0.1
2.8. Electrostatic Energy a s a Function of Field Intensity.-Let
us
imagine for a moment that, conductors have bccn eliminated from the field
and that all surface discontinuitirs a t the boundaries of dielectrics are
replaced by thin but continuous transition layers. Charge is distributcd
throughout the dielectric with a density p, but me shall assume that this
distribution is confined to a region of finite extent: the potential and
intensities of the field vanish a t infinity. Regions free of matter are, of
course, to be considered as dielectrics of unit inductive capacity. The
work required to increase tho charge density a t every point in the field
by a n amount 6p is
p = w =
su = J rp 6p dv,
(17)
+
where
is the potential due t o th e initial distribution p. NQW the
increment ir? charge density is related to a variation of the vector D
1 The
convergence of these integrals will be demonstrated in Chap. 111.
108
[CHAP.I1
STRESS AND ENERGY
by the equation
6p =
(18)
Furthermore,
-
-
6(V D)
-
= V
-
(19) +V (6D) = V (4 6D) - 6D V+
and hence (17) is equivalent to
6U
(20)
=
E * 6D dv
+
- (6D).
=
V * (4 6D)
+E
*
6D,
V * ( 4 SD)dv,
or, upon application of the divergence theorem, to
6U =
(211
$v E
6D dv
+
4 6D n da,
where S is a closed surface bounding a volume V . This region V need
not include all the charges that contribute to the field. If, however, we
allow the surface S to expand into
a sphere of infinite radius about
some arbitrary origin, the contribution of the surface integral
vanishes; for 4 will be shown later
to diminish as l / r a t sufficiently
large distances from the origin,
and D as l/r2. The surface X increases with increasing radius as
r2, and the surface integral therefore vanishes as l/r. The increment of energy s t o r e d in the
electrostatic field can be calculated
from the integral
FIG. 18.-Conductors
surfaces 5'1, SZ, . .
dielectric medium.
.
,
bounded by the
embedded in a
(22)
6U
=
1
E 6D dv
extended over all space.
I n practice the charge is rarely distributed throughout the volume of a
dielectric but is spread in a thin layer of density w over the surfaces of
conductors. These conductors may be considered as electrodes of
condensers, and the increase in the energy of the field is the result, for
example, of work done by the electromotive forces of batteries in the
process of building up the charge on the electrodes. Let us suppose that
there are n conductors, whose surfaces we shall denote by S,,
embedded
in a dielectric of infinite extent. One of these surfaces, let us say S,, will
be assumed to enclose all the others as illustrated in Fig. 18. To ensure
the continuity of the potential and field necessary for the application of
the divergence theorem, surfaces of abrupt change in the properties of the
dielectric may again be replaced by thin transition layers without in
SEC.2.81
109
FIELD I N T E N S I T Y
ar?y way affecting the generality of the results. The work required to
increase the charge density on the surfaces Si by a n amount 6w is
The sum of these n surface integrals can be represented as a single integral
by the artifice of drawing finlike surfaces from the interior conductors to
S, as indicated in Fig. 18. The intcgration starts a t a point P on S,,
extends on one side of the fin to Q, then over S1 and then returns to P
over the opposite face of the fin. The surface PQ carries no charge and
contributes nothing to the integral. The interior of S, is thus reduced
n
to a simply connected region' bounded by a single surface S =
I:8;.
i= 1
SU
=
C$ 6w
da.
I n See. 1.13 it was shown that a t any surface of discontinuity the
normal and tangential components of the vectors D and E satisfy the
conditions
n. (Dz - D1) = W ,
n x (Ez- El)= 0.
(25)
The positive normals to the surfaces Xi are directed into the dielectric,
and since n in Sec. 1.13 was drawn from medium (1) into medium ( 2 ) ,
the index (1) now denotes the interior of th e conductors whereas ( 2 )
refers t o the dielectric. I n Chap. I11 we shall have occasion to consider
in some detail the electrostatic properties of conductors, but for the
present we need only accept the elementary fact that the field at any
interior point of a conductor is zero. Were it otherwise, a movement of
free charge would occur, contrary to the assumption of a stationary state.
The interior and surface of a conductor are, therefore, a region of constant
potential. The charge density at interior points is zero [Eq. (21), page
151 and whatever charge the conductor carries is distributed on the
surface in such a way as to bring about the vanishing of the interior field.
Since El= DI= 0,
n.D2 = W,
n x E2 = 0 ;
at a point just outside the conductor the tangential component of E
is zero and the normal component of D is equal to the surface density
of charge.
Upon introducing the first of these relations into (24) and dropping
the index. we obtain
(26)
6~ = J 4 6 a . n d a .
(27)
1
See Sec. 4.2, p. 227.
110
[CHAP.I1
STRESS A N D ENERGY
To express this result as an integral extended throughout the volume of
dielectric bounded by the surfaces Si we again apply the divergence
theorem, but must note that n in (27) is directed into the dielectric; the
sign must therefore be reversed.
(28)
6U = -
1.
( 4 SD)dv
=
1E
6D dv -
+V
- (6D) dv.
If the density of charge throughout the dielectric is zero or constant,
(6D) = 0, arid we find as in (22) that the work expended by the
batteries in building up the electrode charges by an amount 6w is
V
(29)
SU
=
s
E 6D dv,
an integral extended over the cntire space occupied by the field, and that
this work is stored in the field as electrostatic energy.
Since the energy SU appears to be stored in the field, as the potential
energy of an extended spring is in some manner stored within it, it is
not unreasonable to suppose that the electrostatic energy is distributed
throughout the field with a density
(30)
-
6~ = E 6D.
joules/meter3
It is difficult either to justify or disprove such a hypothesis. The
transformation from a surface integral to a volume integral ici obviously
not unique, for there might be adtlcd to 4 6D in (27) any vector whose
normal component integrated over S is zcro. Furthermorc, it may be
questioned whether the term “encrgy density” has any physical significance. Energy is a function of the configuration of a system as a
whole. The objection has been stated in a rather quaint way by Mason
and Weavcr,l who suggest that it is no more sensible to inquire about the
location of energy than to declare that the beauty of a painting is distributed over the canvas in a specifird manner. However ingenious,
such an analogy scems not entirely well founded. The energy of an
inhomogeneously stressed elastic medium is certainly concentrated
principally in regions of greatest strain and in this case the elastic energy
per unit volume has a very definite physical sense. Granting that the
analogy of the electrostatic to the elastic field is not a close one, and that
we can be certain only of the correctness of the expression (29) integrated
over the entire field, it is ncvcrtheless plausible to assume that the
energy is localized in the more intense regions of the field in the manner
prescribed by (30).
To find the total energy storrd in a field, the increment 6U must be
integrated from the initial state D = 0 to the final value D.
MASONand WEAVER,“The Electromagnetic Field,” p. 266, University of
Chicago Press, 1929.
SEC.2.91
111
A THEOREM O N VECTOR FIELDS
U
=
svI” -
E 6D dv.
In case the medium is isotropic and linear, such that E in the relation
= EEis a function possibly of position but not of E, we have
D
E 6D = 22 6EZ,
and hence
U
=
1
sv
cE2dv.
2.9. A Theorem on Vector Fields.-Let P and Q be two vector
tions of position which throughout all space satisfy the conditions
(33)
v x P = 0, V ’ Q = 0,
fllnc-
and which are continuous and have continuous derivatives everywhere
except on a closed, regular surface S1. The transition of the tangential
components of P and of the normal components of Q across the surface S1
is assumed continuous, but arbitrary discontinuities are permissible
in the normal components of P and the tangential components of Q.
The prescribed conditions over XI are, therefore,
-
n (Q+ Q-) = 0.
(34)
n x (P+ - P-) = 0,
The unit normal n is drawn outward from S1and vectors in the immediate
neighborhood of this outer face are denoted by the subscript
whereas
those located just inside the surface are denoted by the subscript -.
Finally, i t is assumcd that the sources of the fields P and Q are located
at finite distances from an arbitrary origin and that P and Q vanish a t
infinity such that
lim rQ = 0.
lim rP = 0,
(35)
+,
r+
m
r+ m
Then it can be shown that the integral over all space of the scalar product
of an irrotational vector P and a solenoidal vector Q i s zero, provided P and
Q and their derivatives are continuous everywhere except on a finite number
of closed surfaces across which the discontinuities are as specified in (34).
For since V X P = 0, we may write P = -V+ and, hence,
+
P - Q = -V.(+Q)
4V-Q.
(36)
We shall denote the volume enclosed by the surface 81 by V1, and th a t
exterior to it by V z , so that the complete field of P and Q is Vl
Vz.
The last term of (36) vanishes and therefore
+
(37)
T o apply the divergence theorem to this expression we observe th a t VI
112
[CFIAP.I1
STRESS A N D ENERGY
is bounded by S1, and that Vz is bounded on the interior by S1and on the
exterior by a surface S which recedes t o infinity. In view of the behavior
of P and Q at infinity as specified by (35), the integral of +Q n over S is
zero. The positive normal t o a closed surface is conventionally directed
outward from the enclosed volume; the two integrals of +Q n over S1
are therefore of opposite sign but of equal magnitude in virtue of (34).
Equation (37) transforms to
-
as stated. The extension of this theorem to a finite number of closed
surfaces Xi of discontinuity is elementary.
2.10. Energy of a Dielectric Body in an Electrostatic Field.-The
useful theorem of the preceding section may be applied to the solution of
the following problem. Let us suppose that an electrostatic field El has
been established in a diclectric medium. To simplify matters we shall
assume that this medium is isotropic and linear, and hence characterized
by an inductive capacity e l which is either constant or at the most a
scalar function of position. A nonconducting body whose inductive
capacity is e2, is now introduced into the field El, while the sources of El
are maintained strictly constant. We wish to know the energy of the
foreign dielectric body due to its position in the field.
The initial energy U 1 , representing the total work done in establishing the initial field, is obtained by evaluating
Ul
(39)
=
-
f
E1 DI dv
over all space. After the introduction of the body the modified field at
any point is E, and the difference Ez = E - El is thus the field resulting
from the polarization of the body. The volume occupied by the foreign
body we denote by V1, that of the medium exterior i o it by V z . The
energy of the field in this new state is
U
and the change
u=uz-u
-'s
's
-2
Vl+V2
-
VlfVZ
3
E.Ddv,
( E - D - E l - D 1 )dv
must be the energy of the body in the external field EL, and consequently
equal to the work done in introducing it. Equation (41) is equivalent to
SEC.
113
ENERGY OF A DIELECTRIC BODY
2.101
The curl of E is zero everywhere, and the divergence of D - D1 is zero
since tlhe initial source distrihution is fixed. Across the surface which
bounds the medium e Z from the medium €1 the conditions
(43) n x (E, - E-)
=
0,
-
n [(D - D1)+ - (D
- D1)-]
=
0,
arc satisfied provided the surface carries no charge. Then by the theorem
of See. 2.9 the first integral of (42) is zero, leaving
Since D1
=
ELEL,the second integral of (44) is equivalent to
(45)
The conditions of Sec. 2.9 are satisfied by V x EL = 0, V
so that
(46)
s
EI (D
=
0,
El * (D - DI) dv
v1+v2
and hence
(47)
- DI)dv =
- (D - D1)
f
sv8
El * (D - DI) du
B
= --
Sv1
-
El (D - D1) dv.
Upon introducing (47) into (44), we obtain an expression for the energy
of a dielectric body ernbeddrd in a dielectric medium in terms of an
integral extended, not over all space, but over its own volunae alone.
or, since D1 = ~1E1,D = eZEwithin V1,
In case the external medium is free space, the inductive capacity el
reduces to eG. Then, since the resultant field within the body is relatcd
to its polarization according to Sec. 1.6, by
U may be written
114
STRESS A N D ENERGY
[CHAP.
11
T h e potential energy oj' a dielectric body in free space and in a jixed external
jield El i s equal to -+P El per unit volume.
It is the variation of this potential energy U expressed by (49) or (51)
with respect to a small displacement that will lead us to the mechanical
forces exerted on polarized dielectrics. It will be observed that the
energy is negative if e 2 > €1, and decreases in this case with decreasing €1,
increasing c2, or increasing field intensity El. We may anticipate, therefore, that if e2 > el, the body will be impelled toward regions of intense
field or diminishing inductive capacity €1. On the other hand if € 1 > €2, as
might be the case of a solid immersed in a liquid of high inductive capacity, the forces exerted on the body will tend to expel it from the field.
A direct consequence of (49) is the theorem that a n y increase in the
inductive capacity of a dielectric results in a decrease in the total energy of
the jield. Let us suppose again that El is the field of aJixed set of charges
in a dielectric medium whose inductive capacity is E = e(z, y , 2). If ah
every point E is increased by an infinitesimal amount B E , the consequent
variation in thc electrostatic energy will according to (49) be equal to
for the product 6eE * El will then differ from 6~ E 2 by an infinitesimal of
second order.
The energy of an electrostatic field is now completely determined by
the distribution of charge p and o,and by the inductive capacity E ( X , y , 2).
Equation (52) expresses the variation in energy resulting from a slight
change in the properties of the dielectric, in the course of which the
charges are held constant.
u = -1.2
(53)
The variation
.I
+
E2 de dv,
S
(constant charge).
E . 6D dv
6U =
4 6 p dv
4 6w da =
(54)
expresses, on the other hand, the increment of energy resulting from a
small change in the density of charge, in the course of which the properties of the dielectric are held constant.
2.11. T h o m s o n ' s Theorem.-Charges placed o n a system of $xed conductors embedded in a dielectric will distribute themselves o n the surfaces of
these conductors such that the energy of the resultant electrostatic field i s a
minimum.
The proof of this and the following theorems will be confined to the
case of a linear, isotropic dielectric. Let us suppose that there are n
115
THOMSON’S THEOREM
SEC. 2.111
..
conductors bounded by the surfaces Si, (i = 1,2, .
n), each bearing
a charge 9;. Discontinuities in the properties of the dielectric may be
replaced by thin transition layers without affecting the generality of the
theorem and we shall, therefore, assume the inductive capacity E of the
medium to be a continuous but otherwise arbitrary function of position.
It may also be assumed that there is a density of charge throughout the
volume of the dielectric, although such a condition does not often occur
in practice.
At every point within the dielectric the field of the charges in equi.
librium must satisfy the conditions:
(55)
V*D=p,
VXE=O,
)
E=-V4;
over the surface of each conductor S ;
4i
(56)
=
constant,
L,D - n da
= qi;
a t infinity the potential vanishes as l/r.l
Suppose that 4’, E’, D’ is any other possible electrostatic field; it satisfies the conditions (55), but not necessarily (56), and is known to differ
somewhere, if not everywhere, from 4, E, D. Since the volume distribution p and the total charge on the conductors is fixed we have
(57)
-
V (D’ - D) = 0,
ii
(D’ - D) n da = 0.
If U and U’ are the electrostatic energies of the two fields, their difference
is
U‘
-
U
or, since D’ = cE’,
=
‘S
-
2
‘.I
E’sD‘dv - -,j E . D dv,
The second term on the right vanishes, for we see that on putting E = P,
D’ - D = Q, the conditions of the theorem demonstrated in Sec. 2.9
are satisfied. There remains
which is a n essentially positive quantity. The theorem is proved, for
if E’ differs in any region of space from E the resultant energy U’ will be
greater than U . The condition of electrostatic equilibrium is character‘See Sec. 3.5, p. 167.
116
[CHAP.I1
STRESS AND ENERGY
ized by a minimum value of electrostatic energy and one therefore concludes that for the determination of equilibrium the energy U plays the
same role in electrostatics as the potential energy in mechanics.
2.12. Earnshaw’s Theorem.-A charged body placed in a n electrostatic field cannot be maintained in stable equilibrium under the influence of
the electric forces alone.
We shall suppose that the initial field is generated by a set of charges
qi, (i = 1, . . . , n,), distributed on n .fixed conductors whose bounding
surfaces are denoted by Xi. These conductors are embedded in a dielectric whosc inductive capacity e may be a continuous function of position
but in which there is no volume distribution of charge. We note first
that ncithcr the potential nor any of its partial derivatives can assume a
maximum or a minimum value a t a point within the dielectric; for if 4
is to be an absolute maximum it is necessary that the three partial derivatives d2+/dx2, d2&/dy2, d2+/dz2 shall all be negative at the point in question, but this condition is incompatible with Laplace’s equation
Likewise t,he existence of an absolute minimum requires that these three
derivativcs shall be positive, which again is inconsistent with (61).
The same argument applies also to the derivatives of the potential.
Let us suppose now that a chargc qo is placed on a conducting surface
So. The distributions on all the conductors are momentarily assumed to
be fixed and So is introduced into the field of the other n charges. If wg
is the surface density of charge on So, the energy of this conductor is
wherc 4 is the potential of the initial field. Let x,y, z be the coordinates
of any point which is fixed with respect to SOand. f , 9,
those of any
point on the surface. The potential on So due to the other n charges
may be represented by a Taylor series in terms of its value and the value
of its derivatives a t z, y, z,
r,
(63)
+...hence the energy, too, may be referred to the potential and its derivatives
a t this point. But since 4 cannot be a minimum a t x, y, z, it is always
possible to displace the conductor SOin such a way that the energy Uo
is decreased. If after this displacement the charges, which thus far
have been assumed to be [‘frozen” on the surfaces So, Si,are released,
SEC.
117
UNCHARGED CONDUCTORS
2.131
these surfaces will again become equipotentials and by Thomson’s
theorem the energy of the field will be still further diminished. A
minimum value of the energy function Uo(x, y, x ) does not exist in the
electrostatic field and conscquently the conductor So is never in static
equilibrium.
2.13. Theorem on the Energy of Uncharged Conductors.-The introduction of an uncharged conductor into the Jield of a fixed set of charges
diminishes the total energy of the jield.
The conditions are the same as in the prcvious theorem but the
surface Sonow carries no charge. Let E, D be the vectors charact,crizing
the field before, E’, D’ t,he field after the introduction of the conductor So.
Thc change in energy is
s
‘.I
U-U‘=- 2
(64)
.
E - D ~ V - E’
~ D’ dv.
The volume integrals are to be extended through all space, but since the
field vanishes in thc interior of the surfaces So, S,, the contribution from
these regions is zero. Let V be the volume of the field exterior to the n
conductors S , before the introduction of SO,and V Othe volume bounded
by So. Then
v1 = v - vo
(65)
represents the volume of the region occupied by the field after the introduction of So,and Eq. (64) may be written in the form
(66)
U
-
U‘
=
svl
+ dl,
3 S v E * D dv -
-
= 1 Jvo E D dv
.
= 1 Jv, E D
dv
+f
f
(E D - E’ D’) dv
-
(E - E’) (D - D’) dv
+ Jvl
.
E’ (D - D‘) dv.
The last integral on the right can be shown to vanish. For
E’ (D - D’) = -V+’ (D - D’) = -V * [+’(D - D’)]
+‘ V * (D - D’)#
+
-
and V (D - D’)
(67)
v1
-
=
0. By the divergence theorem,
E’ (D - D’) dv =
V .[+’(D’ - D)] dv
-
2
t=1
4;
L,
-
(D’ - D) n da,
118
STRESS A N D ENERGY
[CHAP.I1
where the 4; are the potentials of the conducting surfaces Xi.
total charge on each surface is constant, so that
The
and hence
The difference between the initial and final energies is, therefore,
an essentially positive quantity.
MAGNETOSTATIC ENERGY
2.14. Magnetic Energy of Stationary Currents.-Let us consider a
stationary distribution of current confined to a finite region of space.
This current may be supported by conducting matter, or result from the
convection of charges in free space. The equation of continuity reduces
to v . J = 0, in virtue of which we may imagine the distribution to be
resolved into current lines closing upon themselves. A current tube or
filament may be constructed from the current lines passing through an
infinitesimal element of area. The tube is bounded by those lines which
pass through points on the contour of the element. At every point
on the surface of a tube the flow is tangential; no current leaves the tube
and consequently the net charge transported across every cross section of the
c,
tube in a given time is the same.
We shall calculate first the potential
c
2
ds
energy of a single, isolated current filament
in the field B of fixed external sources.
n,
in Fig. 19 as C1. The force exerted by the field on a linear element of the
filament is
f = I d s X B.
(1)
Suppose now that the filament C1 is translated and deformed in such a
manner that every element ds is displaced by the infinitesimal amount 6r
into the contour Cz. The displacement 6r is assumed to be a continuous
function of position about C1 but is otherwise arbitrary. The work
done by the force (1) in the displacement of the element ds is
(2)
-
f 6r
=
-
I ( d s x B) 6r
=
-
I B (6r X ds).
119
SEC.2.141 MAGNETIC ENERGY OF STATIONARY CTJRRENTS
Let SI be any regular surface spanning the contour C1. A second
surface Sz is drawn to span Cz, but in such a way as to pass through C1
and then coincide with XI The surface
(3)
IS3
=
SB
-
s1
is, therefore, a band or ribbon of width 6r bounded b y the curves
C2. If daa is a n element of X3 wc have clearly
(4)
Sr X ds
=
C1
and
n3da3,
where n3 is the positive unitj normal to 8 3 . Thc positive faces of X1
and S 2 are determined by thc usual convention that an observer, circulating about the contours in 1he direction 01 thc current, shall have the
positive face a t his left. The magnetic fluxes threading 61 and C z are
(5)
%
=
B . nz daz;
hence the net change in flux resulting from the displacement is
On the other hand the total work done by the mechanical forces is
obtained by integrating (2) around the closed contour C1, whence in
virtue of (4) we find
If now i t be assumed that in the course of the virtual displacement 6r
the external sources and the currcnt I are maintained strictly constant,
then the work done by the mechanical forces is compensated by a decrease
in a potential energy function U .
6U = -SW = - I 64,,
(8)
or
(9)
where X represents any surface spanning the current filament. Inversely,
the mechanical forces and torques on a currcnt filament in a magnetostatic field can be determined from a variation of thc function U whilc
holding constant the current I and the strength of the sources.
If 6r is a real rather than a virtual displacemcnt, work must be
expended t o keep the current constant. The change &@ in the flux
induces a n electromotive force
120
STRESS A N D ENERGY
[CHAP.I1
where 6t is the time required to effect the displacement. This induced
e.m.f. must be counterbalanced by an equal and opposite applied e.m.f.
V'. The work expended by the applied voltage on the circuit in the time
6t is
(11)
V'I 6t = +I 6@.
T h e work done by the transverse mechanical forces in a small displacement
of a linear circuit i s exactly compensated by the energy expended by longitudinal electromotive forcps necessary to maintain the current constant.
The total work done o n the circuit i s zero.
Let us suppose now that the source of the field B is a second current
filament. Thus we imagine for the moment that the current distribution
consists of just two isolated, closed filaments I1 and I z . The fields
generated by these currents are respectively B1 and B2. The potential
energy of circuit 12 in the external field B1 is
(12)
U21 = -I2
h,B1 -
n2 daz,
where S 2is any surface spanning the filament I,.
cnergy of circuit I1 in th e field of I , is
Likewise the potential
U12 and U21 are scalar functions of position and circuit configuration
whose derivatives give the forccs and torques exerted by one filament
on the ot'hcr. From the equalit,y of action and reaction it follows t h a t
(14)
ill12
=
U2l =
u,
and hence the mutual potential energy may be expressed as
lJ = -- &I&%
(15)
where a1is the magnetic flux threading the filament I1.
The mutual energy U , which reduces to zero when the separation
of the filaments becomes infinite, is not equal to the total work th a t must
be done in approaching them from infinity tlo some finite mutual configuration, and therefore does not rcprcsctnt the total energy of the system.
For let us suppose th at I z is allowed a small displacement under the
forces exerted by I I , wit,h a resultant decrease -6U in the potential
energy. To maintain the current I2 constant during the displacement,
an equal amount of work 6W must be done on the circuit I z to compensate
the effect of the induced e.m.f. Thus far the net change in energy is zero.
But now we must note that, the displaccment of I , results in a change
in the flux threading 11and givcs rise, t,herefore, to an induced e.m.f. V ,
opposing the current I1. If I1 is to be maintained constant, a voltage V {
SEC.2.141 MAGNETIC ENERGY OF STATIONARY CURRENTS
121
must be impressed on the circuit I I compensating V l and doing work
at the rate Ill':. The induced e.m.f., however, depends only on the
relative motion of I1 and I,. The work done by the electromotive forces
induced in circuit (1) by the displacement of circuit (2) must equal that
which would be done were ( 2 ) fixed and a n equal and opposite displacement imparted to (1). The work done on (1) is therefore 6W. I n sum,
when the mechanical forces acting on the two filaments are allowed to
do work, there is a decrease in the mutual potcntial energy of amount
-6U, but this is offset by thc work 26W which must be done o n the
system to maintain I1 and I , constant. If the total magnetic energy
of the system be denoted by T , the variation in T associated with a
relative displacement at constant c u r r e n t is
(16)
6T
=
26W -
SLl =
6W
=
-6U,
or, by (151,
+
T = +I1@1 +I2@2.
(17)
In general, if the current system is composed of n distinct current filaments, the magnetic energy is
T
=
-3
1 " I@i,
2
i=l
where @i is the flux threading circuit i due to the other n - 1circuits.
Equation (18) is an expression for the work necessary to bring n
closed current filaments from an initial position a t infinity t o some specified finite configuration. The cross section of any filament is very small
but does not vanish, and a filament in the sense that we have used it here
is not, therefore, a line singularity. Each filamcnt or tube may be subdivided into a bundle of thinner filaments carrying fractions of the initial
current. I n the limit of vanishing cross section, the current carried
by the filamcnt is infinitesimally small, as is also the energy necessary
t o establish it. Consequently the total energy of a current distribution
is simply the mutual energy of the infinitesimal filaments int.0 which the
distribution can be resolved.
Now in practice a currcnt distribution in a conducting medium cannot
be established by the mathematically simple expedient of collecting
together current filaments from infinity, arid although (18) proves t o be
correct for the magnetic energy of n circuits in a medium for which the
relation of H to B is linear, this is not necessarily true in general. I n
place of a continuous distribution let us consider for the moment n
linear circuits embedded in any magnetic makrial. The resistance of
the i t h circuit is Ri and a t any instant the current it carries is I ( . T o each
circuit there is now applied an external e.m.f. V: generated by chemical
122
STRESS AND ENERGY
[CHAP.I1
or mechanical means. These applied voltages give rise t o variations
in the currents and corresponding variations in the magnetic flux threading each circuit. If T’i is the voltage induced by a variation in ail the
relation of current t o total e.m.f. in the circuit at any instant is
or, since V i =
->
d@.i
dt
The power expended by the impressed voltage V: is VIIi, and consequently the work done on the i t h circuit in the interval 6t is
(21)
SWi
=
R J t 6t
+ Ii S@i.
Of this work the amount R,I; 6t is dissipated as heat, while the quantity
I i 6+i is stored as magnetic energy.’ A variation in the magnetic energy
of the n filaments is therefore related to increments in the fluxes by
n
(22)
ST
=
2 Ii 6@i,
i=l
and the total energy expended on the system, apart from that dissipated
in ohmic heat loss within the conductors, as the currents are slowly
increased from zero to their final values, is
(23)
aiUis
the flux threading the ith circuit a t the initial instant when all
currents are zero. This initial flux will be zero if there is no remnant
magnetization of the medium about the circuits. Equation (23) can
be interpreted as available encrgy stored in the magnetic field only when
the relation of Ii to @i is single-valued (no hysteresis), and (23) reduces
to (18) only when the relation of H to B and consequently of I ; to @.,is
6Ii.
linear, such that li 6@i =
It is a simple matter to extend (23) from a finite number of current
filaments to a continuous distribution of current. Within this distribution let us choose arbitrarily a surface element da. The current
lines passing through points of da constitute a current tube, which, in
virtue of the stationary character of the distribution, closes upon itself.
If J is the current density, the scalar product J n da = d l is constant
.
A portion of the energy I , &a,may, however, be made unavailable because of
hysteresis effects. See Sec. 2.16.
Next Page
SEC,2.151
MAGNETIC ENERGY A N D FIELD I N T E N S I T Y
123
over every cross section of the tube. Let B’ be the magnetic field a t any
point within the tube produced by all those current filaments which lie
outside. The increment in the energy of the current dI due t o an
infinitesimal increase in the field of the external filaments is
6(dT) = dI 6@’
(24)
dI
=
s
-
6B‘ n da,
the integral t o be extended over any surface bounded by the contour of
the filament dI. If A’ is the vector potential of the field B’, then
B’ = v x A’ and (24) may be transformed by Stokes’ theorem t o a line
integral following the closed contour of the filament.
6(dT) = J * n du
(25)
$ 6A’ - ds.
Now the total vector potential A along any central line of the tube dI is
equal to A’ plus the contribution of the current dI = J n du itself; but
as the cross-sectional area du -+ 0, the latter contribution vanishes, so
t h a t in the limit A’ may be replaced by A. Since furthermore the current density vector J, the unit vector n normal to du, and the element
of length ds along the tube are parallel to one another within a tube of
infinitesimal cross section, one may write in place of (25) :
.
6(dT) =
(26)
-
J 6Aduds.
The product da d s = dv represents the volume of an infinitesimal length
of tube. The total increment in the energy of the distribution is to be
found by summing the contributions of all the tubes into which it has
been resolved, and this summation is clearly equivalent to an integration
of the product J 6A over the entire volume occupied by current.
-
61’ =
(27)
$ J . 6Adv.
The work required t o set up a continuous current distribution by means
of applied electromotive forces is, therefore, in general
T
=
sc
J . 6Adv.
I n case the relation between the current and the vector potential which
i t produces is linear, this reduces to
2.16. Magnetic Energy as a Function of Field Intensity.-We shall
suppose that discontinuities in the magnetic properties of matter in the
Previous Page
124
STRESS A N D ENERGY
[CHAP.I1
field can be represented by layers of rapid but continuous transition.
The field may be due in part to currents, in part to permanent magnets
or residually magnetized matter, but all sources are located within a
finite radius of some arbitrary origin. It will be demonstrated in Chap.
I V that under these circumstances the vectors A and B vanish a t infinity
as r-l and r2.The current density J a t any point is related to the
vector H at the same point by
(30)
J = V x H .
Furthermore, by a well-known identity,
J * ~ A = ~ A * V X H = V . ( H~ A
X ) + H . V X ~ A
= V * ( H X 6A)
H * 6B.
Upon introducing (31) into (28) and applying the divergence theorem,
(31)
+
we obtain
where V is any volume bounded by a surface S enclosing all the sources
of the field. If the surface S is allowed to recede toward infinity, the
second integral of (32) vanishes, for the integrand diminishes as r3
whereas S grows only as r2. Therefore the work done by impressed
c.m.fs. (such as might, be dcrived from batteries or gcnerators) in building
up a magnetic field from the initial value B, t o the final value B can be
represented by the integral
(33)
extended over all space. It is to be emphasized that (33) is the energy
associated with the establishmentj of a current distribution in the presence
of magnetic materials, and does not include the internal energy of permanent magnets or the mutual energy of systems of permanent magnets.
The magnetic properties of all materials exclusive of the ferromagnetic
group differ but slightly from those of free space; the relation between B
and H is linear within wide limits of field intensity, the factor p in the
relation €3 = pH is very nearly equal to go, and there is no appreciable
remnant magnetism. Under these circumstances the work done in
building up the field to a value B is returned as the field is again decreased
to zero. Equation (33) integrates to
which we interpret as the energy stored in the magnetic field. As in the
corresponding electrostatic case we may suppose this energy to be distributed throughout the field with a density & p H 2 j ~ u l e s / m e t e r . ~
SEC. 2.161
FERROMAGNETIC MATERIALS
125
2.16. Ferromagnetic Materials.-The relation of H t o B for ferromagnetic substances is in general nonlinear and multivalued. I n an
initial, unmagnetized state thc vectors B and H are zero. If the field is
now built up slowly by means of impressed electromotive forces applied
to conducting circuits, the function B = B(H) at any point in the ferro-
magnetic material follows a curve of the form indicated by Fig. 20.
According t o (33) the work done in magnetizing a unit volume of the
substance is represented by the shaded arra in the figure. Were the
function B(H) single-valued, a decrease in the field from H1, Bl t o zero
would follow the same curve and the entire energy (33) would be available for useful work. Actually, the return usually follows a path such
as that indicated in Fig. 21. Starting
a t HI, the field is decreased until
H = 0. The associated value of
B = B,, however, is still positive.
T o reduce B t o zero, negative values
must bc impartcd to H, meaning
physically t h a t H must be increased
in the o p p o s i t e direction. At
H = -H, the vector B is zero, and
as H continues t o increase in negative
value a point is eventually reached
FIG.2l.-Hysteresis loop.
where simultaneously H = - H 11
B = -B1. The return t o the positive values HI, B1 follows t h e symmetrical path through B = -B2,H = 0 and B = 0, H = fH3. At all
points along the segment B1B2 the value of B is greater than its initial
value on the dotted curve for the same value of H ; the change of B lags
behind that of H and the substance is said to exhibit hysteresis.
126
STRESS A N D ENERGY
[CHAP.I1
Let w be the work done per unit volume of magnetic material in changing the field from the value BI to B.
(35)
UI =
H-dB = H .B
-
-
B dH.
If the variation of the field is carried through a complete cycle following
the hysteresis loop from B1 through B2, -B1, -B2 and returning to B1,
the net work done per unit volume is
n
(36)
W =
-$B-dH,
a quantity evidently represented by the enclosed area of the hysteresis
loop illustrated in Fig. 21. The net work done per cycle throughout the
entire field is
(37)
Q = -J~$B.(EH.
Q is the hysteresis loss, an irretrievable fraction of the field energy dissipated in heat.
2.17. Energy of a Magnetic Body in a Magnetostatic Field.-Let us
suppose that a magnetic ficld B1 from fixed sources has been established
in a magnetic medium. We shall
assume that the relation of B1 to H1
is linear and that the medium is isotropic.
v,
Then B1 = pIH1, where p1 is at most a
B = P ~H
scalar function of position which reduces
to a constant in case the medium is
homogeneous. The energy of the field
is
FIG.22.-The region V1 is occupied
by a magnetic body embedded in the
homogeneous, isotropic medium V2.
(38)
=
f
’ B1
dv,
extended over all space. Now reduce the intensity of the sources to
zero and introduce into a suitable cavity formed in the first medium a
body which we shall assume to be unmagnetized but whose magnetic
properties are otherwise arbitrary. AS in Sec. 2.10 the volume occupied
by the embedded body will be denoted by V1, Fig. 22, and the entire
region exterior to it by V2. If the external sources are currents, the
work which must be done in order to restore them to their initial intensity
is
The ultimate field B differs a t every point from the initial field B1 by
an amount B2 = B - B1 arising from the change in polarization of the
SEC.2.171
127
ENERGY OF A MAGNETIC BODY
matter contained within VI. The increase in the work necessary to build
up the current strength to the initial value is
(40) T = Tz - TI
(H * B - H I * B1) dv
=
It has been assumed that throughout the region V , exterior to the body
the relations between B and H are linear:
Bi = ~ i H i ,
(41)
B
=
in Vz.
PIH
The first integral in (40)is therefore equivalent to
By hypothesis the initial and final current distributions in the source are
identical, so t ha t in both V1 and V z the conditions
(43)
V (B
+ Bi)
=
0,
V X (H - Hi)
=
0,
are satisfied, while over the surface bounding V1 we have according to
(18), Page 37,
n [(B BJ+ - (B BI)-] = 0,
(44)
n x [(H - HI)+ - (H - Hi)-] = 0.
-
+
+
The theorem of See. 2.9 may be applied, giving
or
=
-aJv,
- + B1) dv.
(H - HI) (B
The additional work required to build up the currents in the presence of
the body can thus be expressed in terms of integrals extended over the
volume occupied by the body. Upon introducing (46) into (40), we
obtain
128
STRESS A N D ENERGY
[CHAP.I1
If hysteresis effects are negligible (H is a single-valued function of B),
the quantity T may be interpreted as the energy of the body in the magnetic field of any system of constant sources; and the variation of T
resulting from a virtual displacement of the body determines the mechanical forces exerted upon it. If furthermore the magnetic properties of
the material within 171 can be characterized by a permeability p2, so that
B = p2H, then (47) reduces to
The body was assumed to be initially free of residual magnetism; consequently the vectors B and H within V1 are related to the induced magnetic polarization M by Sec. 1.6,
+
B = PO(H M),
(49)
and, when B = p2H, this leads to
If, therefore, the cavity V1 was initially free of magnetic matter we may
put p~ = p o and write (48) in the form
T =
a
Jv, M . B1 dv,
an expression which corresponds to (51), page 113, for the electrical case
in all but algebraic sign. This distinction, however, is fundamental.
Whereas in the electrostatic case th e work done by the external forces in a
virtual displacement is accompanied by a decrease of the potential
energy U , we shall learn soon t h a t the mechanical forces exerted on a
magnetized body are to be determined from the increase in T , so th a t T
in this sense behaves as a kinetic rather than a potential energy.
A useful analogue of Eq. (52), page 114, may be deduced for the
magnetostatic field from (48). Suppose again th a t B1 is the field of
fixed sources-either currents or permanent magnets-and that a t every
point in space the permeability is specified by p = p ( x , y, z ) , a continuous
function of position. If now the permeability is varied by an infinitesimal amount 6p) the consequent change in the magnetic energy is
therefore, the magnetic energy of a distribution of matter in a fixed field is
(53)
SEC.2.181 P O T E N T I A L ENERGY OF A P E R M A N E N T MAGNET
129
provided the matter is initially unmagnetized and p is independent of H.
For paramagnetic material p > po, K~ > 1, but in the case of diamagnetic matter p < po and the magnetic energy of the initial field is diminished by the introduction of diamagnetic bodies.
2.18. Potential Energy of a Permanent Magnet.-A rigorous analysis:
of the energy of material bodies in a magnetic field becomes very much
more difficult when these bodies are permanently magnetized. I n t h a t
case there is a residual field, uliich we shall denote by Bo, Ho, associated
with the magnet even in the absence of external sources. The relation of
Bo t o Ho may still be expressed in the form
Bo
(54)
=
+ Mo),
p*o(Ho
but Mo, the intensity of niagnc+ization, is now quite independent of Ha,
being determined solely by the previous history of the specimen. If an
external field is applied, there will be induced a n additional component
of magnetization which, as in the past, we denote by M. This induced
polarization M depends primarily on the resultant field H within the
magnet (vanishing as H reduces t o Ho), but also on the state of the magnet and its permanent or residual magnetization Mo.
(55)
+ M(H, Mo) + Mol.
B=
The resultant field H a t an interior point is determined furthermore by
the shape of the magnet as well as by the intensity and distribution of the
external sources.
We shall content ourselves with the derivation of a simple and frequently used cxpresion for the potential energy of a system of permanent magnets. The proof rests on assumptions which are approximately
fulfilled in practice; namrly, that the magnetization Mo is absolutely
rigid and t h a t the induced magnetization M in one magnet arising from
the external field of another is of negligible intensity with respect t o Mo.
The field a t all points, both inside and outside a magnetized body occupying a volume V1, is exactly that which would be produced by a stationary
volume distribution of current throughout V1 of density
J
(56)
=
V X Mo,
together with a current distribution on the surface
density1
X
bounding V1 of
K = M0 x n,
(57)
where n is the unit outward normal t o S. As a direct consequence of the
analysis of Sec. 2.14 and in particular of Eq. (9), it follows that the potential energy of the magnet V1 in the field of other permanent magnets or of
1
See Sew. 1.6 and 4.10.
130
STRESS A N D ENERGY
[CHAP.I1
constant currents is
U=
(58)
-s
v1 J - A d v - h K . A d a ,
or, in virtue of (56) and (57),
(59)
U=
-sv,
-
(V X Mo)A dv
-
-
(Mo x n) A da.
Now
(61)
(M x A) - n = (n x Mo) - A = -(Max n) - A ,
so that upon applying the divergence theorem and putting B =
(59) reduces to
U=
(62)
-S
V1
vx
A,
M 0 . B dv.
In (62) one may consider dm = Mo du to be the magnetic moment of
an element dv of the magnet. Its potential energy in the resultant field
B is dU = -B dm. The resultant field B is composed of the initial
field B1 of all ext,ernal sources, phis the field B2 due to all other elements
of the same magnet V1. Therefore the work necessary to construct the
magnet by collecting together permanently magnetized elements in the
absence of an external field should be
-
(63)
U2=
-Ll MO.B2dv.
These elements must be held together by forces of a nonmagnetic character. The work necessary to introduce the magnet as a rigid whole
from infinity to a point within the external field B1 is then
and the force exerted on a unit volume of the magnet by the external
sources is
(65)
f = +v(Mo
*
BI)-
The difference of (64) and (51) is accounted for when we note that
(64) is only the potential energy of the magnetized body in the external
field, while (51) includes the work involved in building up the magnetization from zero to M, and is based on the assumption that there exists a
linear relation between B1and H1,
and consequently between B1 and M.
SEC.
POYNTING’S THEOREM
2.191
131
ENERGY FLOW
2.19. Poynting’s Theorem.-In the preceding sections of this chapter
it has been shown how the work done in bringing about small variations in
the intensity or distribution of charge and current sources may be
expressed in terms of integrals of the field vectors extended over all space.
The form of these integrals suggests, but does not prove, the hypothesis
that electric and magnetic energies are distributed throughout the field
with volume densities respectively
u = JDE.dD,
w
=
JBH.dB.
The derivation of these results was based on the assumption of reversible
changes; the building up of the field was assumed to take place so slowly
that it might be represented by a succession of stationary states. It is
essential that we determine now whether or not such expressions for the
energy density remain valid when the fields are varying a t an arbitrary
rate. It is apparent, furthermore, that if our hypothesis of an energy
distribution throughout the field is a t all tenable, a change of field intensity and encrgy density must be associated with a $ow of energy from
or toward the source.
A relation between the rate of change of the energy stored in the field
and the energy flow can be deduced as a general integral of the field
equations.
aB
(I) V X E +% = 0,
(111) V B = 0,
-
We note that E J has the dimensions of power expended per unit
volume (watts per cubic meter) and this suggests scalar multiplication of
(11) by E.
(2)
aD
E * V X H - E . x = E . J.
I n order that each term in (I) may have the dimensions of energy per
unit volume per unit time i t must be multiplied by H.
(3)
H * VX E
+ H e -dBat = 0.
Upon subtracting (2) from (3) and applying the identity
(4)
V * (E X H)
=
-
H V X E - E * V X H,
132
[CHAP.I1
STRESS A N D ENERGY
we obtain
(5)
dD
dB
V . ( E X H ) + E * J = -E.--H*--.
at
at
Finally, let us integrate (5) over a volume V bounded by a surface S.
This result was first derived by Poynting in 1884, and again in the
same year by Heaviside. Its customary interpretation is as follows.
We assume t h a t the formal expressions for densities of energy stored in
the electromagnetic ficld are the same as in the stationary regime. Then
the right-hand side of (6) represents the rate of decrease of electric and
magnetic energy storcd within the volume. The loss of available stored
energy must be accounted for by the terms on the left-hand side of (6).
Let u be the conductivity of the medium and E’ the intensity of impressed
electromotive forces such as arise in a region of chemical activity-the
interior of a battery, for example. Then
(7)
J
=
a(E
+ E’),
E
J
= - U
E’,
and hence
The first term on the right of (8) rcprcscnts t,he power dissipated in
Joule heat-an irreversible transformation. The second term expresses
the power expended by t,he fiow of charge against the impressed forces,
the negative sign indicating that, these impressed forces are doing work
on the system, offsetting in part the Joule loss and tending to increase
the energy stored in the field. If, finally, all material bodies in the field
are absolutely rigid, thercby excluding possible transformations of
electromagnetic energy into elastic energy of a stressed medium, the
balance can be maintained only b y a flow of electromagnetic energy
across the surface bounding V . This, according to Poynt,ing, is the
significance of the surface integral in (6). The diminution of electromagnetic energy stored in V is partly accounted for by the Joule hcat
loss, partly compensated by energy introduced through impressed forces ;
the remainder flows outward across the bounding surface S , representing
a loss measured in joules per second, or watts, by the integral
(9)
S .nda
=
(E
x
H) . n d a .
The Poynting vector S defined by
(10)
S = E X H
watts/meter2,
POYNTING'S THEOREM
SET.2.191
133
may be interpreted as the intensity of energy flow a t a point in the field;
i e . , the energy per second crossing a unit area whose normal is oriented
in the direction of the vector E x H.
It has been tacitly assumed that the medium is free of hysteresis
effects. I n case the relation between B and H is multivalued, energy to
the amount Q [Eq. (37), page 1261 is dissipated in the medium in the
course of every complete cycle of the hysteresis loop. If the field is
harmonic in time with a frequency V , there will be v cycles/sec. and consequently the hysteresis will also participate in the diminution of magnetic energy a t the rate of vQ joules/sec.
I n the absence of ferromagnetic materials the relations between the
field intensities are usually linear, and if the media are also isotropic
Poynting's theorem in its differential form reduces to
As a general integral of the field equations, the validity of Poynting's
theorem is unimpeachable. Its physical interpretation, however, is open
to some criticism. The remark has already been made that from a
volume integral representing thc total energy of a field no rigorous conclusion ran be drawn with regard to its distribution. The energy of the
electrostatic field was first expressed as the sum of two volume integrals.
Of these one was transformed by the divergence theorem into a surface
integral which was made t o vanish by allowing the surface to recede to
the farther limits of the field. Inversely, the divergence of any vector
function vanishing properly a t infinity may be added to the conventional expression u = +E D for the density of electrostatic energy without affecting its total value. A similar indefiniteness appears in the
magnetostatic case.
A question may also be raised as t o the propriety of assuming that
-
aD
dB
E - - and H - represent t,he rate of change of energy density for
at
at
rapid as well as quasi-static changes. Such an assumption seems plausible, but we must note in passing that a transformation of the energy
function expressed in terms of thc field vectors to an expression in terms
of the densities of charge and current leads now to difficulties with surface
integrals. Let
be t'he rate a t which work is done on the system by external forces.
introducing the potentials
Upon
134
(13)
STRESS AND ENERGY
E=-V+--i
i3A
at
[CHAP.I1
B=VXA,
and applying the identities
we obtain
where X is a surface enclosing the entire electromagnetic field. Now in
the stationary or quasi-stationary state the potentials can be shown t o
vanish a t infinity as r-l and the fields as r P 2 ;the integral over a n infinite
surface then vanishes. It will be shown in due course, however, t h a t
t h e fields of variable sources vanish only as r-l and in this case t h e last
term of (15) cannot be discarded by the simple expedient of extending t h e
integral over a n infinikly remote surface. On the other hand we know
that fields and potentials are propagated with a finite velocity. If,
therefore, the field was first established a t some finite instant of the past,
a surface may be imagined whose elements are so distant from the source
that the field has not yet arrived. The intensity over S is then strictly
zero and under these circumstances
Finally, it must be granted that even though the total flow of energy
through a closed surface may be represented correctly by (9), one cannot conclude definitely t h a t the intensity of energy flow a t a point is
S = E x H ; for there might he added to this quantity any vector
integrating t o zero over a closed surface without affecting the total flow.
The classical interpret,ation of Poynting’s theorem appears to rest t o
a considerable degree on hypothesis. Various alternative forms of the
theorem have been offered from time t o time,’ but none of these has t h e
advantage of greater plausibility or greater simplicity t o recommend it,
and i t is significant that thus far no other interpretation has contributed
anything of value to the theory. The hypothesis of an energy density
M A C D O N A L D , “Electric Waves,” Cambridge Iiniversity Press, 1902. LIVENS,
“The Theory of Electricity,” Cambridge University Press, pp. 238 $., 1926. MASON
and WEAVER,“The Electromagnetic Field,” University of Chicago Press, pp. 264 $.,
1929.
SEC.
2.201
T H E COMPLEX POYNTING VECTOR
135
in the electromagnetic field and a flow of intensity S = E x H has, on
the other hand, proved extraordinarily fruitful. A theory is not a n
absolute truth but a self-consistent analytical formulation of the relations
governing a group of natural phenomena. By this standard there is
every reason to retain the Poynting-Heaviside viewpoint until a clash
with new experimental evidence shall call for its revision.
2.20. The Complex Poynting Vector.-If
we now let h = u w
represent the density of electromagnetic energy at any instant and
1
Q = - J 2 - E’. J the power expended per unit volume in thermo-
+
(T
chemical activity, the Poynting theorem for a ficld free from hysteresis
effects may be written
V . S + Zah+ Q = O .
(17)
I n a stationary field h is indepondent of the time, so th a t (17) reduces to
V-S
(18)
+ Q = 0.
Q may be positive or negative as the work done by the impressed electromotive forces E‘ is less or greater than the energy dissipated in heat.
Accordingly the energy flows from or toward a volume element depending
on its action as an energy source or sink.
The sources and their fields in most practical applications of electromagnetic theory are periodic functions of the time. The mean value of
the energy density h is constant and &/at = &./at = 0, the bar indicating a mean value obtained by averaging over a period. I n the case of
periodic fields, therefore,
When there are no sources within V , the energy dissipated in heat
throughout V is equal to the mean value of the inward flow across the
surface S.
The advantages of complex quantities for the treatment of pcriodic
states are too well known to be in need of detailed exposition. The
reader may be reminded, however, that certain precautions must be
observed when dealing with products and squares. Throughout the
remainder of this book we shall usually represent a harmonic variation
in time as a complex function of the coordinates multiplied by the factor
e-wt . Thus if A is the quantity in question, we write
+
+
(20) A = Aoe-kt = ( a ip)e-iwt = ( a ip)(cos w t - i sin w t ) ,
where a and /3 are real functions of the coordinates x, y, z. The conjugate
of A is obtained by replacing i =
by -i, and is indicated by the
47
136
STRESS A N D ENERGY
[CHAP.I1
sign A':
A
(21)
= (a - iP)ebt.
Although it is convenient to employ complex quantities in the course of
analytical operations, physical entities must finally be represented by
real functions. If A satisfies a linear equation with real coefficients,
then both its real and its imaginary parts are also solutions and either
may be chosen a t the conclusion of the calculation to represent the
physical state. However, in the case of squares and products, we must
first take the real parts of the factors and then multiply, for the product
of the real parts of two complex quantities is not equal to the real part of
their product. We shall indicate the real part of A by Re(A) and the
imaginary part by I m ( A ) .
Re(A) =
(22)
a cos wt
~ r n ( =~ p) cos wt
+ p sin at
4-
-
a sin wt =
AA
+ +),
+ cos (at+ 41,
sin (at
~-
The square of the amplitude-or
plication with its conjugate.
(23)
=
d a 2
~2
A is obtained by multi-
magnitude-of
= a2
+ Pz.
The real part of A is also given by
&(A) =
-A. +2 A
The product of the real parts of two complex quantities A , and A z is,
therefore, equal to
(25)
+
Re(A1) . Re(A2) = +(A1
"),(A,
= $(AiAz
AiAz
+
+ dz)
+ A1Az + AZAl).
The time average of a periodic function A is defined by
where T is the period. If A is a simple harmonic function of the time,
its mean value is of course zero. The mean value of such functions as
cos wt sin at also vanishes, and we have consequently from (25) the result
(27)
+ AZ;?1)
Re!A1) . Re(A2) = f ( A l A 2
=
+(am
+ PlP2),
SEC.
137
BODY FORCES I N F L U I D S
2.211
According to the preceding formulas the mean intensity of the energy
flow in a harmonic electromagnetic field is
-
S
(29)
=
Re(E) x Re(H)
=
+Re(E x
B),
the real part of the complex vector +E x fi. The properties of this
so-called complex Poynting vector are interesting. We shall denote it by
S* = +E
(30)
x G.
Let us suppose that the medium is defined by the constants E, pl u and
t h a t the field contains the time only in the factor e-wt. Then the
Maxwell equations in regions free of impressed electromotive forces E' are
V X E
(31)
= iwpH,
V X H
=
(U
- ioe)E.
The conjugate of the second equation is
v x I? = (u + iO€)E,
(32)
which when united with the first as in Sec. 2.19 leads to
(33)
or in virtue of (28) and (23),
+
v * s* = -0 i2w(.ii,- a).
(34)
The divergence of the real part of S* determines the energy dissipated in
heat per unit volume per second, whereas the divergence of the imaginary
part is equal t o 20 times the difference of the mean values of magnetic and
electric densities. Throughout any region of the field bounded by a
surface S we have
Re
(35)
s
S
-
S* n da = total energy dissipated,
and
(36)
Im
$s S* - n da = 20 X difference of the mean values of magnetic
and electric energies.
FORCES ON A DIELECTRIC IN AN ELECTROSTATIC FIELD
2.21. Body Forces in Fluids.-Let us consider an electrostatic field
arising from charges located on the surfaces of conductors embedded in
an isotropic dielectric. To simplify the analysis we shall assume for the
moment that throughout the entire field the dielectric medium has no
discontinuities other than those occurring a t the surfaces of the conductors. It will be assumed furthermore that D = eE, where e is a
138
STRESS A N D ENERGY
U=
11
The total electrostatic energy of the
continuous function of position.
field is therefore
(1)
[CHAP.
12
.I
eE2 dv
extended over all space.
The position of any point in the dielectric with respect t o a fixed
reference origin is specified by the vector r. Every point of the dielectric
is now subjected t o an arbitrary, infinitesimal displacement s(r). It is
supposed, however, t h a t the conductors are held rigid and the displacement of dielectric particles in the neighborhood of the conducting surfaces
is necessarily tangential. The displacement or strain results in a variation in the paramcter e with a corresponding change in the electrostatic
energy equal t o
At the same time a slight readjustment of the charge distribution will
occur over the surfaces of the conductors. Initially the surface charges
were in static equilibrium and the initial energy (I), according to Thomson's theorem, Sec. 2.11, is a minimum with respect t o infinitesimal variations in charge distribution. The variation in energy associated with
the redistribution is, therefore, an infinitesimal of second order which
may be neglected with respect to (2).
Consider the dielectric material contained initially in a volume dvl.
The displacement is accompanied by a deformation, so t h a t after the
strain this element of matter may occupy a volume Eq. (54), page 92.
dvz
(3)
= (1
+ v - s) dv1.
The mass of the element is conserved and hence if the density of matter
is denoted by r we have
(4)
71
+
dvl = ~ z ( 1 V S) dvl,
or, for an infinitesimal change,
(5)
87 =
7-2
-
r1
=
-7v
- s.
The inductive capacity e is a function of position in the dielectric and
also of the density r. The element which after the displacement finds
itself at the fixed point r was located before the displacement a t the
point r - s, and the contribution t o Se arising from the inhomogeneity
of the dielectric is therefore -s Ve. If it be assumed that E depends
only on r and T, the total variation is
-
ae
& = --s.Ve+-&
dT
= -s - V e
ae
- 7ar
-v-s.
SEC.
BODY FORCES I N FLUIDS
2.211
139
The change in electrostatic energy associated with the deformation is
(7)
su = 1
2 J (EZve.s+ E 2 a7
7 e v . s ) dv.
Now
(8)
and hence
These volume integrals are t o be extended over the entire field. The
surfaces of the conductors, witkin which the field is zero, shall be denoted
by S1, Sz,. . . , S,. A surface So is drawn to enclose within it all
conductors and all parts of the dielectric where the field is of appreciable
intensity. Within the volume bounded externally by So and internally
by XI, S z ,
. . . , S,
ae
the function E % - s
d7
is continuous.
Thus the
divergence theorem may be applied to the second integral of (9), giving
Over So the intensity of E is zero, while over the rigid conductors S1,
. . . , 8, the normal component of s has been assumed zero. The
integral (10) therefore vanishes.
If one neglects the effect of gravitational action, it may be assumed
that the only body force f is that exerted by the field on elements of
dielectric. The work done by this force per unit volume during the
displacement is f s, and hence according to the principle of conservation
of energy,
-
The displacement s is arbitrary and we find
In case there are also charges distributed throughout the dielectric, a
term pE must be added.
The last term of (12) is associated with the deformation of the dielectric. The assumption that E can be expressed as a function of position
and density alone is admissible for liquids and gases but is not necessarily
140
[CHAP.I1
STRESS A N D ENERUY
valid in the case of solids. To a very good approximation the relation
of dielectric constant K , (t = K Q ) to density in gases, liquids, and even
some solids is expressed by the Clausius-Mossotti law,
(13)
where Cis a constant determined by the nature of the dielectric. Through
a simple calculation this leads to
a€ = -(.
€,
(14)
7-
dr
3
-
The force exerted by the field on a unit volume of a liquid or gas is,
therefore,
(15)
f
=
€0
-E2VK
2
+ 36 v [EyK -
1)(K
+ 2)1.
2.23. Body Forces in Solids.-The volume force (12) exerted on a
dielectric by an electrostatic field has been derived on the assumption
that the variation in the inductive capacity E during an infinitesimal
displacement can be accounted for by the inhomogeneity of the dielectric
and the change in density associated with the deformation. It is clear,
however, that deformations in solids may occur without any accompanying change of volume, and that a rigorous theory must therefore express
the variation of e in terms of the components of strain. Since thc
dielectric is presumed t o be in static equilibrium, the mechanical forces
exerted by the field will be balanced by elastic forces induced during the
deformation. Our problem is to find these forces and to determine the
resultant deformation of the dielectric due to the applied field. We shall
confine the investigation to media whose electric and elastic properties in
the unstrained state are isotropic. Since the variations in E are dependent
on the components of strain, it is hardly t o be expected that a solid will
remain electrically isotropic after the strain is applied. Our first task
is to set up an expression for the electrostatic energy of an anisotropic
dielectric.
In Sec. 2.8 it was shown that the electrostatic energy density within
a dielectric is
without regard to the relation of D to E. We shall now assume that in
an anisotropic medium the components of D are linear functions of the
components of E.
SEC. 2.221
BODY FORCES IN SOLIDS
141
Since the quantities Ek, Dj transform like the components of vectors it is
clear that the E j k are the components of a tensor of second rank. By a
rotation of the coordinate system this tensor may be referred to principal
axes defined by the unit vectors a, b, c such that
The energy density in an anisotropic, linear medium is, therefore,
We have next to calculate the change in the energy resulting from a
variation of the parameters q k while holding the charges fixed. A review
of the proof of Sec. 2.10 shows t h a t it may be adapted directly to our
present needs. To avoid confusion of subscripts we shall replace the
index 1 in See. 2.10 by a prime to denote initial conditions. Thus the
dielectric in the field was characterized initially by the coefficients E;~.
Within a region V1 these values are changed to ejk. The total change in
the energy of the field, according to Eq. (49), page 113, is
or in virtue of (17)
I n case the variations in ejk are infinitesimal, this becomes
Finally, the parameters e j ~must be related t,o the components of
strain. For sufficiently small deformations we may put
The 81 coefficients a$,, are the components of a tensor of rank four.
virtue of the relations e j k = e k j , e l , = emz, we have
(25)
uik
Im
=
.ki
Zm
=
uik
mi’
In
142
STRESS A N D ENERGY
[CHAP.I1
Then of the 81 - 9 = 72 nondiagonal terms, only 36 are independent and
the total number of independent coefficients is reduced to 36
9 = 45.
To reduce thc system still further use must be made of symmetry conditions and we now introduce the limitation to a medium which is initially
isotropic,' though not necessarily homogeneous.
The variation in the density of the electrostatic energy due to a small
deformation is
+
3
6u =
3
3
3
3333
-!
2
j=l
a{iEiEs 6el,.
k = l Z=1 m = 1
The coefficients a$k have fixed values characteristic of the dielectric at
each point, but varying from point to point in case the medium is inhomogencous. Now since the dielectric is assumed to be initially isotropic,
Eq. (26) must be invariant to a reversal in the direction of any coordinate
axis and to the interchange of any two coordinate axes. Thus a reversal
of the axis xj reverses the signs of E i and e j k , k # j , but leaves all other
factors unchanged. As a consequence, certain coefficients must vanish
if 6u is to be unaffected by the reversal or exchange of axes. In fact it is
evident that all but three classes of coefficients are zero, namely:
Now 6u must also be invariant to a rotation of the coordinate axes
and this implies a further relation between the parameters al, a2, a3.
The condition is easily found by rewriting (28) in the form
3
(al - a2 - 4a3)
2 E:
3
6ejj
+ a2E22 6eij
i=l
in which E 2 = E2,
+ E i + Ei.
3
The sum
2 eii is, according to (54),
j=1
page 92, the cubical dilatation, a quantity not dependent on the coordinate system. The middle term of (29) is, therefore, invariant to a rotation of the reference system. The same is true also of the last term; for
if the strains e j k are replaced by the coefficients 2aik of Sec. 2.2, this last
The anisotropic case is discussed by POCKELS,
Encyklopadie der mathematischen
Wissenschaften, Vol. V, Part 11, Teubner, 1906.
SEC.
143
BODY FORCES I N SOLIDS
2.221
term is an invariant quadric similar in form to the strain quadric of
Eq. (36), page 89. Only the first term of (29) is variant with the coordinates and it is necessary that’
(30)
a1 - u2 -
4%
=
0,
a3
=
$(a1 - az).
The variation in the density of electrostatic energy due t o the local
deformation, or pure strain, of a n isotropic dielectric is finally
(31)
6u, =
+
+ +
+ +
+
+
-&[(ulE2, uzEi uzE2,)6ell
(azE? UIE;f a&) 6e22
(azE9 azE2, aiE2,) F e 3 3
(a1 - az)(E~Ez
6e12
E2E3 6 e ~ 3
E3E1 6 e d 1 .
+
+
+
The subscript s has been added t o emphasize the fact t h a t this is only
t h a t portion of the total variation which arises from a pure strain. T h e
most general deformation of an elastic medium, i t will be remembered,
is composed of a translation, a local rotation defined by (33), page 88,
and the local strain defined by (a<)),page 88, of which we have taken
account in (31). I n an anisotropic dielectric the local rotation gives
rise t o variations in the tensor component,s e j l ~and consequently a system
of torques must act on each volume clcment.2 In the initially isotropic
medium considered here these rotational variations are absent. The
variation in the density of electrostatic energy associated with a n infinitesimal translation 6s of an inhomogeneous dielectric was calculated i n
Sec. 2.21.
(32)
6ut = + E 2 V € .6 ~ 9
where E is the inductive capacity in the unstrained, isotropic state.
A deformation of the dielectric occasions a variation in the elastic
energy as well as in the elect’rostaticenergy. According to (68),page 94,
(33)
6u,= (XIV
- s + 2Xzell)
6e11
+ (X1V - s + 2 X m )
+ (X1V - s + 2X2ezz) 6ezz
6e33
+ hz(e1z
6e12
+
e23
6ez3
+
e31
6ed.
Now the work done by the mechanical forces acting on the dielectric
within the volume V1 and over the surface X1 bounding V1 during a n
infinitesimal displacement 6s must equal the decrease i n the total available
energy, elastic plus electrostatic. We write, therefore, for the energy
balance :
The procedure followed here in reducing the constants of electrostriction to two
is identical with that which leads to the reduction of the elastic constants cik, Sec. 2.3,
t o the two parameters XI and X1. See, for example, LOVE,“Treatise on the Mathematical Theory of Elasticity,” Chap. VI, 4th ed., Cambridge Press, 1927. The
determination of these constants for various classes of anisotropic crystals was made
by Voigt, “Kompendium dcr thcorctischcn Physik,” Vol. I, pp. 143-144, Leipzig,
1895. His results are cited by Love, Zoc. cit.
POCKELS,
loc. cat., p. 353.
144
(34)
STRESS A N D ENERGY
-
f 6s dv
Jl
+ hlt - 6s da = - 6
[CHAP.I1
SVI(u,+
u8
+ ut)dv.
It will be convenient to resolve the total body force per unit volume f
arbitrarily into the two components f’ and f”, where
f‘ = -+E2Ve
(35)
is the contribution resulting from the inhomogeneity of the dielectric
and where f ” is the force associated with a pure strain. The latter must
satisfy the relation
(36)
jvl
f” . 6s dv + J,t”
*
6s da =
-6
SV1
+
(u, u,)dv.
The resolution into translational and strain components of body and
surface force is such that
(37)
SvIf’ dv + J, t’
da
=
0,
SV
f” dv 4-
L,t“
da = 0.
Then by Sec. 2.3, the left-hand side of (36) may be transformed to
(38)
6W”
=
lVl
(Ti; 6ell -I- Ti; 6e2, 4-5”;;
6e33
+ 2’::
+ Ti: 6eZ3
+ T:: dv.
6e12
6e31)
The variations in the strain components are arbitrary and hence on
equating coefficients of corresponding terms from (31) and (33) we obtain
These are the components of a stress tensor whose negative divergence
gives us the resultant body force associated with a strain. If the body
force is solely of electrical origin, as will be the case when gravitational
action is neglected, the divergence of the elastic stresses will vanish and
we obtain for the x-component of force:
SEC.
2.221
145
BODY FORCES I N SOLIDS
with analogous expressions for f y and fy. To these are added the components of f’ from ( 3 5 ) to obtain the total body force exerted by an
electrostatic field in an isotropic dielectric.
The parameters a1 and a2 must in general be determined for a given
substance by measurement,. Physically the parameter a1 expresses
the increment of E corresponding to a n elongation parallel to the lines of
field intensity, while a2 determines this increment for strains at right
angles to these lines.
If the dielectric within Ti1 is homogeneous and contains no charge, the
gradient of E is zero and the vector E satisfies V E = 0, V x E = 0.
In this case f’ = 0 and the resultant force f” = f reduces t o
-
+
f = --;(CL~
az)VE2.
(41)
I n a liquid or gas the shearing strains e 3 h are all zero and hence by
Eqs. (24) and (27) a3 = a!; = 0, a1 = a2 = a. Furthermore there can
be no prvferred directions in the dielectric properties of a fluid and consequently Ejk = 0 when j # k , €11 = € 2 2 = €33 = E. I n place of (24) we
may write:
Denoting the cubical dilatation by A
a€
(43)
Now by ( 5 )
(44)
a=-=dejj
= ell
aE
= a 6A.
dA’
ae -_ -ae
_
- ar aA
aTaa
4-ezz + e33, we have
-7-
aE
-1
a7
and consequently the total force per unit volume exerted by an electrostatic field on a fluid dielectric is
(45)
as was deduced directly in Sec. 2.21.
The derivation of the volume forces from an energy principle as in
the foregoing paragraphs seems to have been proposed first by Kortewegl
and developed by Helmholtz2 and others. A complete account of the
theory with references to the older literature is given by pock el^.^ The
energy method has been criticized by Larmor4 and Livens5 who propose
KORTEWEG,
W i e d . Ann., 9, 1880.
HELMHOLTZ,
Wied. Ann., 13, 385, 1881.
3 POCKELS,
Arch. Math. Phys., (2) 12, 57-95, 1893, itnd in the ICncyklopddie der
mntheinntischen Wissrmschaftcn, Val. V, Part 11, pp. 350-392, 1906.
~ L A R ~ I Phil.
O R , Trans., A. 190, 280, 1897.
~IAVENS,
Phil. Mag., 32, 1G2, 1916, and in his text “The Theory of Electricity,”
2
p. 93, Cambridge University Press, 1926.
146
STRESS A N D ENERGY
[CHAP.
11
alternative expressions for the forces. These criticisms, however, do not
appear to be well founded. Objections to the particular form of the
energy integral employed by Helmholtz are satisfied by a more careful
procedure. Livens has undertaken a generalization of the energy method
to media in which the relation between D and E is nonlinear in order to
show that it leads to absurd results; in so doing he has omitted the essential term associated with the deformation. The alternative expressions
for the volume force proposed by Livens may on the other hand be derived
very simply, in both the electric and magnetic cases, by assuming that
polarized matter is equivalent to a region occupied by a charge of density
p’
=
-v P
and current of density J
ap
at
=-
+v x M
(cf. See. 1.6).
The forces may then be calculated exactly as in Sec. 2.5. Such a procedure would be justifiable were the medium absolutely rigid. According
to the Livens theory one calculates the force exerted by the field on the
polarized matter, and then introduces these forces into the equations of
elasticity to determine the deformation. The deformation, however,
affects the polarizations and the two parts of the problem cannot, in
general, be handled separately in any such fashion. Under certain
conditions, particularly in fluids, the two theories lead to identical results,
but in most circumstances they differ. There appears to be little reason
to doubt that the energy method of Korteweg and Helmholtz is fundamentally sound.
2.23. The Stress Tensor.-We shall suppose again that S1 is a closed
surface drawn within an isotropic dielectric under electrical stress. The
properties of the dielectric are assumed to be continuous across this
surface and at all interior points; it is not, in other words, a surface of
discontinuity bounding a whole dielectric body. The total force exerted
on the matter and charge within SI is
F
= Jvl
(pE -
We wish to show now that the force F can be represented by a surface
integral over S1. To this end it is only necessary to express the integrand
as the divergence of a tensor. The term f” appears already in this form
in (40)and therefore needs no further transformation. I n the first term
we replace p by v D and then make use of the identity
E2Ve = V(e E2)- 2(D V ) E ,
(47)
which holds only when V X E = 0. It follows without difficulty that the
components of the vector p E - +EWe are
-
(48)
pEi
1
a€
- - E2 - =
z
axi
3
2a
k=l
i a
(EjDh) - 2axi (E
D),
SEC.2.241
SURFACES OF DISCONTINUITY
147
and the integrand of (46) can thus be expressed as the divergence of a
tensor 2S,
+
pE - ~ E ' V E f f f
(49)
whose components
Sz2=
Sjk
=
v - 'S,
are tabulated below.
1
(Ei - E: - ET) - 2 (azE:
+ alE; + azE:),
......................................
Upon applying the divergence theorem expressed in Eqs. (22) and
(23), page 100, it appears that the force F exerted on the charge and
dielectric matter within 81 is equivalent t o the integral over 81 of a
surface force of density
t
=
(C
+ y)
E(E - n) - + E2n,
a2
6
2
where n is as usual the outward unit normal to an elcment of 81. To
maintain equilibrium an equal and opposite force per unit area must be
exerted by the material outside 81 on the particles composing X I .
The tensor components (50) and the surface force t differ from the
corresponding expressions developed for free space in See. 2.5 in that eo
is replaced by 6 and the deformation is accounted for by the constants al
and a', thus giving rise t o additional terms which may well be important.
At any point on the surface X1 the vector t lies in the plane of E and n,
but is no longer oriented such that E bisects the angle between n and t
as was the case illustrated by Fig. 17, page 102.
2.24. Surfaces of Discontinuity-Throughout
the previous analysis
it. has been assumed t h a t abrupt discontinuities in the properties of the
medium, such as must occur across surfaces bounding dielectric bodies,
may be replaced by layers of rapid but continuous transition. As the
transition layer becomes vanishingly thin, a discontinuity is generated
that gives rise t o infinite values for the gradient of e. But the volume
within the layer also vanishes and we must seek the limit of the force f
per unit volume t o learn what total force is exerted on the resultant
surface of discontinuity.
I n Fig. (23) the crosshatched area represents a section of a transition
layer between dielectric media (1) and (2). The force exerted by the
148
[CHAP.I1
STRESS A N D ENERGY
field on the matter and charge within this layer can be found by integrating (51) over the two faces. The two essential parameters that
characterize an isotropic dielectric will be denoted as
(52)
The force acting on unit area in the direction from medium (1) towards
(2) as the thickness of the layer approaches zero is then
(53)
t
=
-
[aE(E n)lz
+ [aE(E- n)ll - [pE2nlz- [PE2nll,
the subscripts indicating that the values are to be taken on either side
of S. It should be noted that nl = -n2.
Very few reliable measurements of the constants al and u2 for solids
have been reported in the literature. The traction t may, however, be
computed in the case of an interface scparating two fluids to which the Clausius-Mossotti
law can be applied. According to Eqs. (43)
and (14),
(54) al
=
az =
-7-
a€ = -€0
(K
Then for a fluid,
0=
a7
3
- 1)(K
+ 2).
-'
( K 2 - 2 K - 2).
6
At the interface the tangential components of
E are continuous, Etl = Et2, and in the
FIG. 23.-Section
of a tran- absence of surface charge we have also
sition layer separating two
qEnl = e2En2. The field E in (53) may, theredielectric media.
fore, be expressed in terms of its intensity in
either medium. The values of a and from (55) introduced into (53)
lead to
(55)
OL
= E,
+
[K2(Kz
- 2) -
Kl(K1
}
- 2)]E,"1 n2.
The traction exerted by the field on the interface of two fluids is normal
to the surface, directed from medium (1) toward (2), a neccssary consequence, of course, of the fact that a fluid supports no shearing st,ress.
In case medium ( 2 ) is air, K~ may be placed approximately equal to
unity. Equation (56) then reduces to
(57)
SEC.2.253
ELECTROSTRICTION
149
The traction is a maximum when E is normal to the surface, and is in
the direction of diminishing inductive capacity (cf. page 114). However,
if E lies in the surface, Enl = 0, and the tract,ion becomes negative,
implying a pressure exerted by the field on dielectric K ~ .
Another case of interest is that in which medium (1) is a conductor.
Then El = 0, Etz = 0 and (53) reduces to
all quantities having their values just outside the metal surface in the
dielectric (1). I n case this dielectric is a fluid,
(59)
These expressions are valid whether or not the conductors are charged,
for E', is related to the surface charge by EE, = w . But Eqs. (58) and
(59) do not represent the resultant force per unit area exerted o n the surface,
for they will be compensated to a certain degree by stresses or pressurc:;
generated by the field within the dielectric. These we shall now investigate. Only in case medium (1) is free space or a gas of negligible inductive capacity can this internal pressure be ignored. We have then
de
7-
d7
=
0 and (59) becomes
t = l+,,E;n
(60)
where
=
+wEnn,
is the charge per unit area.
elastic deformation of a dirlectric under
the forces exerted by an elcctrostatic fit:ld is called electrostriction. The
displacement s a t any interior point of an isotropic substance must
satisfy Eq. (70), page 95:
w
2.25. E1ectrostriction.-The
+ +
+
f
(XI
X2)VV * s
X2V2s = 0,
(61)
where now the body force f = f'+ f" is given in general by ( 3 5 ) and
(40). To these there may be added, when the occasion demands, a
force E p to account for a volume charge, and a gravitational force 79.
Solutions of (61) must be found in an appropriate system of coordinates.
These solutions are subject t o boundary conditions over a surface bounding the dielectric. On this surface either the applied stresses or the
displacement must be specified. Thus if the surface is free, stresses
will be exerted upon it by the field and from these may be determined the
boundary values of s.
I n solids the effect is small and easily masked by deformations arising
from extraneous causes. The attractive forces between metal electrodes
in contact with the dielectric may bring about deformations within the
150
STRESS AND ENERGY
[CHAP.
11
volume of the dielectric which have nothing to do with electrostriction.
Experimental data on the subject are not abundant and frequently
contradictory. The theory of electrostriction in a cylindrical condenser,
however, has been developed in considerable detail. The method
described above was applied by Adams,' and his results were later
reconciled by Kemble2 with the earlier work of S a ~ e r d o t e . ~An extensive
list of references is given by Cady in the International Critical table^.^
The electrostriction of fluids is more amenable to calculation, for the
shearing modulus A:! is then zero and A1 is equal to the reciprocal of
the compressibility. According to (73), page 95, the pressure within the
fluid is related to the volume dilatation by p = -A1v s. This expression, together with the body force f from Eq. (45), introduced into (61),
leads to the relation
-
between pressure and field intensity at any point in the fluid. To integrate this equation the functional dependence of both pressure and
inductive capacity on the density 7 must be specified. I n a chemically
nomogeneous liquid or gas, one may assume that p = p ( ~ ) e, = E ( T ) ,
and that the dependence on position is implicit in the independent variable
r . Then
and hence (62) reduces to
A scalar function P ( p ) of pressure is now defined such that
P(P> =
1:
1
.(p> dp,
1
7vp,
VP = -
where po is the pressure a t a point in the fluid where E = 0. If ds is an
element of length along any path connecting two points whose pressures
are respectively p and PO, then V P . ds = dP, and (64) is satisfied by
ADAMS,Phil. Mag., (6) 22, 889, 1911.
KEMBLE,
Phys.Rev., (2) 7,614, 1916.
SACERDOTE,
Jour. physique, (3) 8,457, 1899; 10, 196, 1901.
4 International Critical Tables, Vol. VI, p. 207, 1929.
SEC.2.261
FORCE ON A BODY I M M E R S E D I N A F L U I D
151
d7
In liquids d p = XI T, Eq. (77), page 96, or
P-PO
(67)
7
= roe
x1
.
The parameter X,I the reciprocal of the compressibility, is a very large
quantity and hence r cv ro. For liquids, therefore,
or, upon applying the Clausius-Mossotti law,
RT
where R is the
M
In gases (67) is replaced by the relation p = 7-,
gas constant, T the absolute temperature, and M the molecular weight.’
2.26. Force on a Body Immersed in a Fluid.-The results of our
analysis may be summed up profitably by a consideration of the following
--..
/---
Fluid
Fluid
E-
FIQ.24.--A solid body immersed in a fluid dielectric.
problem. In Fig. 24, a dielectric or conducting body is shown immersed
in a fluid. An electrostatic field is applicd and we desire an expression
for the resultant force o n the entire body. Let us draw a surface S1 enclosing
the body and located in the fluid just outside the boundary X. On every
volume element of the solid there is a force of density
+
f = pE - $E2Ve f”,
(70)
where f” is given by Eq. (40), and on each element of the surface of discontinuity there is a traction given by (53). Rather than evaluate the
integrals off and t over the volume and bounding surface X, we need only
1 ABRAHAM,
BECKER,
and DOUGALL,
“The Classical Theory of Electricity and
Magnetism,” p. 98, Blackie, 1932.
152
[CHAP.I1
STRESS A N D ENERGY
calculate the integral of t from Eq. (51) over the surface
3€
lies in the fluid, a1 = a2 = -T - and (51) reduces t o
81.
Since
S
1
aT
-
t = eE(E n) - 5 E2n
2
a€ E2n,
+ -2 di7
where n is t h e unit normal directed from the solid into the fluid and E
and T apply t o the fluid. Now the integral of this traction over XI givrs
the force exerted directly b y the field on the volume and surface of thc
body. B u t the field, as we have just seen, also generates a pressure
(68) in t h e fluid which a t every point on SIacts normally inward, i.~.,
toward the solid. The resultant force per unit area transmitted across
S1 is, therefore, equal t o (71) diminished by (68), and the net force
exerted b y the field on the solid-the force which must be cornpensated
b y exterior supports-is
Thus t o obtain the resultant force on a n entire body we need not know
the constants a1 and a2 within either the solid or the fluid, for the forces
with which they are associated are compensated locally by elastic stresses.
Furthermore, since the fluid is in equilibrium, i t is not essential t h a t S1
lie in the immediate neighborhood of the body surface S. The net forcc
on the liquid contained between XI and a n arbitrary enclosing surface Sz
is zero and consequently
if we adhere t o t h e convention that n is directed outward from the enclosed
region. The surface Sz may, therefore, be chosen in any manner that
will facilitate integration, provided only t h a t no foreign bodies are
intersected or enclosed.
I n case the solid is a conductor, the field E is normal a t the boundary.
The resultant force on a n isolated conductor is then
(74)
where w is t h e charge density on S. This last is true, of course, only for a
surface S1just outside the conductor.
It should be evident now t h a t the force on one solid embedded in
another can be calculated only when the nature of the contact over their
common surface is specified; the problem is complicated by the fact t h a t
153
NONFERROMAGNETIC M A T E R I A L S
SEC. 2.271
tangential shearing stresses as well as normal pressures must be compensated by elastic stresses across the boundary.
FORCES IN THE MAGNETOSTATIC FIELD
2.27. Nonferromagnetic Materials.-The analysis of Sec. 2.22 may
be applied directly t o the magnetostatic field i n all cases where the relation between B and H is of the form
3
with the components pik of the permeability tensor functions possibly of
position but independent of field intensity. The change in magnetic
energy of an isotropic body resulting from a variation in p is given by
Eq. (52), Sec. 2.17.
‘I’
6T = -2
6pH2dv,
and the generalization of this expression t o a n anisotropic body occupying
the volume V1 is
(3)
6T
=
aJvl
+ ~ P z J+G
(8~1lH21
+ 26~12H1Hz
+ 26Pz3HzH3 + 26p31H3H1) dv.
8~33H2
This variation may be expressed in terms of the components of strain,
assuming a linear relation between the components of the permeability
and strain tensors.
(4)
If we assume further that in the unstrained state the medium i s isotropic,
the coefficients @% reduce t o two,
(5)
so that the variation in magnetic energy due t o a pure strain is
(6)
6T
sv,
+ +
+ + +
=
[(blHq
(bzHI
bzH;
bzH;
b ~ H 3Jell
blH2,) 6e33
+ (bzHq + b ~ H +i b2W3
+ ( b l - bd(H1Hz
6czz
6e12
4- HZf13 6 e 2 3 4-H3H1 6c3d1 dv.
Equation (6) differs formally from (31), page 143, only in algebraic
sign. Now the electrostatic energy U represents the work done in
154
STRESS AND ENERGY
[CHAP.I1
building up the field against the mutual forces between elements of
charge. The magnetic energy T , on the other hand, represents work
done against the mutual forces exerted by elements of current, plus
the work done against induced electromotive forces. The work required
to bring about a small displacement or deformation of a body in a magnetic field is done partly against the mechanical forces acting on the body,
partly on the current sources to maintain them constant.' When the
work done against the induced electromotive forces is subtracted from
the total magnetic energy, there remains the potential energy of the
mutual mechanical forces, and this we found in See. 2.14 to be equal and
opposite to T .
SU
(7)
=
-ST.
The forces exerted by the jield on the body within V1 are to be calculated
from ( 6 ) with sign reversed.
I n complete analogy with the electrostatic case we now find that
the body force exerted by a magnetic field on a medium which in the
unstrained state is isotropic, whose permeability is independent of field
intensity, which is free of residual magnetism, but which may carry a
current of density J, is
If the medium is homogeneous and carries no current, (8) reduces to
f =
(9)
-:(bi
I n a gas or liquid bl = b, =
-7
+ b2)VH2.
aP
-,
a7
and (8) can be written in the form
As in the electrostatic case, the body force whose components are
given by (8) can be expressed as the divergence of a tensor 2S,
(11)
f
=
v * 2s,
whose components are
l This remark does not apply if the source is a permanent magnet; in that case the
energy of the field is not given by T = +JB. H dv.
SEC.2.281
FERROMAGNETIC MATERIALS
155
(12)
The force exerted by the magnet.ic field on the matter within a closed
surface S1is, therefore, the same as would result from the application over
S1 of a force of surface density
(13)
Likewise the force per unit area exerted on a surface of discontinuity
can be obtained by calculating (13) on either side of the surface. The
result is the magnetic equivalent of (53). A case of particular importance
is that of a magnetic body immersed in a fluid whose permeability to a
sufficient approximation is independent of density and equal to PO, the
value in free space. Let the body be represented by the region (1) in
Fig. 23 and the fluid by the region (2). The force per unit area on the
bounding surface is then found to be
where the constants bl, bz apply to the magnetic body and H1 is measured
just inside its surface.
Not very much is known about the parameters b l and bz, although
there are indications that they may be very large. They must, of course,
be determined if the elastic deformation of a body is to be calculated.
I n most practical problems, fortunately, one is interested only in the net
force acting on the body as a whole. The forces arising from deformations are compensated locally by induced elastic stresses and consequently terms involving bl and b, will drop out. As in the electric case,
the resultant force exerted by a magnetostatic field on a nonferromagnetic
body immersed in any fluid is obtained by evaluating the integral
pH(H.n) -
1
H2n da
over any surface enclosing the body.
2.28. Ferromagnetic Materials.-The
preceding formulas apply to
ferromagnetic substances in sufficiently weak fields. If, however, the
permeability p depends markedly on the intensity of the field, the energy
of a magnetic body can no longer be represented in the form of Eq. (2)
or (3) and we must use in their place the integral derived in Sec. 2.17.
156
STRESS A N D ENERGY
[CHAP.I1
An analysis of the volume and surface forces for such a case has been
made by Pockels,' who also treats briefly the problem of magnetostriction.
The phenomena of magnetostriction are governed, however, b y several
important factors other than the simple elastic deformation considered
here. A specimen of iron, for example, which in the large appears
isotropic, exhibits under the microscope a fine-grained structure. The
properties of the individual grains or microcrystals are strongly anisotropic. Of the same order of magnitude as these grains but not necessarily
identified with them are also groups or domains of atoms, each domain
acting as a permanent magnet. I n the unmagnetized state the orientation of the magnetic domains is random and the net magnetic moment
is zero. A weak applied field disturbs these little magnets only slightly
from their initial positions of equilibrium. A small resultant moment is
induced and under these circumstances the behavior of the iron may be
wholly analogous to a polarizable dielectric in an electric field. As the
intensity of the field is increased, however, the domains begin to flip over
suddenly t o new positions of equilibrium in line with the applied field,
with a consequent change in the elastic properties of the specimen. A
dilatation in weak fields may be followed by a contraction as the field
becomes more intense, quite contrary to what would be predicted by the
magnetostriction theory when applied to strictly isotropic solids. We
are confronted here with a problem in which the macroscopic behavior of
matter cannot be treated apart from its microscopic structure.
FORCES IN THE ELECTROMAGNETIC FIELD
2.29. Force on a Body Immersed in a Fluid.-The
expressions
for the energy of a body in a stationary electric or magnetic field have
been based in Secs. 2.10 and 2.17 on the irrotational character of the
vectors E and H - HI as well as upon their proper behavior a t infinity.
In a variable field these conditions are not satisfied and therefore (1)
cannot be applied t o the determination of the force on a body or an element of a body without a thorough revision of the proof. At best the
analysis will contain some element of hypothesis, for our assumption that
dD
dB
the quantities E - and H - represent the snergy densities in an
-
at
- dt
electromagnetic field is, after all, only a plausible interpretation of
Poynting's theorem.
POCKELS,
loc.
cit., p. 369.
SEC.2.291
157
FORCE ON A BODY I M M E R S E D I N A F L U I D
In Sec. 2.5 it was shown that the total force transmitted by an electromagnetic field across any closed surface Z in free space is expressed
by the integral
F
(2)
=
z
[eo(E- n)E + 1 (B .n)B - 1(eoE2+ 1 B z ) n ] da.
-
-
PO
PO
Subsequently we were able to demonstrate for stationary fields that if
the surface Z: is drawn entirely within a fluid which supports no shearing
stress, the net force is given by (2) if only we replace e0 and po by their
appropriate values in the fluid.
This is the resultant force on the charge, current, and matter within 2.
The matter within B need not be fluid and there may be sharp surfaces of
discontinuity in its physical properties; the relations between E and D,
and B and H, however, are assumed to be linear.
Now since (2) is valid in the dynamic as well as in the stationary
regimes, there is reason to suppose that (3) may be applied also to variable
fields. It is not difficult to marshal support for such a hypothesis,
particularly from the theory of relativity. However, the right-hand side
of (3) must now be interpreted in the sense of See. 2.6 not as the force
exerted by the field on the matter within B but as the inward flow of
momentum per unit time through 2 . We denote the total mechanical
momentum of the matter within 2 , including the ponderable charges, by
G,,,, and the electromagnetic momentum of the field by G,. Then
(4)
+ p(H - n)H - 1 (eE2+ pW)n
+
d
- (Gmech G,) =
e(E n)E
dt
Sz
I
da.
The surface integral (4) may be transformed into a volume integral.
For in the first place:
eEZnda
(5)
Then
(6)
(7)
*V(eE2) = +E2Ve
(D V)E =
a
(DS)
=
:L
-
+ (D
+ dY
*
v(eEZ)do.
+ D X V X E.
+ a (D,E) - E V
V)E
d
- (DUE)
*
D.
and this obviously cancels the 2-component of the vector e(E n)E in
158
STRESS A N D ENERGY
[CHAP.
I1
(3). Proceeding similarly with the magnetic terms and then making
aD
use of the relations V X E = --aB/at, V x H = - J, V - D = p, we
at
+
are led t o
(9)
+
dt (Gmech G,)
=
V
[pE
+ J X B - -21 E2Ve - -21 H 2 V p
+ -ata (D x B)] dv.
The increase in the mechanical momentum is the result of the forces
exerted by the field on charges and neutral matter.
F
(10)
d
= - Gmeoh.
dt
If, therefore, the right-hand side of (9) can be split into two parts identified with Gmechand G,, the force can be determined. Just how this
resolution is to be made is by no means obvious and various hypotheses
have been suggested.' According to Poynting's theorem the flow of
energy, even within ponderable matter, is determined by the vector
(11)
S = E X H
joules/sec.-meter2.
Abraham and von Laue take for the density of electromagnetic
momentum
1
g, = /.Lo&
=-s
kg. /sec.-meter2.
(12)
c2
Then according to this hypothesis, the resultant force on the charges,
currents, and polarized matter within 2 is
(13) F =
or
(14) F =
s
s[
V
(pE
+ J x B - 21 E2Vc - 21 H 2 V p + ___
-
-
c2
-
E(E n)E
-
dt
+ p(H - n)H - 1 (eE2 + pH2)n] da
-2.s
c2dt
p
E x Hdv.
Practically, the exact form of the electromagnetic momentum term
is of no great importance, for the factor 1/c2 makes it far too small to be
easily detected.
1 PAULI,Encyklopadie der mathematischen Wissenschaften, Vol. V, Part 2,
pp. 662-667, 1906.
SEC.2.291
FORCE ON A BODY I M M E R S E D I N A FLUID
159
On the grounds of (13) it is sometimes stated t h a t the force exerted
by an electromagnetic field on a unit volume of isotropic matter is
(15)
f = pE
+J x B -
Such a conclusion is manifestly incorrect, for (15) does not include the
forces associated with the deformation. As previously noted, these
strictive forces are compensated locally by elastic stresses and do not
enter into the integrals (13) and (14) for the resultant force necessary
to maintain the body as a whole in equilibrium.
CHAPTER I11
THE ELECTROSTATIC FIELD
From fundamental equations and general theorems we turn our
attention in the following chapters to the structure and properties of
specific fields. The simplest of these are the fields associated with
stationary distributions of charge. Of all branches of our subject, however, the properties of electrostatic fields have received by far the most
adequate and abundant treatment. In the present chapter we shall
touch only upon the more outstanding of these properties and of the
methods which have been developed for their analysis.
GENERAL. PROPERTIES OF AN ELECTROSTATIC FIELD
3.1. Equations of Field and Potential.-The equations satisfied by
the field of a stationary charge distribution follow directly from Maxwell's equations when all derivatives with respect to time are placed
equal to zero. We have, then, at all regular points of an electrostatic
field :
(I) V X E = 0,
(11) V * D = p .
According to (I) the line integral of the field intensity E around any
closed path is zero and the field is conservative.
The conservative nature of the field is a necessary and sufficient condition for the existence of a scalar potential whose gradient is E.
(1)
E = -V4.
The algebraic sign is arbitrary but has been chosen negative to conform
with the convention which directs the vector E outward from a positive
charge. Equation (1) does not define the potential uniquely, for there
might be added to 4 any constant Cb0 without invalidating the condition
+
v x v(4 40) = 0.
(2)
I n Chap. I1 it was shown that the scalar potential of an electrostatic
field might be interpreted as the work required t o bring a unit positive
charge from infinity to a point (z, y, z ) within the field:
(3)
We shall show below that the field of a system of charges confined to a
finite region of space vanishes at infinity. The condition that $ shall
vanish at infinity, therefore, fixes the otherwise arbitrary constant $ 0 .
160
SEC.3.11
E QUAT I ON S OF FIELD A N D P O T E N T I A L
161
+
Those surfaces on which is constant are called equipotential surfaces, or simply equipotentials. At every point on an equipotential the
field intensity E is normal to the surfact:. For let
+(x, y , z )
(4)
=
constant
be an equipotential, and take the first differential.
The differentials d z , d y , d z , are the components of a vector displacement dr along which we wish to determine the change in +, arid since
d+ = 0, this vector must lie in the surface, = constant. The partials
a+/ax, a+/ay, a+/&, on the other hand, are rates of change along the
x-, y-, z-axes respectively and as such have been shown to be the components of another vector, namely the gradient
+
d,$ is manifestly the scalar product of these two and, since this product
vanishes, the vectors must be orthogonal. An exception occurs a t those
points in which the three partial dcrivativcs vanish simultaneously.
The field intensity is zero and the points are said t o be points of
equilibrium.
The orthogonal trajectories of the equipotential surfaces constitute a
family of lines which a t every point of the field are tangent to the vector
E. They are the lines of force. It is frequently convenient t o represent
graphically the field of a given system of charges by sketching the projection of these lines on some plane through the field. Let d s represent a
small displacement along a line of force, where
+
+
d s = i dx’ j dy‘
k dz’,
(7)
the primes being introduced to avoid confusion with a variable point
(2, y, z ) on an equipotential.
Then, since by definition the lines of
force are everywhere tangent to the field-intensity vector, the rectangular
components of ds and E(x’, y’, 2 ’ ) must be proportional.
(8)
E, = X dx‘,
E,
= X
dy‘,
E,
=
X dz‘.
The differential equations of the lines of force are, therefore,
(9)
The relations between the components of D and those of E are almost
invariably linear. If the medium is also isotropic, one may put
(10)
D
=
EE= -cV+,
162
T H E ELECTROSTATIC FIELD
[CHAP.111
whence, by (11),4 must satisfy
+
v * (ev4) = €V2+ V €* v4 = -p.
(11)
In case the medium is homogeneous, 4 must be a solution of Poisson’s
equation,
1
(12)
v24 = - - p .
At points of the field which are free of charge (12) reduces to Laplace’s
equation,
v24
(13)
=
0.
The fundamental problem of electrostatics is to determine a scalar
function +(x, y, z ) that satisfies at every point in space the Poisson equation, and on prescribed surfaces fulfills the necessary boundary conditions. A much simpler inverse problem is occasionally encountered :
Given the potential as an empirical function of the coordinates reprcsenting experimental data, to find a system of charges that would produce
such a potential. The density of the necessary continuous distribution
is immediately determined by carrying out the differentiation indicated
by Eq. (12). There will in general, however, be supplementary point
charges, whose presence and nature are not disclosed by Poisson’s
equation. At such points the potential becomes infinite and, inversely,
one may expect to find point charges or systems of point charges located
a t the singularities of the potential function. The nature of these
systems, or multipoles as they are called, is the subject of a later section,
but a simple example may serve to illustrate the situation.
Let the assumed potential be
where r is the radial distance from the origin to the point of observation
and a is a constant. By virtue of the spherical symmetry of this function,
Poissons’s equation, when written in spherical coordinates, reduces to
on differentiation one finds for the density of the required continuous
charge distribution
The charge contained within a sphere of radius r is obtained by integrating p over the volume, or
(17)
f
p
dv = e-=r(ar
+ 1) - I .
SEC.3.21
163
B O U N D A R Y CONDITIONS
If the radius is made infinite, the total charge of the continuous distribution is seen to be
(18)
L m p d v = -1.
But this is not the total charge required to establish the potential (14).
For a t T = 0 the potential becomes infinite and we must look for a point
charge located a t the singularity. To verify this we nced only apply
(11) in its integral form. If q is a point charge a t the origin,
the surface integral extending over a sphcrc S bounding the volume V .
D, da
(21)
and, hence,
q
(22)
=
(ar
=
+ 1)e*',
+l.
The potential defincd by (14) arises, therrfore, from a positive unit
charge located a t the origin and arq {equal negative charge distributed
about it with a density p , the system as a whole being neutral.
3.2. Boundary Conditions.-Thc transition of the field veciors across
a surface of discontinuity in the medium was investigated in See. 1.13
and the results of that section apply directly to the electrostatic case.
The two media may be supposed to meet a t a surface S and the unit
normal n is drawn from medium (1) into medium (2), so that (1) lies
on the negative side of S, medium (2) on the positive side. Then
(23)
where
n
w
x (E2 - El)
=
n . (Dz - D1)
0,
=
W,
is the density of any surface charge distributed over S.
It will be convenient to introduce the unit vector t tangent to the
surface S. The derivativcs &$/an and
represent respectively the
rates of change of q5 in the normal and in a tangential direction. Then
the boundary conditions (23) can be expressed in terms of the potential by
(24)
($)2
-
(2)
- el($)
e.($),
= 0,
= -w.
From the conservative nature of the field it follows also that the potential
itself must be continuous across S, for the work required t o carry a small
charge from infinity to either of two adjacent points located on opposite
sides of S must be the same Hence
(25)
41
=
42.
164
THE ELECTROSTATIC FIELD
[CHAP.
111
The two conditions
are independent.
Conductors play an especially prominent role in electrostatics. For
the purposes of a purely macroscopic theory it is sufficient to consider
a conductor as a closed domain within which charge moves freely. If
the conductor is a metal or electrolyte, the flow of charge is directly
proportional t o the intensity E of the electric field: J = uE. Charge is
free to move on the surface of a conductor but can leave it only under
the influence of very intense external fields or at high temperatures
(thermal emission). If the conductor is in electrostatic equilibrium all
flow of charge has ceased, whence i t is evident t h a t at every interior
point 07 a conductor in an electrostatic field the resultant field intensity E
i s zero, and at every point on its surface thP tangential component of E i s zero.
Furthermore the electrostatic potential 4 within a conductor i s constant and
the surface of every conductor i s an equipotential. Let us suppose that an
unchargcd conductor is introduced into a fixed external field Eo. I n the
first instant there occurs a transient current. According to Sec. 1.7,
no charge can accumulate a t an interior point, but a’redistribution will
occur over the surface such t h a t the surface density a t any point is w ,
subject t o the condition
Ss
w
da
= 0.
This surface distribution gives rise t o an induced or secondary field of
intensity El. Equilibrium is attained when the distribution is such t h a t
at every interior point
Eo
(28)
+ El = 0.
Likewise, if a charge q is placed on an isolated conductor, the charge
will distribute itself over the surface with a density w subject t o (28)
and the condition
Ss
(29)
w
da
= q.
We shall denote the interior of a conductor in electrostatic equilibrium
by the index (1) and the exterior dielectric by (2). Then a t the surface S
(30)
El
=
D1 = 0,
n x Ez = 0,
n.Dz=
or in terms of the potential,
(31)
C$
=
constant,
a4
dn
e2-=
-w.
O,
SEC.3.31
165
GREEN’S THEOREM
If a solution of Laplace’s equation can be found which is constant over
the given conductors, the surface density of charge may be determined
by calculating the normal derivative of the potential.
CALCULATION OF THE FIELD FROM THE CHARGE DISTRIBUTION
3.3. Green’s Theorem-Let
V be a closed region of space bounded
by a regular surface S , and let 4 and $ be two scalar functions of position
which together with their first and second derivatives’ are continuous
throughout V and on the surface 8. Then the divergence theorem
applied to the vector $V+ gives
J,v - ( W 4 >dv
(1)
=
($v4> n da.
+
Upon expanding the divergence to
(2)
v - (*v4) = v+. vlp + 1c.v * vlp
= V1c.e vlp
+ 1c.V2&,
and noting that
(3)
where &$/an is the derivative in the direction of the positive normal,
we obtain what is known as Green’s first identity:
(4)
If in particular we place $= 4 and let 4 be a solution of Laplace’s
equation, Eq. (4) reduces to
Next let us interchange the roles of the functions 4 and
the divergence theorem to the vector 4v$.
1c.; ie., apply
Upon subtracting (6) from (4)a relation between a volume integral and a
surface integral is obtained of the form
(7)
known as Green’s second identity or also frequently as Green’s theorem.
This condition is more stringent than is necessary. The second derivative of one
function Ic. need not be continuous.
166
T H E ELECTROSTATIC FIELD
[CHAP.111
3.4. Integration of Poisson’s Equation.-By
means of Green’s
theorem the potential a t a fixed point (d,
y’, z’) within the volume V can
be expressed in terms of a volume integral plus a surface integral over X.
Let us suppose that charge is distributed with a volume density p(x, y, z).
We shall assume that p(x, y, z ) is bounded but is an otherwise arbitrary
function of position. An arbitrary, regular surface S is now drawn
enclosing a volume V , Fig. 25. It is not necessary that S enclose all
the charge, or even any of it. Let 0 be an arbitrary origin and x = x’,
y = y‘, z = z’, a fixed point of observation within V . The potential
a t this point due to the entire charge distribution is c$(x’, y’, 2’). For
FIQ.26.-Application of Green’s theorem t o a region V bounded externally by the surface S
and internally by the sphere 81.
the function I) we shall choose a spherically symmetrical solution of
Laplace’s equation,
$(z, y, 2; z’, y’, 2’)
(8)
=
1
r
-7
where r is the distance from a variable point (z, y, z ) within V to the
fixed point (x’,y’, 2’).
(9)
r
=
d ( x ’ - x)2
+ (y’ -
y)2
+ (z’
-
z)2.
This function $, however, fails to satisfy the necessary conditions of
continuity a t r = 0. To exclude the singularity, a small sphere of
radius r1 is circumscribed about (z’, y’, z’) as a center. The volume V is
then bounded externally by S and internally by the sphere S1. Within
B both 4 and I) now satisfy the rcquirernents of Green’s theorem and
furthermore V21) = 0. Thus (7) reduces to
the surface integral to be extended over both S and S1. Over the sphere
S1 the positive normal is directed radially toward the center (d,
y’, z’),
SEC.
167
BEHAVICIR AT I N F I N I T Y
3.51
since this is out of the volume V . Over 81,therefore,
Since r1 is constant, the contribution of the sphere to the right-hand side
of (10) is
If
is
6 and s / a r denote mean values of
+ and a+/ar on S1,this contribution
which in the limit rl-+ 0 reduces to - 4 ~ + ( z ‘ , y’, 2 ’ ) . Upon introducing
this value into (10) the potential a t any interior point (d,y‘, 2’) is
or in terms of the charge density when the medium is homogeneous,
1
v2+ = -- p
€
(13)
’
+(d,y’,
2’) =
1
47r€
.(1- f dv + I
47r
(‘)I
[Ar a29n - 4 % 7
da.
I n case the region V bounded by S contains no charge, (13) reduces t o
+(2’, y’, 2 ’ ) =
--
47r
It is apparent that the surface integrals in (13) and (14) represent the
contribution to the potential a t (d,
y’, 2 ’ ) of all charges which are exterior
t o S. If the values of + and its normal derivative over S are known, the
potential a t any interior point can be determined by integration. Equation (14) may be interpreted therefore as a solution of Laplace’s equation
within V satisfying specified conditions over the boundary. The integral
c#l(X’,
y’,
1
4T€ Jv
2’) = -
f dv
is a particular solution of Poisson’s equation valid a t (z’, y’, 2 ’ ) ; the
general solution is obtained by adding the integral (14) of the homogeneous equation v2+= 0. If there are no charges exterior t o s, the
surface integral must vanish.
3.6. Behavior at Infinity-Let us suppose that every element of a
charge distribution is located within a finite distance of some arbitrary
168
[CHAP.I11
T H E ELECTROSTATIC FIELD
origin 0. We imagine this distribution to be confined within the interior
of a surface Sl which also contains the origin 0. The distances r, r1 and
R are indicated in Fig. 26.
r
(16)
=
( R 2f r: - 2rlR cos 0)h.
The potential a t any point P(z’, y’,
2’)
outside S1 is then
As P recedes to infinity, the terms r; and 2rlR cos 6 become negligible
/
/
’,Z‘
/
I
I
I
I
\
\
\
\
\
‘\
---
‘-
/
/
,
---___-*
FIU.%---Figure t o accornpany Sec. 3.5.
with respect to R2, and in the limit as R -+ co we find
where q is the total charge of the system. A potential function is said
to be regular at injinity if R4 is bounded as R --;r 0 0 . The field intensity
E = -V+ at great distances is directed radially from 0 and the function
R21El is bounded.
All real charge systems are contained within domains of finite extent
and their fields are, therefore, regular a t infinity. Frequently, however,
the analysis of a problem is simplified by assuming the external field to
be parallel. Such a field does not vanish a t infinity and can only arise
from sources located a t an infinite distance from the origin.
It is important to note that a closed surface S divides all space into
two volumes, an interior V 1and an exterior V 2 ,and that if the functions
4 and $ are regular a t infinity Green’s theorem applies to the external
region V2 as well as to V1. For certainly the theorem applies to a closed
region bounded internally by S and externally by another surface S2.
a4 and 4 all. vanish
If now S2recedes towards infinity, the quantities $ an
an
169
COULOMB F I E L D
SEC. 3.61
as l/r3 and the integral over S z approaches zero.
Consequently
but the positive normal a t points on S is now directed out of V z and
therefore into V1. Let us suppose, for example, that a chargesystem is
confined entirely to the region V1 within S. The potential a t any point
(z’, y’, x ’ ) in V z outside S may be calculated from (17), or equally well
from a knowledge of 4 and &$/an on S , applying (14) as indicated in
Fig. 27.
FIG.27.-Application
of Green’s theorem to the exterior Vz of a closed surface S.
3.6. Coulomb Field.-According to (18) thc potential of a charge q in
a homogeneous medium a t distances very great relative to the dimensions
of the charge itself approaches the value
+(z’, y’, 2’) =
1 -.
q
-
47re r
If q is located a t (z, y, x ) , the distance from p to the point of observation
(d,g’, z’) is
(21)
T
= d(s’ -
.)Z
+ (y’
-
y)2
+ (d -
2)2.
Let now ro represent a unit vector directed from the source; Le., from
(2,y, z ) towards (z‘, y’, 2’).
Thcn
where the prime above the gradient operator denotes differentiation
with respect to the variables (d,
y’, 2’) a t the point of observation.
The field intensity a t this point is
The field of a point charge i s inversely proportional to the square of the distance and i s directed radially outward when q i s positive. This is the law
170
T H E ELECTROSTATIC FIELD
[CHAP.
111
established experimentally by Coulomb and Cavendish which is usually
taken as a point of departure for the theory of electrostatics. The term
“point charge” is employed here in the sense of a charge whose dimensions are negligible with respect to r. Mathematically one may imagine
the dimensions of q to grow vanishingly small while the density p is
increased such that q is maintained constant. I n this fashion a point
singularity is generated and (23) is then valid a t all points except r = 0.
There is no reason to believe that such singularities exist in nature, but
it is convenient to interprct a field at sufficient distances as that which
might be generated by systems of mathematical point charges. We
shall have occasion to develop this concept in the subsequent section.
The potential a t (z’, y’, 2’) is obtained by integrating the contributions of charge elements dq = p dv over all space.
(24)
The field intensity due to a complete charge distribution in a homogeneous, isotropic medium is, therefore,
E(d, y’,
2’)
1
=-
4ae
p(z, y, z)V
(:)
dv.
3.7. Convergence of Integrals.-The proof of convergence of the
integrals for potential and field intensity is implicit in the method by
which they have been derived, but an alternative treatment will be
described which may serve as a model for proofs of this kind. Since the
element of integration at (2,y, z ) can coincide with (z’, y r , z’), with the
result that the integrand becomes infinite at this point, it is not obvious
that the integral has a meaning. It will be shown that, although an
improper integral of this type cannot be defined in the ordinary manner
as the limit of a sum, it can by suitable definition be made to converge
absolutely to a finite value.
Let the point &(z’, y’, z r ) at which the field is to be determined be
surrounded by a closed surface S of arbitrary shape, thus dividing the
total volume V occupied by charge into two parts: a portion Vl representing the volume within S and a portion Vz external to S. Throughout
V z the integral of Eq. (26) is bounded and consequently the charge
outside X contributes a finite amount to the resultant field intensity at Q ,
a contribution E2,
SEC. 3.71
C'ONVKRGENCE OF' INTEGRALS
171
The contribution of the charge within X to the field intensity at Q may
be called El. If now the field intensity E = E l
Ez at Q is t o have a
definite significance, it is necessary that El vanish in the limit as S
shrinks about Q. A moment's reflection will indicate t h a t this is indeed
the case for, although the denominator of the integrand vanishes as the
square of the distance, the element of charge in the numerator is proportional to the volume V1 which vanishes as the cube of a linear dimension.
It has been tacitly assumed that thc charge density p is finite a t every
point in the volume V . There exists, therefore, a number m such t h a t
IpI < m, and Ip/rl < m / r , a t every point in V . It follows, furthermore,
for this upper bound m that
+
The surface S bounding the volume V , is of arbitrary form, and so, to
avoid the awkwardness of evaluating the integral over this region, a
sphere of radius a concentric with Q is circumscribed about 81. The
volume element of integration is positive; consequently the integral
extended throughout V1 must be less than, or a t the most equal to, the
integral ex tended throughout the volume enclosed by the circumscribed
sphere.
which evidently vanishes wit,h a. The contribution of the charge within
S to the field a t Q becomes vanishingly small as S shrinks about Q and
hence (26) converges for interior as well as exterior points of a charge
distribution.
The potential integral
is also an improper integral when the point of observation is taken within
the charge but, since the denominator of the integrand vanishes only
5ts the first, power of r , the proof above holds here a fortiori.
If p is a
bounded, integrable function of position, (31) is a cont,inuous function
of the coordinates x', y', z', has continuous first derivatives, and satisfies
the condition E = - V ' + everywhere. It can be shown, furthermore,
172
T H E ELECTROSTATIC FIELD
[CHAP.I11
that i f p and all its derivatives of order less than n are continuous, the potential 4 has continuous derivatives of all orders less than n
1.l
+
EXPANSION OF THE POTENTIAL I R SPHERICAL HARMONICS
3.8. Axial Distributions of Charge.-We shall suppose first t h a t an
element of charge q is located a t the point z = 1 on the z-axisof a rectangular coordinate system whose origin is at 0. We wish t o express the
potential of q a t any other point P in terms of the coordinates of P with
respect to the origin 0. The rectangular coordinates of P are 2, y, z,
but since the field is symmetric about the z-axis i t will be sufficient t o
locate P in terms of the two polar coordiz
nates r and 6, Fig. 28. T h e distance from
q to P is r2and the potential at P is, therefore,
+(r, 6 )
(1)
Q -> 1
= ~4ac r 2
the medium being assumed homogeneous
and isotropic.
r2 = ( r 2
(2)
+ l 2 - 2r{ cos e)*.
28.-Figure
to
Sec. 3.8.
There are now two cases to be considered.
The first and perhaps less common is t h a t
in which P lies within a sphere drawn from 0 as a center through
Then T < { and we shall write
r.
(3)
7.2
=
Ai-[]+
(;y -
2TCOS
i0
I-’
.
The bracket may be expanded by the binomial theorem if
2);(I
r
- 2 ? cos
01
< 1.
< 1, the rcsultant series converges absolutely and consequently the various powers may be multiplied out and
the terms rearranged a t will. If the terms of the series are now ordered
in ascending powers of r / { , we find
See for example KELLOGO,“Foundations of Potential Theory,” Chap. VI,
Springer, 1929, or PHILLIPS,
“Vector Analysis,” pp. 122 f.,Wiley, 1933.
SIX. 3.81
173
A X I A L DISTRIBUTIONS OF CHARGE
which shall be written in the abbreviated form
(5)
The coeficients of r /p are polynomials in cos 0 and are known as the
Legendre polynomials.
(6)
PO(cose) = 1,
P~(COS
e) = COS
e,
+
o - 1) = +(3 cos 28 11,
P3(cos e) = + ( 5 c0s3 e - 3 COY e) = + ( 5 cos 38
3 cos
...................................................
p Z ( c O s e) = +(3 cos2
+
e).
The absolute value of the coefficients P, is never greater than unity;
hence the expansion convergcs absolutely provided r < /(I.
I n the second case P lies outside the sphere of radius <, such that
r > {. The corresponding expansion is obtained by interchanging r
and { in ( 3 ) and (5).
This last result may be obtained in a slightly different manner.
Consider the inverse distance from the point z = [ t o P as a function of {
and expand in a Taylor series about the origin, { = 0.
f(s) = -1 = (r2 + c2 - Zr{ cos e)-*,
(9)
r2
Now in rectangular coordinates, r2 is
r2
(1 1)
and
=
[x2
(12)
Hence.
and, sincef(0)
=
l / r , we have
+ y2 + (z - ~7~1+,
174
T H E ELECTROSTATIC FIELD
[CHAP.111
The pwkntial a t a point P outside the sphere through p can be written
in either of the forms
from which it is apparent that
Finally, let us suppose that charge is distributed continuously along
a length 1 of the z-axis with a density p = p @ ) . The potential a t a
sufficiently great distance from the origin is
this expansion,
where p is now the total charge on the line, is evidently the Coulomb
potential of a point charge q located at the origin. However, the density
may conceivably assume negative as well as positive values such that
the net charge
(19)
is zero. The dominant term approached by the potential when r >> 1 is
then
is called the dipole moment of the distribution.
write
a s an axial rnultipole of nth order
I n general, we shall
SEC.3.91
175
T H E DIPOLE
3.9. The Dipole.-In order that the potential of a linear charge distribution may be represented by a dipole i t is neccssary that the net
charge be zero-the system as a whole is neutral-and that the distance
to the point of observation be very great relative t o the length of the line.
We have seen that the potential dois that which would be generated by a
mathematical point charge located a t the origin. do has a singularity
a t r = 0, for a true point charge implies an infinite density. We now
ask whet,her a configuration of point charges can be constructed which
will give rise to the dipole potential 41.
Let us place a point charge + q a t a point z = 1 on the z-axis and an
equal negative charge - q a t the origin. According to Eq. (8) thc
potential a t a distant point is
The product p = ql is evidently the dipole moment of the configuration
Suppose now that 1-3 0, but a t the same time q is increascd in magnitudt:
in such a manner that the product p remains constant. Then in the
limit a double-point singularity is generated whose potential is
(25)
everywhere but a t the origin. A direction has been associated with a
point. The dipole moment is in fact a vector p directed, in this case,
along the z-axis. The unit vector directed along r from the dipole
towards the point of observation is again ro, and the potential is,
theref ore, 1
1 p-ro
4(x, y, 2) = - __ =
4ae r2
4ae
The field of a dipole is cylindrically symmetrical about the axis;
hence in any meridian plane the radial and transverse components of
field intensity are
(27)
+
1 I n (26) r = 4 z 2 $- y2
z2 and V ( l / r ) implies differentiation a t the point of
observation. If the dipole wcre located at (z, y, z ) and the potential measured at
(d,y’,
z ’ ) , we should have
Eq. (22), p. 169.
+(d,y’,
2’)
=
-- p * V’
47rs
1
176
THE ELECTROSTATIC FIELD
[CHAP.I11
The potential energy of a dipole in an external field is most easily
determined from the potential snergies of its two point charges. Let a
charge + q be located a t a point a, and a charge - q at b displaced from
t,he first by an amount 1, in an external field whose potential is +(x, y, 2).
The potential energy of the system is then
i
Y
(28)
or as b
= Y+(U> -
Y+(b),
3 a,
(29)
li
= qd+ = pl-V+ = -p.E
=
-pE cos 0,
where 0 is the angle made by the dipole with the external field E.
J?IQ. 29.-Lines
of force in a meridian plane passing through the axis of a dipole p .
The force exerted on the dipole by the external field is equal to the
negative gradient of U when the orientation is fixed.
(30)
F
= v(p
E>B
oomtant.
On the other hand a change in orientation at a fixed point of the field also
leads to a variation in potential energy. The torque exerted on a dipole
by an external field is, therefore,
or vectorially,
(32)
T=pXE.
3.10. Axial Multipo1es.-Let us refer again to Eq. (17) for the potential of a linear distribution of charge. This expansion is valid at all
points outside a sphere whose diameter is the charged line. Now the
first term +o of the series is just the potential that would be produced
by a point charge y located at the center of the sphere. The second term
represents the potential of a dipole p located at the same origin. We
shall show that the remaining terms
may likewise be interpreted
as the potentials of higher order charge singularities clustered at the
center.
+,,
SEC.
3.101
A X I A L MULTIPOLES
177
The dipole, whose moment we now denote as p ( l ) , was constructed
by placing a negative charge - q a t the origin and a positive charge +q
at a point z = ZO along the axis. The two are then allowed t o coalesce
such that p ( l ) = qZo remains constant. The potential of the resultant
singularity is
(33)
The singularity of next higher order is constructed by locating a dipole
of negative moment - p ( l ) a t the origin and displacing from it another
equal positive moment by the small amount 11. For the present we
confine ourselves to the special case wherein the axes of both dipoles
as well as the displacement 11 are directed along the z-axis. The potential
of the resultant configuration is
(35)
An axial quadrupole moment is drfined as the product
(36)
p(2) = Z(p(ljZ1) = Z(qZ011).
The mathematical quadrupole is generated by letting Z0 --f 0, Zl - 0 ,
q 3 co such that tJhe product (36) remains finite. The potential of this
configuration is then strictly
(37)
a t all points excluding r = 0.
By induction one constructs charge singularities, or multipoles, of
yet higher order. I n each case a multipole of order n - 1 and negative
moment --P("-~) is located at the origin and a n equal positive multipole
p(n-1) displaced from it by 1,.
I n the gcnrral case t o he dealt with below,
the displacements are arbitrary in direction; i f , as in the present special
case, all are along t,he same straight line, the multipole is said t o be axial.
The potential is given approximately by the first term of a Taylor series,
which again by (12) may be written
The n-pole moment is defined as the limit of the product
p(n) = np (n-1)l n-1
(39)
178
when
finite.
[CHAP.I11
T H E ELECTROSTATIC FIELD
I, -+ 0 and
~ ( n - ~-+
)
in such a manner that p("-')ln remains
m
The potential of a point charge of nth order is then
&=---
4ae
P
(
<
#
G
,'
'I
<el,/"' I'b
6Fc'I
J/ p
/
( x , y , 2)
q
I
I
I
n!
(n = 0,
8zn
1,
*
).
in terms of the coordinates of P with
respect to 0.
The rectangular coordinates of P
are x, y, z and its polar coordinates
r, 8, I), where
+ +
The expansion converges provided r1 = dE2 v2 c2 < R , r > R.
Next we integrate (43) over the entire charge distribution; i.e., with
respect to the coordinates El 7, r, nobing that r = d x 2 y 2 z 2 is
+ +
179
GE.NERAL THEORY OF MULTIPOLES
SEC. 3.121
independent of this integration. Once again we shall write Cp
as the
m
sum of partial potentials
#n,
4=
2 #*.
Then the first term of the
n=O
expansion is
(44)
is the total charge. At sufiiently large distances relative to R the potential
of a n arbitrary distribution of charge m a y be represented approximately
by the Coulomb potential of a point charge located at the origin.
The second term, which is dominant when the net charge is zero, is
(46)
where
(47)
p(1’ is
2;’;’)
a vector dipole moment whose rectangular components are
=
s
pE
dv,
pi1) =
s
pq
dv,
p!”
=
s
p{
dv.
T h e potential of a n y distribution whose net charge i s zero m a y be approximated at large distances by the potential of a dipole located at the origin
whose cowLponents are determined by (47).
I n like manner the partial potential
arises from a quadrupole whose
components are
C#J~
Whereas the dipole moment is a vector, the quantities defined in (48)
constitute the components of a tensor of second rank. The multipole
moments of higher order determined from subsequent terms of the
expansion (43) are likewise tensors of higher rank.
3.12. General Theory of Multipo1es.-Let li he a vector drawn from
t h e origin 0 t o the point Q ( 4 , 7, {) whose direction cosines with the coorp(4
dinate axes z, y , z are respectively ai, fit, yi. Let - fi(z, y, z ) be the
4ae
potential a t the point P ( z , y , z ) due t o a charge singularity - p ( a of any
order located at the origin. If now a charge singularity of equal magnitude but positive sign be placed a t the outer extremity of the vector l;,
the potential a t P due to the pair of multiple points will be
p‘ 9
(49)
&+l = -[fi (z - crizi, y - pizi, z - Yili) - f i ( 5 , y , z)],
€0
180
T H E ELECTROSTATIC FIELD
[CHAP.I1 I
for it is to be noted that the change in potential a t (z, y, z) arising from a
displacement of the source from the origin to a point Q whose coordinates
are ( = or&, 7 = Pili, { = yili, is identical with that resulting from an
equal but opposite displacement of P to a point (z - ,$,y - 7, z - {)
during which the source remains fixed at the origin. Expanding by
Taylor's theorem, we have
order in 1,.
The expression in parentheses is evidently the derivative of
respect to the direction specified by the vector li.
4'f . =
-
azi
mi
df
-i + pi
dX
The multipole moment of order i
moment of order i by the limit of
fi
with
af i + y ; -.
afi
-
dY
az
+ 1 is again defined in terms of the
+
pci+l' = (i
l)p("L,
(i = 0,1, * . . ),
(52)
as 1; + 0 and p ( < )-+00 in such a manner that their product remains
finite. The potential at any point (z, y, z ) due to a multipole of moment
pCi+') located a t the origin is then
(53)
A single point charge may be considered as a singularity of zero
order and " moment " p(O) = q.
(54)
The potential of a dipole oriented in any direction specified by the vector
lo is consequently
(55)
The quadrupole is next generated by displacing the dipole parallel to
its axis in a direction specified by the vector l I . For its potential we find
from (53) and (55)
(56)
Thus by induction the potential of the general multipole of nth order is
(57)
&
= (-
1)"p("'
SEC. 3.121
GENERAL THEORY OF MrJLTIPOLES
181
The construction of a quadrupole and a n octupole is visualized in Figs.
31a, 31b.
The result of applying the operator
n times to the function
= (zz
+ y2 + z-)'
-h
must be of the form
(59)
where Y,. is a function of the cosines of the angles p i between r and the
n axes l;, and of the direction cosines ai, pi, yi of the angles made by the
FIG.3la.---Quadrupole.
FIG.3lb.-Octupole.
vectors 1; with the coordinate axes. But the angles p i are related t o
the spherical coordinates 8, $ of P b y the equation
+
+
(60)
cos pi = (ai cos #
Pi sin $) sin e yi cos 8,
which may be demonstrated by calculating the distance QP = r2, Fig.
30, both in terms of T I = Zi, r, and the angle pi, and in terms of its projections on the coordinate axes. Thus Y , is a function of the angles 6 and $,
and hence of position on the surface of a unit sphere. For the dipole
term we lhave
Upon replacing rectangular by spherical coordinates, Eqs. (41), this
becomes
1 p(1'
41 = G~
- T ~[(a"cos #
PI, sin #) sin 6 yo cos 61,
(62)
+
+
or
(63)
41 =
r2 [(a0 cos $ + Pu sin $)P:(cos
1 p(')
0)
+ yoPO,(cos e)],
182
THE ELECTROSTATIC FIELD
[CHAP.I11
where the functions ~ ( C O8) Sare the associated Legcndre functions
defined by
dm
(64)
~ ( C O0 ) S= sinm
P,(cos
'd(Cos e p
Between the four constants
0 ( 0 , 00, yo, there exists one relation,
p," yi = 1, and there remain, consequently, three arbinamely a;
trary parameters which determine the magnitude and orientation of the
dipole.
Proceeding to the quadrupole term, we obtain
+ +
or, with reference to (64),
+ (azzcos 2+ + bzz sin 2+)P;(cos e)].
Between the six direction cosines there are two relations of the form
a:
p:
y: = 1; consequently there are, together with the arbitrary
moment
five completely arbitrary constants which are sufficient to
determine the magnitude and orientation of the quadrupole.
A solution of Laplace's equation is called a harmonic function. Since
the coordinates r, 8, are arbitrary, each tcrm +n in the expansion of 4
must itself satisfy Laplace's cquation and is, therefore, harmonic. Generalizing, we conclude that the harmonic function +n representing the
potential of a multipolc of nth order is a homogeneous polynomial of nth
degree in 5, y, z. Upon transformation to spherical coordinates r, 0, #,
the function
can be expressed in the form (59), where the spherical
surface harmonic Y,(e, +) is now explicitly
+ +
+
+,
c
n
n
2 ( a , cos m+ f b,, sin m+)PT(cos 0).
The harmonic of nth order contains 2n + 1 arbitrary constants sufficient
(67) Y,(8, $)
=
m=O
Ynm(O,$)
=
m=O
to determine the moment and orientation of the axes of the corresponding
multipole. If charge be distributed in arbitrary manner within a sphere
of finite radius, the potential a t all points outside the sphere must be
harmonic and it must furthermore be regular at infinity. These condi-
SEC.3.131
INTERPRETATION OF THE VECTORS P AND
n
183
tions are satisfied by
If the charge distribution is known, the components of the multipole
moments are determined as in Sec. 3.11; if the distribution is unknown
but an external field is specified, these constants are obtained from the
boundary conditions as we shall shortly sec.
DIELECTRIC POLARIZATION
3.13. Interpretation of the Vectors P and n.--Although the two vectors E and D are sufficient t o characterize complctcly an electrostatic
field in any medium, i t is convenient to introducc a third vector P
defined in Sec. 1.6 as the difference of these two fields,
P
(1)
=
D - EOE.
The vector P vanishes in free space and therefore is definitely associated
with the constitution of the dielectric. The divergence equation then
becomes
1
V * E = - ( p - V*P),
(2)
€0
from which one concludes that the effect of a dielectric on the field may
be accounted for by an equivalent charge distribution whose volume
density is
p” =
(3)
-v
*
P.
At every interior point of the dielectric the potential satisfies a modified
Poisson equation,
1
v%#J
= -- ( p
p’).
(4)
EQ
+
The validity of (4) is no longer contingent upon the isotropy and homogeneity of the medium. If, however, there are surfaces of discontinuity,
such as the boundaries of a dielectric, the vcctor D is subject to the
condition
(5)
(6)
n - ( E , - El)=
n . (D,- D1)=
1
1
-w
€0
--
w,
1
- n . (Pz- PI) = - (
€0
€0
w
+
0’).
The primary sources of an electrostatic field are the “real” charges whose
volume and surface densities are respectively p and w. The effect of
rigid dielectric bodies on the potential may be completely accounted for
by distributions of induced, or “bound,” charges of volume density p’
184
THE ELECTROSTATIC FIELD
[CHAP.
111
and a density
w’ = -n
(7)
- (P2 - P1)
over each surface of discontinuity, the unit vector n being drawn from
medium (1) into ( 2 ) . If (1) is dielectric and (2) free space, the induced
surface charge is simply a’ = n - P I , where n is the outward normal.
In the case of a transition from a dielectric (2) to a conductor (1) bearing
a free surface charge a, there results a net surface charge w - n - P z ,
sincc P1 vanishes within the conduckr.
The potential at any fixed point either within a dielectric or exterior
to it can be expressed by
The surface integrals arc to be extended over all surfaces of discontinuity.
It nced scarcely be remarked t h a t this analysis in most cases is purely
formal, for in order t o know P it is usually necessary to calculate first
the potential.
Thc physical significance of the vector P becomes apparent if we transform (8) with the help of the identity
(9)
-v - p- v.(;)
r
-
pv(;).
The dielectric may be resolved into partial volumes Vi bounded by
surfaces Si of discontinuity. For each partial volume
On summing these integrals over the partial volumes and remembering
that ni is directed out of Vi and hence in to a contiguous V k , one finds
that (8) reduces t o
The third integral is now clearly the potential produced by a continuous
distribution of dipole moment, arid P is, therefore, to be interpreted as
the dipole moment per u n i t volume, or polarization of the dielectric.
According to (11) a dielectric body in the primary field of external
sources gives rise t o an induced secondary field such as is generated by a
distribution of dipoles of moment P per unit volume.
I n a purely macroscopic theory this is really all that can be said about
the polarization vector. Macroscopically a charged conductor behaves
SEC.3.141
185
VOLUME DISTRIBUTIONS OF CHARGE
as though the charge were distributed with a continuous density, though
i n fact we know i t to be constituted of discrete electronic charges. Likewise, neutral matter under the influence of an external field acts as though
dipoles were distributed as a continuous function of position. Now we
know that neutral atoms and niolecules are in fact constituted of equal
numbers of positive and negative elementary charges. A microscopic
electrodynamics must show hoR the electrical moments of each atomor a t least their most probable values-can be calculated. Then by a
suitable process of averaging over large-scale volume elements the polarization per unit volume will be determined from the atomic moments
and the transition effectrd from microscopic t o macroscopic quantities.
A word a t t,his point is in order concerning the physical significance
of the Hertz vector n introduced in a more general connection in Sec.
1.11. We shall suppose that the density of free charge p within the
dielectric is zero. If the potential is expressed as the divergence of a
vector function n,
+
(12)
r$
=
-v
.II,
then l’oisson’s cquation assumes the form
this condition will certainly be satisfied if
(
to
)
=
0, and
1
v211 = -- P.
(13)
€0
The vector
intqqral
whence
v - v2n[+ - P
is deterrnincd from the polarization by evaluating the
n is sometimes referred
to as the polarization potential.
DISCONTINUITIES OF INTEGRALS OCCURRING IN POTENTIAL THEORY
3.14. Volume Distributions of Charge and Dipole Moment.-According t o the foregoing analysis, a conductor is simply a region of zero field
bounded by a surface bearing a layer of charge of density w , while a rigid
dielectric may be represented either b y an equivalent volume distribution
of density p’ = -V P bounded by a surface layer w’, or as a region
occupied b y a continuous distribution of dipoles whose moment per
unit volume is P. The boundary conditions, which heretofore have
been deduced from the field equations, must evidently follow directly
from thc analytic properties of the integrals expressing the potential
and its dcrivativcs in terms of volume and surface densities of charge
and dipole moment.
-
186
THE ELECTROSTATIC FIELD
[CHAP.
111
In Sec. 3.7 it was established that the improper integrals
are tonvergent and continuous functions of x’, y’, x’ provided only that the
charge density i s bounded and piecewise continuous; i.e., that the volume
occupied by the charge can be resolved into a finite number of partial
volumes within each of which p is finite and continuous. Thus on either
side of a surface bounding a discontinuous change .in p, 4 and E will have
the same values, although the derivatives of E will in general be dis-
---___
r
a=l
KIG. 32.-Transition
of potential and field intensity a t the surface of a uniform spherical
charge of unit radius.
continuous. Consider, for example, a spherical distribution of constant
density po and radius a. A t a point inside, the field E is found most
simply by taking account of the spherical symmetry and applying the
Gauss law,
r
s-
E n da
=
I
p
dv.
A t a distance r from the origin,
€0
< a, we have
(3)
where q is the total charge contained within the sphere. Likewise at any
external point
(r
(4)
The potential is calculated from the integral
+ = -Ji
E dr, giving
> a).
SEC.3.151
SINGLE-LAYER CHARGE DISTRIBUTIONS
187
I n Fig. 32, C#I and its first derivative E are plotted as functions of r.
Note that although E passes continuously through the surface r = a,
its slope-the second derivative of 9-suffers an abrupt change because
of the failure in continuity of the density p .
Consider next a volume distribution of dipole moment of density P.
The potential a t any exterior point is
+(dly’,
2’) = 4nco
-
.I -
P V(i)dv.
This can be resolved into three integrals of the type
(7)
+ l =
4nto
P,
&);1(
dv
= --- a
4~ a d
pdv.
r
From what has just been proved regarding (1) it follows that (6) converges
within the dipole distribution as well as at points extprior to it, and i s a
continuous function of position provided only P(x, y , z ) i s bounded and
piecewise continuous. Since a dielectric body is equivalent to a region of
dipole moment, we have here proof of the continuity of the potential
across a dielectric surface without recourse to an energy principle.
Across a surface of discontinuity in P, the normal derivative of (6),
and consequently the normal component of E, is discontinuous. The
magnitude of this discontinuity can be determined most readily by
reverting from the volume distribut>ion of moment to the equivalent
volume and surface-charge distribution dcfined in Sec. 3.13. The vector
E passes continuously into the volume charge, but we shall now show t h a t
its normal component suffers an abrupt change in passing through a
layer of surface charge.
3.16. Single-layer Charge Distributions.-Let charge be distributed
over a surface S with a density w which we shall assume to be a bounded,
piecewise continuous function of position on S. The potential a t any
point not on S is
where T as usual is drawn from the charge element w(z, y, z) da t o the
point of observation. If now (d,y’, 2’) lies o n the surface, the integral
(8) is improper and its finiteness and continuity must be examined.
About the point (d,y’, 2’) on S let us circumscribe a circle of radius a.
If the radius is sufficiently small, the circular disk thus defined may be
assumed plane. Now the potential a t (z’, y’, z’) due t o surface charges
outside the disk is a bounded and continuous function of position in the
vicinity of (z’, y’, 2’). Call this portion of the potential 42. There
remains a contribution 41 due t o the charge on the disk itself. The
188
[CHAP.I11
T H E ELECTROSTATIC FIELD
surface density w is bounded, and we proceed as in Sec. 3.7. There exists
a number m such that at every point on the disk Iw[ < m, Iw/rl < m/r.
(9)
where S1 indicates that the surface int>egralis to be extended over the
y’, z’) on the
disk. The resultant potential at an arbitrary point (d,
surface is, hence,
4
(10)
=
41
+ 42.
Both 41 and 42have been shown to be bounded, and +2 is also continuous.
But 41 vanishes with the area of the disk and consequently, since 4
differs as little as desired from a continuous function, it is itself continuous. The specialization of the disk t o circular form is no restriction
on the generality of the proof, since the circle may be considered to be
circumscribed about a disk of arbitrary shape. T h e potential due to
a surface distribution of charge i s a bounded, continuous function of position
at all points, both o n and o$ the surface. The function defined by (S),
therefore, passes continuously through the surface.
The integral expression of the field intensity,
is continuous and has continuous derivatives of all orders a t points not
on the surface, but suffers an abrupt change as the point ( d , y’, 2 ’ )
passes through S. The nature of the discontinuity may be determined
directly from ( l l ) , I but we shall content oursolves here with the simple
method employed in Sec. 1.13 based on the divergence and rotational
properties of E. The transition of the vector E through the layer S is
subject to a discontinuity defined by
E+ - E-
(12)
=
1
-wn,
€0
where n is the unit normal drawn outward from the positive face of the
surface. If now by w we understand the true plus the bound charge,
w - n (P+ - P-), then (12) is equivalent t o
+
.
+
.
-
(13)
(eOE+ P+) n - (e0E- P-) n = (D, - D-) n = w.
This specifies at the same time the transition of E at a surface of discontinuity in a dipole distribution.
3.16. Double-layer Distributions.-I t frequently happens that the
potential of a charge distribution is identical with that which might be
Cf. KELLOGG,
Zoc. cit.; PHILLIPS,
Zoc. cit., Chap VI.
189
DOUBLE-LAYER DISTRIBUTIONS
Snc. 3.161
produced by a layer of dipoles distributed over a surface. We imagine
such a surface distribution to be generated by spreading positive charge
of density +w over the positive sitlr of a regular surface S , and an identical
distribution of opposite sign on its negative side. The result is a double
layer of charge separated by thc infinitesimal distance 1. The dipole
moment per unit area, or surface density 7 , is a vector directed along the
positive normal t o S and is dcfined as the limit
(14)
T =
n lim (wZ) as Z -+ 0,
w
+ cc.
The dipole moment corresponding t o an clcmcn 1 of area d a on the double
layer is T da, and its contribution l o the potcntial at a fixed point (x’,y’, 2’) not on the surface is
Now
cos
e
- da measures exactly the solid angle
r2
d n a t the point of observation (s‘,y’, z‘) subtendcd by the element of area da. This angle
FIG. 33.-The element of
is positirc if the radius vector r drawn from area da subtends a positive
(s’,y’, z ’ ) to the clement du makes an acute solid angle a t P I and a
negative angle a t 1’2.
anglc with the positive normal n. Thus in Fig.
33 thc clement du subtends a positive angle a t P1 arid a negative one a t Pz.
The potential due to the entire distribution may be written
(16)
-
+(x’, y’,
1
47rq
2’) = --
-
~ ( x y,, z ) V
(:)
da =
4aeo
where Q is the solid angle subtended a t (2’)y’, z’) by the surface S. T h a t
the second integral must be precrded by a negative sign is evident, if one
notes that do is positive when the laycr is viewed from the lower or
negative side, where the potential is certainly negative.
The potential $.I has distinct v:zlues on opposite sides of a double layer.
Suppose first t h a t the surface S is closed and that the distribution is of
constant density, so t h a t r may be taken outside the integral sign. The
positivc layer lies on the outer side of S so that z has the same direction
as the positive normal. There is a well-known theorem on solid angles
which applies to this case: “If S forms the complete boundary of a threedimensional region, the total solid angle subtended by S a t P is zero,
i f P lies outside the region, and 47r if P lies inside.”’ The potential
1
a t any interior point of S is, therefore, 4- = -- r ; a t any exterior point
€0
On the arialytic properties of solid angles see Phillips, Zoc. cit., p. 112.
190
++ = 0.
T H E ELECTROSTATIC FIELD
[CHAP.I11
The difference in potential on either side of the double layer is
++
(17)
-
+-
1
= - 7.
€0
Next we observe that (17) represents correctly the discontinuity in +
as one traverses the double layer also when S is an open surface. For S
may be closed by adding an arbitrary surface S’. At every point within
or without this closed surface the resultant potential can be resolved into
two parts, a fraction due t o the distribution on S , and a fraction +’, due
to that on S’. Inside the closed surface the resultant potential is again
1
-- 7; outside, it vanishes. Thus, as we traverse the surface S,
+’
+
++
€0
1
+’,due to the layer S‘,
is certainly continuous across S and hence the entire discontinuity is in +,
changes discontinuously by an amount
2 T ; but
€0
as specified by (17).
It remains to show that (17) is also valid when T is a function of
position on S. About an arbitrary point P on the surface S draw a circle
of radius a. Let the radius a be so small that, over the area enclosed by
the circle, 7 may be assumed to have a constant value 7 0 . The potential
C$
in the neighborhood of P may again be resolved into two parts, a
fraction 4’ due to the infinitesimal circular disk and a fraction 4‘’ due
t o that portion of the dipole layer lying outside the circle. 4‘’ is con1
tinuous at P. 4‘ on the other hand suffers a discontinuous jump of - T,,
€0
+
on crossing the circular disk. The resultant potential + = +‘
4’’
must, therefore, exhibit a discontinuity of the same amount and, in the
limit as a -+0, we find that (17) holds for variable distributions if for T we
take the value at the point of transition through S.
These results might be interpreted to mean that the work done in
moving a unit positive charge around a closed path which passes once
through a surface bearing a uniform dipole distribution of density r0 is
1
f - ro, depending on the direction of circuitation.
The potential in
€0
presence of such a double layer is a multivalued function; for to its value
1
at (x’, y’, 2’) one may add - mr, where m is any positive or negative
€0
+
one need only return to the
integer. To obtain a different value of
initial point after traversing the surface S. All this appears to be in
contradiction to the conservative nature of the electrostatic field. As a
matter of fact, the mathematical double layer constitutes a singular
surface which has no true counterpart in nature. We shall show below
by means of Green’s theorem that the potential within a closed domain
SEC.
191
DOUBLE-LA YER D I S T R I B U T I O N S
3.161
bounded by a surface S due to external charges is identical with that
which might be produced by a certain dipole distribution over S. This
equivalent double layer, however, does not lead to correct values of 4
outside S.
1
The application of the integral
E n d a = - q to a small right
§
€0
cylinder, the ends of which lie on cither side of the double layer, after the
manner of Sec. 1.13, indicates at, once that there is no discontinuity in
the normal component of the field vector E across the surface since the
total charge wit.hin the cylinder is now zero.
(E+ - E-). n
(18)
=
0.
The line integral of E around a closed path, however, is no longer necessarily zero; consequently we anticipate a possible discontinuity in the
tangential components. Let t,he
./-at+
contour start at 1, Fig. 34, at which
point the dipole density is r. The
difference in potential between
R E t p0int.s 1 and 2 is, according to (17), s
FIG.34.-Transition of the tangential com1
41 - 42 = - T. (Note th at t.he
ponents of E across a double layer.
7-=
-
€0
intensity E is given b y the derivative of the potential, not the diference in
potential across the discontinuity. The normal derivative is continuous.)
If A1 is the length of the path tangential to t,he surface, t'he density at 3 is
r
V r . Al, and the potential difference between 4 and 3 is
+
$4
-
43
1
=:
-
(T
60
Now the quantity ( 4 2
identically zero; hence,
(19)
1
-- 7
€0
- 41)+ ( 4 3
+ VT
- 42)
*
Al).
+ (44 - + (41- 44) is
43)
+ (43 - + 1 (r + Vr - Al) + (41 - 44) = 0.
Abbreviating 44 -
42)
41
= &b+,
$3
- $2
= A$-,
(19) reduces to
where t is a unit vector tangent, to S. The limit of -A+/Al as 1 -i
0
is the component of E in the direction of t, so that in virtue of the continuity of the normal component of E the transition of the field vector is
specified by
1
E+ - E- = ---vT.
(21)
€0
192
[CHAP.I11
T H E ELECTROSTATIC FIELD
Clearly V r signifies here the gradient of 7 in the surface and is, therefore,
a vector tangent t,o 8. The proof is subject to the assumption t h a t r and
its first derivatives are continuous over S.
3.17. Interpretation of Green’s Theorem.-In Sec. 3.4 it was shown
that the potential a t any interior point of a region V bounded by a closed,
regular surface S could be expressed in the form
(22)
+(d,y‘,
2’)
da -
=
1
&.(I4 ana (-)
da.
r
-
From the analysis of the preceding paragraph we are led t o interpret this
result in the following way. The volume intcigral represents, of course,
the contribution of the charge within S , and the surface integrals account
for all charges exterior to it. However, the first of these surface integrals
is also equivalent t o the potential of a single layer of charge distributed
over S with a density
(23)
a+
a = e--t
dn
and the second can rvidently be interpreted as the potential of a double
layer on S whose density is
(24)
7
=
€4,
T h e charges outside S m a y be replaced by a n equivalent single and double
layer, the densities of which are specified by (23) and (24), without modifying
in a n y way the potential at an interior point. The potential outside S
produced by these surface distributions corresponds in no way, however,
t o that arising from the true distribution. On the contrary, it is most
important t o note that these equivalent siirface layers, which give rise t o
the proper value of the potential a t all interior points of S , are just those
required to reduce both potential and field E to zero at every point outside.
We observe first t h a t the single layer givcs rise t o a discontinuity in the
normal derivative of 4 equal t o
The normal derivative in Eq. (23) must be calculated on the inner or
negative side of S , since this alone belongs t o V. Replacing (a4/dn)by its value (23)) it follows that (a+/&)+ = 0. Furt,hermore, the discontinuity in potential due to the double layer is specified by (17), which
together with (24) shows that 4+ = 0. T h a t and its derivatives vanish
everywhere outside S is readily shown by applying (22) to the volume V 2
exterior to S. Since there are now no charges in V2, V24 = 0. At
infinity the potential is regular, and consequently the potential within
C#I
Next Page
SEC.3.181
IMAGES
193
V z is dctermined solely by the values of 4+ and (a+/&)+ on S. But,
these have just been shown to vanish. The function 4(d, y’, 2’) of (22)
is continuous and has continuous derivatives. The field intensity E
outside X is, thtrcforc, also zero.
From the foregoing discussion it is clear t h a t one may always close
off any portion of a n elwtro5taf ic ficld by a surface, reducing the field
and potential outside to zero, and taking account of the effect of external
charges on the field within by proper single- and doublc-layer distributions on the bounding surface. 1t is instructive to consider these results
from the standpoint of the ficld intensity E. The single layer introduces
the proper discontinuity in the normal component E,, but does not affect
the transition of the tangential component. The double layer, on the
othrr hand, in no wise affects the transition of the normal component, but
may be adjusted to introduce the proper discontinuity in Et, or according
t o (211,
Et+
=
0,
where 1 is any direction tangent to 8.
If in particular the surface X is an
equipotcntial, n X E = 0, and no
dipole distribution is neccssary; the
field inside S due to external chargvs
can then be accounted for by a single
a4 on the
layer of density (J = E dn
equipotential.
3.18. Images.-An important application of these principles is t o be
found in the theory of images. The
equipotential surfaces of a pair of
equal point charges, one positive and FIG. BB.-Application of Green’s theorem
to the theory of images.
the other negative in a homogeneous
dielectric of inductive capacity E , form a family of spheres whose centers
lie along the line joining t h e charges. Let t h e surface S be represented by any equipotential about - q , Fig. 35. This surface thus
divides all space into two distinct regions, and in view of the regularity
of the potential a t infinity, Green’s theorem applies t o both, S being t h e
bounding surface in either case; i~.,
either region may be taken as t h e
“interior” of S. If, therefore, the charge - q be removed, the field in
the region occupied b y +q is unmodified if a charge of density w is spread
over S as specified b y (23). Inversely, if the charge + q be located as
Previous Page
194
THE ELECTROSTATIC FIELD
[CHAP.111
shown with respect to a conducting sphere, a surface charge w will be
induced upon it such that it becomes an equipotential. The contribution
of this induced charge to the field outside the Conductor S is now determined most simply by replacing the surface distribution by the equivalent
point charge -q. The charge - q is said to be the image of + q with
respect to the given sphere. In case the equipotential surface is the
median plane x’ = 0, the calculation is extremely simple. The potential
at any point (x’,y’, 2’) to the right of x’ = 0 is
(27)
+(XI,
y’,
2’)
=
1
9
The normal derivative a t x’ = 0 must be taken in the direction of the
negative x’-axis, since the region on the right of x‘ = 0 is to be considered
the interior of the equipotential.
hence by (23) the charge density is
w =
where r2 = a2
+ yf2 + z ‘ ~ . ~
1 aq,
--29 r3
BOUNDARY-VALUE PROBLEMS
3.19. Formulation of Electrostatic Problems.-The analysis of the
preceding sections enables one to calculate the potential at any point in
an electrostatic field when the distribution of charge and polarization is
completely specified. In practice, however, the problem is rarely so
elementary. Ordinarily only certain external sources, or an applied
field, are given from which the polarization of dielectrics and the surface
charge distribution on conductors must be determined such as to satisfy
the boundary conditions over surfaces of discontinuity.
Among electrostatic problems of this type are to be recognized two
classes : the homogeneous boundary-value problem and the inhomogeneous
problem. To illustrate the first, consider an isolated conductor embedded
in a dielectric. A charge is placed on the conductor and we wish to know
its distribution over the surface and the potential of the conductor with
respect to earth or to infinity. At all points outside the conductor the
“Mathematical Theory of Electricity and
* On the method of images, see JEANS,
Magnetism,” 5th ed., Chap. VIII, Cambridge University Press; or MASONand
WEAVER,“The Electromagnetic Field,” pp. log#., University of Chicago Press,
1929.
SEC.3.191
FORMULATION OF ELECTROSTATIC PROBLEMS
195
potential must satisfy Laplace’s equation. At infinity it must vanish
(regularity), and over the surface of the conductor it must assume a
constant value. We shall show that these conditions are sufficient t o
determine uniquely. The density of surface charge can then be determined from the normal derivative of +, subject to the condition t h a t
JW da over the surface of the conductor must equal the total charge.
An inhomogeneous problem is represented by the case of a dielectric
or conducting body introduced into the $xed field of external sources.
A charge is then induced on the surface of a conductor which will distribute itself in such a manner that the resultant potential is constant over
its surface. The integral J w d u is now zero. Likewise there will be
induced in the dielectrics a polarization whose field is superposed on the
primary field t o give a resultant field satisfying the boundary conditions.
A schedule can be drawn up of the conditions which must be satisfied
in every boundary-value problem. To simplify matters we shall assume
henceforth that the dielectrics are isotropic and homogeneous except
across a finite number of surfaces of discontinuity.
+
V2+ = 0 at all points not o n a boundary surface or within external
sources;
$I i s continuous everywhere, including boundaries of dielectrics or of
conductors, but excluding surfaces bearing a double layer;
+ i s finite everywhere, except at external point charges introduced as
primary sources;
e2
fg)2-
el
= 0
across a surface bounding two dielectrics;
a+
at the interface of a conductor and dielectric;
an
O n the surface of a conductor either
( a ) + i s a known constant +i, or
(b) + i s a n unknown constant and
e - = -w
+ i s regular at infinity provided all sources are within a finite
distance of the origin.
In (4) it is assumed t h a t the interface of the dielectrics bears no charge,
as is almost invariably the case. The normal is directed from (1) to (2),
and in ( 5 ) from conductor into dielectric.
An electrostatic problem consists in finding among all possible solutions of Laplace’s equation the particular one that will satisfy the conditions of the above schedule over the surfaces of specified conductors
and dielectrics.
196
THE ELECTROSTATIC FIELD
[CHAP.
111
3.20. Uniqueness of Solution.-Let 4 bc a function which is harmonic
(satisfies 1,aplacc’s equation) and which has continuous first and second
derivatives throughout a region V and ovcr its bounding surface S.
According t o Green’s first identity, (5), page 165, 9 satisfies
Suppose now that 9
=
0 on the surface 8. Then
sv
dv
(v+)2
= 0.
The
integrand is an essentially positive quantity and consequently V 4 must
vanish throughout 8. This is possible only if 4 is constant. Over the
boundary 4 = 0 and, since by hypothesis 4 is continuous throughout V ,
it follows that 6 = 0 over the ent,ire region.
Now let 41 and 4 2 be two functions that arc harmonic throughout the
closed region V and let
Then if
and 42are equal over the boundary S, their difference vanishes
identically throughout V . A function which i s harmonic and which
possesses continuous first- and second-order derivatives in a closed, regular
region T7 i s uniquely determined by its values o n the boundary X.
Consider next a system of conductors embedded in a homogeneous
delectric whose potentials are specified. We wish t o show t h a t the
potential a t every point in space is thereby uniquely determined. The
reasoning of the preceding paragraph is applied t,o a volume V which
is bounded interiorly by the surfaces of the conductors, and on the
exterior by a sphere of very large radius R. Lct us suppose that there are
two solutions, 41 and &, that satisfy the prescribed boundary conditions.
Then on the surfaces of the conductors, 4 = $1 - $2 = 0. Since $1
and $2 are assunied t o be solut,ions of the problem, they must bot,h satisfy
the conditions of See. 3.19; hence their differcnce 4 is harmonic, has the
value zero over the conductors, and is rcgular a t infinity. Thc surface
integral on the right-hand sidc of (1) must now be extended over both
the interior and exterior boundaries. Over the interior boundary 9 = 0 ;
hence the integral vanishes. On the outer sphere 8 4 / 8 n = &$/ah!. If
€2 approaches infinity, 4 vanishes as 1/R and a+/aR as 1/R2. The
integrand 9 89
- thus vanishes as 1 / R 3 , whereas the area of the sphere
an
becomes infinite as R2. The surface integral over the exterior boundary
is, therefore, zero in the limit R-+ 00. It follows too t h a t the volume
integral in (1) must vanish when extended over the entire space exterior
to the conductors, and we conclude as before t h a t if the two functions $1
SEC.
3.211
SOLUTION OF LAPLACE’S EQUATION
197
and 4 2 are identical on the boundaries they are identical everywhere:
there is only one potential function that assumes the specified constant
values over a given set of conductors.
The left-hand side of (1) m a y also be made to vanish b y specifying
that d4ldn shall be zero over th(1 enclosing boundary S. Then throughout V we h a w again v+ = 0, whence i t follows that is constant everywhere, although not necessarily zero since the condition &plan = 0 does
not imply the vanishing of 4 on S . One concludes, as above, that if the
normal derivatives d & / a n and d & / d n of two solutions are identical on the
boundaries, the solutions themselves can differ only by a constant. I n
other words, the potential i s uniquely determined except for a n additive
constant by thp values of the normal derivative o n the boundaries. But the
normal derivative of the potential is in turn proportional to the surface
charge density and, consequently, there is only one solution corresponding
to a given set of charges on the conductors.
In case there are dielectric bodies present in the field the requirement
t h a t the first derivatives of +, as well as itself, shall be continuous is no
longer satisfied and (1) cannot be applied directly. However, the region
outside the conductors may be rcsolved into partial volumes V , bounded
by the surfaces S , within which the dielectric is homogeneous. To each
of these regions in turn, (1) is then applied. The potvntial is continuous
across any surface S , and the derivatives on one side of S, are fixed in
terms of the derivatives on the other. It is easy to see that also in this
more general case the electrostatic problem i s completely determined by the
values either of the potentials or of the charges specified on the conductors of
the system.
3.21. Solution of Laplace’s Equation.-It
should be apparent t h a t
the fundamental task in solving an electrostatic problem is the determination of a solution of Laplace’s equation in a form that will enable onc to
satisfy the boundary conditions by adjusting arbitrary constants. There
are a certain number of spccial methods, sucli as t h r method of images,
which can sometimes be applied for this purpose. Apart from the theory
of integral equations, the only procedure that is bolh practical and
general in character is the method known as “separation of the variables.”
Let us suppose that the surface S bounding a conductor or dielectric body
satisfies the equation
+
+
We now introduce a set of orthogonal, curvilinear coordinates ul, u2,u3,
as in See. 1.14, such that one coordinate surface, say u1 = C, coincides
with the prescribed boundary ( 3 ) . If then a harmonic function +(ul,
u2,u3)can be found in this coordinate system, it is evident that the normal
derivative a t any point on the boundary is proportional to the derivative
198
THE ELECTROXTATIC FIELD
[CHAP.
111
of 4 with respect to u l , and th at the derivatives with respect to u2and u3
are tangential.
According to (82), page 49, Laplace's equation in curvilinear coordinates is
(4)
Let us suppose th at the scale factors hi satisfy the condition
9
- = Mif,(u1)f,(u2)f3(u3).
(5)
hi
Each of the product functions fi(ui) depends on the variable ui alone,
but the factor M i does not contain ui. It may, however, depend on the
other two. We next assume that likewise may be expressed as a product of three functions of one variable each.
+
+ = F~(U')F,(U~)P~(U~).
(6)
Then (4) can be written in the form
(7)
If finally the M ; are rational functions, Eq. (7) can be resolved into three
ordinary differential equations. The method is best described by
example.
1. Cylindrical Coordinates.-From (I), page 51, we have
whence by inspection
(9)
fl
=
r,
f2
= f3 =
1,
MI
=
M3
=
1,
M2
=
1
7.
Equation (7) becomes
The first two terms of (10) do not contain z ; the last term is independent of
r and 4. A change in x cannot affect the first two terms and therefore
the last term must be constant if (10) is to be satisfied identically for any
range of z.
1 d2F,
- -c,.
Fa dz2
SOLUTION OF LAPLACE'S EQUATION
SEC. 3.211
199
The arbitrary constant CI, called the separation constant, has been chosen
negative purely as a matter of convenience and the partial derivative
has been changed to a total derivative since FS is a function of z alone.
Upon replacing the third term of (10) by -C1 and multiplying by r2
one obtains
It is again apparent that the second term of (12) is constant, leading to
two ordinary equations,
d2F,
(13)
+ CzFz
=
0,
The equations for Fz and F3 are satisfied by exponential functions of
real or imaginary argument depending on the algebraic sign attributed
to the separation constants; F1 is a Besselfunction. In the not uncommon
case of a potential 9 which is independent of z, we find F3 constant,
C1 = 0, and in place of (14)
2. Spherical Coordinates.-From
whence
(17)
fl
=
r2,
.f2
(2), page 52, we have
= sin 8,
f3
=
1,
where $, has in this case been employed in place of 9 to represent the
azimuthal angle. Separation of the variables leads to the three ordinary
equations
1 d (rz?)
= cl,
F 1 dr
sin 6 d
___ - (sin 0
= cz - c1sin2 0,
F z d0
s)
E3
+ C2F3 = 0.
dlL2
Of these three, only the Leqendre equation (19b) is of any complexity.
200
T H E ELECTROSTATIC FIELD
3. Elliptic Coordinates.-According
[CHAP.I11
t,o (3), page 52, we have
which by inspcction lead t o
The Laplace equation for t,hc elliptic cylinder is
which upon separation gives
1 d2F3
(23a)
-
-c1,
Both F1 and F , are Mathieu junctions, but simplify notably when C1 = 0
as is the case when 4 is uniform along thc lcngth of the cylinder.
4. Spheroidal Coordinates.-According t o (6), page 56, we have
di c (_
P_
-_
VZ)_ _
(24) -T
4 --c(tZ- l),
- __
hl
fl = .p - 1,
fz = 1 - 7 2 ,
f 3 = 1,
(25)
C
C
C(t2 - -~
M , = - - . ~ ,M 3 == ______
.
Mi =
(2 - 1
1 - 172,
[(t2- 1)(1- s"1'
Laplace's equation rcduces to
~
~
ded in a homogeneous, isotropic dielectric
of inductive capacity € 2 . At z = .( > r1
on the z-axis, there is located a point
-- -32=t
..-
---;?(r,
e)
1931.
3 WATSON,
“Treatise on the Theory of Hessel Functions,” Cambridge Ilniversity
Press, 1922.
WHITTAKER
and WATSON,“ Modern Analysis,” Cambridge University Press,
1922.
6 JEANS,
loc. cil.
6 SMYTHE,
“Static and Dynamic Electricity,” McGraw-Hill, 1939.
202
THE ELECTROSTATIC FIELD
[CHAP.
111
+
~ ( C OO),S imposing on C1 the value C1 = n(n l), n = 0, 1, 2, . . . .
Under these circumstances (19u) is satisfied by either rn or r-%--l. The
condition that the potential be single-valued is fulfilled by the function
where an,, b,, are arbitrary constants. But 41 must also be regular a t
infinity, which necessitates our placing unm= 0. Furthermore, the
primary potential +o is symmetric about the z-axis; consequently m = 0
in this case. The potential of the induced distribution is, therefore,
represented by the series
m
The expansion of the primary potential 40 in spherical coordinates
was carried out in Sec. 3.8. When r < {,
(3)
and the resultant potential on the surface r =
r1
is
Now is a constant, and since (4) must hold for all values of 8, it follows
that the coefficients of P,(cos 0 ) must vanish for all values of n greater
than zero. The coefficients b, are thus determined from the set of
relations
(5)
At any point outside the sphere
To determine the charge density, we compute the normal derivative
on the surface.
(7)
SEC.3.221
203
CONDUCTING SPHERE I N FIELD
and the induced charge density is
The total charge on the sphere is
q1 =
(9)
I"d""
ur: sin 0 d0 d#.
Now a well-known property of the Legendre functions is their orthogonality.
6"
(10)
~ , ( c o se)P,(cos
e) sin e ae
= 0,
n # rn.
We may take m = 0, Po(cos 0) = 1, and learn that P,(cos 0) vanishes
when integrated from 0 to T if n > 0.
q1
(11)
=
-q-
rl
l
+4 ~ ~ ~ ~ 1 4 . .
The potential of the sphere is, therefore,
q1 representing an excess charge that has been placed on the isolated
~
be put equal to zero.
sphere. If the sphere is grounded, $ Jmay
It is interesting to observe that the potential 91 of the induced distribution at any point outside the sphere is that which would be produced
by a charge 4ae2bo = q1, a dipole moment 4n& = - q r:
51
etc., all located
l
a t the origin and oriented along the z-axis (Sec. 3.8). There is, however,
another simple interpretation. The point x = l', Fig. 36, where ll' = r;,
is said to be the inverse of z = { with respect to the sphere. The reciprocal distance from this inverse point to the point of observation is, by
(8), Page 173,
Thus the resultant potential ( 6 ) may be written
Outside the sphere the potential is that of a charge q at z = l, an image
r1
charge q' = -q - located a t the inverse point z =
l
b', a charge q1 (which
204
T H E ELECTROSTATIC FIELD
[CHAP.
111
is zero when the sphere is uncharged) located a t the origin, and a charge
q
r
at the origin which raises the potential of th e floating sphere to the
proper value in th e external field.
3.23. Dielectric Sphere in Field of a Point Charge.-At any point
outside the sphere, whose conductivity is zero and whose inductive
capacity is €1, the potential is
(15)
The notation ++ mill be employed to denote a potential or field a t points
outside, or on the positive side, of a closed surface. The expansion of +1
in inverse powers of r does not hold within the sphere, for the potential
must be finite everywhere. We resort, therefore, t o a n alternative solution of Laplace's equation obtaincd from (1) by putting the coefficients
b,, equal to zero. At any interior point the resultant potential is
m
4- includes the contribution of the charge q as well as that of the induced
polarization, for the singularity occasioned by this point charge lies
outside thc region to which (16) is confined. I n the neighborhood of the
surface, r < jcj, so that +o can be expanded as in (3). Just outside the
sphere,
Across the surface
(r
=
r1).
A calculation of the coefficients from these boundary conditions leads to
2n+1
,
47rp+1nE1 (n i)Z
(19)
q r?+l
-_
€1
n _
b, = .
L
_
47r {"+I
€2
nE1
(n 1)ez
The potential a t any point outside the sphere is
a,
4= -
+ +
+ +
while a t a n interior point
205
SPHERE I N A PARALLEL FIELD
SEC.3.241
It is important to observe that a region of infinite inductive capacity
behaves like an uncharged conductor. As e l becomes very large, it will
be noted that the first term in the series (20) corresponding to n = 0
vanishes, so that in the limit one obtains (6) for the case q 1 = 0.
3.24. Sphere in a Parallel Field.-As the point source q recedes from
the origin, the field in the proximity of the sphere becomes homogeneous
and parallel. We shall consider the case of a sphere embedded in a
dielectric of inductive capacity e2 under the influence of a uniform,
parallel, external field Eodirected along the positive z-axis. The primary
potential is then
(22)
(Po =
-Eox = --or
cos 0 = -EorPl(cos 8).
Note that 4 0 is no longer regular a t infinity, for the sourcc itself is infinitely
remote. The potential outside the sphere due to either induced surface
charge or polarization is again
P,(cos 8)
n=O
If the sphere is conducting, the resultant potential on its surface and
throughout its interior is a conshnt &.
is independent of 0; whence it follows that
bo
=
bl
Eor cos 8
=
r!Eo,
b,
cos 8
= 0
whenn
> 1.
+ Ear: __+ 4*.:
r2
The charge density and the total charge are respectively
(27)
w =
3e2E0cos 8 f
€24
)'
r1
q1 = 4 ~ r ~ e ~ 4 ~ .
The potential of the induced surface charge is, therefore, that of a dipole
of moment p = 4mzEor:; a moment, in other words, which is proportional
to the volume of the sphere. To this is added the potential of q1 in case
the sphere is charged.
If the sphere is a dielectric of inductive capacity e l , the potential at
an interior point is of the form (16). To satisfy the boundary conditions
(18), the coefficients must now be
206
THE ELECTROSTATIC FIELD
[CHAP.I11
The resultant potential is then
Within the sphere the field i s parallel and uniform.
The dielectric constant ~1 of the sphere may be either larger or smaller
than K ~ . Thus the field within a spherical cavity excised from a homogeneous dielectric K Z is
Next we note that the induced field outside is that of a dipole oriented
along the z-axis whose moment is
Apparently even a spherical cavity behaves like a dipole. This effect
may be readily accounted for by recalling that thc walls of the cavity
bear a bound charge of density w’ = -n Pz, wher3 P2 is the polarization
of the external medium.
In the case of a dielectric sphere in air, € 2 = €0. The polarization of
the sphere is then
K1 - 1
Pi = E O ( K ~1)E- = 3 ___
(33)
K1
2 roEo’
and its dipole moment
-
+
The energy of this polarized sphere in the external field is
The dielectric polarization modifies the field within the sphere. It
will be convenient to express this modification directly in terms of P1.
A depolarixing factor L is defined by
-
E- = Eo LP1.
(36)
From (30) and the relation PI= C O ( K I - l)E-, one calculates for a sphere
in a parallel external field
SEG.3.251
FREE CHARGE O N A CONDUCTING ELLIPSOID
207
(37)
I n the case of a sphere immersed in air K~ = 1, L
FIG.37a.-Conducting sphere in a
parallel field. The external medium
is air.
=
-.1
3Eo
FIG.37b.-Dielectric sphere in a
parallel field. The external medium
is air.
PROBLEM OF T H E ELLIPSOID
3.26. Free Charge on a Conducting Ellipsoid.-In ellipsoidal coordinates Laplace’s equation reduces by virtue of Eq. (135), page 59, to
The properties of the ellipsoidal harmonics that satisfy this equation
have been extensively studied, but we shall construct here only certain
clementary solutions that will prove sufficient for the problems in view.
Consider first a conducting ellipsoid embedded in a homogeneous
dielectric Q. The semiprincipal axes of the ellipsoid are a, b, c. It
carries a total charge q, and we assume initially that there is no external
field. We wish t o know the potential and the distribution of charge over
the conducting surface.
T o solve this problem a potential function must be found which
satisfies (I), which is regular a t infinity, and which is constant over the
given ellipsoid. Now 4 is the parameter of a family of ellipsoids all
confocal with the standard surface .$ = 0 whose axes have the specified
values a, b, c. The variables 9 and { are the parameters of confocal
hyperboloids and as such serve to measure position on any ellipsoid
4 = constant. On the surface ( = 0, therefore, q5 must be independent of
q and {. If we can find a function depending only on E which satisfies (1)
and behaves properly a t infinity, it can be adjusted to represent the
potential correctly a t any point outside the ellipsoid C; = 0.
208
T H E ELECT ROST A TIC FIELD
Let us assume, then, that
+ = +(E).
[CHAP.111
liaplace’s equation reduces to
which on integration leads to
(3)
where C1 is a n arbitrary constant. The choice of the upper limit is such
as to ensure the proper behavior a t infinity. When t becomes very
large, REapproaches t $and
2c1
(4)
-77
On the other hand, the equation of an ellipsoid can be written in the form
+ +
If r2 = x 2 y2 z 2 is the distance from the origin to any point on the
ellipsoid t , it is apparent that as ( becomes very large 4 -+ r2 and hence
a t great distances from the origin
The solution (3) is, therefore, regular a t infinity. Moreover (6) enables
us to determine a t once the value of C1; for it has been shown that,
whatever the distribution, the dominant term of the expansion a t remote
points is the potential of a point charge at the origin equal to the total
charge of the distribution-in this case p. Hence C1 = -,9 and the
8aez
potential at any point is
(7)
The equipotential surfaces are the ellipsoids t = constant. Equation
(7) is an elliptic integral arid it,s values have been tabulated.‘
To obtain the normal derivative we must remember that distance
along a ciirvilincar coordinate u’is measured not by du’ but by hl du’
(See. 1.16). In ellipsoidal coordinates
-a _
+ -- - l_a + - -
(9)
dn
1
hldt
4-
~
~ T C Z
_
-
1
~
_
~
~- {)~
-
.
~
)
(
See for example Jahnke-Emde, “Tables of Functions,” 2d ed., Teubner, 1933
~
SEC.3.261
209
CONDUCTING ELLIPSOID I N A PARALLEL FIELD
The density of charge over the surface 5 = 0 is
If now in the three equations (132), page 59, defining x, y, z in terms of
(, 7, we put E = 0, it may be easily verified that
r,
Consequently, the charge density in rectangular coordinates is
Several special cases are of interest. If two of the axes are equal, the
body is a spheroid and (7) can be integrated in terms of elementary
functions. Thus, if a = b > c, the spheroid is oblate and
When c = 0, the spheroid degenerates into a circular disk. On the other
hand, if a > b = c the spheroid is prolate and we find for the potential
The eccentricity of a prolate spheroid is e =
J1 -
c>’
-
.
&e--+1,
the spheroid degenerates into a long thin rod.
3.26. Conducting Ellipsoid in a Parallel Field.-We assume first that
a uniform, parallel field Eo is directed along the x-axis, and consequently
along the major axis of the ellipsoid. The potential of the applied field is
the value of x in ellipsoidal coordinates being substituted from Eqs.
(132), page 59. This primary potential is clearly a solution of Laplace’s
equation in the form of a product of three functions,
It is not, however, regular a t infinity.
210
THE ELECTROSTATIC FIh’LD
[CHAP.
111
Now if the boundary conditions are to be satisfied, the potential 61
of the induced distribution must vary functionally over every surface
of the family f = constant in exactly the same manner as $0. It
differs from b0 in its regularity at infinity. We presume, therefore, that
41 is a function of the form
bl = CZGI(f)F2(7)F3({),
(17)
where
(18)
Fz(7) = d
7
+ a2,
Z73G-1
=
d m .
To find the equation satisfied by Gl(f) we need only substitute (17) and
(18) into (l),obtaining as a result
(19)
Equation (19) is an ordinary equation of the second order and as such
possesses two independent solutions. One of these we know already to be
F1 = z / E
u2. There is a theorem’ which states that if one solution
of a second-order linear equation is known, an independent solution can
be determined from it by integration. If y l is a solution of
+
then an independent solution yz is given by
In the present instance
The limits of integration are arbitrary, but G1(f)is easily shown to vanish
properly at infinity if the upper limit is made infinite. The potential of
the induced charge is, therefore,
(24)
* See
1927.
for example Ince, “Ordinary Differential Equations,” p. 122, Longmans,
SBC.3.271
DIELECTRIC ELLIPSOID I N PARALLEL FIELD
21 1
The constant Cz is determined finally from the condition that on the
ellipsoid [ = 0 the potential is a constant
At any external point the potential is
As in the analogous problem of the sphere, the constant +a can be calculated in terms of the total charge on the ellipsoid. The integrals
occurring in (26) are elliptic and of the second kind.’
In case the field is parallel to either of the two minor axes it is only
necessary to replace the parameter a2 above by b2 or c2. Thus the
potential about a conducting ellipsoid oriented arbitrarily with respect
to the axis of a uniform, parallel field Eo can be found by resolving Eo
into three components parallel to the principal axes of the ellipsoid and
then superposing the resulting three solutions of the type (26).
3.27. Dielectric Ellipsoid in Parallel Field.-It is now a simple matter
to calculate the perturbation of a uniform, parallel field due to a dielectric
ellipsoid. We shall assume that the inductive capacity of the ellipsoid
is ~ 1 ,and that it is embedded in a homogeneous medium whose inductive
capacity is again €2 The applied Eo is directed arbitrarily with respect
t o the reference system and has the components Eo,, Em, E o along
~
the
axes of the ellipsoid.
Consider first the component field E g Z . Outside the ellipsoid the
resultant potential must exhibit the same general functional behavior
as in the preceding example, and will differ from it only in the value of
the constant Cz. In this region, therefore,
The variable s has replaced [ under the integral to avoid confusion with
the lower limit.
The interior of the ellipsoid corresponds to the range -c2 5 I0 if
a 2 b 1 c. In this region 3- must vary with 77 and < as determined by
the function Fz(v)F3(<)and, since (19) has only two independent solutions,
the dependence on 4 must be represented by either F1(E) or GI([) to
satisfy Laplace’s equation. But the function G1(E) is infinite a t [ = -c2,
See for example Whittaker and Watson, “Modern Analysis,” 4th ed., pp. 512fl.,
Cambridge University Press, 1927.
212
[CHAP.111
T H E ELECTROSTATIC FIELD
whereas F1([) is finite a t all points within the surface ,$
potential within this region must have the form
=
0. Thus the
(28)
'#'-= C81(,$)F2(7)F3(0,
where CBis an undetermined constant.
The constants C2 and C3 are to be adjusted to satisfy the boundary
conditions
The first of these leads to
Since $0 = -Eo,z,
the ellipsoid is
(32)
4- = 1
one finds that the potential a t any interior point of
+
EO~X
abc
'
A1
-dA1
(€1
=
J
ds
*
(s
+ a2)Rll'
and the field intensity is
(33)
The components along the other two axes are found in an identical
manner. The potential of the applied field is in total
40 = - E o ~ x- EoUy - Eotz,
(34)
and the resultant potential a t an interior point is
(35)
4- = -
[ +2
Eozx
1
(€1
- e2)A1
+
1
EOUY
abc
+2 ~ 2
(€1
-Q)A~
+
-
in which the constants
A2
1
+ 2abc
ez
and Aa are defined by
as
+
as
+
(36)
=
(s
bZ)R,'
83 =
(s
c2)R,'
The remaining components of the perturbed field are
(€1
- €,)A3
SEC.3.281
213
C A V I T Y DEFINITIONS OF E A N D D
We are led to a conclusion of great practical importance: If the applied
field i s initially uniform and parallel, the resultant field within the ellipsoid
i s also uniform and parallel, whatever the orientation of the axes. The vector E-, however, is not in general parallel to Eo, for the constants A1,
A 2 , A S are equal only in degenerate cases.
Outside the ellipsoid the resultant potential associated with the
primary component Eo, is given by (27),
abc
'4
(38)
= +o
1
--
€1
- €2
+ -2~- €2
SLGT%pZ)
ubc €1 1 +8-€2
€2
.I
ds
(s a2)R8
+
with corresponding potentials for the fields Eo, and Eoz. If in (38) we
let e l - + a ,we find that ++ reduces to (26) for the case +8 = 0. T h e
potential outside u body of infinite inductive capacity i s the same as that
outside a grounded conductor of the same shape.
The induced polarization PI within the ellipsoid tends to decrease the
applied field. The depolarizing factors L1, L2, La are defined by the
relations
(39)
EZ
=
Eoz - LiPi,,
i7;
=
Eo, - Lgi,,
=
Eoz - Ld'iz.
Putting P1= (€1 - eO)E- and introducing the components of E- from
(33) and (37) leads t o
L 1. -- - -abc u1 - u2 A it
( j = 1, 2, 3).
2€2 ti1- I
Most commonly the medium outside thc ellipsoid is free space, t i 2 = 1.
I n that case the depolarizing factors depend only on the form of the
ellipsoid.
( j = 1, 2, 3).
The polarization may then be determined from the simple formula
with corresponding expressions for P, and P,.
3.28. Cavity Definitions of E and D.-The field inside an ellipsoidal
cavity is given by (33) and (37) when €1 = e0. There are two eases of
considerable interest: that of a disk-like cavity whose plane is normal t o
the direction of the applied field, and t h a t of a needle-shaped cavity
parallel to the field.
2 14
T H E ELECTROSTATIC FIELD
[CHAP.I11
We shall consider first the applied field to be oriented along the z-axis.
Then as c becomes very small the ellipsoid degenerates into the disk
illustrated in Fig. 38a. Thcre is no loss of generality in assuming the
disk circular, a = b, and in this case the elliptic integral A Breduces to a
more elementary function.
The limit of the product a2cA3as c -+ 0 is 2. The field inside the cavity
is purely transverse, and since ~1 = 1,
1
E- = K~EO= -Do.
(44)
€0
Apart from the proportionality constant €0, the field intensity E- within
the disk-shaped cavity i s equal to the vector Do of the initial field in the
dielectric.
FIG.38a.
FIG.3%.
the cavity definitions of the vectors E and D.
indicate the direction of the applied field.
FIGS.3% and b.-Illustrating
The arrows
In Fig. 3% the field is directed along the major axis of the prolate
spheroid a > b = c.
(45)
where e
=
=
lm
+
J-T
1-
(s
1
(2, +
+ a2)a - -~
a3e3
ds
b2)(s
is the eccentricity.
As e + 1, the spheroid degen-
erates into a long, needle-shaped cavity and the product ab2A1 approaches
zero. A t any point inside this cavity E- = Eo: the field intensity E- i s
exactly the same as that prevailing initially in the dielectric.
These cavity definitions of the vectors E and D in a ponderable medium
were introduced by Lord Kelvin. Obviously a direct measurement of
E or D in terms of the force or torque exerted on a small test body is not
SEC. 3.291
215
TORQUE EXERTED ON A N ELLIPSOID
feasible within a solid dielectric, and if a cavity is excised the field within
it will depend on the shape of the hole. Our calculations have shown,
however, that the electric field intensity within the dielectric is exactly
that which might be measured within the needle-shaped opening of
Fig. 38b, whereas the field measured within the flat slit of Fig. 3% differs
from D within the dielectric only by a constant factor.
3.29. Torque Exerted on an Ellipsoid.-The intensity of an electrostatic field may be measured by introducing a small test body of known
shape and inductive capacity suspended by a torsion fiber and observing
the torque. Inversely, if the intensity of the field is known, such an
experiment may be employed to determine the inductive capacity or
susceptibility of a sample of dielectric matter. In general the field and
polarization throughout the interior of the probe are nonuniform and an
accurate computation is difficult, or impossible. On the other hand, the
advantages of an ellipsoidal test body for these purposes are obvious.
The polarization of the entire probe is constant and the applied torque
depends essentially only on its volume and its inductive capacity.
According to Eq. (49), page 113, the energy of a dielectric body in an
external field is
where as above E- denotes the resultant field inside the body and Eo
the initial field. If the body is ellipsoidal and Eo is homogeneous, we
have by (33) and (37):
+
E,"z
abc
1+ 2
and, since the volume of an ellipsoid is +rabc,
U
(48)
=
~1
-
K~
(a)
A3
.
&abc(E2 - el)E- Eo.
This energy depends not only upon the intensity of the initial field but also
upon the orientation of the principal axes with respect to the field. Let
the vector 60 represent a virtual angular displacement of the ellipsoid
about its center and T the resultant torque exerted by the field. Both T
and 60 are axial vectors (page 67), and the components SW,, SW,, SW, are
the angles of rotation about the axes x, y, z, respectively. The work done
in the course of such a rotation is
(49)
SW
=
-
T 60
=
T , SW,
+ T,SW, + T,6
~ .
216
[CHAP.I11
THE ELECTROSTATIC FIELD
This work must be compensated by a decrease in the potential energy U .
I n virtue of the homogeneous, quadratic character of (47), we may write
6U = $ T U ~ C ( ~Z ~1)6(E* Eo) = + T u ~ c ( E
-~EI)E- * 6Eo.
(50)
The reference axes have been chosen to coincide with the principal
axes of the ellipsoid. Relative t o this system fixed in the body the
variation of Eo corresponding to a virtual rotation 60 is
6Eo = Eo X 6 ~ 3
(51)
whence for the energy balance we obtain
(52)
6br = + T u ~ c (c ~e1)E-
- (Eo X 60)
= + T U ~ C ( E~
ei)(E- X Eo) * 60
=
-6W,
and from the arbitrariness of the components of rotation i t follows that
the torque exerted by the field is
T = gTabc(E1 - EZ)E- X ED.
(53)
Thc components of this torque are
T,
=
+ r ( ~ b c (E*
)~-
E2I2
€2
EoyEoz
A3 - A ,
l+X--abCKi - K 2 A 3 ) ] 7
1+-------
K2
(54)
T,
€2
= % ( u ~ c ) ’___- EoJoz
(€1
-
€2
A1
[(
I + abc
r - KI K2
T,
=
+ r ( ~ b c (€1
)~-
€212
€2
K2
- A3
(
abc K i - Kz
A 3 ) 1’T- K 2
EoSoU
r
A, - A,
1
T o investigate the stability of the ellipsoid we must determine the
relative magnitudes of the constants A1, Az, AB. I n the first place, it is
clear from their definition, (32) and (36), that all three are positive,
whatever the values of a, 6, c. It is easy t o show, furthermore, that the
order of their relative magnitudes is thc inverse of the order of the
three parameters. That is, if a > b > c , A1 < A 2 < A S . Next, one
finds that the sum of the three integrals can be reduced to a simple integral
when u = R: is introduced as a new variable, giving
(55)
A1
+ +
A2
A3
=
2 .
Sc,
217
PROBLEMS
whence from the essentially positive character of the constants it follows
that
2
0 I
A - < -9
( j = 1, 2, 31,
(56)
’ - abc
a b c ~ 1- K~
(j = 1, 2, 3).
1 f - Z - P Ai > 0,
(57)
KZ
The denominators of (54) are, therefore, positive both when € 1 > ez and
when e l < €2, and the direction of the components of torque i s independent
of the relative magnitudes of € 1 and e z . If the applied field is parallel to
any of the three principal axes, all components of torque arc zero, so
that these constitute three directions of equilibrium.
The stability of equilibrium depends on the direction of the torque
and this, we see, depends solely on the sign of A i - A k , which in turn
is determined by the relative magnitudes of the axes a, b, c. Thus
A t - A z is positive if b > c, negative when c > b. The components of
torque are such as t o rotate the longest axis into the direction of the field
by the shortest route. An ellipsoid whose major axis i s oriented along
the applied jield i s in stable equilibrium; the equilibrium positions of the
minor axes are unstable.
Problems
1. The coordinates 6,7,< are obtained from the rectangular coordinates 5, y, z by
the transformation
5
+ i?
= f(z
+ iy),
r
= 2,
where f is any analytic function of the complex variable z
following properties of this transformation:
+
+ iy.
+
Demonstrate the
a. The differential line element is ds2 = h2(,dt2 dq2) dr2, where h = l/lj’l,
f’ denoting differentiation with respect to the variable z
iy;
b. The system 5, 7, is orthogonal;
c. The transformation is conformal, so that every infinitesimal figure in the zyplane is mapped as a geometrically similar figure on the .$?-plane.
Show that in these coordinates the Laplacian of a scalar function 4 assumes
the form
+
and find expressions for the divergence and curl of a vector.
2. With reference t o Problem I, discuss the coordinates defined by the following
transformations:
218
[CHAP.111
THE ELECTROSTATIC FIELD
+
4
(5)
~ T J=
In
3. A set of ring or toroidal coordinates A,
x
= r cos
+,
y = r sin
+ iY + a ,
5
x+iy-u
p,
+ are defined by the relations
+,
z =
sin p
cosh h
cos
+
9
p
where
r =
cosh
sinh X
cos
+
1.1
Show that the surfaces, $ = constant, are meridian planes through the z-axis, that
the surfaces, h = constant, are the toruses whose meridian sections are the circles
+
22
and t h a t the surfaces,
circles
- 22 coth X
22
+ 1 = 0,
constant, are spheres whose incridian sections are the
p =
x2
+ z2 + 22 t a n p - 1 = 0.
Show that the system is orthogonal apart from ccrtain exceptional points, and find
these points. Find the expression for the differential element of length and show that
Is Laplace's equation separable in these coordinates?
4. Let F ( x , y, z ) = h represent a family of surfaces such that F has continuous
partial derivatives of first and second orders. Show that a neccssary and sufiicient
condition that these surfaces may be equipotentials is
Show that if this condition is fulfilled the potential
wherej(h) is a function of X only.
is
where c1 and c2 are constants.
6. Show that $ = F ( z
iz cos u
+
an+
ax2
+ i y sin u ) is a solution of Laplace's equation
+
2 %* =
+
ap
a22
in three dimensions for all values of the parameter u and for any analytic function F .
Show further that anv linear combination of 2n
1 independent particular solutions
can be expressed by the integral
+
$ =
J:
(z
T
+ iz cos u + iy sin u ) " f , ( u ) du,
219
PROBLEMS
where fn(u) is a rational function of e”, and finally that every solution of Laplace’s
equation which is analytic within some spherical domain can be expressed as an
integral of the form
J. =
s”
--*
f(z
4-ix cos u + iy sin u,u) du.
(See Whittaker and Watson, “ Modern Analysis,” Chap. XVIII.)
6. Charge is distributed along an infinite straight line with a constant density of q
coulombs/meter. Show that the field intensity a t any point, whose distance from
the line is r, is
and that this field is the negative gradient of a potential function
1
ro
+(z, y) = - q In -1
2se
r
+
where ro is an arbitrary constant representing thc radius of a cylinder on which = 0.
From these results show that if charge is distributed in a two-dimensional space
with a density w(x, y) the potential a t any point in the zy-plane is
where r = d ( z ’
-x
+
) ~ (y’ - y)’ and that +(x, y) satisfies
a26
a’+
$. - =
ax2
ay2
1
-4x9 Y).
Show furthermore t h a t this potential function is not in general regular at i n h i t y ,
m.
but that if the total charge is zero, so that J w du = 0, then r+ is bounded as r
7. Charge is distributed over a n area of finite extent in a two-dimensional space
with a dcrisity w(x, y). Express the potcntinl a t an external point z‘, y‘ as a power
serics in r, the distance from a n artiitrary origin to the fixed point x‘, y‘. Show t h a t
the successive terms are the potentials of a series of two-dimensional multipoles.
8. Let C be any closed contour in the xy-plane bounding a n area S and let + b e a
two-dimensional scalar potential function. By Green’s theorem show that
---f
where u
formula
=
4(x’, y’) if z‘, y‘ is an i n h i o r point, and u = 0 if d,y’ is exterior.
a*+ + a’+
ax2
ag
V Z =~-
-9
r = 2/(xf
- 2)’ + (y’ - y
I n this
) ~
and 7~ is the normal drawn outward from the contour.
9. A circle of radius a is drawn in a two-dimensional space. Position on the circle
is specified by an angle 8‘. The potential on the circle is a given function +(u, 6 ’ ) .
220
T H E ELECTROSTATIC FIELD
[CHAP.TI1
Show that a t any point outside the circle whose polar coordinates are r, e the potential
is
This integral is due to Poisson.
10. An infinite line charge of density q per unit length runs parallel to a conducting
cylinder. Show that the induced field is that of a properly located image. Calculate
the charge distribution on the cylinder.
11. The radii of two infinitely long conducting cylinders of circular cross section
b. The
are respectively a and b and the distance between centers is c, with c > a
external medium is a fluid whose inductive capacity is B . Thc potential difference
between the cylinders is V volts. Obtain expressions for the charge density and the
mechanical force exerted on one cylinder by the other per unit length. Solve by
introducing bipolar coordinates into Laplace's equation and again by application of
the method of images.
12. An infinite dielectric cylinder is placed in a parallel, uniform field Eo which is
normal to the axis. Calculate the induced dipole moment per unit length and the
depolarixing factor L (p. 206).
13. Two point charges are immersed in an infinite, homogeneous, dielectric fluid.
Show that the coulomb force exerted by one charge on the other can be found by
evaluating the integral
+
F
=
[
-
BE(E n) -
2
[Eq. (72), p. 1521 over any infinite plane intersecting the line which joins the two
charges.
14. Two fluid dielectrics whose inductive capacities are € 1 and c 2 meet a t an infinite
plane surface. Two charges p l and qz are located a distance u from the plane on opposite sides and the line joining them is normal to the plane. Calculate the forces acting
on q1 and qz and account for the fact that they are unequal.
16. A conducting sphere of radius T I , carrying a charge Q, is in the field of a point
charge q of the same sign. Assume q
Q. Plot the force exerted by the sphere
on q as a function of distance from the center and calculate the point at which the
direction reverses.
16. A charge q is located within a spherical, conducting shell of radius b a t a point
whose distance from the center is a, with a < b. Calculate the potential of the spherr,
the charge density on its internal surface, and the force exerted on q. Assume the
sphere to be insulated and uncharged.
17. Find the distribution of charge giving rise to a two-dimensional field whose
potential a t any point in the zy-plane is
<<
+
18. Show that the electrostatic potential is uniquely determined by the values
of either + or a+/an on the surfaces of conductors embedded in a n isotropic, inhomogeneous dielectric.
19. A set of n conducting bodies is embedded in a dielectric medium which is isotropic but not necessarily homogeneons. If the charges q,, q2, . . . , q. are placed on
these conductors the corresponding potentials on the conductors are + I , $2, . . .
22 1
PROBLEMS
4,,. Likewise the charges qi, q;,
. . . , dngive rise to the potentials +:,#I;,
. . , 4;.
Show that
n
20. Find the charge in coulombs on a metal sphere of radius a meters necessary to
raise the potential one volt with respect to infinity.
21. An infinite plane surface divides two half spaces, the one occupied by a diclectric of inductive capacity E , the other b y air. A charge p is located on the air side a
distance a from the surface. Obtain expressions for the potential a t points in the air
and in the dielectric and show that the potential in either space can be represented in
tcrms of images.
What is the density 0’of induced, or bound, charge on the surface of the dielectric?
Calculate the force acting on q and the work done in withdrawing the charge to
infinity.
22. The radii of two mctal spheres are respectively a and b and the separation of
b. A charge pa is placed on the first, thus raising its
their centers is c, where c > a
potential to a specified value 4.. The potential 4 b of the second sphere is maintained
a t zero by grounding. Obtain expressions for the charges qa and &f and their distribution over the spheres b y building up a series of images at the inverse points of the
spheres (p. 203).
The method is due to Lord Kelvin. See the discussion by F. Noether in RiemannWeber’s “Differential und Integralgleichungen der Physik,” Vol. 11, p. 281, 1927.
23. Two infinite, parallcl, conducting planes coincide with the surfaces = 0,
x = a. The planes are grounded. A charge q is located on the z-axis a t the point
x = c, where 0 < c < a. Show that the density of induced charge on the plane
x = Ois
+
@(X, y) =
_II_
4lr
5 {-
n=-
m
2na -
[(211a - c)2
+ r2]$
-
+c
+ c)z +
2na
I(2na
1.
r2]#
where r is the distance from thc origin to the point z, y on the plane. What is the
density on the plane 1: = a? Show that the total charge on the two planes is -q.
(Kellogg.)
24. Prove that the surfacc charge density at any point on a charged ellipsoidal
conductor is proportional to the perpendicular distance from the center of the ellipsoid
to the plane tangent to the ellipsoid a t the point.
26. Equation (14), p. 209, gives the potential of a charged prolate spheroid. Show
that as the lmgth -+ 00 and the eccentricity -+ 1, this function approaches the
logarithmic potential characteristic of a two-dimensional space. Apply the theorcm
of Problem 24 to find the charge per unit length.
26. Charge is distributed with constant density within a thin ellipsoidal shell
bounded by the two similar concentric ellipsoidal surfaces whose principal axes are a,
b, c and a
( + ;) ( + 3..( + :)
1
-
7
b
1
-
1
-
. Show that the field is zero a t all internal
points and obtain an expression for thc potential a t external points.
27. The major axis of a conducting, prolate spheroid is 20 em. in length and its
eccentricity is e. The spheroid is suspended in air and carries a total charge of q
coulombs. The maximum field intensity a t the surfacc shall at no point, exceed
3 x 106 volts/meter. Plot the maximum value of charge t h a t can be placed on the
spheroid subject to this condition as a function of eccentricity.
222
T H E ELECTROSTATIC FIELD
[CHAP.I11
28. The effect of lightning conductors has been studied by Larmor (Proc. Roy.
Soc., A, 90, 312, 1914) through the following problem. A hemispheroidal rod projects
~ the charged cloud above is uniabove a flat conducting plane. The potential Q , of
form in the neighborhood of the rod. Find a potential function Q, which is constant
over the spheroidal surface and over the plane. Plot the ratio of potential gradient
at the tip of the rod to intensity of the applied field against eccentricity of the rod.
Assume the length of the rod (semimajor axis) to be fixed and equal to one meter.
IXscuss the lateral influence of the rod on the field as a function of eccentricity.
29. A very thin, circnlar, metal disk of radius a carries a charge p. Show that the
density of charge per unit area is
Q
1
w = __
%a
- r2
daz
where r is the radial distance from the center on the disk.
Calculate the field intensity a t the surface of the disk and plot volts per meter per
coulomb against r/a. Assume the disk t o be suspended in air.
30. A spheroidal, dielectric body is suspended in a uniform electrostatic field.
The length of the major axis of the spheroid is twice that of its minor axis, and the
specific inductive capacity is 4. The external medium is air. For what orientation
with respect t o the axis of the applied field is the torque a maximum? Calculate the
value of this maximum torque per unit volume of material, per volt of applied field
intensity.
31. A stationary current distribution is established in a medium which is isotropic
but not necessarily homogeneous. Show that the medium will in general acquire a
volume distribution of charge whose density is
1
p = --(
UVE
a
-
EVU) *
VQ,,
where u and e are respectively the conductivity and inductive capacity of the medium.
32. Two homogeneous, isotropic media characterized by the constants 61, ui, and
eZ, uZ, meet a t a surface S . A stationary current crosses S from one medium t o the
other. If the angles made by a line of flow with the normal a t the point of transition
and 42, show that each line is refracted according to the law
are
cz cotan 42 =
61
cotnn 61.
Show also t h a t a charge appears on S whose surface density is
w =
(..-:)pn.
33. I n many practical current-distribution problems it can be assumed that a system of perfectly conducting electrodes is embedded in a poorly conducting medium
which is either homogeneous or in which discontinuities occur only across specified
surfaces. The electrodes are maintained at fixed potentials and either these potentials
or the electrode currents are specified. Show that the current distribution is uniquely
determined by a potential function Q, satisfying the following conditions:
( a ) V2+ = 0 a t all ordinary points of the medium;
( b ) Q, = Q,;, a constant on the zth electrode;
( c ) At a surface of discontinuity in the medium the transition is governed by
223
PROBLEMS
( d ) Regularity a t infinity;
(e) Condition ( b ) may be replaced by I ,
-$
a+
u - da,
where u is the conducan
tivity of thc mcdium just outside the electrode whose bounding surface is Si
and where I , is the total current leaving this electrode.
=
S,
34. A system of electrodes is embedded in a conducting mcdium. The electrodes
are maintained a t the Fxctl potentials + I , 9 2 , . . , +,, and the currents leaving the
electrodes are 11,Iz, . . . , I,. Show that the total Joule heat generated in the
medium is
.
n
36. Prove that for a system of stationary currents driven by electromotive forces
the current will distribute itself in such a way that the heat generated will be less for
the actual distribution than far any other which coincides with the actual distribution
in the region occupied by the driving forces and which is solenoidal everywhere else.
Assume that the medium is isotropic and that the relation between field intensity and
current density is linear.
36. A circular disk electrode of radius a is in contact with a n infinite half space of
conductivity U , such as the surface of the carth. The distance to all other electrodes
is large relative to the radius a. Show that the current distribution is determined by
the conditions:
( a ) V2+ = 0 within the conducting half space;
( b ) &$/an = 0 a t the boundary excluding the disk;
I
a+
___ on the disk, where r is the radial distance from the
(c) - = an
2snal/az - r2
center.
37. A stationary current is carried by a curved, conducting sheet of uniform thickness. The sheet is assumed to be so thin t h a t the current distribution is essentially
two-dimensional. If 5 and 7 are curvilinear coordinates on the surface, the geometry
of the surface is defined by the element of arc
ds2 = g i l dEz
+ 2912 d € d? +
922
d?,
the properties of the gib having been defined in Sec. 1.14. The scalar potential determining the current distribution satisfies the Laplace equation
Define a conjugate function $ by the relation
34
gn -
a?
- Q ' 2 a+
-
= -ug11g22
- gTz
a+
at
gzz-
lit
+ 1/
- g12-a+
a?
Qll922
dll
- 912
and show that $ satisfies the same equation as +. The functions 4 and $ are potential
= 4
i+ will be called a complex
and streamlinc functions on the surface, a n d
+
224
T H E ELECTROSTATIC FIELD
function on the surface. Now choose and q so t h a t b = [
tion. Show that in this case + satisfies the simple equation
and that
[CHAP.
111
+ iq is itself such a func-
+ and $ are related by
a + -*,
at
a7
% = - -a$
a<
a7
+
while the line clement reduces to ds2 = gll(d[2
dq2). Thus the complex functions of the surface are analytic functions of each other and the transformation
[
ill = x
i y maps the surface conformally on to the complex z-plane. If the
conformal mapping of a surface on to thc plane is known, the current distribution
problem reduces to the solution of Laplace's cquation in rectangular coordinates.
Inversely, if a current distribution over a curved surface can be determined experimentally, a conformal mapping of the surface on the plane is found. (Kirchhoff.)
38. Referring to Problem 37, show t h a t
+
+
R
z = aeia t a n -
2
is a conformal, stereographic projection of a sphere of radius a on the complex plane,
where z = x
i y and p is the equatorial angle or longitude and e the colatitude on
the sphere. Show further that
+
is a complex potential function on the sphere where f is any analytic function of z.
39. A steady current I enters a conducting spherical shell of radius a and surface
conductivity u a t a point on the surface and leaves a t a similar point diametrically
opposite. Find the potential at any point and the equation of the streamlines.
40. A cylindrical condenser is formed of two concentric metal tubes. The space
between the tubes is occupied by a solid dielectric of inductive capacity e which, however, is not in direct contact with the tubes, but separated from them b y t.hin layers
of a nonconducting intermediary fluid of inductive capacity 6'. The solid dielectric
extends so f a r beyond the ends of the metal tubes that the stray field is of negligible
intensity. A constant potential difference is maintained between the metal electrodes. Obtain an expression for the change in length per unit length of dielectric.
(Kemble, Phys. Rev., 7, 614,1916.)
CHAPTER IV
THE MAGNETOSTATIC FIELD
The methods that have been developed for the analysis of elcctrostatic fields apply largely to the magnctostatic field as well. Every
magnetostatic field can be represented by an clectrostatic field of identical
structure produced by dipole distributions and fictive double layers.
The equivalence, however, is purely formal. There is no quantity in
magnetostatics corresponding to free charge, and the surface singularity
generated by a double layer of electric charge does not actually exist. A
double layer leads in fact to a multivalued potential and consequently to
a nonconservative field. Whatever the analytical advantages of the
electrostatic analogy may be, it is well to remember that the physical
structure of a field due to stationary distributions of current differs
fundamentally from that of any configuration of electric charges.
GENERAL PROPERTIES OF A MAGNETOSTATIC FIELD
4.1. Field Equations and the Vector Potential.-The equations satisfied by the magnetic vectors of a stationary field are obtained by placing
the time derivatives in Maxwell’s equations equal to zero.
(I) V X H
=
(11) V * B = 0.
J,
To these must be added the equation of continuity which now reduces t o
(111)
V-J=O.
T h e current distribution in a stationary Jield i s solenoidal; all current lines
either close u p o n themselves, or start and terminate at infinity. From (11)
it follows likewise that all lines of the vector B-lines of magnetic flux, as
they are commonly called-close upon themselves. Suppose that the
flux density of the field produced by a current filament I l is B1. All lines
of this field link the circuit 11. A fraction of the flux B1 may, however,
thread also a second current filament I,. The concept of “flux linkagcs”
commonly employed in the practical analysis of electromagnetic problems
is based on these solenoidal properties of current and flux in the stationary state. The extension to slowly varying fields (quasi-stationary
state) is justifiable only so far as a p / a t remains negligible with respect
to V * J.
Upon applying Stokes’ theorem to (I), we obtain the equivalent
integral equation,
225
226
THE MAGNETOSTATIC FIELD
[CHAP.
Iv
where S is any surface spanning the contour C , and I is the total current
through this surface. The line integral of H around a closed path is
equal to the current linked: the magnetic field i s nonconservative.
Every solenoidal field admits a representation in terms of a vector
potential. As in Sec. 1.9, Eq. (11) is identically satisfied by
(1)
B=VXA.
The vector A is now chosen to satisfy (I), and the relation H = H(B)
between the magnetic vectors must t]hereforr be specified. If the
medium is nonferromagnetic, the relation is linear; if moreover i t is
homogeneous and isotropic, we put B = p H and arrive a t
(2)
V X V X A
= ,uJ
for the vector potential.
I n See. 1.9, i t was shown t,hat there may be added to A the gradient
of any scalar function IJ without, affecting the definition (I). By a proper
choice of IJ i t is always possible to effect the vanishing of the divergence
of A. The vector potential is uniquely defined b y (2) plus the condition
(3)
V-A=Q.
I n rectangular coordinates, (2) then reduces to
(4)
v2A
= - PJ)
where i t is understood that the Laplacian operates on each rectangular
component of A.
4.2. The Scalar Potential.-The existence of a scalar potential function associated with the electrostatic field is a direct consequence of the
irrotational character of the field vector E. If double-layer distributions
of charge are excluded, the curl of E is everywhere zero; hence one may
express E as the negative gradient of a scalar 4. Furthermore, since the
line integral of E vanishes about every closed path of the region, the potential is a single-valued function of t h e coordinates.
A region of space, enclosed by boundaries, is said t o be connected
when it is possible to pass from any one point of the region to any other
by a n infinite number of paths, each of which lies wholly within the region.
Any two paths which can by continuous deformation be made t o coincide
without ever passing out of the region are said t o be mutually reconcilable. Any closed curve or surface is said to be reducible if b y continuous variation it can be contracted to a point without passing out
of the region. Two reconcilable paths when combined form a reducible
circuit. If finally the region is such that all paths joining any two points
SEC.4.31
THE SCALAR POTENTIAL
227
are reconcilable, or such t h a t all circuits drawn within i t are reduciblc,
it is said to be simply connected. The electrostatic field of a volume
distribution of charge constitutes a simply connected region of space
within which every closed curve is reducible.
Contrast this geometrical property of the electrostatic field structurc
with the magnetic field of a stationary distribution of current. We shall
call the region of space occupied by current V1, while the region within
which J = 0 will be denoted as Vz. Figure 39 represents a plane section
through the field, the shaded arcas indicating, for example, the cross
Q
I
FIQ. 39.-Illustrating
,’ /’
--_--,’
PL=-----’
a doubly connected space.
sections of conductors carrying current. I n the region VI the curl of H
does not vanish and conscqucntly no scalar potential exists. Throughout
Vz we find
(5)
VXH=O,
and so within this region H may be expressed as the negative gradient
of a scalar function +*(z, y, 2).
(6)
H
=
-V+*.
The line integral of H along any contour wholly within V z connecting
two points P and Q is
(7)
ds = +*(P) - 4 * (Q)
The paths by which one may reach Q from P are not, however, all mutually rcconcilable. A closed curve composed of two segments such as
PEQ and QFP is irrcduciblc, since on cont,raction t o a point i t must
necessarily penetrate the region V1. Thus the region Tiz occupied by the
magnctostatic field exterior t o the current distrihut,ion is multiply
connected. The scalar potential +* is a multivalucd function of position,
for to its value a t any point P one may add a factor nI by performing n
complete circuitations about thc current I. However Vt may be rendered
simply connected and the potential single-valued by the introduction
228
T H E MAGNETOSTATIC FIELD
[CHAP.IV
of a cut or barrier which prevents a circuit from being closed in such a
manner as t o link a current. There are, of course, an infinite number of
ways in which such an imagined barrier might be placed but, if the current is constituted of a bundle of current filaments forming a single,
closed current loop, i t is most simple t o think of it as a surface spanning
the loop. The line M N in Fig. 39 represents the trace of such a surface. I n the “ c u t ” spacc the line integral of H about every closed path
vanishes, for the barrier surface excludes current linkages, and consequently the potential function +* is single-valued. At points lying on
either side and infinitely close t o the barrier the values of cp* differ by
I . There is a discontinuity in the potential function thus defined equal
to
4: - 4:
(8)
=
I,
where the subscripts refer to the positive and negative sides of the surface.
According to Xec. 3.16 the discontinuity in 4” across the barrier i s equivalent
to that of a surface dzpole layer of constant moment I per unit area. Only
whcn all currents are zero and the sources of the field are permanent
magnets does the potcntial become a single-valued function of position.
4.3. Poisson’s Analysis.-At
every point where the current density
is zero, and in particular a t interior points of a permanent magnet, we
have
B
(9)
= -poV+*
+ P O ( M+ Mo).
Since B is divergenceless, the scalar potential satisfies a Poisson equation
(10)
v2+*
*
= -P l
p* =
- V . (M
+ Ma).
The quantity p* is the magnet,ic analogue of “bound charge” density,
which in the older literature was frequently called the density of Poisson’s “ideal magnetic matter.”
Across any surface of discontinuity in the medium the magnetic
vectors satisfy, according to Sec. 1.13, the conditions
(11)
-
n (B2 - B,)
=
0,
n X (H, - HI)
=
0.
The equivalent conditions imposed on the scalar potential are, therefore,
W i t h i n a n y closed region containing permanent magnets and polarizable
matter, but throughout which the conduction current density J i s zero, the
magnetostatic problem i s mathematically equivalent to a n electrostatic
problem.
SEC.4.31
229
POISSON’S A N A L Y S I S
The calculation of the potential 4* from the densities
been described in the previous chapter.
(14)
+*(XI,
Y’, z’)
=
‘S
-
4a
p * ( x , y, z )
___.
+
1
;dv
+
p*
and w* has
w * ( x , y, z )
1
;da,
+
where as usual r = d ( z ’ - x ) ~ (y’ - y)z (z’ - z-)~. To evaluate
the integrals one must know both the permanent magnetization Mo
and the induced magnetization M ; but the induced magnetization is a
function of the field intensity and consequently of +* it,self. If, however,
the permanently magnetized bodies are not too close to one another, the
induced magnetization ran usually be neglected with respect to Mo.
Some assumption is then made regarding the magnetization M, and the
fixed or primary field of the sources calculated by means of (14). This
primary field induces a magnetization M in neighboring polarizable
bodies-soft iron, for example. The calculation of the induced or secondary field is a boundary-value problem to be discussed later.
Let us calculate now the magnetic moment of a magnetized piece of
matter. By analogy with the electrostatic case, Sec. 3.11, this moment
is defined by the integral
+ +
jq
k{, Fig. 30, page 178, is the vector drawn from the
where r1 = it
origin to the element of ((magneticcharge,’’ and S the boundary of a body
whose volume is V . Upon replacing p* and w* by their values defined
in (10) and (13), then transforming by means of the identity
-
lV M
=
-
V * (EM)- M VE,
which applies t o each component of rl, we find that (15) reduces to
The magnetic moment of a body i s equal to the volume integral of it,.
magnetization.
Finally, if Eq. (14) is transformed by (9), page 184, after p* and u’’
have been replaced by (10) and (13), one obtains for the potential
4*(x’, y’,
2’)
=
1
Jv
4a
-
(M + M,) V
(i)
dv.
If the dimensions of the magnetized body are sufficiently small with
respect to the distance to the observer at (x’,y’, x ’ ) , the variation of
230
T H E M A G N E T O S T A T I C FIELD
[CHAP.IV
V l / r within V may be neglected and we are led to a n expression for the
scalar potential of a magnetic dipole,
CALCULATION OF THE FIELD OF A CURRENT DISTRIBUTION
4.4. The Biot-Savart Law.-It has been shown th a t in a homogeneous,
isotropic medium of constant permeability p the vector potential satisfies
the system of equations
v 2 A . = -p J j ,
(1)
V - A = 0,
( j = 1, 2,3).
The Ai are rectangular components of the vector A, b u t the restriction to
rectangular coordinates does not necessarily apply to the scalar operator
v2.
Considering the Ai and Ji as variant scalars, the theory developed
in Sec. 3.4for the integration of Poisson’s equation may be applied directly
t o (1). If the current distribution can be circumscribed by a sphere of
finite radius each component of the vector potential can be expressed
as a n integral,
1
Aj(Z’, y’, 2’) = 21, z> ;
dv, ( j = 1, 2, 3),
(2)
4a Jh,
J
extended over all space. These components are now recombined to
give a vector.
+
__-
+
where r = d ( z ’ - z ) ~ (y’ (z’ - z ) ~ . Moreover, it follows
from the discussion of See. 3.7 th at if J is a bounded, integrable function,
then A is a continuous function of the coordinatcs (z’, y’, z’) possessing
continuous first derivatives a t every point insidc and outside the current
distribution. If J and all its derivatives of order less than n are finite
and continuous, the vector function A has continuous derivatives of all
1.
orders less than n
The vector potential defined by (3) satisfies the condition V * A = 0
provided the current distribution is spatially bounded. The integral
is regular and consequently may be differentiated under the sign of
integration.
+
(5)
SEC.4.41
T H E BIOT-SAVART LAW
in virtue of the condition V * J
=
0.
23 1
Then
for, since all current lines close upon themselves within a finite domain,
the surface S may be chosen such t h a t there is no normal component
of flow.
Note that the contribution d A t o the vector potential is directed
parallel t o t h e element of current J du. Equation ( 3 ) can be simplified
in the case of currents in linear conductors; i.e., conductors whose cross
section da is so small with respect to the distance r t h a t the density vector
may be assumed uniform over the cross-sectional area and directed
along the wire.
J dv
(7)
=
J da ds
I ds,
=
where I is the total current carrird by the conductor and ds is a n element
of conductor length. Since the steady current I must be constant
throughout the circuit, Eq. (3) reduces to
A(z’, y’, z’)
=
.
I
47
ds,
in which the integral is to be extended around the complete circuit C.
The field can be obtained directly from ( 3 ) by calculating the curl of A
a t the point (z’, y‘, 2 ’ ) . I n virtue of the convergence of the integral and
the continuity of its first derivatives the differential operator may be
introduced under the sign of integration.
(9)
Expansion of the integrand according to elementary rules yields
(10)
V’ X
[
J(z, Y, z ) ]
=
1
;V’ X J(z, Y, z> + V’
(i)J(x, Y,
X
2)
But the current density is a function of the running coordinates x, y, z,
whereas the differentiation applies only to d,y’, 2’; consequently the first
term on the right is zero, giving for the field vector
(11) H(z’, y’,
2’) =
47r
v’
(:)
x J(z,?J,
=
Z) dv
1
-_4T
s
J(x, y, z ) x
v
(i)
d ~ .
232
T H E MAGNETOSTATIC FIELD
[CHAP.IV
If rois a unit vector directed from the current element toward the point
of observation, (11) is equivalent to
H(z’, y’,
2’) =
1
4a
v d v ,
which for linear circuits assumes the form
H(z’, y‘,
2’) =
cr$
ds,
where so is a unit vector in the direction of the current element I ds.
The law expressed by (13) is frequently formulated with the statement
that each linear current element I ds contributes an amount
dH
1 so x ro
I ds
4a r2
=--
to the total field. This is essentially the result
deduced from the experiments of Biot and Savart in 1820. There is no very cogent reason
to reject the formula (14), other than the fact
that this resolution of (13) into differential
elements is not unique; for there may be added
t o (14) any vector function integrating to zero
around a closed circuit. All stationary currents are composed of closed filaments, so that
it is difficult to imagine an experiment to determine the contribution of an isolated element.
Obviously no error can r e s u l t from an
application of (14) to the analysis of a field
when it is understood that these contribuFIG. 40.-Solenoidal
current
tions are eventually to be summed around the
distribution.
circuit.
Whereas the vector potential due to a differential element of current
is directed parallel to the clement, the field vector H is oriented normal
to a plane containing both the currcnt element and the line joining it
with the point of observation; i.e., to the plane defined by the unit
vectors so and ro. If 0 is the angle which ro makes with so, the intensity
of the field is
I sin e
dH=-ds.
47r r2
7
dx
1
The field along the axis of a solenoid, Fig. 40, may, for example, be
calculated approximately by assuming an equivalent current sheet of
SEC.4.51
233
EXPANSION OF THE VECTOR P O T E N T I A L
negligible thickness. If there are n turns/meter carrying a current of
I amp., the current density of the sheet is K = In. The cont,ribution
of a solenoidal element of radius a and length dx a t a distance x along
the axis from the point of observation is
These contributions are directed along the axis. At any point on the
axis, the field is
At the center this reduces to
H
(18)
=
1
nI
.\/Fq$
Another elemcntary ease of some interest is that of a straight wire of
infinite length and circular cross section. The current I is assumed to be
distributed uniformly over the cross section. The simplest solution is
obtained by taking advantage of the cylindrical symmetry of the field t o
-
apply the integral law $H ds = I . If r is the radial distance from the
axis and a the radius of the wire, it is obvious that
2arH
=
ar2J
r2
a2
= -I
2arH = I ,
I
H = - r
2aa2
'
H
=
-7
I
2ar
'
(r
> a).
The field outside the wire, which is independent of its radius, can also be
obtained from the Biot-Savart law. If, howcvcr, one attempts to calculate the vector potcntial by means of (8), it will be noted that th e
integral diverges owing to the fact th at the current distribution is in
this case not confined to a region of finite extent. The vector potential is
actually
PI 1
A , = -1n-7
A , = A0 = 0,
(r > a),
(20)
2a r
as may be verified by calculating the curl in cylindrical coordinates
(page 51), and this function becomes infinite as r -+ m .
4.6. Expansion of the Vector Potential.-Following
the example set
in Sec. 3.11 for the scalar potential, we shall determine a n expansion of
the vector potential of a stationary current distribution in terms of the
coordinates of a point relative to a fixed origin.
To ensure regularity a t
234
[CHAP.I V
THE MAGNETOSTATIC FIELD
infinity i t will be assumed that the entire distribution can be circumscribed by a sphere of finite radius R drawn from the origin; we shall
consider here only points of observation exterior t o this sphere. Thc
current distribution is completely arbitrary, but i t must be assumed that
the permeability p is everywhere constant and uniform. Practically
this excludes only ferromagnetic materials and admits conductors of any
other nature and arbitrary shape embedded in dielectrics of any sort.
The notation will be essentially that adopted in Sec. 3.11 and illustrated in Fig. 30; for convenience in summing we shall let the coordinates
of the fixed point P be 2 1 , x2, 2 3 , and those of the current element f l , f z , f 3 .
The distance from the origin 0 t o P is r = *-T
xi
xi and that from
the element J(&, &, 18)dv t o P is
+
At P the vector potential is, therefore,
extended over the volume V bounded by the sphere of radius R. If rl
is the vector whose components arc ti,the expansion of 1/r2 in a Taylor
series about 0 leads to
(22)
1 1
=- r2 r
-
2
3
3
3
3=1
j=1
k=l
Consequently the vector potential can be represented by the expansion
Note that J and rl are functions of the with respect to which the integration is to be effected; thc function l / r depends only on the xi, and the
operator v applies to these coordinates alone.
I n virtue of its stationary character the current distribution may be
resolved into filaments, all of which close upon themselves within V .
Let us single out any one of these filaments. We suppose the infinitesimal
cross section to be dal and the current carried by this linear circuit to be
I = IJj dal. A point on the circuit is located by the vector rl drawn
SEC.4.51
235
E X P A N S I O N OF THE VECTOR P O T E N T I A L
from the origin, Fig. 41, and an element of length along the circuit in
the direction of the current is drl. Then for each filament or tube
where C is the closed filament contour. (A(") would not necessarily
vanish were V t o contain only a portion of the current distribution.)
The dominant term of the expansion is, therefore, the second, which
__-- ?
_----i
7P
4,:
we transform as follows.
[rl - V (s>] drl 1 (rl x
x v (i)
+ f {[ r~ - v
(25)
=
d
C
:'5,
//"
drl)
2 5
//r
/
/
/
/
(;)]rl]-
0
current filament followAgain the total differential vanishes FIG.41.-Closed
ing the contour C'.
when i n t e g r a t e d about a closed
filament, while $rl X drl = n d a is the vector area of the infinitesimal
triangle shown shaded in Fig. 41.
The quantity
(27)
m
=
I
J n d a f Jcrl
=
x drl
is by definition the magnetic dipole moment of the circuit. Since the
surface integral may be extended over any regular surface spanning the
contour, the magnetic moment depends only on the current and the form
of the contour.
The dipole potential of the total distribution is obtained by summation
over all the current filaments. By definition the magnetic dipole moment
of a volume distribution of current with respect to the origin 0 is
The magnetization M of a region carrying current can likewise be defined
as the magnetic moment per unit volumc.
It is clear that magnetization with respect to a giv-cn origin will occur
whenever the current density has a component transverse to the vector
236
[CHAP.IV
T H E MAGNETOSTATIC FIELD
rl; t h a t is to say, whenever the charge has a motion of rotation about 0
as a center. If pis the charge per unit volume and v its effective velocity,
so t h a t J = pv, the magnetization is
M = +rl X pv,
(30)
which is identical with angular momentum per unit volume when p is
interpreted as the density of mass.
The remaining terms of the expansion (23) are the vector potentials
of magnetic multipoles of higher order. Two distinct definitions of
magnetic moment have been presented: the one in terms of Poisson’s
equivalcnt magnetic charge, wholly analogous t o t)he electrostatic case,
Sec. 4.3; the other in terms of current distribution. The identity of
these two viewpoints must now be demonstrated.
4.6. The Magnetic Dipole.-According
t,o Sec. 4.3, a magnetized
piece of matter has a dipole moment defined by (15), page 229, and at
sufficient distances its scalar potential is given by (18) of the same section.
Our first task is t o determine the equivalent vector potential; i.e., a
vector function whose curl leads t o the same field as the gradient of the
scalar 4”. We shall again let (z’, y’, 2’) be the fixed point of observation
and locate the dipole a t (2, y, z). Then
To expand the right-hand side of (32), note t h a t m is a constant vector
or at most a function of x, y, z, whereas v’ applies only to the coordinates
z’, y’, z’. Furthermore
(33)
v’ x
.I(;)
= 0,
v’. v’
(3
-
=
0.
It may be verified now without difficulty that
(34)
hcnce,
(35)
(36)
where f is an undetermined scalar function. The divergence of the left
side vanishes, and consequently f may be any solution of Laplace’s
SEC.4.71
MAGNETIC SHELLS
237
cquat,ion. However, f contributes nothing to the field H and we are a t
liberty to place it equal t o zero. The vector potential of a magnctie
dipole is, therefore,
Equation (37) is identical with the vector potential attributed to a
current dipole moment in the preceding paragraph. At a suficienb
distance the field of a n y source-magnetic matter or stationary currentreduces to that of a magnetic dipole. This amounts to no more than a
restatement of the fact that a magnetized region is fully equivalent l o a
current dstribution of density J’ = V x M, and gives mathematical
support to Ampike’s interpretation of magnetism in terms of infinitesimal
circulating currents.
4.7. Magnetic Shells.-We have shown thus far that the potential of
any closed linear current can be expanded into a series of rnultipole
potentials. By allowing the circuit contour to shrink to infinitesimal dimensions
about some point (2,y, z), the potential
can be represented everywhere but a t the
singularity ( 2 , y , z) by the dipole term
alone. The magnetic moment of the
resultant elementary linear current is
Ii
(38)
dm = I n d a ,
where n is the positive normal to the
infinitesimal plane area da. The elementary moment depends on the current
and the area enclosed by the circuit, not
upon its particular form.
FIG. 42.-Resolution
of the
current I about the contour C
Any linear current of arbitrary size into elementary currents.
and c o n f i g u r a t i o n can be resolved
into a system of elementary cmrcnts. Let S, Fig. 42, be any regular
surface spanning the circuit contour C and draw a network of intersecting
lines terminating on C. Imagine now that about the contour of each
elementary area there is a current I equal in magnitude to the current
in C and circulating in the samc sense. The magnetic field due to this
network of currents is everywhere identical with that of the current I in
the simple contour C ; for the currents along the common boundaries of
adjacent surface elements cancel one another except on the outer contour.
As the fineness of the division is increased, the field of each mesh current
approaches more closely that of a dipole whose axis is oriented in the
direction of the positive normal. Clearly in the limit the field of the
238
T H E MAGNETOSTATIC FIELD
[CHAP.
Iv
network, and consequently that of the current I about C, is identical
with that of a dipole distribution, or double layer, over the surface S.
The density, or moment per unit area, of the equivalent surface distribution is constant and equal to
dm
2 = - = In.
(39)
da
In effect every linear current acts then as a magnetized shell. The
scalar potential of such a shell has been determined in Sec. 3.16 in connection with double layers of charge.
4*(x’, y’,
2’) =
- 4a
q . d Q =
-
where 0 is the solid angle from (x’,y‘, 2’) subtended by S. Moreover the
potential of a double layer has been shown to bc multivalued. The
discontinuity suffered by +* as one traverses X is by (17), page 190,
(41)
:
4
-
:4
=r =
I;
consequently, the line integral of H about any closed path piercing
single time is
(42)
$H-ds
=
§ d+*
-
Xa
=I.
The surface S bearing the double layer or magnetic shcll is completely
arbitrary and evidently equivalcntj mathematically to the barrier or
‘r c ut ” by means of which the field was reduced in Sec. 4.2 to a simply
connected region.
A DIGRESSION ON UNITS AND DIMENSIONS
4.8. Fundamental Systems.-In
Sec. 1.8 a dimensional analysis of
electromagnetic quantities was based directly on Maxwell’s equations
and the adoption of the m.k.s. units was advocated on the grounds that
they constitute a system which is both “absolute” and practical. Historically the development of the various systems followed another course.
As long as electrostatic and magnetostatic phenomena were considered
to be wholly independent of one another, it was natural that two independent, absolute systems should exist for the measurement of electric
and magnetic quantities. Discovery of the law of induction by Faraday
and proof much later that current is no more than charge in motion established a connection between the two groups of phenomena and imposed
certain conditions on the choice of otherwise arbitrary constants.
The units and dimensions of electrostatic quantities have been commonly based on Coulomb’s law: the force between two charges is pro-
SEC.4.81
FUNDAMENTAL SYSTEMS OF U N I T S
239
portional t o the product of their magnitudes and inversely proportional
to the square of the distance between them.
Since the presence of matter docs not affect the dimensions of charge,
we assume henceforth that all forres are measured in free space. I n
“rationalized” units the proportionality constant k, is chosen equal to
1/4m0; in “unrationalieed” systems the factor 47r is dropped. The
constant is arbitrary and not necessarily dimensionless, so that the
dimensions of charge can be expressed only in terms of eO. From (1)
we obtain
(2)
[p] =
E”;M+LIT-’;
from the equation of continuity it is evident that the dimensions of
current are those of charge divided by time.
(3)
[ I ] = €,:M*LaT-2.
From these it is a simple matter t o deduce the dimensions of all other
purely electrical quantities.
The properties of magnetic mattcr can be described more naturally
in terms of magnetic moment m than by the introduction of fictitious
“magnetic charges.” The torque exerted on a magnet or current loop
of moment m2 in the field B produced by a source of moment ml is
(4)
T
=
mz x B,
and the vector B is expressed in terms of the source ml by (31), page 236.
From these two relations one may readily verify that the dimensions
of magnetic moment arc
(5)
[m]= pO-SM+LBT-l.
Equation (5) is a relation deduced directly from an expression for the
mutual torques exerted by two magnetic dipoles, the proper magnetic
counterpart of Coulomb’s law. But this is not the only experimental
law involving magnetic moment. It is a matter of observation t h a t the
moment m of a small loop or coil carrying a current I is proportional to
the current and to the area of thc loop. The constant of proportionality,
which we shall call l/r, is, like €0 and po, arbitrary in size and dimensions
so that m is subject also to the dimensional equation
(6)
[m] = r-’[I]Lz= y-1e0iM+LXT-2.
The relation of magnetic moment t o current and of current to electric
charge thus imposes a condition upon the otherwise arbitrary constants
240
T H E MAGNETOSTATIC FIELD
[CHAP.IV
eo, pa and y. Whatever the choice of these three factors, their dimensions
must be such that (5) and (6) are consistent. Upon equating (5) and
(6), it follows that
The magnitude of the characteristic velocity c must be determined by
experiment. I n Chap. I it appeared as the velocity of propagation of
fields and potentials in free space; now it occurs as a ratio of electric and
magnetic units. To determine c without recourse to a direct measurement of velocity, a quantity is chosen that can be calculated in one system
of units and measured in another. The capacity of a condenser, for
example, can be calculated if its geometrical form is sufficiently simple,
and the electrostatic capacity of any other condenser may be measured in
terms of this primary standard by means of a capacity bridge. On the
other hand a ballistic galvanometer measures the charge on the condenser in terms of the magnetic effects produced when the condenser is
discharged through the galvanometer. The ratio of the two values of
capacity can be shown to be
where C , is the calculated electrostatic value and C, the measured value
in magnetic units. The accepted value of c is approximately 3 x lo8
meters/sec.
The various dimensional systems which a t one time or another have
been employed in the literature on electromagnetic theory are obtained
by ascribing arbitrary values to any two of the three constants € 0 , po and
yo; the third is then determined by ( 7 ) . We shall review the most
important of these systems briefly.
1. The Electrostatic System.-If
€0 = 1, y = 1, all quantities, both
electric and magnetic, are expressed in electrostatic units. For pa one
obtains from ( 7 ) the value
(9)
po =
+ x 10-20
(sec./cm.)2.
The c.g.s. units in this system are based on a definition of charge from
Coulomb’s law: two unit charges of like sign concentrated at points 1 cm.
distant repel one another with a force of 1 dyne.
2. The electromagnetic system is obtained by putting
(10)
po =
1,
y = 1,
co =
&
X
The ratio of charge measured in the two systems is
(sec./cm.)z.
SEC.4.91
24 1
COULOMB’S LAW FOR MAGNETIC M A T T E R
where qe is the magnitude of a charge measurcd in electrostatic units,
qm the magnitude of the same charge measured in e.m.u.
3. The m.k.s. system, described in detail in Sec. 1.8, is essentially an
electromagnetic system in which the units of mass and length have been
adjusted such that the units of electric and magnetic quantities conform
to practical standards. I n the form adopted in the present volume, the
coulomb is introduced as an independent unit, leading to
(12)
po =
y = 1,
4.rr X lo-’ henry/ineter,
to
=
8.854 X 10-12 farad/meter.
An alternative form of the m.k.s. system puts
(13)
PO =
4~ X lo-’,
y =
1,
eo =
8.854 X
(sec./meter)2,
which is dimensionally identical with the elcctrornagnctic system.
4. The Gaussian system has until recently becn most commonly
employed in scientific literature. It is a mixture of the electrostatic
and electromagnetic systems obtained by putting
(14)
EO =
1,
po =
1,
y = c.
The factors e0, po drop out of all equations and electrical quantities are
expressed in electrostatic units, magnetic quantities in magnetic units.
The advantages of all this are rather dubious. It is fully as difficult to
recall the proper position of the factor c entering into electromagnctic
equations written in Gaussian units as i t is to retain the factors E,, and po
from the outset. The m.k.s. system is quitc as much a n “absolute” or
“scientific” system as the Gaussian, and there are few t o whom the terms
of the c.g.s. system occur more readily than volts, ohms, and amperes.
All in all i t would seem that the sooner the old favorite is discarded, t h r
sooner will a n end be made to a wholly unnecessary source of confusion.
Any equation appearing in this book can be transformcd to Gaussian
units by replacing € 0 and po each by l/c; J and p by J/c and p / c
respectively.
4.9. Coulomb’s Law for Magnetic Matter.-In
the older literature
the magnetic system of units was commonly based on a law of force
between magnetic “polcs.~’ The “charge” or pole strength q* was
defined such that the product of q* and the length of a dipole was equal to
its moment m. The dimensions of q* based o n (5) are then
while for the law of force corresponding to Coulomb’s law we obtain
242
THE MAGNETOSTATIC FIELD
[CHAP.IV
This expression differs from the onc commonly assumed in t h a t po
appears in the numerator rather than the denominator.’ At the root of
this puzzling result is the fact t h a t the force on an element of current,
as deduced from Maxwell’s equations, Sec. 2.4, is J X B, not J X H.
Consequently the torque on a dipole is m x B and the force on a magnetic
“charge” is F = p*B.
To make (16) eonform strictly t o the electrostatic model it has been
customary to write the factor 1.1 in the denominator and t o define the
vector H by the equation F = q*H. This procedure, however, leads t o
dimensional inconsistencies in the field equations unless 1.10 is assumed to
be a numeric and there is little t o be said in its favor. The parallel
nature of the vectors E and B and the vectors D and H has been recognized by most recent writers.
MAGNETIC POLARIZATION
4.10. Equivalent Current Distributions.-The equivalence of a magnetized body and a distribution of current has been noted previously
on various occasions. Let us see just how this distribution in a body of
volume V bounded by a surface S is to be determined.
We shall suppose that the intensity of magnetization, or polarization,
M includes also the permanent or residual magnetization Mo if any is
present. At any point outside the body the vector potential is by ( 3 7 ) ,
page 237, equal to
By virtue of the vector identity
and the fundamental formula for the transformation of the volume
integral of the curl of a vector to the surface integral of its tangential
component,
(3)
Jv
vx
(M) dv
=
J M+
da,
we see that the right-hand side of (1) may be resolved into two parts.
1 SOMMERFELD,
“Ueber die elektromagnetischcn Einheiten,” p. 161, Zeeman, Verhandelingen, Martinus Nijhoff, The Hague, 1935.
SEC.4.111
FIELD OF M A G N E T I Z E D RODS A N D SPHERES
243
The vector potential due to a magnetized body is exactly the same as
would be produced by volume and surface currents whose densities are
J’=VXM,
(5)
K’=Mxn.
The validity of this result a t interior as well as exterior points is
easily demonstrated. For the surface integral certainly converges a t
all points not on S, and the volume integral can be resolved into thrcc
scalar components, each of which is equivalent to the scalar potential
of a charge distribution.
We recall that the magnetic polarization was introduced initially,
Sec. 1.6, as the difference of the vectors B and H.
1
Mz-B-H.
PO
At a surface of discontinuity between two magnetic media the field
vectors satisfy the boundary conditions of SCC.1.13; namely,
n (Bz- BI)
(7)
=
n
0,
x
(Hz - HI) = K,
in which the surface current density is zero whenever the conductivity
is finite. Then by (6) these conditions are equivalent to
(8)
.
n (Bz- BI)= 0,
n
x (Bz- Bl)= po(K + K’),
where
(9)
K’
=
(MI - Mz) x n.
4.11. Field of Magnetized Rods and Spheres.-A steel rod or cylinder
of circular cross section uniformly magnetized in the axial direction will
serve as a n example. The length of the cylinder is I, its radius a,and
the magnetization is MO. Since Mo is assumed constant, thc volume
density of equivalent current J’ is zero. On the cylindrical wall the
current density is K’ = Mo x n, so th at the magnetized rod is equivalent
to a solenoidal coil of the same dimensions. The vector potential and
field a t any internal or external point can now be calculated from the
series expansion derived in See. 4.5. The value of B along the axis has
already been computed in Sec. 4.4. In particular, we find a t the center
For an actual magnet the validity of our assumption of uniform
magnetization is dubious, particularly in the vicinity of the edges. If,
however, the length of the cylinder is either very large or very small with
244
T H E MAGNETOSTATIC F I E L D
[CHAP.I V
respect to the radius, the approximation is usually a good one. Thus, if
1 >> a so that the rod is needle-shapcd, the field at the center reduces to
B
(11)
= poMo.
l n case 1 << a, i t degenerat,es to a disk, with
B
( 12)
=
0.
If the cylinder is soft iron instead of steel in an external field Ho,
the permanent polarization M, must be replaced by the induccd magnetization M and (10) is to be interpreted as the induced field B1. At the
center of a long needle in a parallel field Ho, the resultant fields are
(13)
B
= Bo
+ B1 = Bo + poM;
H
1
= -13
PO
-M
=
H0 ;
the field H in the interior of the needle is the same as that which existed
initially. In the case of the disk
B
(14)
=
Bo,
H = Ho - M.
The assumption of a uniform magnetization of a soft-iron cylinder
with squarely cut ends is obviously a dircct violation of the boundary
conditions on the vectors B and H. On the other hand the induced
magnetization of a sphere or ellipsoid in a parallel external field must be
uniform, for the problem is wholly analogous to the electrostatic case
considered in Sees. 3.24 and 3.27. The induccd or secondary field B1
can now be attributed to a surface current circulating in zones concentric
with the axis of magnetization, but the density will evidently vary with the
latitude. If 0 is the angle made by the outward normal with the direction
of M,
K’
(15)
=
/ M X nl = M sin 8.
By the Biot-Savart law the component of BY in the direction of M is
When integrated this leads to
for the induced field a t the center, while for the total field one obtains
(18)
B-
=
Bo
+ By = po(Ho + QM),
H- = Ho - QM.
.an important corollary to this result is the value of the field at the
center of a spherical cavity cut out of a rigidly and unijormly magnetized
SEC.4.121
245
SURFACE DISTRIBUTIONS OF CURRENT
medium. The surface current distribution is the same but, since the
normal is now directed into the sphere, the circulation is in the opposite
direction. The induced field a t the center is -$poMO, and the resultant
field is
+
+
~ M o ) , H- = Ho +Mo,
(19) B- = Bo - 3poMo = po(Ho
where Ho and Bo are the initial fields within the medium. Obviously
these fields do not satisfy the customary boundary conditions on the
normal and tangential components, and thry are possible only when the
cavity is excised after the rigid magnetization of the material.
DISCONTINUITIES OF THE VECTORS A AND B
4.12. Surface Distributions of Current.-It follows directly from the
investigation of Sec. 3.14 that the vectors A and B are everywhere continuous functions of position provided the current is distributed in volume with a density which is bounded and piecewise continuous. When,
however, the conductivity of a body is assumed infinite, or the magnetized
state of the body is to be represented in terms of an equivalent current
distribution, it becomes necessary t o take account of surface currents
across which the transition of the vector B is discontinuous. Let us
imagine a current with a finite volume density J confined to a lamellar
region of thickness t. Suppose now that the thickness is allowed t o
shrink, but that throughout the process the total current traversing its
cross section is maintained at a constant value. As t -+0, the volume
density J necessarily approaches infinity. The surface current density is
defined as the limit of a product:
(1)
K
=
lim t J ,
as
t -+ 0, J + 00.
Because of the infinite value of J, this distribution constitutes a surface
singularity.
We shall assume that K is a bounded, piecewise continuous function
of position on a regular surface X. The vector potential a t any point
(d,y’, 2’) not on S is
in which r is, as usual, drawn from the element K(z, y, z ) du t,o the fixed
point (z’, y’, 2’). Each rectangular component of A is an integral of the
form representing the electrostatic potential of a surface distribution of
charge. I n Sec. 3.15 i t was shown that such integrals are bounded, continuous functions of the coordinates (d,y’, z’) a t all point,s in space,
including those lying on S. Both the normal and tangential components oj
the vector potential pass continuously through a single surface current layer.
246
T H E MAGNETOSTATIC F I E L D
[CHAP.IV
Turning our attention now to t,he vector B, we have according to
Eq. (ll),page 231, at a point not on S
Upon resolving K into its rectangular components,
+
K = iK, jK,
it appears that (3) is equivalent t,o
+ kK,,
Now an integral of thc type
may be interpreted, apart from a dimensional factor c0, as the field due to
a charge of density w spread over the surface X; and, as was shown in
Stc. 3.15, it exhibits a discontiriuitjy as the point (d,
y’, z’) traverses the
surface equal to
E+ - E-
(6)
=
wn,
where w is the local value of the surface density a t the point of transition.
Upon applying this result to (4)it follows immediately that
B, - B-
(7)
=
poK x n.
T h e normal component of B passes continuously through a surface layer of
current.
-
n (B, - B-)
(8)
=
0.
T h e tangential components o j B experience a discontinuous j u m p dejined by
(9)
n
x (B, - B-)
=
pon x (K x n).
The right-hand side, by a well-known identity, transforms to
(10)
n
x (K x n)
=
(n n)K - (n - K)n.
Since, however, K lies n the surface and is hence orthogonal to the unit
normal n, (9) reduces to
(11)
n x (B, - B-) = poK.
The significance of this analysis is made clearer by comparison with
the boundary conditions deduced directly from Maxwell’s equations in
S U R F A C E D I S T R I B U T I O N S OF M A G N E T I P A 1 0 1/BN?’
SEC.4.131
247
See. 1.13. According to Sec. 4.10 a magnetized medium is equivalent t o
a volume distribution of current of density J’ = V x M, but discontinuities in the magnetization must be accountcd for by surface layers of
current whose density is K’ = n x (M+- M-). The discontinuity in
the field across such a surface is
n x (B+ - B-)
(12)
1
or, since - B - M
PO
(13)
=
= pdK
3- n x (M+ - M-)I,
H,
n
x (H, - H-) = I(.
The continuity of the normal comporicnts of B across the boundary is
ensured by (8).
4.13. Surface Distributions of Magnetic Moment.-The equivalence
of a linear circuit and a surface bearing a magnetic dipole distribution was
demonstrated in Sec. 4.7. We shall give some further attention to the
discontinuities exhibited by the potential and field in the neighborhood of
such singular surfaces without restriction t o the case of normal magnetization. The vector M shall now represent the surface density of magnetization, the magnetic moment of the surface S per unit arca. The direction
of M with respect t o the positive normal n is arbitrary.
At any point not on X the vector potential of the distribution is by
( 3 7 ) , page 237, equal to
We need but not,c the similarity of form which this expression bears to
Eq. (3) t o conclude that the vector potential experienccs a discontinuity
on passing through a surface boaring a distribution of magnetic moment,
the amount of the discontinuity being
(15)
A+ - A-
=
poM x n.
The normal component of A is continuous,
(16)
-
n (A+ - A_)
=
0,
but the tangential component i s continuous only in case the magnetization i s
normal to the surface. Otherwise
(17)
n x (A+ - A_)
=
-
poM - pc(n M)n.
To investigate the discontinuity exhibited by the magnetic vector B
as it passes through a surface layer of magnetic moment i t will be advantageous to employ an expression of the form of Eq. (31), page 236, which
we shall now write as
248
T H E MAGNETOSTATIC FIELD
[CHAP.I V
The surface polarization M may be resolved into a normal and a tangential component, by means of the identity
M = ( n . M)n
(19)
so that in place of (18) onc has
(20) B(x’, y’,
2’)
=
+ (n x
-
(n M)n v
-E V‘
4n
x n,
M)
1
- da
r
On comparison with (16), page 189, it is clcar that the first integral may
be interpreted as the scalar potential due t o a double layer of moment
T = (M n)n, while the ncgativc gradient of this potential a t the point
(x’,y’, 2’) gives the field intensity. Now it was shown in Scc. 3.16 th a t
the normal components of the field intensity approach the same limit on
either side of such a distribution, but that the tangential components
cross it discontinuously as expressed by Eq. (21), page 191, the amount of
the jump in the present case being
-
B L+- B,-
(21)
-
d
(M n).
dl
= -po -
Thus the component of B along any direction tangent to a surface bearing
a distribution of magnetic moment expcriences a discontinuity on traversing the surface which is proportional to the rate of change of the normal
component of the polarization in th at same direction.
Turning our attention next to the contribution of the second integral
in (20), we reduce it to a more familiar form by certain transformations of
the integrand. The identity
1
[(n x M) x n] .V ;= (n x M)
(22)
-
can easily be verified. Next let us resolve (n x M) into scalar comjf2
kf3. Wc have then
ponents, so that n x M = ifl
+ +
(23) [(n x M)
x
n]
-
SEC.4.131
249
SURFACE DISTRIBlJTIONS OF MAGNETIC M O M E N T
Now if j ( x , y, z ) be any function which is continuous and has continuous
first and second derivatives over a regular surface S and on its contour
C , then
l n x Vjda
=
ifds.
It follows, therefore, th at if the tangential components of M satisfy the
prescribed conditions over S , the three surface integrals whose integrands
are of the type i
can be transformed to integrals along any
*
closed contour on S , so that
(25)
i
-
(n x
):V
da
+j -
(n )
x:V
da
The remaining three terms of the integrand may be reduced by the
relation
n
(i x V j l
r
= -
-
+ j x vf2+ k x vf3)
=
-
n - V X (n
x M)
r
The second term on the right-hand side of (20) has, therefore, been
resolved into a contour integral plus a surface integral, or
(27)
-:
V’
[(n x M) x n]
- v r-1 da
=
- l 47r
o V’ Jn<y
- ds
Now the contour integral is continuous and continuously diff erentiable at points on S which are interior to the contour. On the other
hand, the last term of (27) might be interpreted as the field intensity a t
a point (z’, y’, 2’) due t o a surface charge distribution of density
-n.
vx
(n x M).
Consequently, one must conclude that the tangential component of
surface magnetization leads to a discontinuity in the normal components
of B a t the surface S of amount
(28)
Bn+
-
Bn-= - p 0 n . v x (n x M).
250
[CHAP.IV
T H E MAGNETOSTATIC FIELD
Differentiation is restricted to directions tangent to the surface. Consequently the vector v(M n) lies in the surface, while the vector
v x (n x M) is normal to it. T h e discontinuities exhibited by the vector
B in its transition through a surface distribution of magnetic moment can
be expressed by the single formula:
-
(29)
B+ - B-
=
-- d V ( n
- M ) + v x (n x M)I.
INTEGRATION OF THE EQUATION V X V X A =
pJ
4.14. Vector Analogue of Green’s Theorem.-The classical treatment
of the vector potential is based on a rosolution into rectangular components. On the assumption that v A = 0, each component can be
shown t o satisfy Poisson’s equation and the methods developed for the
analysis of the electrostatic potential are applicable.
The possibility of integrating the equation V x V x A = pJ directly
by means of a set of vector identitics wholly analogous to those of Green
for scalar functions appears to have been overlooked. Let V be a closed
region of space bounded b y a regular surface S , and let P and Q be two
vector functions of position which together with their first and second
derivatives are continuous throughout V and on the surface S. Then,
if the divergencc theorem be applied to the vector P x V x Q, we have
-
(1)
J v v - ( P xV x Q ) d v = ~ ( P X V x Q ) . n d a .
Upon expanding the intcgrand of the volume integral one obtains the
vector analogue of Green’s first identity, page 165,
(2) ~ ~ ( V X P . V X Q - P ~ V X V X Q ) d v
=
(P x
vx
Q) Onda.
The analogue of Green’s second identity is obtained by an interchange
of the roles of P and Q in ( 2 ) followed by subtraction from ( 2 ) . As a
result
(3)
(Q-v x
vx
P - P - V x V x Q) dv
=
4
(P x
v x Q -Q x v x
P).nda.
4.16. Application to the Vector Potential.-We shall assume th a t the
volume density of current J(z, y, z ) is a bounded but otherwise arbitrary
function of position. The regular surface S bounding a volume V need
not necessarily contain within it the entire source distribution, or even
any part of it. As in Sec. 3.4 we shall choose 0 as a n arbitrary origin
and x = x’, y = y‘, x = z’ as a fixed point within V .
SEC.4.151
251
APPLICATION TO T H E VECTOR POTENTIAL
Now let P represent the vector potential A subject to the conditions
VXVXA=wJ,
(4)
V.A=O,
where p is the permeability of a medium assumed homogeneous and isotropic. The choice of Q may be made in either of two ways. It will
be recalled that in the scalar case the Green’s function $ satisfied Laplace’s
equation v2#= 0 and could be interpreted as the potential a t (z’, y’, z’)
due to a charge 47rc located a t (z, 1 ~ ,z). Likewise the vectorial Green’s
function Q may be chosen to represent the vector potential a t (d,y’, z’)
produced by a current of density 47r/p located a t (2,y, x ) and directed
arbitrarily along a line determined by the unit vector a.
( 5 ) Q(s, y,
Z ; z’, y’, 2 ’ ) =
5,
r
r
= d(2’- 2)Z
+ (y’ - y)Z +
(2’
- 2)“
However the divergence of this function is not zero and consequently
( 5 ) fails to satisfy the condition V x V x Q = 0. On the other hand,
the vector potential arising from a distribution of magnetic moment has
been shown t o satisfy V x V x A = 0 everywhere and an appropriate
Green’s function for the present problem is, therefore,
Obviously (6) may be interpreted as the vector potential of a magnetic
47r
dipole of moment - a.
I.1
Either (5) or (6) may be applied to the integration of (4)but, the
necessary transformations turn out to be simpler in the case of (5) in spite
of the divergence trouble. We have in fact
(7)
(8)
V x Q = V
xa,
V x v x Q = v
P . V x v x Q = A . V
I n these transformations and those that follow it is to be kept in mind t h a t
a is a constant vector. The left-hand side of the identity (3) can now
be written
=a.
’
(/IIdv - a - 1( A . (k)
n)V
da.
252
[CHAP.IV
T H E MAGNETOSTATIC FIELD
Proceeding to the transformation of the surface integrals, we have
(10)
(P x
vx
x [v(:)
~ > . =n {A
x a]>.n
=
(Q X v x P ) . n =
(11)
in which B replaces
(12) p J
1dv =
V r
v x A.
S
a-v
(9
x ( A X n),
(: x V x A ) - n = a - - 9B xr n
The identity (3) becomes
(A n)v
(i)+ v (+)x
da
(A
x n) da
+
*n
r
da.
Now the validity of this relation has been established only for regions
within which both P and Q are continuous and possess continuous first
and second derivatives. Q, however, has a singularity at r = 0 and
y', 2') a
consequently this point must be excluded. About the point (d,
small sphere of radius r1 is circumscribed. The volume V is now bounded
by the surface 81 of the sphere and an outer enveloping surface S as
indicated in Fig. 25, page 166. Since V(l/r) = ro/r2, the surface integrals
over S1 may be written
L
L L l r o ( A . n ) d a + T, I
r o x(Axn)da+r:
The integrand of the middle term is transformed to
(13)
ro X (A
x n)
=
-
-
(ro n)A - (A n)rO
and since on the sphere roe n
over SI reduce to
=
L
nxBdu.
+ A x (ro X n),
1, ro x n = 0, the surface integrals
If A and n x B denote the mean values of the vectors A and n X B over
the surface of the sphere, these integrals have the value
Upon introducing
which in the limit as rl + 0 reduces to 4aA(x', y', 2').
this result into (12) and transposing, we find the value of the vector
potential at any fixed point expressed in terms of a volume integral
SEC.4.151
APPLICATION TO T H E VECTOR POTENTIAL
253
and of surface integrals over an outer boundary which is again denoted
by S.
The proof that the divcrgcnrc of (14) a t the point (d,
y', 2') is zero
is lcft to the reader. Clearly the surface integrals represent the contribution to the vector potential of all sources that are exterior t o the
surface S. At all points with n V , the vector A(z', y', 2 ' ) defined by (14)
is continuous and has continuous derivatives of all orders. Across the
surface X,however, it is apparent from the form of the surface integrals
t h a t A and its derivatives will exhibit certain discontinuities. We shall
show in fact that outside S the vector A is zero everywhere.
The first surface integral in (14) may be interpreted as the contribution t o the vector potential of a surface current
K
(15)
=
1
--n
P
x B-,
in which the subscript of B- emphasizes t h a t this value of B is taken
just inside the surface S. Now in Sec. 4.12 it was shown that a surface
layer of current does not affect the transition of the vector potential, but
gives rise to a discontinuity in B of amount
(16)
n x (B, - B-)
= pK.
Upon replacing I( by its value from (15) it is clear t h a t just outside S
nxB+=O
(17)
I n like manner the second surface integral is equivalent to the vector
potential of a distribution of magnetic surface polarization of density
M = - 1A - x n.
(18)
P
The normal component of A passes continuously through such a layer,
but the tangential component is reduced discontinuously to zero, as we
see on substituting (18) into (17), page 247
(19)
n x (A+ - A_)
=
-n x A-
+ [n - (n x A-)]n,
whence
(20)
n x A+
= 0.
254
T H E MAGNETOSTATIC FIELD
[CHAP.I V
The mathematical significance of the last term of (14) is apparent,
but it is difficult to imagine a physical distribution of current or magnetic
moment of the type called for. This is the form of integral which we
have associated with the field intensity of a surface charge of density
(A- n), and which leads to a discontinuity in the normal component
specified by
-
-
n (A+ - A_) = n A_,
(21)
as a consequence of which we conclude that
n.A+
(22)
=
0.
Thus far we have demonstrated that on the positive side of the closed
surface S the tangential and normal components of A and the tangential
component of B are everywhere zero. It follows a t once, however, that
the normal component of B must also vanish over the positive side of S ;
for the normal component of the curl A involves only partial derivatives in
directions tangential to the surface. Furthermore, we need but apply
(14) itself to the region external to S to prove that A, and consequently
B, must vanish everywhere. Current and magnetic polarization are
absent outside V , since their effect is represented by the surface integrals.
Then, since n x B+, n x A+ and n A+ are all zero, it follows from (14)
that A ( d , y’, 2’) must vanish at all points outside S.
When Q = v ( l / r ) x a is chosen in place of ( 5 ) as a Green’s function,
it can be shown without great difficulty that
.
(23) B(z’, g’, 2 ’ ) = .E
47r
sv
J x V
(:)
dv
-
& Ss (n x B) x v (:)
-471.!
Ss-(n (:)
B)V
da
da.
This is the extension of the Biot-Savart law to a region of finite extent
bounded by a surface X. The contribution of currents or magnetic
matter outside S to the field within is accounted for by the two surface
integrals.
BOUNDARY-VALUE PROBLEMS
4.16. Formulation of the Magnetostatic Problem.-A homogeneous,
isotropic body is introduced into the constant field of a fixed and specified
system of currents or permanent magnets. Our problem is to determine
the resultant field both inside and outside the body. In case the current
density at all points within the body is zero, the secondary field arising
from the induced magnetization can be represented everywhere by a
single-valued scalar potential 4: and the methods developed for the
SEC.4.161
FORMULATION OF T H E M A G N E T O S T A T I C PROBLEM
255
treatment of electrostatic problems apply in full. A schedule for the
solution may be drawn up as follows.
The scalar potential of the primary field is 4:.
I n case the primary
source is a current distribution 4; is multivalued but this in no way
affects the determination of the induced field @. The resultant potential
4:. The permeability of the body will be denoted by p1
is &* = 4:
and that of the homogeneous medium in which it is embedded by p2.
Thcn a function 4: must be constructed such that:
+
vZ+T= 0, at
all points not o n the boundary;
4: i s jinite and continuous everywhere including the boundary;
Across the boundary the normal dcrivatives of the resultant potential
4* satisfy the condition
p2
($)+
-
kLl(””>
an - = 0,
the subscripts iand - implying that the derivative i s calculated
outside or inside the boundary surface respectively. T h e induced
potential 4: i s , therefore, subject to the condition
in which f i s a known function of position o n the boundary satisfying
the condition
J,j d a = 0 ;
At infinity &? must vanish at least as l/r2, so that rz4T remains
finite as r-+ 00, f o r there i s no free magnetic charge and consequently 4: must vanish as the potential of a dipole or multipole of
higher order.
In case the body carries a current, the interior field cannot be represented by a scalar potential and the boundary-value problem must be
Solved in terms of a vector potential. Such a case arises, for example,
when a n iron wire carrying a current is introduced into an external magnetic field. The distribution of the current in the stationary state is
unaffected by the magnetic field. Its determination is in fact an electrostatic problem. The vector potential of the primary sources is Aa while
the potential of the induced and permanent magnetization of the body
and of the current which it may carry will be denoted b y A1. This
function A1 is subject to the following conditions:
(1) V X V X A1
density i s J;
= plJ,
at points inside the body where the current
256
T H E M A G N E T O S T A T I C FIELD
CHAP. I V
(2) V X V X A1 = 0, at all other points not o n the boundary S ;
(3) V A1 = 0, at all points not on S ;
(4) A1 i s finite and continuous euerywhere and passes continuously
through the boundary surface (Secs. 4.10 and 4.12);
( 5 ) Across the boundary the normal derivatives of the potentzal Al-as
well as those of the total potential A-satisfy
-
i n which n i s the outward normal and where M- i s the polarization
of the body and M+ that of the medium j u s t outside the boundary.
Since M+ and M- are determined a t least in part by the field itself this
relation is usually of no assistance in the determination of A1. I n its
place the customary boundary condition on the tangential components
of the total field must be applied.
B+ Lz B0, which imposes a rela(
)
tion between the derivatives A in a specified coordinate system;
(6) n x (H+ - H-)
=
n x
-
=
of
(7) A s r -+
00
the product r A l remains finite.
4.17. Uniqueness of Solution.-The proof that there is only one function 4: satisfying the conditions scheduled above was presented in
Sec. 3.20. A corresponding uniqueness theorem for the vector potential
may be deduced from the identity ( 2 ) of page 250. Let us put
P = Q = A and assume first t h a t within V l bounded by S the current
density is zero. Then V x V x A = 0 and
svl
(V x A)2dv
= -
ss
A (n x V x A) da.
From the essentially positive character of the integrand on the left it
follows that if A is zero over the surface S, then B = V x A is zero
everywhere within the volume V1. Hence A is either constant or a t
most equal to the gradient of some scalar .)I But since A is zero on S ,
t,he normal derivative d + / d n is also zero over this surface and it was
shown in Sec. 3.20 t h a t this cnndition entails a constant value of 1c.
throughout V1. Consequently, if A vanishes over a closed surface, it
vanishes also a t every point of the interior volume. It is also clear that
the vector function A is uniquely dekrmined in 8 1 by its values on S.
For if there existed two vectors A1 and A:! which assumed the specified
values over the boundary, their difference must vanish not only over S
but also throughout 8 1 .
The condition v x A = 0, A = v+, throughout 'Irl can be established also by the vanishing of V x A, or of the tangential vector
SEC.4.181
FIELD OF A UNIFORMLY MAGNETIZED ELLIPSOID
257
n X v X A over S. I n this case, however, it does not necessarily follow
that A is everywhere zero. If the two functions v x A1 = B1 and
V x A2 = Bz are identical on S, then B1 and B, are identical at all
interior points and A1 and Az can differ a t most by the gradient of a
scalar function.
I n case there are currents present within V1 the vector potential A
is resolved into a part A' due t o these currents and a part A" due t o
external sources. The vector A' is uniquely determined by the current
distribution, while the values of A" or its curl over S determine A" at
all interior points.
The vector potential is regular at infinity and consequently the proof
applies directly to tjhe region V z exterior to S. The vector B i s uniquely
determined within a n y domain by the values of its tangential component
n X B over the boundary.
PROBLEM OF THE ELLIPSOID
4.18. Field of a Uniformly Magnetized Ellipsoid.-An ellipsoid whose
semiprincipal axes are a, b, c is uniformly and permanently magnetized.
The direction of magnetization is arbitrary, but since the magnetization
vector can be resolved into three components parallel to the principal
axes we need considcr only the case in which Mo is constant and parallel
t o the a-axis.
I n view of the uniformity of magnetization p * = -V Mo = 0 a t
all points inside the ellipsoid. The potential 4* of the magnet is due t o a
"surface charge'' of density w* = n . M o . The external medium is
assumed in this case to be empty space. The problem is now fully
equivalent t o t h a t of the polarized dielectric ellipsoid treated in Sec.
3.27. From Eqs. (27), (32), and (42) the potential 41 due to the polarization P, may be found in terms of P, and the parameters of the ellipsoid.
Upon dropping the factor e 0 and replacing P, by Moz one obtains
-
as the magnetic scalar potential a t points inside the ellipsoid; a t all
external points
The field inside the ellipsoid is
(3)
258
TIIE MAGNETOSTATIC FIELD
[CHAP.IV
Consequently if the magnetization is parallel to a principal axis, the field
H inside the ellipsoid is constant, parallel to the same axis, but opposed
to the direction of magnetization.
4.19. Magnetic Ellipsoid in a Parallel Field.-An ellipsoid of homogeneous, isotropic material is introduced into a fixed and uniform parallel
field Ho. The ellipsoid is free of residual magnetism and we shall suppose
that for sufficiently weak fields its permcability p1 is constant, as is also
the permeability p z of the external medium.
The determination of the field can again be adopted directly from the
electrostatic case, for upon reference to Sec. 4.16 it will be noted that
the boundary conditions imposed upon $I* are identical with those to be
satisfied by 4 when p is substituted for E . The resultant magnetic
potential at an interior point of the ellipsoid is, therefore,
outside the ellipsoid (b: is obtained from (38), page 213, by replacing
by P and adding the contributions induced by H w and Ho,.
The torque exerted by the field on the ellipsoid may likewise be
determined directly from (54), page 216, after a similar substitution.
The importance of these results is fundamental. There are experimental
methods of measuring small torques with great precision. The magnetic
field near the center of a long solenoid is very nearly parallel and uniform,
and its intensity in terms of the current can be calculated. Thus the
permeability or susceptibility of the test sample can be determined with
great accuracy.
E
CYLINDER IN A PARALLEL FIELD
4.20. Calculation of the Field.-As
an example of the application of
the vector potential, let us consider the following simple problem. A
cylinder or wire of circular cross section and permeability p1 is embedded
in a medium of permeability pz. The wire is infinitely long, has a radius
a, and carries a current I . The external field Bo is directed transverse
t o the axis of the wire and is everywhere parallel and uniform. We
wish to calculate first the resultant field at points inside and outside
the cylinder.
The x-axis of the reference system will be chosen parallel to the
vector Ba, while the z-axis is made to coincide with the axis of the cylinder
SEC.4.201
CALCULATION OF T H E FIELD
259
and the direction of the current, Fig. 43. A vector potential is to be
found which satisfies the conditions
The current density has only a z-component.
(3)
The vector potential may be resolved into two parts: the potential A0
of the applied field, and the secondary potential A1 due in part to the
FIG.43.-Cylinder
in a uniform magnetostatic field.
current I, in part to the induced magnetization. Now clearly Bo can
be derived from a vector potential directed along the z-axis. If ia is
the unit vector in the positive z-direction, then
(4)
A. = iaBoy= i3BOrsin 8.
Moreover Al is also oriented along this same axis, for the magnetization
can be represented by equivalent volume and surface currents. Certainly the induced magnetization lies in the transverse plane and consequently the equivalent currents J' = V x M Iand
K' = n x (M2 - M1)
of Sec. 4.10 are directed along the cylinder. Since the vector potential
is always parallel to the current, A1, = A,, = 0. The expmsion of (2)
is now a simple matter. From (85), page 50, we have
(5)
in which the subscript z has been dropped.
A general solution must first be found for the homogeneous equation
obtained by putting the right-hand member of (5) equal to zero. The
separation of such an equation was discussed in Sec. 3.21, and since the
vector potential is a single-valued function of 8 we write down at once
260
T H E MAGNETOSTATIC F I E L D
(6) A' =
2 (a, cos
n0
n=O
[CHAP.IV
+ b, sin n0)r" + 2 (cIpcos no + d, cos ne)r-".
n=O
Moreover the vector potential must be finite everywhere in space and
consequently inside the cylinder th e coefficients c, and d, are zero.
The general solution of ( 5 ) is obtained by the usual method of adding
to (6) any particular solution of (5). Since r2J depends only on r, we
shall assume that this particular solution A" is independent of 8.
Consequently a t all interior points of the cylinder the resultant vector
potential is
A - = -I
' I 1:
(9 1
47T
+
3
(a,cos no
+ b, sin ne)rn.
n=O
Since the vector potential of the induced magnetization is regular a t
infinity, outside the wire we put a, = 0, = 0 in ( 6 ) . To this must be
added the vector potential of the current I , which in Eq. (ZO), page 233,
was shown to be p z I / 2 a In l / r , and lastly the contribution of the external
field itself from (4). The resultant vcctor potential a t any point outside
the cylinder is, therefore,
(10)
A+
=
Bor sin 0
4- 27T
r
(c, cos n0
+ d, sin n0)r-".
n=O
Next, the coefficients of the series expansions are determined from the
condition that the vector potential and tangential components of H shall
be continuous across the boundary r = a. From the expansion of the
curl in cylindrical coordinates, page 51, we have
so that the boundary conditions imposed on the vector potential are
(r
=
a).
The result of equating coefficients of like terms from (9) and (10) is then
and all other coefficients zero.
A+
(14)
261
FORCE E X E R T E D ON T H E C Y L I N D E R
SEC.4.211
A-
=
=
--In27r
P2
a
r
-!!?(?)
- I*J + ( r
47r
2
+ PI
--)P1
+
p2a2
Bo sin e,
(T
> a),
(T
< a).
P2 T
2Pl Bor sin 8,
+.
P1+ P2
By means of (11) the field may be calculated.
H o sin 8,
The nature of the field inside the cylinder is made somewhat clearer by
a transformation to rectangular coordinates.
(17)
H,
=
H , cos 8 - H o sin 8,
H , = H , sin B
+ H s cos 8.
The magnetization of the cylinder is given by the relation
M
- P1 - P o H-.
PO
Evidently the magnetization induced by the applied field Bo is in the
direction of the positive x-axis. Upon this is superposed the magnetization induced by the current I.
4.21. Force Exerted on the Cylinder.-If the medium supporting the
wire is fluid, or a nonmagnetic solid, the force per unit length exerted by
the field can be computed from (15), page 155. Resolved into rectangular components, this becomes
(20)
The rectangular components of the field outside the cylinder are first
calculated from (15) and (17).
262
T H E MAGNETOSTATIC FIELD
H$
(21)
=
H+ -
Ho
P1
P1
[CHAP.IV
I sin 9,
- ma2
+ ~+ Ho cos 29 - 2n-r
I
cos 8,
+- H o sin 29 + 27rr
PI
PZT2
P1
PPa2
P2T2
while H 2 is equal to the sum of the squares of these two components.
Upon int,roducing (21) and (22) into (20) and evaluating the integrals,
one obtains the components of the total force exerted by the field upon
the wire and the current it carries.
Fz
(23)
FU
= 0,
=
p,HoI.
That the induced magnetic moment of the wire contributes nothing to
the resultant force follows from the uniformity of the applied field. If
this field, on the other hand, were generated by the current in a neighboring conductor, the distribution of induced magnetization would no longer
be symmetrical; a force would be exerted on the cylinder even in the
absence of a current I .
Problems
1. An infinitely long, straight conductor is bounded externally by a circular cylinder of radius a and internally by a circular cylinder of radius b. The distance between
centers is c, with a > b
c. The internal cylinder is hollow. The conductor carries
a steady current I uniformly distributed over the cross section. Show that the field
within the cylindrical hole is
+
H =
CI
2a(a2 - b2)
and is directed transverse to the diameter joining the two centers.
2. Two straight, parallel wires of infinite length carry a direct current I in opposite
directions. The conductivity u of the wires is finite. The radius of each wire is a and
the distance between centers is b.
a . Using a bipolar coordinate system find expressions for the electrostatic potential and the transverse and longitudinal components of electric field intensity a t
points inside and outside the conductors.
b. Find expressions for the corresponding components of magnetic field intensity.
c. Discuss the flow of energy in the field.
3. A circular loop of wire of radius a carrying a steady current I lies in the xy-plane
with its center a t the origin. A point in space is located by t h e cylindrical coordinates
PROBLEMS
263
4, z, where x = r cos 4, y = r sin 4, Show that the vector potential at any point
in the field is
T,
= 7rk
-!!, I
(:)'
-
[(I
kz) K - E ] ,
+ +
+ >>
where k2 = 4 a r / [ ( a r ) 2 91, and K and E are completc elliptic integrals of the
first and second kinds.
Show that when (r2 9)): a, this reduces to Eq. (37), p. 237, for the vector
potential of a magnetic dipole.
4. From the expression for the vector potential of a circular loop in Problem 3
show that the components of field intensity are
T
r
6. The field of two coaxial Helmholtz coils of radius a and sepnrated by a distance
a between centers is approximately uniform in a region near the axis and halfway
between them. Assume that each coil has n turns, that the cross section of a coil is
small relative to a, and t h a t each coil carries the same current. From the results of
Problem 4 write down expressions for the longitudinal and radial components of H at
points on the axis and a t any point in a plane normal to the axis and located midway
between the coils. What is the field intensity a t the mid-point on the axis? Find
a n expansion for the longitudinal component H , in powers of z / a valid on the axis in
the neighborhood of the mid-point, and corresponding expressions for H , and H , in
powers of r / a valid over the transverse plane through the mid-point.
6. Two linear circuits CI and CZ carry steady currents I I and ZZ respectively.
Show that the magnetic energy of the system is
where dsl and dsz are vector elements of length along the contours and r12 the distance
between these elements. The permeability p is constant.
The coeficient of mutual inductance is defined by the relation
7. Show that the mutual inductance of two circular, coaxial loops in a medium of
constant permeability is
264
THE MAGNETOSTATIC FIELD
[CHAP.I V
where a and b are the radii of the loops, c the distance between centers, K and E the
complete elliptic integrals of the first and second kinds, and k is defined by
k2 =
4ab
__-__.
(a
+ + c2
b)2
When the separation c is very small compared to the radii and a
formula reduces to
b show that this
+
where d = [ ( a - bI2 czl*.
8. The coeficient of self-inductance LI1 of a circuit carrying a steady current 11is
defined by the relation
TI = &lz;
where T 1 is the magnetic energy of thc circuit.
either of the expressions
Hence L I I can be calculated from
provided B is a linear function of H. The volume integral in the first case is extended
over the region occupied by current, in the second over the entire field. That portion
of Lll associated with t,he energy of the field inside the conductor is called the internal
self-inductance Lil.
Show that the internal self-inductance of a long, straight conductor of constant
perheability p1 is
L;,
=
PI
8u
henrys/meter.
9. Show that the self-inductance of a circular loop of wire of radius R and crosssectional radius r is
where p1 is the permeability of the wire and p2 t h a t of the cxternal medium, both being
assumed constant.
10. A toroidal coil is wound uniformly with a single layer of n turns on a surface
generated by the revolution of a circle T meters in radius about an axis IZ meters from
the center of the circle. Show that the self-inductance of the coil is
L = pn2(R - d R 2 - r2)
henrys.
11. A circular loop of wire is placed with its plane parallel to the plane face of a
semi-infinite medium of constant permeability. Find the increase in the self-inductance of the loop due to the presence of the magnctic material.
Show that if the plane of the loop coincides with the surface of the magnetic
material, the self-inductance is increased b y a factor which is independent of the form
of the loop.
12. Find an expression for the change in self-inductance of a circular loop of wire
due to the presence of a second circular loop coaxial with the first.
PROBLEMS
265
13. An infinitely long, hollow cylinder of constant permeability p1 is placed in a
fixed and uniform magnetic field whose direction is perpendicular to the generators nf
the cylinder. The potential of this initial field is 4; = -HOT cos 8, where r and e
are cylindrical coordinates in a transverse plane with the axis of the cylinder as an
origin. The cross section of the cylinder is bounded by concentric circles of outer
radius a and inner radius b. The permeability of the external and internal medium has
the constant value p2. Show that the potential a t any interior point is
L
\PI - P Z /
\a
14. A hollow sphere of outer radius a and inner radius b and of constant permeabill
ity PI is placed in a fixed and uniform magnetic field Hn.The external and internal
medium has a constant permeability p z . Show that the internal field is uniform and is
given by
1
Discuss the relative effectiveness of a hollow sphere and a long hollow cylinder for
magnetic shielding.
16. A magnetic field such as is observed on the earth’s surface might be generated
either by currents or magnetic matter within the earth, or by circulating currents
above its surface. Actual measurement indicates that the field of external sources
amounts to not more than a few per cent of the total field. Show how the contributions of external sources may be distinguished from those of internal sources by measurements of the horizontal and vertical components of magnetic field intensity on the
earth’s surface.
16. Assume that the earth’s magnetic field is due to stationary currents or magnetic matter within the earth. The scalar potential a t external points can then be
represented as an expansion in spherical harmonies. Show that the coefficients of
the first four harmonics can be determined by measurement of the components of
magnetic field intensity a t eight points on the surface.
17. A winding of fine wire is to be placed on a spheroidal surface so that the field
of the coil will be identical with that resulting from a uniform magnetization of the
spheroid in the direction of the major axis. How shall the winding be distributed?
18. A copper sphere of radius a carries a uniform charge distribution on its surface. The sphere is rotated about a diameter with constant angular velocity. Calculate the vector potential arid magnetic field a t points outside and inside the sphere.
19. A solid, uncharged, conducting sphere is rotated with constant angular velocity
w in a uniform magnetic field B, the axis of rotation coinciding with the direction of
the field. Find the volume and surface densities of charge and the electrostatic
potential a t points both inside and outside the sphere. Assume that the magnetic
field of the rotating charges can be neglected.
20. The polarization P of a stationary, isotropic dielectric in a n electrostatic field
E is expressed by the formula
P
=x
(C
- Q)E.
266
T H E MAGNETOSTATIC FIELD
[CHAP.
Iv
If each element of the dielectric is displaced with a velocity v in a magnetic field the
polarization is
P = ( E eo)(E v X B),
+
-
-
at least t o terms of the first order in l / c = Z / e o p o .
A dielectric cylinder of radius a rotates about its axis with a constant angular
velocity 0 in a uniform magnetostatic field. The field is parallel to the axis of the
cylinder. Find the polarization of the cylinder, the bound charge density appearing
on the surface, and the electrostatic potential at points both inside and outside the
cylinder.
21. A dielectric sphere of radius a in a uniform magnetostatic field is rotated with
a constant angular velocity w about an axis parallel to the direction of the field. Using
the expression for polarization given in Problem 20, find the densities of bound volume
and surface charge and the potential at points inside and outside the sphere.
22. Show t h a t the force between two linear current elements is
dF
=
P
- IlIZ
4r
p _
I l l
= _
2
4r r2
ds2 X (dsl X ro)
r2
-
-
[(dsz ro) dsl - (dsl ds2)ro],
where ro is a unit vector directed along the line T joining dsl and dsz from dsl toward
dsz.
23. A long, straight wire carrying a steady current is embedded in a semi-infinite
mass of soft iron of permeability p a t a distance d from the plane face. The wire is
separated from the iron by a n insulating layer of negligible thickness. Find expressions for the field a t points inside and outside the iron.
24. A long, straight wire carrying a steady current is located in the air gap between
two plane parallel walls of infinitely permeable iron. The wire is parallel to the walls
and is assumed t o be of infinitesimal cross section. Find the field intensity a t any
point in the air gap. Plot the force exerted on the conductor per unit length for a current of one ampere as a function of its distance from one wall.
26. A long, straight conductor of infinitesimal
cross section lies in a n infinitely deep, parallel-sided
slot in a mass of infinitely permeable iron. The thickness of the slot is I and the conductor is located by the
Air
parameters s a n d h as in the figure. Calculate the
field intensity at points within the slot and the force
on unit length of conductor per ampere. (Cj. Hague,
“Electromagnetic Problems in Electrical Engineering,”
andLinder, J.Am. Znst. Elec. Engrs., 46,614,1927.)
26. An infinite elliptic cylinder of soft iron defined by the equation
b2x2
+ a2y2 =
a2b2
is placed in a fined and uniform magnetic field whose direction is perpendicular to the
generators of the cylinder. Find expressions for the magnetic scalar potential a t
internal and external points and calculate the force and torque exerted on the cylinder
per unit length. Discuss the case of unit eccentricity in which the cylinder reduces
t o a thin slab.
27. A particle of mass m and charge q is projected into the field of a magnetic
dipole.
PROBLEMS
267
a. Write the differential equations for the motion of the particle in a system of
spherical coordinates whose origin coincides with the center of the dipole.
b. Show that the component of the angular momentum vector in the direction of
the dipole axis is constant.
c. Discuss the trajectory in the case of a particle initially projected in the equatorial plane.
28. A small bar magnet is located near the plane face of a very large mass of soft
iron whose permeability is p. The magnet is located by its distance from the plane
and the angle made by its axis with the perpendicular passing through its center.
Considering only the dipole moment of the magnet, find the force and torque exerted
on it by the induced magnetization in the iron.
29. Two small bar magnets whose dipole moments are respectively ml and m 2are
placed on a perfectly smooth table. The distance between their centers is large
relative to the length of the magnets. A t any instant the dipole axes make angles el
and e2 with the line joining their centers.
a. Calculate the force exerted by ml on m 2and the torque on mzabout a vertical
axis through its center.
b. Calculate the force exerted by m2 on ml and the torque on ml about a vertical
axis through its center.
c. Calculate the total angular momentum of the system about a fixed point in the
plane. Is it constant?
30. Show that the force between two small bar magnets varies as the inverse
fourth power of the distance between centers, whatever their orientation in space.
31. A magnetostatic field is produced by a distribution of magnetized matter.
There are no currents a t any point. Show that the integral
B-Hdu=O
2
if extended over the entire field.
32. If J ( T , y, z ) is the current density at any point in a region V bounded by a closed
surface S, show that the magnetic field a t any interior point d , y’, z’ is
+
where r = 1 / ( z ’ - E ) +
~ (y’ - y)2
(2’ - z ) ~ and
,
that a t exterior points the value
of the surface integral is zero.
33. Determine the field of a magnetic quadrupole from the expansion of the vector
potential given by Eq. (23), p. 234. Give a geometrical interpretation of the quadrupole moment in terms of infinitesimal linear currents.
CHAPTER V
PLANE WAVES IN UNBOUNDED, ISOTROPIC MEDIA
Every solution of Maxwell’s equations which is finite, continuous,
and single-valued at all points of a homogeneous, isotropic domain
represents a possible electromagnetic field. Apart from the stationary
fields investigated in the preceding chapters, the simplest solutions
of the field equations are those that depend upon the time and a single
space coordinate, and the factors characterizing the propagation of
these elementary one-dimensional fields determine also in large part
the propagation of the complex fields met with in practical problems.
We shall study the properties of plane waves in unbounded, isotropic
media without troubling ourselves for the moment as to the exact nature
of the charge and current distribution that would be necessary to establish
them.
PROPAGATION OF PLANE WAVES
6.1. Equations of a One-dimensional Field.-It
will be assumed for
the present that the medium is homogeneous as well as isotropic, and
of unlimited extent. We shall suppose, furthermore, that the relations
(1)
D
=
EE,
B
= pH,
J
=
uE,
are linear so that the medium can be characterized electromagnetically
by the three constants E , p , and u. If the conductivity is other than zero
any initial free-charge distribution in the medium must vanish spontaneously (Sec. 1.7). I n the following, p will be put equal to zero in dielectrics
as well as conductors. The Maxwell equations satisfied by the field
vectors are then
(I) V X E
+
dH
p~
= 0,
l3E
(11) V X H - e - - UE = 0,
at
(111) V * H = 0,
(IV) V * E
=
0.
We now look for solutions of this system which depend upon the time
and upon distance measured along a single axis in space. This preferred
direction need not coincide with a coordinate axis of the reference system.
Let us suppose, therefore, that the field is a function of a coordinate
( measured along a line whose direction is defined by the unit vector n.
The rectangular components nz,%, n, of this unit vector are obviously
268
E Q U A T I O N S OF A O N E - D I M E N S I O N A L FIELD
SEC.5.11
269
the direction cosines of the new coordinate axis {. Our assumption
implies that a t each instant the vectors E and H are constant in direction
and magnitude over planes normal to n. These planes are defined by
the equation
-
r n = constant,
(2)
where r is the radius vector drawn from the origin to any point in the
plane as indicated by Fig. 44.
'Y
FIG.44.-Homogeneous
plane waves are propagated in a direction fixed by the unit vector n.
Since the ficlds are of the form
(3 )
E = E(C, 0, H = W, 0,
the partial derivatives with respect to a set of rectangular coordinates
may be expressed by
(4)
From these we construct the operator
sions for the curl and divergence.
(5)
v
=
v and so obtain simplified expres-
a
a
a
a
+
j - + k - = (in, + jn, + kn,) - = n -,
as
ay
az
ar:
ar:
a
i-
vx
(6)
E
=
v.E
(7)
n
=
aE
8
x=
ar: ar: (n x
n.-
-
dE
ai-
=
E),
a
- (n-E).
ar:
In virtue of these relations, the field equations assume the form
aH
(111) n . - = 0,
(11) n
aH
aE
x- aE = 0,
ar: - E at
ar:
aE
(IV) n .- = 0.
ar:
270
PLANE W A V E S I N UNBOUNDED, ISOTROPIC M E D I A
[CHAP.V
This system is now solved simultaneously by differentiating first (I)
with respect to { and then multiplying vectorially b y n. (11) is next
differentiated with respect to t. We have
upon elimination of the terms in H, the vector E is found to satisfy the
equation
a2E
aE
_a2E
_
-p- pta- = 0.
(9)
at2
at
A similar elimination of E leads to an identical equation for H.
a2H
a2H
aH
~
Since n . (n
n - -aH
at
=
x
a!? - p E - at2
g)
- p(T-
at
=
0.
is identically zero, it follows from (I) th a t
0. Taken together with (111),this gives
We are forced to conclude that a variation of the {-component of H with
respect to either t or t is incompatible with the assumption of a field which
is constant over planes normal to the l-axis a t each instant. Equation
(11) does admit the possibility of a static field in the f-direction but, since
at present we are concerned only with variable fields, we shall put Hr = 0.
Likewise, i t follows from (11) th at
(;
n . e-++E
) =O;
this together with (IV) leads to
(13)
n
($dt + f E dt + a< d { )
n - (dE
=
+ f E dl)
=
0.
The component of E normal to the family of planes, therefore,. satisfies
the condition
which upon integration gives
(15)
-_ t
El = Eore
T
SEC.5.11
EQUATIONS OF A ONE-DIMENSIONAL FIELD
271
where Eof is the longitudinal component of E at t = 0 and r = e/u is the
relaxation time defined on page 15. If the conductivity is finite, the
longitudinal component of E vanishes exponentially : a static electric
field cannot be maintained in the interior of a conductor. According to
(15) there may be a component El in a perfect dielectric medium, but
this particular integral does not contain the time.
Equations (11) and (13) prove the transversality of the field. The
vectors E and H of every electromagnetic $eld subject to the condition (3) lie
in planes normal to the axis of the coordinate {.
Let us introduce a second system of rectangular coordinates E, 7, {,
whose origin coincides with that of the fixed system x , y , z and whose
<-axis is oriented in the direction specified by n. With respect to this
new system the vector E has the components E t , E,,, and Er= 0. Both
E Eand E1satisfy (9), which can be solved by separation of the variables.
Let
EE
(16)
Then
= fl(Ofi(t>.
where - k 2 is the separation constant. The general solution of the
equation in f l is
(18)
fl({)
=
Aeikf
+ Be+T,
where A and B are complex constants; for fz we shall take the particular
solution
(19)
j z ( t ) = Ce-Pt.
Then p must satisfy the determinantal equation
p " - - , pu+ - = Ok2
.
EIE
There exists a fixed relation between p and the separation constant k 2 ;
the value of either may be specified, whereupon the other is determined.
From (18) and (19) may be constructed a particular solution of the
form
(21)
E E = EIEeikf-pt + E 2Ee--ikf-Pt*
likewise, for the other rectangular component,
(22)
E,, = El,,eikr-Pt
+ EZ,,e-ikt-Pf.
272
P L A N E W A V E S I N U N B O U N D E D , ISOTROPIC M E D I A
[CHAP.V
The four complex constants Elt . . . Ez, are the components of two
complex vector amplitudes lying in the 4s plane. Combining (21) and
(22), one obtains
(23)
E = Elgikl-pt
+ EZe-ikl-Pt.
This vector solution of (9) in turn is introduced into the field equations
t o obtain the associated magnetic vector H. Since H must have the
FIG.45.-Relative
directions of the electric and magnetic vectors in positive and negative
waves.
same functional dependence on { and t , we can write
(24)
H = HleikS-Pt
and then determine the constants
+ H,e-ikl-pt
HI,
H2 in terms of El and E2.
These derivatives are introduced into (I) and lead to
(26) (ikn x El - ppHl)ei"f-Pt - (i k n X Ez
+ ppHz)e-ikl-pt= 0.
The coefficients of the exponentials must vanish independently and,
hence,
(27)
ik
H1 = -- n x El,
PEL
ik
Hz = -- n X Ez.
I n terms of the rectangular components,
PP
PLANE W A V E S H A R M O N I C I N T I M E
SEC.5.21
.
273
.
Since E n X E = 0, it follows that E H = 0. The electric and magnetic
vectors of a$eld of the type defined by Eq. (3) are orthogonal to the direction n
and to each other. These mutual relations are illustrated in Fig. 45.
6.2. Plane Waves Harmonic in Time.-There
are now two cast’s t o
be considered, dependent upon the choice of p and k , which it will be well
to treat separately. Let us assume first that the field is harmonic in
time, and that therefore p is a pure imaginary. The separation constant
is then determined by (20).
(29)
p
=
iw,
k2
= p€w2
+ ipuw.
I n a conducting medium k 2 , and hence k itself, is complex. The sign o j
the root will be so chosen that the imaginary part of k is always positive.
k
(30)
= a
+ ip.
The amplitudes E ~ .E. . Ex, arc also complex and will be written now in
the form
Ell = aleis’,
E21 = a2e292,
(31)
El,, = het*’,
E,, = b2eZ#2,
where the new constants a1 . . . b2, 81 . . . $xare real. In virtue of
these definitions we have for the (-component of the vector E
Et = ale-Bf+i(.f’Wt+S1)
(32)
+ a2eBf-t(af+wt--8z).
Since (32) is the solution of a linear equation with real coefficients, both
its real and imaginary parts must also be solutions. The real part of (32)
is
(33)
E,
=
ale-ar cos (wt - a{ - 0,)
+ azeaf cos (wt + a{ - ox).
The phase angles B1 and Ox are arbitrary and consequently a choice of the
imaginary part of (32) does not, lead to a solution independent of (33).
I n the same way the 7-component of E is obtaincd from (22).
(34)
Ev = ble-@f cos
(wt - a[
- $1) 3- hear cos
(wt
+ a[ - +J.
The components of the associated magnetic field are found from (28).
We now have
Thc complex t-componcnt of H is
upon substitution of the appropriate values of IilE and H z t , we obtain for
274
P L A N E W A V E S I N UNBOUNDED, ISOTROPIC M E D I A
[CHAP.
Y
the real part
(37) HE =
d
2
2
-b1
a
+
cos (wt - a t - $1 - y )
e-81
w
Similarly for the 11-component,
(38)
H,
= UI
67 e-81
Pw
cos (wt - a t - 81 - y)
These solutions correspond obviously to plane waves propagated
along the {-axis in both positive and negative directions. Consider first
the most elementary case in which all amplitudes except a1 are zero in a
nonconduct,ng medium. Then Q and consequently p are zero and the
field is represented by the equations
(39)
E~ = al cOS (wt - at - el),
H,
=
a
- al COS (wt - at - el).
Pw
This field is periodic in both space and time. The frequency is w / 2 ~= v
and the period along the time axis is 2 ~ / w= T . The space period is
called the wave length and is defined by the relat,ion
Ly.
=
2T
-,
x
=
2T
-1
a
The argument $1 = wt - a[ - B1 of the periodic function is called the
phase and the angle el, which will be determined by initial conditions, is
the phase angle. At each instant the vectors E and H are constant over
the planes { = constant. Let us now choose a plane on which the phase
has some given value at t = 0, and inquire how this plane must be displaced along the {-axis in order that its phase shall be invariant to a
change in t. Since on any such plane the phase is constant, we have
(41)
41 = w t - a[
-
el = constant,
d& = w dt
- adt
=
0.
The surfaces of constant phase in a field satisfying (39) are, therefore,
planes which displace themselves in the direction of the positive {-axis
with a constant velocity
v is called the phase velocity of the wave. It represents simply the
velocity of propagation of a phase or state and does not necessarily
coincide with the velocity with which the energy of a wave or signal is
SEC.5.21
275
P L A N E W A V E S HARMONIC I N T I M E
propagated. In fact v may exceed the critical velocity c without violating
in any way the relativity postulate.
In a nonconducting medium
1
c
Ly=wl&,
v=___
(43)
fi€
= d
KK
, m’
where c is the velocity of the wave in free space and K ~ K,,,, are the electric
and magnetic specific inductive capacities. In optics the ratio
n = c/v = a c / w is called the index of refraction. Since in all but ferromagnetic materials K~ is very nearly unity, the index of refraction should
be equal to the square root of K ~ . This result was first established by
Maxwell and was the basis of his prediction that light is an electromagnetic phenomenon. Marked deviations from the expected values of
n were observed by Maxwell, but these were later accounted for by the
discovery that the inductive capacity does not necessarily maintain a t
high frequencies the value measured under static or quasi-static conditions. A functional dependence of n on t,he frequency results in a
corresponding dependence of the phase velocity and leads to the phenomena known as dispersion.
At a given frequency the wave length is determined by the properties
of the medium.
(44)
where Xo is the wave length at the same frequency in free space. I n all
nonionized media n > 1, and consequently the phase velocity is decreased
and the wave length is shortened.
Referring again to (39), we see that the relation of the magnetic
to the electric vector is such that the cross product E x H is in the
direction of propagation. The magnetic vector is propagated in the
same direction with the same velocity, and in a nonconducting medium
is exactly in phase with the electric vector. Their amplitudes differ
by the factor
(u =
(45)
0).
If in Eqs. (33) to (38) all amplitudes except a2 are put equal to zero,
another particular solution is found which in nonconducting media
reduces to
(46)
E~ = a2 cos (wt
+ a!: - e,),
H, = a2$
cos (wt
+
- e2).
This field differs from the preceding only in its direction of propagation.
a( - 82 and the surfaces of constant phase are
The phase is 4 2 = wt
+
276
P L A N E WAVES I N UNBOUNDED, ISOTROPIC M E D I A
[CHAP.V
propagated with the velocity v = - w / a in the direction of the negative
{-axis.
The two particular solutions whose amplitudes are bl and b2 represent
a second pair of positive and negative waves, both characterized by
electric vectors parallel to the 7-axis.
Let us remove now the restriction t o perfect dielectric media and
examine the effect of a finite conductivity. It is apparent in the first
place that both electric and magnetic vectors are attenuated exponentially in the direction of propagation. Waves traveling in the negative
direction are multiplied by the factor e+@r,but since { decreases in the
direction of propagation this too represents an attenuation. Not only
does a conductivity of the medium lead to a damping of the wave, but
i t affects also its velocity The constants a! and p can be calculated in
terms of p, E, and u by squaring (30) and equating real and imaginary parts
respectively to the real and imaginary terms in (29).
(47)
Upon solving these relations simultaneously, we obtain
(48)
(49)
Ambiguities of sign that arise on extracting the first root are resolved
by noting that a and must be real.
The planes of constant phase are propagated with a velocity
which increases with frequency so long as the constants K,, K ~ and
,
u are
independent of frequency. Attenuation of amplitudes in a surface of
constant phase is determined by the attenuation factor /3 which also
increases with increasing frequency. The complex factor k will be referred
to as the propagation constant, and its real part a! may be called the phase
constant, although this term is applied also to the angles 0 and +.
The effect of frequency and conductivity on the propagation of plane
waves is most easily elucidated by consideration of two limiting cases.
Examination of (48) and (49) shows that the behavior of the factors
the
Q and
is essentially determined by the quantity U ~ / E ~ W Now
~ .
PLANE WAVES H A R M O N I C I N T I M E
SEC.5.21
27 7
total current density a t any point in the medium is
J
=
UE + E-
aE
at
=
(U
- iw~)E,
whence it is apparent that U / U is equal to the ratio of the densities of
conduction current to displacement current.
Case I .
<< 1.
U2
~
€2W2
The displacement current is very much greater
than t,he conduction current. This situation may arise either in a
medium which is but slightly conducting, or in a relatively good conductor such as sea water through which is propagated a wave of very
high frequency. Expansion of (48) and (49) in powers of u2/c2w2 gives
(52)
r
r-
(53)
One will remark th at t o this approximation the attenuation factor is
independent of frequency. I n a given medium the attenuation approaches
asymptotically a m a x i m u m dejined by (53) as the frequency is increased.
U2
Case 11. T2>>
1. The conduction current greatly predominates
E W
ovcr the displacement current. This is invariably the case in metals,
where u is of t he order of lo7 mhos/meter. Not much is known about the
inductive capacity of metals but there is no reason t o believe th a t it
assumes large values. Since the magnitude of e is probably of the order
of
the displacement current could not possibly equal the conduction
current a t frequencies less than 10’7, lying in the domain of atomic
phenomena to which the present obviously does not apply. For CY and /3
we obtain the approximate formula
(54)
a =
p
=
J
?!
2
=
1.987 X lO-’.\/VaK,.
An increase of frequency, permeability, or conductivity contributes in
the same way t o an increase in attenuation. The phase velocity also
increases with frequency, but decreases with increasing u or K ~ . Thus
the higher order harmonics of a complex periodic wave are constantly
advancing with respect to those of lower order.
The amplitudes of the electric and magnetic vectors of a plane wave
are related by
(5.5)
278
PLANE W A V E S I N UNBOUNDED, ISOTROPIC M E D I A
[CHAP.V
(56)
I n poorly conducting media expansion of (56) gives
whereas in good conductors,
In a perfect dielectric the electric and magnetic vectors oscillate in
phase; if the medium is conducting, the magnetic vector lags by an
angle y.
If
L>>
1, this ratio reduces to unity; hence the magnetic vector
2W2
of a
plane wave penetrating a metal lags behind the electric vector by 45 deg.
6.3. Plane Waves Harmonic in Space.-The assumption that p in
Eq. (20) is a pure imaginary leads to complex values of k and to electromagnetic fields which are simple harmonic functions of time. A field
which at any given point on the {-axis is a known periodic function of t
can be resolved by a Fourier analysis into harmonic components propagated along the {-axis as just described. The time variation at any other
fixed point can then be found by recombining the components at the
point in question. I n place of the variation of the field with t at a certain
point c, one may be given the distribution with respect to at a specified
value of t and asked to determine the field a t any later instant. A
harmonic analysis must then be made with respect to the variable 1.
Let the separation constant k 2 be real. Then p is a complex quantity determined by
p =
a+ i
2E -
If for the moment we write q
in (23) assumes the form
=
JF$.
,,/x2,
the electric vector defined
279
POLARIZATION
SEC.5.41
If now a2/42 < P / p , the quantity iq is a pure imaginary and the field
may be interpreted as a plane wave propagated along the {-axis with a
phase velocity
The amplitude of oscillation a t any point { decreases exponentially with t,
the rate of decrease being determined essentially by the relaxation time
7
= e/u.
When a2/4e2 > k 2 / p e , the quantity iq is real and there is no propagation in the sense considered heretofore. The field is periodic in { but
decreases monotonically with the time. There is no displacement along
the space axis of an initial wave form: the wave phenomenon has degenerated into d i f u s i o n .
6.4. Polarization.-Inasmuch
as the properties of the positive and
negative waves differ only in the direction of propagation, we shalI confine our attention a t present to the positive wave alone. Furthermore,
since the attenuating effect of a finite conductivity in a n isotropic,
homogeneous medium enters as an exponential factor common to all
field components, it plays no part in the polarization and will be neglected.
Let us suppose then that the amplitudes and phase constants of the
rectangular components of E have been specified and investigate the
locus of IEl = d E t 2 EV2in the plane p = constant.
To determine this locus one must eliminate from the equations
+
(63)
Ec
=
a cos (4
+ 0),
the variable component 4
is written in the form
(64)
$
=
cos (4
+
E,
=
b cos (4
+ +),
- wt of the phase.
= cr{
7
Er = 0,
To this end (63)
+ 0) cos 6 - sin (4 + e) sin 6,
Upon squaring, the periodic factor cos (4 + 0) can
+ O),
= cos
(4
where 6 = - 0.
be elirninat>edand it is found that the rectangular components must satisfy
the relation
The discriminant of this quadratic form is negative,
4
cos2 6 - 4 = -__
4
,
sin2 6 5 0,
a2b2
a2b2
a2b2
and the locus of the vector whose components are Ec, E, is, therefore,
an ellipse in the Eq-plane.
280
P L A N E W A V E S I N UNBOUNDED, ISOTROPIC M E D I A
[CHAP.
V
I n this case the wave is said to be elliptically polarized. The points
of contact made by the ellipse with the circumscribed rectangle are found
from (65) to be ( + a , & b cos 6) and ( + a cos 6, Ab). I n general the
principal axes of the ellipse fail to coincide with the coordinate axes [, 7 ,
but the two systems may be brought into
coincidence by a rot,ation of the coordinate
system about the {-axis through a n angle 6
defined b y
tan 26
(67)
46.--Po1ari2ation
(68) Et.2
= __2ab
a2 - b2
cos 6.
When the amplitudes of the rectangular
components are equal and their phases differ
by some odd integral multiple of ~ / 2 ,the
polarization ellipse degenerates into a circle.
+ EV2= a2,( a = b,
6
m?r
2 j
=-
The wave is now said to be circularly polarized. Two cases are recognized according to the positive or negative rotation of the electric vector
about the {-axis. It is customary to describe as right-handed circular
polarization a clockwise rotation of E when viewed in a direction opposite
to that of propagation, looking towards
the source.
By far the most important of the
special cases is th at in which the polarization ellipse degenerates into a straight
line. This occurs when 6 = I m n , where Right-handed
Left-handed
FIG. 47.-Circular
polarization.
m is any integer. The locus of E in the
the circles indicating the locus of E.
t7-plane then reduces to a straight line The direction of propagation is normaking an angle 6 with the &axis de- mal to the page and toward the
observer.
fined by
Ev b
t a n 6 = - - (-l)m--
00
J%
Q
The wave is linearly polarized. The magnetic vector of a plane wave is
a t right angles to the electric vector and H oscillates, therefore, parallel
to a line whose slope is (- l ) % / b . It is customary to define the polarization in terms of E and to denote the line (69) as the axis of linear polarization. In optics, however, the orientation of the vectors is specified
traditionally by the “plane of polarization,” by which is meant the plane
normal to E containing both H and the axis of propagation.
SEC.5.51
28 1
ENERGY FLOW
6.6. Energy Flow.-The rate a t which energy traverses a surface in an
electromagnetic field is measured by the Poynting vector, S = E x H
ik
defined in Sec. 2.19. Since by (27) H = rt - n
PP
x E, the sign depending
upon the direction of propagation, it is apparent that the energy flow is
normal to the planes of constant phase and in the direction of propagation.
To calculate the instantaneous value of the flow one must operate
with the real parts of the complex wave functions. The mean value of
the flow can be quickly determined, on the other hand, by constructing
the complex Poynting vector S* = +E x fi as in Sec. 2.20. For simplicity lct us consider a plane wave linearly polarized along the x-axis
and propagated in the direction of positive x . Then
The complex flow vector is, therefore,
S:
=
1 E,H,
-
-
2
=
a2
e-.@*-iy
2uw
7
whose real part represents the energy crossing on the average unit area
of the zy-plane per second.
The factor cos y arising from the relativc phase displacement between
E and H may be expressed in terms of a and 0.
(73)
tan
y =
P
->
a
cos y
=
~ CY .\/,yz+pz
.
(74)
According to Poynting's theorem the divergence of the mean fiow vector
measures the energy transformed per unit volume per second into heat.
I n the present instance
(75)
which in turn is obviously equal t o the conductivity times the mean
square value of the electric field vector E.
282
P L A N E WAVES I N U N B O U N D E D , ISOTROPIC M E D I A
[CHAP.v
6.6. Impedance.-In
a world which outwardly is all variety, it is
comforting t o discover occasional unity and tempting t o speculate upon
its significance. T o the untutored mind a vibrating set of weights
suspended from a network of springs appears t o have little in common
with the currents oscillating in a system of coils and condensers. But
the electrical circuit may be so designed t h a t its behavior, and the vibrations of the mechanical system, can be formulated b y the same set of
differential equations. Between the two there is a one-to-one correspondence. Current replaces velocity and voltage replaces force; and
mass and the elastic property of the spring are represented by inductance
and the capacity of a condenser. It would seem that the L‘absolute
reality,” if one dare think of such a thing, is an i n w t i a property, of which
mass and inductance are only representations or names.
Whatever the philosophic significance of mechanical, electrical, and
chemical equivalences may be, the physicist has made good use of them
to facilitate his own investigations. The technique developed in the last
thirty years for the analysis of electrical circuits has been applied with
success to mechanical systems which not long ago appeared too difficult
to handle, and mechanical problems of the most complicated nature are
represented by electrical analogues which can be investigated with
ease in the laboratory. Not only the methods but the concepts of
electrical circuits have been extended t o other branches of physics.
Certainly the most important among these is the concept of impedance
relating voltage and current in both amplitude and phase. This idea
has been applied in mechanics to express the ratio of force to velocity, and
in hydrodynamics, notably in acoustics, to measure the ratio of pressure
to flow.
The extension of the impedance concept to electromagnetic fields is
not altogether new, but it has recently been revived and developed in a
very interesting paper by Sche1kunoff.l The impedance offered by a
given medium t o a wave of given type is closely related t o the energy
flow, but t o bring out its complex nature we may best start with an
analogy from a one-dimensional transmission line, as docs Schelkunoff.
Let z measure length along a n electric transmission line and let
V = VoePiwt,I = Ifleczwtbe respectively the voltage across and the current in the line at any point z. The quantities V0, loare functions
of z alone. The resistance of the line per unit length is €2 and its inductance per unit length is L. There is a leakage across the line a t each
point represented by the conductance G and a shunt capacity C. The
series impedance Z and the shunt admittance Y are, thercfore,
SCHELKUNOFF,
Bell System Tech. J., 17, 17, January, 1938.
SEC.5.61
283
IMPEDANCE
while t,hc voltage and current are found to satisfy the relations'
These equations are satisfied by two independent solutions which represent waves traveling respectively in the positive and negative directions.
I 1 -I, =
(78)
where
Alekkz-iwt
v1 = ZOI',
Vf = -Z,I2,
>
Aze-~kz--iW1,
(79)
k is the propagation constant and 2 0 the characteristic impedance of the
line.
Consider now a plane electromagnetic wave propagated in a direction
specified by a unit vector n. Distance in this direction will again be
measured by the coordinate { arid we shall suppose that the time enters
only through the factor e--zwt. Whereas voltage and current are scalar
quantities, E and H are of course vectors. To establish a fixed convention for determining the algebraic sign, the field equations will be
written in such a way as t o connect the vector E with the vector H x n
which i s parallel to E and directed in the same sense.
(80)
aT
=
a
i w p (x~n),
-
aT
(H x n)
=
i(we+ w ) ~ .
By analogy with (77)
(81)
z=
-i
Y = -i(w€
WPC,
+ iu).
The propagation constant is
k
(82)
=
i
dY2
= dw%p
+ iwpu
(B),
page 273, whilc the intrinsic impedance of the medium f o r
plane waves is defined by Schelkunoff as the quantity
as in
___
ZO++
(83)
W€
+ xu =
e-iy*
i+p*
1 See for example, Guillemin, " Communication Networks," Vol. 11, Chap. IT,
Wiley, 1935.
2 These results differ in the algebraic sign of the imaginary component from those
usually found in the literature on circuit theory due to our choice of e--rwt rather than
e"f. Obviously this choice is wholly nrbitrary. Although it is advantageous t o
use erwt in circuit theory, we shall bf concerned in wave theory primarily with the space
rather than the time factor. Thr eupansions in curvilinear coordinates to be carried
out in the next chaptcr justify the choice of e-twt.
284
PLANE W A V E S I N UNBOUNDED, ISOTROPIC M E D I A
[CHAP.V
I n free space this impedance reduces to
(84)
Zo =
t
-=
376.6 ohms.
Assuming n to be in the direction of propagation so that the distinction
between positive and negative waves is unnecessary, lbe relation between
electric vectors becomes
(85)
n x E
=
Z,H,
E
=
Z,H x n.
There is an intimate connection bet,wcen the intrinsic impedance
and the complex Poynting vector.
and, consequently,
GENERAL SOLUTIONS OF THE ONE-DIMENSIONAL WAVE EQUATION
I n the course of this chapter we have investigated certain particular
solutions of the field equations which depend on one space variable and
the time. Owing to the linear character of the equations these particular solutions may be multiplied by arbitrary constants and summed to
form a general solution, about which we now inquire. I n virtue of the
infinite set of constants a t our disposal it must be possible to construct
solutions that satisfy certain prescribed initial conditions. We shall
show, for example, that the distribution of a field vector as a function of l
may be specified a t a stated instant t = t o , and that the field is thereby
uniquely determined a t all subsequent times; or the variation of a field
vector as a function of time may be prescribed over a single plane { = lo
and the field thus determined a t all other points of space and time.
The means a t our disposal for the investigation of general integrals of a
partial differential equation of the second order fall principally into two
classes: the methods of Fourier and Cauchy, and those of Riemann and
Volterra. The latter have come to constitute in recent years the most
fundamental approach to the theory of partial differential equations, and
their application to the theory of wave propagation has been developed
in a series of brilliant researches by Hadamard.l But although the
method of characteristics, first proposed by Riemann, affords a deeper
insight into the nature of the problem, it proves less well adapted as yet
HADAMARD,
“Lepons sur la propagation des ondes,” A. Hermann, Paris, 1903;
“Lectures on Cauchy’s Problem,” Yale University Press, 1923
285
ELEMENTS OF FOURIER ANALYSIS
SEC.5.71
to the requirements of a practical solution than the methods of harmonic
analysis which will occupy our attention in the present section.
6.7. Elements of Fourier Analysis.-For
the convenience of the
reader, who presumably is to some degree conversant with the subject,
we shall set forth here without proof the more elementary facts concerning
Fourier series and Fourier integrals.
Consider the trigonometric series
(1)
2+
(ul cos x
+ bl sin x) + . -
(a,cos nx
+ b, sin nx) + -
..,
with known coefficients a, and b,, and let it be assumed that the series
converges uniformly in the region 0 I
x I27r. Then (1) converges
uniformly for all values of x and represents a periodic function f(x) with
period 2 ~ .
+ 27r)
f(x
(2)
= fb).
The coefficients of the series may now be expressed in terms of f(x).
In virtue of the assumed uniformity of convergence, (1) may be multiplied by either cos nx or sin nx and integrated term by term. I n the
domain 0 < x I
27r, the trigonometric functions are orthogonal; that
is to say,
cos nx cos m x dx
=
6’“
sin nx sin m x dx
0,
cos nx sin m x dx
=
0,
=
0,
(m #
n).
Since
cos2nx dx
(4)
it follows that
1
1
1
a, = ;
(5)
b,
=
1
=
2r
JzT
sin2 m x dx =
7r,
j(x> cos nx cis,
(m= 0, 1 , 2
--
. ),
2*
; f(x) sin nx dx.
Inversely let us suppose that a function f(x) is given. Then in a
purely formal manner one may associate with f(x) a “Fourier series”
defined by
+ 2 (a, cos nx + b, sin ax),
00
f(x>
=
n=l
286
P L A N E W A V E S I N U N B O U N D E D , ISOTROPIC M E D I A
[CiwP. V
the coefficients to be determined from ( 5 ) . The right-hand side of (6),
however, will converge to a representation of f(s) in the domain
0 5 x 5 2~ only when the function f(x) is subjected to certain conditions. The whole question of the convergence of Fourier series is an
extremely delicate one, and even the continuity off (2) is not sufficient to
ensure it. It turns out, fortunately, that under such circumstances
the series can nevertheless be summed to represent accurately the function within the specified interva1.l A statement of the least stringent
conditions to be imposed on an otherwise arbitrary function in order
that it may be expanded in a convergent or summable trigonometric
series would necessitate a lengthy and difficult exposition; since we are
concerned solely with functions occurring in physical investigations, we
can content ourselves with certain suficient (though not necessary)
requirements. We shall ask only that in the interval 0 5 x 5 2r the
function and its first derivative shall be piecewise continuous. A function
f(s) is said to be piecewise continuous in a given interval if it is continuous throughout that interval except at a finite number of points. If
such a point be so,the function approaches the finite valuef(zo 0) as
so is approached from the right, and f(s0 - 0) as it is approached from
the left. A t the discontinuity itself the value of the function is taken to
f b o 0) f(.o - -.
0)
be the arithmetic mean -___
2
The Fourier expansion is particularly well adapted to the represrntation of functions which cannot be expressed in a closed analytic form,
but which are constituted of sections or pieces of analytic curves not
necessarily joined a t the ends. I n view of the admissible discontinuities
in the function or its derivative, the fact that two Fourier series represent
the same function throughout a subinterval does not imply at all that
they represent one and the same function outside that subinterval.
I t is thus essential that one distinguish between the representation of a
function and the function itself. The expansion in a trigonometric
series indicated on the right-hand side of Eq. (6) represents and is
equivalent to the piecewise continuous function f(x) within the region
0 5 x I2 r . Outside this domain the values assumed by the series
are repeated periodically in x with a period 2r, whereas the functionf(x)
may behave in any arbitrary manner. Only when f(s) satisfies the
additional relation (2) will it coincide with its Fourier representation
over the entire domain - co < x < 00.
If within a specified interval the Fourier series of f(s)is known to be
uniformly convergent, it may be integrated term by term and the series
thus obtained placed equal to the integral of f(x) between the same limits.
+
+ +
WHITTAKER
and WATSON,“Modern Analysis,” 4th ed., Chap. IX, Cambridge
University Press, London.
SEC. 5.71
28 7
ELEMENTS OF FOURIER ANALYSIS
The uniform convergence of a Fourier series is in itself not sufficient to
justify its differentiation term by term and the equating of the derived
series to dfldx. If furthermore f(z) is discontinuous a t some point
within an interval, its Fourier series certainly is not uniformly convergent
everywhere within t h a t interval, and consequently the Fourier expansions
of its integral and of its derivative demand special attention.’
I t is usually advantageous to replace (6) by an equivalent complex
series of exponential terms of the form
m
f(x)
(7)
=
2
n=-
c,eim,
m
whose coefficients arc determined from
2T
cn
=
_I_
f(a)e-i*a da,
2iT
(n= 0,
1, 5 2
. . ).
Since
(9)
eanx =
cos nx
+ i sin nx,
it is clear that the complex coefficients cn are related to the real coefficients a, and b, by the equations
2cn
=
a,, - ib,,
2c0 = ao,
2cn = a-%
+i L ,
By an appropriate change of variable the Fourier expansion (7)
may be modified to represent a function in the region -1 5 x 5 1.
Now this stretching of the basic interval or period from 27r to 21 suggests
strongly the possibility of passing to the limit, as the basic interval
becomes infinite and thereby obtaining a Fourier representation of a
nonperiodic function for all real values of x lying between - w and 03.
We shall assume t h a t throughout the entire region - < x < w , f(x)
and its first derivative are piecewise continuous, that a t the discontinuities the value of the function is to be determined by the arithmetical
+
0 ~ )
mcan, and further that the integral Jm- w
or in other words, that the integral
1
J
m
-m
f(x) dx is absolutely convergent,
If(x)l dx exists. Let all
= Au.
These questions are thoroughly treated by Carslaw, “Introduction to the Theory
of Fourier’s Series and Integrals,” Chap. VIII, Macmillan, 1921.
288
PLANE WAVES I N U N B O U N D E D , ISOTROPIC M E D I A
[CRAP.~
Then (11) may be written
j-mm
+(u)d u is defined as the
On the other hand the definite integral
limit of a sum as Au -+ 0:
As 1 approaches infinity, Au approaches zero and we may reasonably expect
the limit of (12) to be
(14)
f(.)
=
& s_-* J:=
du
f(a)eiu(z--a) da.
This is the Fourier integral theorem according to which an arbitrary
function, satisfying only conditions of piecewise continuity and the
$p
If(z)l dz, may be expressed as a double integral.’
existence of
I t will be noted that the result has been obtained by a purely formal
transition to the limit which makes the existence of the representation
appear plausible but does not confirm it. A rigorous demonstration is
beyond the scope of these introductory remarks.
If f(z) is real, the imaginary part of the Fourier integral must vanish
and (14) then reduces to
Since
(16)
cos u ( x - a ) = cos ux cos ua
+ sin ux sin ua,
it is apparent that the Fourier integral of a real, even function f ( -z)
$(XI is
(17)
L
f(z) =
cos ux
1
f(a) cos ua d a du,
and that of a real, odd functionf(-z)
(18)
f(x)
=
27r
O0
sin u z
=
=
1
-f(z) is
f(a)sin ua d a du.
As a general reference on thc subject the reader may consult “Introduction to
the Theory of Fourier Integrals,” by E. C. Titchmarsh, Oxford University Press, 1937.
ELEMENTS OF FOURIER ANALYSIS
SEC. 5.71
289
From these formulas we see that the Fourier integral of a function may
be interpreted as a resolution into harmonic components of frequency
u/2~
over the continuous spectrum of frequencies lying between zero
and infinity, In (18), for example, one may consider the function
;1
as the amplitude or spectral density of f(z) in the frequency interval u
to u du. Then
+
_I
g(u) sin usdu.
The relation between f(x) and g ( u )
is reciprocal. g ( u ) is said to be the
Fourier transform of f ( x ) , and f(z) is
FIG.48.-Step function.
likewise the transform of g ( u ) . In
the more general case of Eq. (14) one may write for the spectral density
of f(x) the function
and hence for f(z) the reciprocal relation
An extensive table of Fourier transforms has been published by Campbell
and Foster.’
The application of the Fourier integral may be illustrated by several
brief examples of practical interest. Consider first the discontinuous
step function defined by
when 121 < I ,
Ax) = 1,
(23)
f(x> = +,
when 1x1 = I,
when 1x1 > 1.
f(x> = 0,
The function is real and even, so that we may employ (17). The transform is
2 sin ul
g(u) =
cos uz d x =
$1’
and the Fourier integral
(25)
f(x)
=
$ km
g ( u ) cos ux d u = 2
r
sorn
sin u l y ux du.
1 CAMPBELL
and FOSTER:“Fourier Integrals for Practical Applications,” Bell
Telephone System Tech. Pub., Monograph R-584, 1931. Appeared in earlier form in
Bell System Tech. J., October, 1928, pp. 639-707.
290
P L A N E WAVES I N U N B O U N D E D , ISOTROPIC M E D I A [CHAP.V
Let us take next for f(z) the “error function”
f(z)
=
c
-a2x2
_
2
.
To find its Fourier transform we must calculale the integral
which is clearly equal to the real part of the complex integral
On completing the square, (28) may be written
The path of integration follows the imaginary axis from -iu/a2 to 0
and then runs along the real axis from 0 to co. The imaginary part of
the integral arises solely from the interval -iu/a2 5 /3 < 0, and since
we are concerned only with the real part the lower limit may be taken
equal to zero. The result is a definite integral whose value is well known:
a222
so that the transform of e-T
is
Then f(x), which is reciprocally the transform of ~ ( u )is,
I n the particular case a
=
1, Eq. (32) becomes a homogeneous integral
_ XZ
_
equation satisfied by the function e 2 . The pair of functions defined
by (31) and (32) have other properties more important for our present
needs. Let us modify (31) slightly to form the function
(33)
SEC.5.71
ELEMENTS OF FOURIER ANALYSIS
29 1
The area under this curve is equal to unity, whatever the value of the
parameter a ;
J--*S(Z, a) ax = 1,
(34)
as follows directly from (30). Now let a become smaller and smaller.
The breadth of the peak grows more and more narrow, while a t the same
time its height increases in such a manner in the neighborhood of x = 0
as to maintain the area constant, as indicatcd in Fig. 49. I n the limit
as a+O, the curve shrinks to the line II: = 0 where it attains infinite
amplitude. A singularity has been generated, an impulse function
bounding unit area in the immediate neighborhood of x = 0. A unit
X
FIG.49-Thc
impulse function S(s, a ) .
impulse function which vanishes everywhere but a t the point x = xo is
represented by
(35)
Any arbitrary function F ( $)) subject to the usual conditions of continuity,
can now be expressed as an infinite integral.
It is apparent from (32) that the transform of the impulse function S o ( x )
is a straight line displaced by an amount 1/45from the horizontal axis.
As a final illustration of the Fourier integral theorem consider a
harmonic wave train of finite duration. Such a pulse might result from
closing and then reopening a switch connecting a circuit to an alternating
current generator, or represent the emission of light from an atom in the
course of an energy transition. Let
(37)
f(t>= 0,
when
It1
T
> 2’
f(t)
when
It1
T
< 3.
=
cos mot,
292
P L A N E WAVES I N IJNBOUNDED, ISOTROPIC M E D I A
CHAP.^
To facilitate the integration we shall take f(t) equal to the real part of
T
eiwotin the domain It/ < - and use Eqs. (21) and (22).
2'
The Fourier integral of f(t) corrcsponds now to the real part of (22), or
(wo -
_-
sin
2
(0°
+
wo - w
'
wo+w
w ) T ] cos wt dw.
Equation (39) may be interpreted as a spectral resolution of a function
which during a finite interval T is sinusoidal with frequency wo. The
amplitude of the disturbance in the neighborhood of any frequency w is
determined by the function
sin
(40)
A(u)
=
- -
7r
(a0
-
w)T
2
wo-w
The amplitude vanishes a t the points
w = wo -
%n
-7
T
(n = *l, +2
*
*
*
),
in the manner indicated by Fig. 50, and has its maximum value a t
w = w0. As the duration T of the wave train increases, the envelope
of the amplitude function is compressed horizontally until in the limit,
as T ---f W , the entire disturbance is confined to the line w = w,, on the
frequency spectrum. The simple harmonic variations that enter into
so many of our discussions are mathematical ideals; the oscillations of
natural systems, whether mechanical or electrical, are finite in duration,
and the associated waves are periodic only in an approximate sense.
We shall discover shortly that in the presence of a dispersive medium the
entire character of the propagation may be governed by the duration
of the wave train.
6.8. General Solution of the One-dimensional Wave Equation in a
Nondissipative Medium.-To acquire some further facility in the use of
the Fourier integral it will repay us to pause for a moment over the elementary problem of finding a general solution to the wave equation in a
SEC.5.81
293
G E N E R A L SOLUTION
nonconducting medium. Let II. represent either the x- or y-component
of any electromagnetic vector. Then satisfies
+
According to Sec. 5.2, a particular solution of (42) is represented by
.W
+ = ( A e r + Be-*Tz)e-&t.
1-2
(43)
The coefficients A and B are arbitrary and may depend on the frequency
w ; that is to say, we associate with each harmonic component an appropriate amplitude which may be indicated by writing A(w) and B(w).
FIG.50.-The amplitude function A ( w )
- ~(wo_ _- -w~) T-/ -2 .
1 sin
=- 7r
wo-w
Now the general solution of (42) is obtained by summing the particular
solutions over a range of w . In case # is to be a periodic function, the
sum will extend over a discrete set of frequencies. I n general the wave
function is aperiodic in both space and time, and the frequency spectrum
is consequently continuous.
(44)
Let us suppose that over the plane z = 0 the values of the function 4
and of its derivative in the direction of propagation are prescribed functions of time.
(45)
294
PLANE WAVES' I N UNBOUNDED, ISOTROPIC MEDIA [CHAP.V
The problem is to find the coefficients A ( w ) and B(w) such that these
two conditions are satisfied, and thereby to show that the prescription
of the function and its derivative at a specified point in space (or time)
is sufficient t o determine $(z, t ) everywhere.
If we assume provisionally th at the integral on the right of (44) is
uniformly convergent, it may be differentiated under the sign of integration with respect to the parameter z, yielding
On placing z
=
0 in (44)and (46), we obtain
upon comparison wit,h (21) and (22) it is immediately evident th a t the
coefficients of the factor ePiwt in the integrands of (47) are Fourier
transforms.
A
(48)
iw
+B =zt,
-(A 21
B) =
Lmm
1
J
2a -
f(t)eiwtdt,
F(t)eiwt dt.
m
Solving these two relations simultaneously for A and B and substiiuting'
(Y for t as a variable of integration, we obtain
and on substitution into (44)there results:
Now the first of these double integrals is the Fourier expansion of a
function which on reference to (14) may be written down a t once. I n
1 The variables (Y and 6 appearing in the next few pages obviously have no connection with the real and imaginary parts of k = CY
ip defined on p. 273.
+
295
G E N E R A L SOLUTION
SEC.5.81
the second let us invert the order of integration.
The imaginary part of the last integral vanishes because
1
-
w
sin
w
2,
z sin w ( a
- t)
integrates to an even function. Therefore,
=
Jn"
[sin w ( a
+
- 1) - sin w ( a
-
g.
- 1):
By a slight modification of the conditions which defined the step function
(23), i t is easy t o show that each of the last two integrals represents a
function with a single discontinuity. In fact
a
when
p
0,
when
p = 0,
2
sin px
(53)
dx
=
Hence (52) will vanish whenever the arguments a
are of the same sign.
> 0,
-1
Changes of sign occur a t a
+ z - t and
-
LY
U
= t -
z
-
U
and LY
- -z - i
U
=t
+ Z-.U
The integral (52) has, therefore, a nonvanishing value only in the interval
2
t -21
<a <t
+z
-J
2,
where it is equal to a. The general solution of (42)
is now established in terms of the prescribed initial conditions.
Let us define a function h(P) by the relation
B
(55)
h(P) = - u
F ( a ) da.
296
P L A N E W A V E S I N UNBOUNDED, ISOTROPIC M E D I A
[CAAP.V
Then
dh
d p -- h'(P)
(56)
=
-vF(p).
I n place of (54) one may write
(57)
l ) ( z , t ) = - f 2l ( t - -
:) +-f; ( t + -:) ; (
+-h
t - -
t>
;(
--h
t + ; .
When z = 0, it is evident, that +(O, t ) reduces to f(t), and (d+/dz),=o to
1
- h', or F ( t ) . It may also be verified b y differentiation and substitution
V
that (42) i s satisjied by a n y function whose argument i s of the form t
z
-.
V
z
The replacement of the variable t in f ( t ) by t - - results in a displacement
V
or propagation without distortion of the function t o the right along the
positive z-axis with a velocity v. The initial distribution therefore
splits into two waves, the one traveling to the right and the other to
the left. Upon these two waves are superposed the h waves, so chosen
that a t z = 0 they annul one another but such that the derivative of
assumes the prescribed value F ( t ) .
Prescription of the two functions f(t) and F ( t ) on the plane z = O is
sufficient to determine completely the electromagnetic field. Let us
suppose, for example, th at the electric vector of the plane wave is polarized
along the x-axis. The Maxwell equations are then
+
Let E , be represented by the function +(z, t ) in either (54) or (57); then
on differentiating with respect to both t and z we find expressions for
aH,/at and aH,/az with the aid of (58) ; from these in turn it is a simple
matter to deduce the expression for H,.
(59)
E" - 2
H,
=
;
If :) +
z)
$1,
+ + h (t - E) - h ( t +
[f ( t - ;) - f ( 1 + ;) + h ( t - ;) + h ( 1 +
(1 -
f (1
These fields are clearly such that over the plane x
=
0,
31.
SEC.5.91
DISSIPATIVE MEDIUM
297
The electromagnetic field is, therefore, determined by the specification,
as independent functions of time on the plane z = 0, of any pair in (60)
containing both f ( t ) and F(t). Thus the electric ficld and i t s normal
derivative may be prescribed, or the electric and magnetic fields without
restriction on the derivatives.
6.9. Dissipative Medium ; Prescribed Distribution in Time.--We
pass on now to the more difficult problem of finding an electromagnetic
ficld in a conducting medium which reduces to certain prescribed funcwe shall allow I) t o
tions of the time on a plane z = constant,. Again
rcprescnt any rectangular component of a n electromagnetic vector
satisfying the equation
(61)
It will be convenient in what follows to introduce two new constants.
(62)
Note that in a dissipative medium a differs from the phase velocity
v = c / n of a harmonic component. I n this notation the wave equation
assumes the form
Particular solutions of (63) that are harmonic in time were investigated
in Sec. 5.2 and found to be
$, = ( A e i k z + B e - - i k z ) , - i u t .
(64)
Assuming the coefficients A arid 6’ as well as the complex quantity
k to be functions of the frequency, we construct a general solution by
summation of the harmonic components.
Again, let us suppose that the wave function and its normal derivative
arc prescribed functions of time over the plane z = constant. For
convenience the origin of the reference system is located within the
plane so that the constant is zero.
I n physical problems the character of the function #(z, t ) is such that the
298
PLANS IVA Y E S I N U N B O U N D E D , ISOTROPIC M E D I A
[CHAP.V
infinite integral convergra uniformly. Assuming this convergence, it.
may be differentiated with rcspcct t o the parameter 2 . l
(67)
2
=
&[A ( u ) e z k z B ( ~ ) e - ~ ~ ~do.
]e--i~~
Jewrn
On the initial plane, (65) and (67) reduce t o
f(t) =
[A(u)
J-m
+B ( ~ ) ] e - ~ ~ d w ,
m
(68)
F(t)
=
[: i k [ A
(w) -
.
dw.
m
Again the coefficients of c z w
may
t bc interpreted as the Fourier trans..
forms of f ( t ) and P ( t ) respectively.
(69)
Solving for A and R :
When these values are reinserted into (65) one obtains, after inversion
of the order of integrat,ion, the expression
Thus far the analysis has not deviated formally from that of the
problem in a nondissipative medium. At this point, however, we encounter a difficulty due to the complexity of k and we shall have to digress
momentarily t o discuss the representation of the function sin k z l k as a
definite integral. The basis of this representation is an integral due to
Gegenbauer2 which in the simpler case a t hand reduces to
See for example Carslaw, “Introduction to the Theory of Fourier’s Series and
Integrals,” Chaps. IV arid VI, Mncmillan, 1921.
WATSON,“ A Treatise on the Theory of Bessel Functions,” p. 379, Cambridge
University Press, 1922. Equation (72) is a special case of (691, p. 411, derived in
Sec. 7.7.
SEC.
299
DISSIPATIVE M E D I U M
5.91
J o ( u sin 4 sin 6) denotes the zcro-order Bessel function of argument
u sin 4 sin 6.
Let us make the following substitutions:
(73)
where X and
p
are real or complex constants.
.u
u sin 4 = z + q~),
2
u cos 4 = u ( p
cos 0 =
B
-)
21
- p),
sin 6 =
sin Od6
=
--.doV
I n terms of these new parameters, (72) becomes
(75)
sin
d ( P
+ X)(P +
Y)
Observe that the function to the left of Eq. (75) is the Fourier transform
of a function g(p) defined as follows:'
g(P) = 0, when
1/31 > v,
Finally, let us assign t o the parameters in this general formula the particular values
n
21
6
p
= ->
a
= w,
X
=
2bi,
p
= 0.
The desired result is then
After this small excursion we return once more to Eq. (71). I n the
second integral on the right-hand side, which for convenience may be
CAMPBELL
and
FOSTER,
loc. cit., No. 872.2.
300
P L A N E W A V E S I N UNBOUNDED, ISOTROPIC M E D I A [CHAP.V
denotcd G2, introduce (77) and then invert the order of integration.
where
(79)
The double integral within the brackets is the Fourier expansion of the
function +(t - p, p); consequently,
or, upon a slight change of variable,
To calculate the first integral to the right of Eq. (71), which shall be
denoted $1, we need only replace F ( a ) in $2 by f ( a ) and differentiate
partially with respect t o z. Since the limits in (81) are functions of z,
this differentiation must be effected according to the formula
T h e wave function which in a conducting m e d i u m reduces o n the plane
= 0 to f ( t ) and whose normal derivative reduces to F ( t ) i s determined everywhere by the expression
z
DISSIPAT IVE M E D I U M
SEC.5.101
301
It is apparent from (84) that the character of the propagation is profoundly modified by the presence of even a slight conductivity. A t
z = 0 the integrals vanish due to the identity of the limits. The init,ial
pulsef(t) then splits into two waves of half the initial amplitude as in the
nondissipative case. These partial waves are propagated to the right
and to the left with a velocity a which is necessarily independent of
frequency, and also independent of the conductivity. They are furthermore attenuated exponentially in the direction of propagation, as might
have been anticipated. Note that the attenuation factor is that
approached by a harmonic component in the limit as u2/e2u2+ cc).
a
=
5
2
8
=
188.3 u
$,
which is identical with (53), page 277. The initial function is no longer
propagated without change of form, for the integrals represent contributions to the field persisting for an infinite time a t points which have been
traversed by the wave front. The wave now leaves in its wake a residue or
tail which subsides exponentially with the time. The nature of these contributions will be clarified by the numerical example to be discussed in
Sec. 5.11.
6.10. Dissipative Medium ; Prescribed Distribution in Space.-The
initial conditions are frequently prescribed in another fashion. Let us
suppose that a t the instant 1 =: 0 the field and its time derivative a+/at
are specified as functions of the space coordinate z.
+
A harmonic analysis is called for in space rather than in timr arid a
general solution may be constructed from particular solutions of the type
discussed in Sec. 5.3.
(87)
J,
= e -3
2r (Aeiqt + Be--iqt)eikz,
where
The constants a and b are defined as in Sec. 5.9 but k is now a real variable.
To eliminate the common exponential factor it is convenient to take
302
PLANE WAVES I N U N B O U N D E D , ISOTROPIC MEDIA
[CHAP.V
The general solution is to be constructed by assuming the amplitudes
A and B to depend on k and then summing over all positive and negative
values of k.
(90)
u(2,
t)
=
j-mm
[A
+ B ( k ) ~ ~ * ~dk.] e ~ ~ "
(k)eiqt
For the derivative, wc have
The procedure is now essentially the samc as that described in the foregoing paragraph. A t t = 0 ,
G(z) =
+ B ( k ) ] J e i kdzk ,
J-
( i p [ A ( k )- B ( k ) ] - b [ A ( k )
and the coefficients are readily determined by the Fourier transform
theorem.
A(k) = (93)
B(k)
=
4aq
[(Q -
I:"
[(q
i b ) g ( a ) - i G ( a ) ] e ~d~a~, a
+ i b ) g ( a ) + i G ( a ) ] ~da.
-~~~
m
Substitution of these valucs into (90) leads, after simple reductions, to
as a definite integral.
v = at,
To thc parameters in (75) are assigned the values
q
=
-k,
X
=
b/a,
p =
-b/a,
SEC.5.101
DISSII’A T I V E
1l.I
EDI U M
303
whence it follows that
Let us denote the threc tctrms on the right-hand side of (84) by ul,u z ,
and u3 respectively. Then
where
(97)
By the Fourier integral theorcni, the double integral within the brackets
represents the expansion of + ( z - /?,/?), and conscqucntly
or, after a change of variable,
I n like manner, onc obtains
from this u1 is derived by differentiahn with respect to t.
(101)
1
2
u1 = - g ( z
+ at) + 21 g ( z - at)
-
The wave function which in a conducting medium reduces at t = 0 to g ( z )
and whose time derivative at the same instant i s G(z) i s determined everywhere
by the expression
(102)
+(z, t ) = e--bt
g(z
+ at) + 1 g ( z - at)
304
P L A N E W A V E S I N UNBOUNDED, ISOTROPIC M E D I A
[CHAP.V
In its essential characteristics this solution does not differ markedly
from that derived under thc assumptions of SCC.5.9. The initial field
distribution along the z-axis splits into two waves which arc propagatcd
to the right and left respectively. If the conductivity is other than
zero, a residue or tail remains after the passage of the wave front, and the
field subsides exponentially with the time.
6.11. Discussion of a Numerical Example.-The physical significance
of these results can be most easily elucidated in terms of a specific
example. Fresh water is a convenient medium for such a discussion,
for a t radio frequencies it occupies a position midway between the good
conductors and the good dielectrics. Let us assume the following
constants :
K~
= 1,
K~
=
81,
u =2
X lop4 mho/meter.
From these values we calculate
a
=
+ X lo8 meters/sec.,
b
b
- = 4.2 X
=
1.4 X lo5 SCC.-~,
meter-'.
U
It is usually easier to visualize a distribution and propagation along
a n axis in space than along one in time; for this reason we shall consider
(102) first. Let us suppose that a t t = 0 the field vanishes everywhere
except within the region -25 < z < 25, where its amplitude is unity.
when Iz/ < 25,
when IzI > 25,
for all values of z.
g(z) = 1,
g(z) = 0,
G(z) = 0,
+ +
I n Fig. 51a +[g(z
at) g ( z - at)] is plotted as a function of z for
various values of t , and in Fig. 51b the same as a function of t for various
values of z. These two figures represent the propagation of the initial
pulse in time and space in a medium of zero conductivity. The function
g(z - at) vanishcs if
z
< at
t
< -U1 ( z
- 25
+ 25,
or
z
> at
or
t
> a1 (z + 25).
or, if
- 25)
Likewise the negative wave g(z
z
<
-at
t
<
-a1 (25 + z )
- 25
+ at) is zero at all points for which
or
z >- -at + 25,
or
or
t
> -U1 (25
- 2).
305
DISCUSSION OF A N U M E R I C A L E X A M P L E
SEC.5.111
In Figs. 52a and 52b the same wave functions are multiplied by the
attenuation factor e-bt. Nothing has been stated regarding the origin
g
I
(2+at)
.
I
g(z-at)
z=o
n'"
1
t=0.45x1t6 sec.
t = 3.75 x 1 6 sec.
~
2=15 meters
z = 125 meters
- 1.0
-6
t = 8.25 x 10 sec.
z= 275 meters
- 1.0
I
I
I
1
I
1
1
1
-10-8 -6 -4 -2 0 2 4 6
t in micro-seconds
FIG. 51b.
8 10
-bt
e2 g(r-at)
t = 0.45 x 10-6 sec.
1
z=15 meters
I
t = 3.75 x
I
z = 125 meters
sec.
t = 8.25 x 10 sec.
-200
-100
t=O
i
t
-300
I
0
100
z in meters
FIG.52a.
200
300
nta
z = 275 meters
-10-8 -6 -4 -2
0 2 4
6
8 10
t in micro-seconds
FIG. 52b.
of the field, but i t is supposed that at some anterior time it was generated
by a systcm of appropriately disposed sources from whence the field is
propagated and attenuated in such a manner that at t = 0 it is distributed
306
PLANE W A V E S I N UNBOUNDED, ISOTROPIC M E D I A
[CHAP.V
as shown a t the top of Fig. 52a. I n tlie prescnt instance it is clear that
the wave startcd as two pulses a t some value of t < -10 X
sec.,
thc one a t a distant point on the positive x-axis, propagated to the left,
and the othcr at a point on the negative x-axis, propagated to the right.
I n the ricighborhood of t = 0, z = 0, they meet, pass through one another
with consequent reinforcement, and then continue on their way. Their
subsequent progress is indicated at t = 0.45, t = 3.75, and t = 8.25
microsec. The succession of events occurring a t a fixed point in space is
illustrated in the series of Fig. 526. Take, for example, z = 275 meters.
At t = -9 microsec. the front of a negative wave arrives a t this point,
traveling toward smallcr valucs of x. An intcrval of 1.5 see. is now
required for this wave to pass, in the course of which its amplitude is
diminished exponcntially. From t = -7.5 to t = 7.5 microsec. all is quiet;
then arrives the front of the positive wave with greatly reduced amplitude, for it has passcd through unit value at the origin and traveled on to
the positive point of observation.
This simple history of a field must now be modified t o take intcl
account the contributions of thc integral terms in Eq. (102). I n most
cases the evaluation of thcsc tcrnis necessitates numerical or mechanical
processes of intcgration, but thanks to the simple functional form of
g(z) in the present instance the intcgration can be carried through
analytically for sufficicnlly small values of x and t. From the theory of
Bessel functions we take the following relations :
Introducing tlie appropriate valuc of z lradn to
An investigation of the convergence of these series indicates that in the
present instance no serious error will be committed if we limit ourselves
t o the first two terms, provided that x does not exceed 300 meters. The
SEC.5.111
307
DISCUSSION OF A N U M E R I C A L EXAAlPLE
Bessel functions will, therefore, be represented approximately by
These funct>ionsarc now to be multiplied by g(P) and integrated with
respect to P. With regard to the limits we note the following. Suppose
that t is positive. If z - at > 25 or z
at < -25, the resultant field
+(z, t ) is zero. If -at
25 < z < at - 25, the partial fields g ( z
at)
and g(z - at) are zero but the integrals do not vanish, and this no matter
+
+
+
t0.G
z in meters
FIG.53n.
how large t may become. A residual field lingers in the region previously
traversed by wave pulses. If z
at > 25 and z - at < -25, thc limits of
integrat,ion may be fixed a t k25, for beyond thcsc values g ( P ) is zero.
Such limits are, therefore, appropriate for the entire region between
the pulses, -at
25 < z < at - 25.
The results of the integration are plotted in Figs. 53a and 538. The
effective field +(z, t ) is indicated by a hcavy black outline; the partial
+
+.
-1
10
t in micro-seconds
FIG.53h.
fields expressed by the various terms of (102) are shown dotted. Figure
53a illustrates the distribution along the z-axis a t the instant t = 3.75
microscc. We note the beginning of a deformation of the initially rcctangular pulse, the top sloping forward and the residual tail trailing
308
P L A N E W A V E S I N UNBOUNDED, ISOTROPIC M E D I A
[CHAP,V
behind. As the wave progresses, the sharp front decreases in height,
so that tail and pulse eventually merge into a rounded contour. I n
Fig. 5% observations are made a t the fixed point z = 125 meters. At
times anterior to t = -3 microsec. a field existed here, a field which was
wiped out by the passage of the negative pulse. At t = 3 microsec. the
positive wave arrives, followed by its tail which persists after the passage
of the pulse.
The essential difference between the solutions represented in Eqs.
(84) and (102) is due to an interchange of the roles of space and time.
To illustrate the behavior of the field, let it be assumed that the initial
time function f ( t ) in Eq. (84) is a rectangular pulse defined by
when Jtl < 1 X
sec.,
when It/ > 1 X
sec.,
for all values of t.
f(t) = 1,
f(t) = 0,
F(t)
= 0,
-_ bZ
The functions -f t
- and -f t - - are plotted in Figs. 54a
ea
2
e 2a
and 54b at various values of z and t. In virtue of the definition off@)
i t is clear that the third term on the right-hand side of (84) contributes
b
-2
( + :)
( :)
*
8-;
fct-i)
;2
4
f(t+S)
,
I
-.=lo meters
I
z=100 meters
t=O
1.0
t=0.5 x 1f6 sec.
t = 2 x 1 C 6 sec.
- 1.0
t = 8 x 1 f 6 sec.
t
,300 -200
t in micro-seconds
FIG.54a.
z
nothing if t - a
> 1 or
t
0
-100
2
+ -az < -1.
100
200
300
in meters
FIG.54b.
2
If 1 - a
< t < -az - 1, thc
vanish but the integral does not.
func-
In this
interval, which lies between the pulses on the t-axis, the limits of integra-
309
THEORY OF LAPLACE TRANSFORMATION
SEC.5.121
tion are +1. In Figs. 55a and 55b are shown the deformations of an
initially rectangular pulse a t fixed points in time and space due to the
tail of the wave.
t = 3.75 x 10-6
Sec.
- 0.4
- 0.2
,
I
f
,
I
,
,
,
,
,
,
,
FIG.55a.
When studying these figures it must be kept in mind that two preexisting waves coming from sources at infinity have been assumed
which superpose to give just lbe correct distribution at t = 0 or z = 0.
,I
: ;p,,
11.0
z = 125 meters
, , , , , , ,
-12
-10
-8
-6
,
-4
-_
~,
b , Ib,
;2
_-
-2
t in micro-seconds
FIG.55b.
A single wave whose amplitude is zero for all values of t and z less than
zero can be generated only by a source a t the origin constituting a
singularity in the field-a problem we have not yet considered.
5.12. Elementary Theory of the Laplace Transformation.-Among
the oldest and most important of the various methods devised for the
solution of linear differential equations is that of the Laplace transformation. If in the equation
the dependent variable w is transformed by the relation
(104)
W(Z) =
J u ( t ) o P dt,
it will be found in many cases that u(t) satisfies a differential equation
which is simpler than (103), and in fact that u(t) is frequently an elementary function whereas w(z) cannot, in general, be exprcssed directly
in terms of elementary transcendentals. 1 When the coefficients p ( z )
and q ( z ) are functions of the independent variable z, it is necessary to
1 INCE,“Ordinary Differential Equations,” Chaps. VIII and XVIII, Longmans,
London, 1927.
310
P L A N E WAVES I N UNBOUNDED, ISOTROPIC M E D I A
[CHAP.V
choose the path of integration properly in the complex plane of t in
order thsl; (104) shall be a solution of (103);the independent solutions are
then distinguishcd by the choice of path. The result is a representation
of the particular solutions as contour integrals. If, however, the coefficients p and q are constant, the equation has no other singularities than
the essential one at infinit,y and the paths of integration can be chosen t o
coincide with the real axis. The solutions are t'hen expressed in terms
of infinite irilegrttls intimately related, if not equivalent, to a Fourier
integral representation.
I n recent years the application of the Laplace transformation to
linear equations with constant coefficients has been considered with
growing interest, on the part of physicist,s and engineers, and its orderly
and rigorous procedure appears to be rapidly replacing the quasi-empirical
methods of the Heaviside operational calculus. Although the Laplace
met'hods lead to no results unobtaiiiable by direct applicat,ion of Fourier
integral analysis, they do offcr certain definite rtdvaritagcs from the
st'andpoint of convenience. They are particularly well adapted t o the
treatment of functions which do not, va'nish a t
~0 a,nd which consequently fail to satisfy the condition of absolute convergence, and they
lend in a simple and direct manner t o the soliit>ionof a n equation in terms
of its initial conditions. On the other hand, they can be applied only t o
problems in which the field may be assumed to be zero for all negative
values of the independent variable; in short, the Laplace integral is
applicable to problems wherein all the future is of importance, but the
past of no consequcnce.
We shall approach the theory of the Laplacc transform from the
standpoint of the Fourier integral atid suggest rudimentary proofs for
the more important theorems. Let us consider a function f(t) which
vanishes for all negative values of t . Then apart from a factor 1,'the Fourier transform of f(t) is
+
F(w) =
provided only that the integral
f ( t ) e c i W tdt,
6"if(t)l
dt exists.
In case f(t) does not
vanish properly a t infinity thc integral fails to converge, but under
certain circurnstanccs absolute conwrgerice can be restored by introThe Fourier transform of e - Y t f ( t ) is then
ducing a factor e-Yt.
If there exists a real number y,such t h a t
Next Page
SEC.
5.121
311
T H E O R Y OF LAPLACE T R A N S F O R M A T I O N
then f(t) will be said to be transformable. The lower bound y a of all
the 7’s which satisfy (107) is called the abscissa of absolute convergence
Suppose, for example, that f(t) = eht when t 2 0 and f(t) = 0 when
t < 0. Thenf(t) is transformable in the scrise of (106) and y > b = y a .
On the other hand a function f ( t ) = et2 is not transformable, for there
exists no real number y leading to the corivergence of (106).
The inverse transformation follows from thc reciprocal propertics of
the Fourier transforms, Eqs. (21) and (22).
where y > y a . The function defined by thc intcgrd on the right is
equal to f(t)e-Yt only for positive \ d u e s of t , and vanishcs whenever
t < 0. To give explicit cxprrssion to this €act wc introduce the unit
step function u(t) which is zero when t < 0 and c.qi,:tl to unity when t > 0.
The inverse transformation may thus be writ tcn
If y a 5 0, one may pass to the limit as y -+ 0 aftcr the integration3
indirated in (106) or (109) have bccn effected and thus determine the
Fourier transforms of functions not othcrwise integrable. Such a step
is unessential, and the Laplace transform contains this convergence
factor implicitly. Let us introduce the complcx variable s = y
iw.
Then the Laplace transform of f ( t ) , henceforth dcsignsted by the operator
L[f(t)I, is
+
(110)
~ [ f ( t )=
l
f(t)e-’i at
=
~(s),
Ws)
> Ya,
where Re(s) is the usual abbreviation for “ t h e real part of s,” and y a
is determined in each case by the functional properties of f ( t ) . The
inverse transformation is reprcwntcd by a n integral taken along a path
in the complex plane of s.
(111)
1
L-1[F(s)l = 27Fi
1
-+zm
--z
F(s)ets as = f ( t > . u(t),
where y > ya. The Laplace transformation can be interpreted as a
mapping of points lying on the positive real axis of t onto that portion
of the complex plane of s which lics to the right of the abscissa y a . The
domains of f(t) and its transform F ( s ) are indicated in Fig. 56.
From the linearity of the operators it follows that if L[f,(t)]= F l ( s ) ,
U fs(t)l = F ~ ( s )then
,
(112)
L[fl(t) -t
fi(t)l
= Fl(S)
k Fds).
Previous Page
312
P L A N E W A V E S I N U N B O U N D E D , ISOTROPIC M E D I A
[CHAP.V
If, furthermore, a is any parameter independent of t and s, then
L[uf(t)]= uiqs).
(113)
The application of the Laplace transform theory to the solution of
differential equations is based on a theorem concerning the transform of
d
a derivative. Assume that f(t) and its derivative -f(t) are transformable
dt
in the sense of (107), and that f(l) is continuous at t = 0.
Then by partial integration
hence, if L[f(t)]
=
F ( s ) , it follows that
L
K2)I
--
=
sF(s) - f(0).
The importance of this thcorem lies in the fact that it introduces the
initial valuef(0) of the functionf(t). Integrating partially a second time
t
Donlain of f ( t )
FIG.56.-A
Laplace transformation effects a mapping of points on the positive real axis of
t onto the shaded portion of the s-plane.
leads to a theorem for the transform of a second derivative which involves
not only the initial value f(0) but also the value of the derivative
(df / d t ) t=o.
Thus the transform of the dependent variable of a second-order differential
equation is expressed in terms of the initial conditions. These results
have no analogue in the theory of the Fourier transform.
The “ Faltung ” or “folding” theorem is a particular application of
a well-known property of the Fourier integral.’ Suppose again that
1 See for example Bochner “Vorlesungen uber Fouriersche Integrale,” and Wiener,
“The Fourier Integral,” p. 45, Cambridge University Press, 1933.
313
THEORY OF LAPLACE TRANSFORMATION
SEC. 5.121
L[f1(2)] = F l ( s ) , L[fz(t)]= Fz(s), and that we wish to determine the
Laplace transform of the product function fl(i!) fz(t).
(116) L[fl(l) * fdt)l
=
fz(t)e-"f dt
=
fl(1)
fl(t) .
[-L
2 ~ i
1-
y2
1
Fz(a)eutda e-*tdt
Jy*'im
-i m
y2+i
m
da F,(a)
=2?ri
LJ
O1
fl(t)e-(*-u)L
dt,
y2-j-
and, hence, the first theorem:
yz+i
(117)
L[fl(t) .f 4 ) l
YZ
2 ~J
i
= --
> 7aZl
F1(s
- u ) . Fz(a) du,
y?--im
Re(s - a)
> Yale
Of more frequent use is a second Faltung theorem on the inverse transformation of a product of transforms. We wish to determine a function
whose Laplace transform is F,(s) . Fz(s).
=
-!L
27ri
Y+i
-
Fl(s) [Jmfz(7)e--sr d ~ esf
] ds
y--im
=
d7 fi(T) .
=
som
Fl(s)e*(f-r)ds
fz(.) . fl(t
-
T)
. u(t
-
7)
dr.
Since the unit function vanishes for negative values of the argument,
the upper limit of 7 must be t and hence
or
Geometrically, the product F l ( s ) F z ( s ) represents the transform of an
area constructed as follows: The function f l ( 7 ) is first folded over onto
the negative half plane by replacing T by -7, and then translated to the
right by an amount i!. The ordinate fl(t - T ) is next multiplied by
fZ(.) dr and the area integrated from 0 to t .
314
PLANE W A V E S IN UNBOUNDED, ISOTROPIC MEDIA
[CHAP.V
A set of operations applied repeatedly in the course of our earlier
discussion of the Fourier integral may be referred t o as translations. Let
a be a parameter independent of s and t.
(121)
L[eatf(t)] =
f(t)e--(s--a)tdt
=
F ( s - a).
Equation (121) is obviously valid for both positive and negative values
of a. Next consider the transform of f ( t - a).
(122)
L[f(t - a)]
f ( t - a)e-st dt =
=
e-OS
J-:j(7)ecsr d7.
Now i t has been postulated throughout that the function f ( t ) shall vanish
for all negative values of the argument arid hence the lower limit -a
may be replaced b y zero. As a consequence, we obtain
L [ f ( t - a)] = ecaaF(s),
Likewise, we may write
a > 0,
f(t>= 0
when t
< 0.
when t
< a;
but one must note with care that this last result applies only to functions
f ( t ) which by definition vanish for t < a. A replacement of t by t - a or
t a rcprescnts a translation of f ( t ) a distance a to the right or left
+
respectively.
I n addition t o thc foregoing fundamental theorems there exist a
number of clernentary but useful relations which may be deduced
directly from (110). The following formulas are set down for convenience and their proof is left t o the reader.
f(0)
=
lim f ( t )
t
i
m
lim s F ( s ) ;
s
im
=
lim sF(s).
s+o
Evaluation of the contour integrals that occur in the inverse transformations (111) is most easily effected by application of the theory of
residues. The functions f ( t ) with which one has t o deal are analytic
SEC.5.121
THEORY OF LAPLACE TRANSFORMATION
315
functions of the complex variable t except at a finite number of poles.
It will bc recalled that in the neighborhood of such a point, say t = a,
the function may be expanded in a series of the form
where @(t)is analytic near and a t a and, therefore, contains only positive
powers of t - a. The singularity is said to be a pole of order m if m is
a finite integer. If, however, an infinite number of negative powers is
necessary for the rcprcsentation of j ( t ) , the singularity is said to be essential. If now the function f ( t ) is integrated about any closed contour C
in the t-plane which encircles the pole a but no othcr singularity, it can be
shown that all the terms in the expansion (131) vanish with the exception of the one whose coefficient is bkl. The result is
f(t) dt
(132)
=
27rib-1.
The coefficient bkl is called the residue a t the pole. In case the contour
encloses a number of poles, thr: value of the integral is equal to 27ri times
the sum of the residues. In particular, if f(t) is analytic throughout the
region bounded b y the contour C, the integral vanishes. T h w e results
are a consequence of a fundamental theorem of function theory. If j ( t )
is analytic on and within the (<ontourC,-which is to say that it may be
expanded in terms of positivcl powers of t only-then the value of f(t)
at any interior point a may b r expressed as
(133)
This is known as Cauchy's throrem.
If the function f ( t ) vanishes properly a t infinity, a n integral which is
extended, as in (111)' along an infinite line may be closed by a circuit of
infinite radius and thus made subject to the theorems of the preceding
paragraph. The restrictions t,hat govern the application of the theory of
residues to the evaluation of infinite integrals are specified in what is
known as Jordan's lemma quoted as follows:
If & ( z ) --+ 0 uniformly with regard t o arg z as IzI 3 m when 0 5 arg z 5 a,
and if Q ( z ) is analytic when both Iz/ > c (a constant) and 0 2 arg z Ia,then
lim
P+
where
e-Q(z)
dz
=
0,
=
r is a semicircle of radius p above the real axis with center at the origin.'
WHITTAKER
and WATSON,lac. cit., p. 115. In Chaps. V and V I of this reference
the reader will find all that is essential for the application of function theory to the
solution of physical problems.
316
PLANE W A V E S I N UNBOUNDED, ISOTROPIC M E D I A [CHAP.V
As stated, this lemma applies to the evaluation of integrals along
the real axis from - 00 to
0 0 , but it may be modified to include
integrations along an imaginary axis ; by appropriate changes the circuit
may be transferred from the half plane lying above the real axis to that
below, or from the right half plane to the left. I n case &(z) has poles
within the closed contour a t distances IzI < c from the origin, the value
of the integral is equal not to zero but to the sum of the residues.
The bearing of these theorems on the theory of the Laplace transform
may be illustrated by two elementary examples. Consider first the step
function defined by
+
f(t) = 0,
j ( t ) = 1,
(134)
when t
when t
< 0,
> 0.
It will be noted in passing that (134) does not fall within the province of
functions admitting a Fourier analysis, for the integral
is nonconvergent. The Laplace transform of (134) is
J- If(t)l dt
(135)
We verify this result by applying the inverse transformation (111).
Now F ( s ) has a pole of first order at s = 0. The abscissa of absolute
convergence is zero, whence it is necessary that Re(s) > 0. The integration is to be extended along any line parallel to and to the right of the
imaginary axis. The integrand of (136) satisfies the conditions of
Jordan’s lemma and the path of integration may be closed by an infinite circuit. If t > 0, the path may be deformed to the left so that
Re(s) -+ -a. The closed contour y - i.o t o y
i m to - 00
im to
- m - iw to y - im contains the pole at s = 0. The residue is unity,
and hence by (132) the right-hand side of (136) is also unity for t > 0.
Since F ( s ) is only defined in the right half plane, a question may arise as
to the justice of extending the integration to the left half plane. One
must be careful to distinguish between a function and the representation
of that function in a restricted domain. l/s is a representation of F ( s )
when Re(s) > 0, but in view of possible discontinuities in F ( s ) the two
functions need not coincide a t all when Re(s) < 0. Now along the line
y .- i
to y
i m we are integrating the function l / s ; for purposes of
integration we may make use of any of its analytic properties. The
analytic continuation into the left half plane is of the function l / s , and
not of F ( s ) .
+
+
+
317
THEORY OF LAPLACE TRANSFORMATION
SEC. 5.121
When t < 0, the vanishing of the integrand may be assured by
deforming the path to the right. Within the closed contour y - im to
y
i co to m i to co - ia) to y - i.o there are no singularities and
the right-hand side of (136) is zero.
We have verified that the unit step function may be expressed
analytically by
+
+
u(t) =
(137)
- est ds,
~
y
> 0.
It is interesting to apply the translation theorem (123) to this result,
putting a = z/v.
[(
31
L u t-or
-_
2
= e
V'FF(S)=
1
-;s,
-e
S
(139)
From the preceding discussion it is apparent that (139) vanishes when
2
t --
21
< 0, and is equal to unity when t - 22-, > 0.
I n Figs. 57a and 57b
is represented as a function of t and z respectively.
u (t -Z/U)
u (t - Z / U )
t'= z/u
u(t-z/u)=O, t < Z / U ,
~
u ( t - z / u ) = l , t>z/u.
FIG.67a.
t
z'= ut
I
u(t--Z/U)=O, z> ut,
u ( t - z / v ) = l , z< ut.
FIG.57b.
As a second example let us imagine that a harmonic wave is switched
on suddenly a t t = 0 and continues indefinitely.
when t
when t
< 0,
> 0.
As in the previous case we can verify (141) by applying the inverse
transformation
w
2 0.
318
P L A N E WAVES I N UNBOUNDED, ISOTROPIC M E D I A
[CHAP.V
If t > 0, the path is again deformed to the left. The enclosed pole is
located a t s = - i w ; on applying the Cauchy theorem we see that (142)
is indeed equal to rtwt.
On the other hand if t < 0, the path is deformed
t o the right and the integral vanishes. I n general,
6.13. Application of the Laplace Transform to Maxwell's Equations.Let E, = f(z, t ) and 11, = g(z, t ) be the components of a plane electromagnetic field. The functions f(z, t ) and g(z, t ) are assumed to be transformable in the sense of (107) a d t o vanish for all negative values of t ;
moreover, they are related t,o one another by the field equations
3.f
dz
(144)
+ p %d t
=
0,
ddz! ' + € - +df
+f=O.
dt
Treating z as a parameter, the Laplace transform of this system with
respect to t proves to be
Upon elimination of G(z, s) an ordinary, inhomogeneous equation for
F ( z , s) is obtained in which s enters only as a parameter.
Since the derivative of g(z, t ) with respect to x a t the instant t = 0 can
be expressed in terms of f(z, 0) and (dj/dt)t30 through (144), this last
result is equivalent to
Byf(z, 0) one means strictly the limit of f(z, t ) as t
fl(z)
(148)
=
lim
f(z, t),
t-0
---f
0.
The functions
8.f (2, t ) ,
f 2 ( z ) = lim t-+O
dt
represent the initial state of the field and are assumed to be known.
writing h2 = pes2 pus, wc obtain finally
+
(149)
d2F - h2F =
~
dz2
_ h2
_ fd.4 - pdz(z)
s
=
Z(Z, s>.
Upon
SEC.5.131
319
A P P L I C A T I O N OF T H E L A P L A C E TRANSFORM
The general solution of (149) consists of a solution of the homogeneous equation
containing two arbitrary constants, t,o which is addcd a particular solution of (149). One can verify casily enough that (149) is in fact satisfied
by
(151) F ( z , s) = Aeh8
+ Be-hz +
in which h is that root of v'hz which is positive when h2 is real and positive. The constants A and B are to be determined such as to satisfy
specified boundary conditions at the points z = z1 and x = z2. After
A and B have been evaluated in terms of s, a n inverse transformation
Scads from (151) back to a solution f(z, t ) of (144) satisfying both
initial and boundary condition?.
T o illustrate, consider first the elementary case in which the field is
zero everywhere a t the initial instant t = 0. Then Z = 0 and we are
concerned only with the solution of th e homogeneous cquation. Let us
suppose that a t z = 0 there is a source switched on a t the instant t = 0
which sends out a wave in the positive z-direction and whose intensity
at the origin is f(0, t ) = f 3 ( t ) . Thus the second boundary condition a t
z = z 2 is replaced in th e present example by the stipulation that the field
shall be propagated to the right along the z-axis. The transform of the
boundary value is represented by
(152)
F ( 0 , s)
=FdS) =
L[fa(t)l.
Since the wave is to travel to the right, the constant A must be placed
equal to zero. This is clear nhcn u = 0, for then h = s/a, and by the
translation theorem of page 314
(1Ti3)
( t 2 0,
(154)
2
> 0).
If the conductivity is not equal to zero, the solution is more difficult
In this case we shall write h2 =:
(s
~ +G-
$
')u
-
$ and apply th e
320
PLANE WAVES I N UNBOUNDED, ISOTROPIC M E D I A
N CHAP.^
integral
-~
____
Upon substitution of 2 / p z
+ z2 = at, this reduces to
where u = 1 / 6 , b = a / 2 ~ as
, defined on page 297. Finally, by differentiation we obtain
where d(z, t ) is a function defined by
=
(159)
0,
when 0
<t <
:,
dz,t )
With this transform at our disposal, we now write
-_bz
(160)
F ( z , s ) = F3(.s)e-hz = L[j3(t)]ea . e
-2
a
- uL[f3(t)]
. L[+].
To this we apply the translation theorem and the Faltung theorem of
page 313, followed by the inverse transformation of the complete equation.
or, with a minor change of variable,
(162) f(z, t ) = e-%'f,(t -
:)
WATSON,loc. n't., p. 416, (4).
SEC.5.141
DISPERSION I N DIELECTRICS
321
‘This function, as the reader should verify, satisfies the wave equation in a
dissipative medium and represents a wave traveling in the positive direction which at z = 0 reduces to f3(t) for all values of t > 0.
DISPERSION
6.14. Dispersion in Dielectrics.-A pulse or “signal” of any specified
initial form may be constructed by superposition of harmonic wave
trains of infinite length and duration. The velocities with which the
constant-phase surfaces of these component waves are propagated have
been shown to depend on the parameters E , p , and CT. In particular, if
the medium is nonconducting and the quantities E and p are independent
of the frequency of the applied field, the phase velocity proves to be
constant and the signal is propagated without distortion. The presencb
of a conductivity, on the other hand, leads to a functional relation
between the frequency and the phase velocity, as well as to attenuation.
Consequently the harmonic components suffer relative displacements in
phase in the direction of propagation and the signal arrives at a distant
point in a modified and perhaps unrecognizable form. A medium in
which the phase velocity is a function of the frequency is said to be
dispersive.
At sufficiently high frequencies a substance may exhibit dispersive
properties even when the conductivity u due to free charges is wholly
negligible. In dielectric media the phase velocity is related to the index
At frequencies
.
less than
of refraction n by v = c/n, where n = Gm
lo8 cycles/sec. the specific inductive capacities of most materials are
substantially independent of the frequency, but they manifest a marked
dependence on frequency within a range which often begins in the ultrahigh-frequency radio region and extends into the infrared and beyond.
Thus, while the refractive index of water at frequencies less than los
is about 9, it fluctuates at frequencies in the neighborhood of loLo
cycles/sec. and eventually drops to 1.32 in the infrared. Apart from
solutions or crystals of ferromagnetic salts, the dispersive action of a
nonconductor can be attributed wholly to a dependence of K~ on the
frequency.
All modern theories of dispersion take into account the molecular
constitution of matter and treat the molecules as dynamical systems
possessing natural free periods which are excited by the incident field.
A simple mechanical model which led to a strikingly successful dispersion
formula was proposed by Maxwell and independently by Sellmeyer.
A further advance was accornplished by Lorentz who extended the
theory of the medium as a fine-grained assembly of molecular oscillators
and who was able to account at least qualitatively for a large number of
electrical and optical phenomena. According to Lorentz, however, these
322
P L A N E W A V E S I N U N B O U N D E D , ISOTROPIC M E D I A
N CHAP.^
molecular systems obeyed the laws of classical dynamics; it is now known
that they are in fact governed by the more stringent principles of quantum
mechanics. Following upon the rapid advances of our knowledge of
molecular and atomic structure revisions were made in the dispersion
theory which a t present may be considered to be in a very satisfactory
state.
Both the classical and the quantum theories of dispersion undertake
to calculate the displacement of charge from the center of gravity of an
atomic system as a function of the frequency and intensity of the disturbing field. After a process of averaging over the atoms contained
within an appropriately chosen volume element, one obtains an expression
for the polarization of the medium; that is to say, the dipole moment per
unit volume. The classical result corresponds closely in form to the
quantum mechanical formula and leads in most cases to an adequate
representation of the index of refraction as a function of frequency. We
shall confine the discussion, therefore, to the case in which the electric
polarization in the neighborhood of a resonance frequency can be
expressed approximately by the real part of1
a2
P= -
w ; - a2 -
iwg
eoE.
By the electric field intensity, we shall now understand the real part of
the complex vector
(2)
E
=
Eo e-iut.
The constant a2 is directly proportional to the number of oscillators per
unit volume whose resonant frequency is WO. The constant 20 is related
to this resonant frequency by
(3)
2;
= W; - +a2,
such that WO + wo a t sufficiently small densities of matter. The constant
g takes account of dissipative, quasi-frictional forces introduced by
collisions of the molecules. The constants Wo and g which characterize
the molecules of a specific medium must be determined from experimental data.
At sufficiently low incident frequencies W, the polarization P according
to (1) approaches a constant value
(4)
1 For the derivation of this result, see Lorentz, “Theory of Electrons,” or any
standard text on physical optics such as for example Born, “Optik,” Springer, 1933,
or Forsterling. “Lehrbuch der Optik,” Hirzel, 1928.
SEC.5.141
DISPERSlOfi I N DIELECTRICS
3 2.1
and, since the specific inductive capacity is related to the polarization by
P = ( K - l)~oE,
(5)
one may express K in terms of the molecular constants.
When, however, the incident frequency is increased, further neglcct
of tlie two remaining terms in the denominator becomes inadmissible.
In that case we shall define by analogy a complex inductive capacity K'
through either of the equation:.;
(7)
P
= (K'
- l)eoE,
D
=
K'QE,
whence from (1) we obtain
In terms of this complex parameter the Maxwell equations in a medium
whose magnetic permeability i;i po are
(9)
as a consequence of which the rectangular components of the field vectors
satisfy the wave equation
A plane wave solution of (10) is represented by
9
(11)
where
(12)
k
=
=
+oeik"-iwt
,
--*+
w
c
=a
iB,
so that
(13)
The wave is propagated with :a velocity v = w / a = c/n, but a and the
refractive index are now explicit functions of the frequency obtained by
introducing (8) into (13).
I n gases and vapors the density of polarized molecules is so low that
K' differs by a very small amount from unity.
The constant a2 is thercfore small, so that 60 differs by a negligible amount from the natural
324
P L A N E W A V E S I N UNBOUNDED, ISOTROPIC M E D I A
[CHAP.
V
frequency wo and the root of K' can be obtained by retaining the first two
terms of a binomial expansion. Thus
When the impressed frequency w is sufficiently low, the last two terms
of the denominator may be neglected so that
The index of refraction and consequently the phase velocity in this case
are independent of the frequency; there is no dispersion.
If the impressed frequency w is appreciable with respect to the
resonant frequency wo but does not approach it too closely, the damping
term may still be neglected. \WE - w21 >> wg,
The attenuation factor is zero and the medium is transparent, but the
refractive index and the phase velocity are functions of the frequency.
If w < wo, n will be greater than unity and an increase in w leads to a n
increase in n and a decrease in v. If w > wo, the refractive index is less
than unity but an increase in w still rcsult,s in an increase in the numerical
value of n. The dispersion in this casc is said t o be normal.
Finally let w approach the resonance frequency w 0 . Upon resolving
(14) into its real and imaginary parts one obtains
InqFig. 58 are plotted the index of refraction and the absorption coefficient /?c/o of a gas as functions of frequency. The absorption coefficient
exhibits a rather sharp maximum a t wo, so that in this region the medium
is opaque to the wave. As w rises from values lying below wOl the index
n reaches a peak at w1 and then falls off rapidly to a value less than unity
at 02,
whence it again increases with increasing w ) eventually approaching
unity. When-as in the region wlw2-an increase of frequency leads to a
decrease in refractive index and an increase of phase velocity, the dispersion
i s said conventionally to be anomalous. According to this definition
dispersion introduced by the conductivity of the medium and discussed
in the earlier sections of this chapter is anomalous. In fact it might
SEC.5.151
DISPERSION I N METALS
325
almost be said that the “anomalous” behavior is more common than the
<I
normal.”
In the case of liquids and solids these results must be modified to some
degree since the assumption of a small density of matter is no longer
tenable, but the general form of the dispersion curves is not greatly
affected. Discussion has been confined, moreover, to the behavior of the
propagation constants in the neighborhood of a single resonance frequency.
Actually a molecule is a complicated
dynamical system possessing infinite
series of natural frequencies, ea,eh affecting the reaction of the molecule to the
incident field. The location of these
natural periods cannot be determined
0
by classical theory; by proper adjustment of constants to experirncntal data,
an empirical dispersion formula can be
w2
set up, of which (8) is a typical term
and which is found to satisfy the observ- cuy:I;
ed data over an extensive range of fre- Bonance frequency.
quencies.
6.16. Dispersion in Metals.-A surprisingly accurate theory of the
optical properties of metals may be deduced from the following rather
crude model. One imagines the fixed, positive ions of a metallic conductor to constitute a region of constant electrostatic potential. Within
this region a cloud or “gas” of freely circulating electrons has attained
statistical equilibrium, so that on the average the force acting on any
particular conduction electron is zero, and apart from normal fluctuations
the total charge within any vcllume element is zero. The application of
an cxt,crnal field E results in a general drift of the free electrons in the
dirertion of the field. The motion is opposed by constantly recurring
collisions a t the lattice points occupied by the heavy ions; the consequent
transfer of momentum from the drifting electrons to the lattice points
results, on the one hand, in thermal vibrations of the ions, on the other in a
damping of the electron motion. The exact nature of the local force
exerted on a conduction electron is a somewhat delicate question, but in
so far as one can ignore the contribution of bound electrons there appears
to be justification for the assumption that the effective field intensity is
equal to the macroscopic field E prevailing within the conductor.
Let r be the displacement from the neutral position of a free charge e
whose mass is m. The charge is acted on by an external force eE and its
dr
motion is opposed by a force -mg - proportional to the velocity which
dt
~ ~ ~ e ;
326
PLANE WA V E S I N CNBOUNDED, ISOTROPIC M E D I A
[CHAP.V
accounts empirically for the dissipative effect of collisions. The equation
for the average motion of e is, therefore,
m d2r
- + mg dr = eEoe-wt,
dt
at2
whose steady-state solution is
r
(19)
=
1
e i E -&t
oe
g - i w wm
.
If there are N free electrons in unit volume, the current density is
Then by analogy with the static conductivity defined by J = aE, one
is led to introduce a complex conductivity a' which in virtue of (20) is
e2
N= -. m
g - iw
The propagation constant and attenuation of a plane wave are now
obtained in the usual way from the relation k 2 = p e w 2 iu'pw. The
resonance frequencics of the bound electrons of metallic atoms arc known
t o lie far in the violet or ultraviolet region of the spectrum, so that in
the visible red, the infrared and certainly a t radio frequencies the inductive capacity of a metal can be safely assumed equal t o e,,. Moreover
if the conductor is nonferromagnetic, p is also approximately cqual t o
P O and
+
k2
= (a
+
ip)2
$ (1 + i&).
=C
Upon rcplacing u' by (21) and separating into real and imaginary parts,
one obtains
(23)
At sufficiently low frequencies the inertia force m'i in Eq. (18) is
negligible with respect to the viscous force mgi. Within this range the
conductivity u' defined by (21) reduces t o a real and constant value,
SEC.5.161
PROPAGATION IX A N IONIZED ATMOSPHERE
327
which must be identical with the static conductivity u. I n a series of
researches of fundamental value, it was demonstrated by Hagen and
Rubens' that the values of conductivity measured under static conditions
may t)c employed without appreciable error well into the infrared. A t
wave lengths shorter than about 25 X
em., however, the conductivity exhibits a pronounced dependence on frequency and the
observed dispersion can be accounted for roughly by the relations set
down in ( 2 3 ) .
The sum and substance of these investigations is to show that dispersion formulas expressing the dependence of the parameters E, p , and u
on frequency permit the extension of classical electromagnetic theory
far beyond the limits arbitrarily established in Chap. I.
6.16. Propagation in an Ionized Atmosphere.-Recent investigations
of the propagation of radio waves in the ionized upper layers of the
atmosphere have led to some very interesting problems. In these
tenuous gases the mean free path of the electrons is exceedingly long and
the damping factor consequently negligible, so that the apparent conductivity turns out to be purely imaginary,
. Ne2
u' = z
(25)
-1
mu
whence
An electromagnetic wave is propagated in an electron atmosphere without attenuation and with a phase velocity greater than that of light in
free space. A t frequencies less than a certain critical value determined
by the relation
(27)
the index of refraction becomes imaginary, and it can be shown that at the
boundary of such a medium the wave will be totally reflected.
Actually the problem of propagation in the ionosphere is complicated
by the presence of the earth's magnetic field. Let us suppose that a
plane wave is propagated in the direction of the positive z-axis of a
rectangular coordinate system. For the sake of example we shall
assume only that there are N particles per unit volume of charge e and
mass m in otherwise empty space. There is also present a static magnetic field that may be resolved into a component in the direction of
propagation and a component transverse to this axis. Only the longiAnn. Physik, 11, 873, 1903. S(xe Chap. IX, p. 508.
328
PLANE W A V E S IN UNBOUNDED, ISOTROPIC MEDIA
[CHAP.V
tudinal component, whose intensity will be denoted by N o , leads to an
effect of the first order.
The equations of motion of a particle are
Now the ratio of the force exerted by the magnetic vector of the traveling
wave to that exerted by the electric vector is equal to the ratio of the
velocity of the charge to the velocity of light. The terms involving
H , and H , in (28) may, therefore, be neglected so that the equations of
motion reduce to the simpler system
Since the motion takes place entirely in the xy-plane and since the
vectors of the incident wave are likewise confined to this plane, complex
quantities can be employed to advantage. Let us write
u =x
(30)
+ iy,
E
=
E,
+ iE,,
H
=
H,
+ iH,.
Then the equations of motion are expressed by the single complex
equation
while for the field equations one obtains
dE - . aH
_
az
w o= 0
at
aH
--
an
+
,
dE
Z E~
=
at
-iNeu,
in which the charge per unit volume times the average velocity has been
introduced for current density.
A solution must now be found for this simultaneous system of three
equations. Let us try
(33)
E
= Aek i(hs-4
1
H
=
Be+i(hz-wt)
= Cef i ( h a - w t ) .
These are harmonic waves traveling in the positive direction with a
propagation constant h ; whereas the algebraic sign of the exponent is
immaterial in the absence of an applied field Ho, its choice in the present
SEC.5.161
PROPAGATION I N A N I O N I Z E D A T M O S P H E R E
329
instance gives rise to two distinct solutions as we shall immediately see.
For upon introducing (33) into (31) and (32) one obtains
rowA
+ ihB + NewC = 0.
In order that this homogeneous set of equations for the amplitudes
A , B , C have other than a trivial solution, it is necessary that the determinant vanish.
(35)
Expansion of this determinant leads to an equation for the propagation
constant h.
Ne2
W'
+
m
The phase velocity v = w/h, and consequcntly (36) is equal to the square
of the index of refraction n.
It turns out, thereforc, that an electron atmosphere upon which is
imposed a stationary magnetic field acts like a n anisotropic, doublerefracting crystal in that there are two modes of propagation with two
distinct velocit,ies. It is apparent, moreover, that by an appropriate
choice of w the index of refraction of the medium with respect to one of
these waves becomes infinite.
Likewise, at another frequency the index of refraction
Ne2
reduces to zero. The remarkable optical properties of the KennellyHeaviside layers can be largely accounted for by these formulas. A
330
P L A N E WAVES I N U N B O U N D E D , ISOTROPIC M E D I A
[CHAP.V
linearly polarized wave entering the medium is resolved into a righthanded and a left-handed circularly polarized component, the one
propagated with the velocity v+ and the other with t,he velocity v-.
Since the reflection occurring at the layer is determined by the index of
refraction, the polarization of the wave returning to earth may be greatly
altered and frequently contains a strong circularly polarized component
which contributes to “fading” phenomena.‘
VELOCITIES OF PROPAGATION
6.17. Group Velocity.-The concept of phase velocity applies only to
fields which are periodic in space and which consequently represent wave
trains of infinite duration. If a state of the medium is represented by
the function $(z, t ) , where
(1)
+@,t ) =
Aetbz--ut
then the surfaces of constant state or phase are defined by
(2)
kz - wt = constant,
and these surfaces are propagated with the velocity
(3)
A wave train of finite length, on the other hand, cannot be represented in
the simple harmonic form (l), and the term phase velocity loses its
precise significance. One speaks then rather loosely of the “velocity of
light,” or of the “wave-front velocity.” The necessity of a n exact
formulation of the concept of wave velocity became apparent some years
ago in connection with certain experiments fundamental to the theory
of relativity, and the subject a t present has a n important bearing upon
the problem of communication by means of short waves propagated
along conductors. Since an electromagnetic field can never be completely localized in either space or timc, there must be a n essential
arbitrariness about every definition of vclocity. T h e slope and height
of a wave front vary in the course of its progress, a n d the concept of a
“center of gravity” grows vague as the pulse diffuses. It will probably
be of more practical importance t o the reader to know what certain
common terms do not mean, than to attach to them a too precise physical
significance.
A more complete account of propagation in ionized media with bibliography
will be found in an article by Mimno, Rev. Mod. Phys., 9, 1-43, 1937. See also the
“Ergebnisse der exakten XTaturwissenschaften,” Vol. 17, Springer, 1938.
33 1
GROUP VELOCITY
SEC.5.171
Next in importance to the idea of phase velocit,y is that of group
velocity. Let us consider first the superposition of two harmonic waves
which differ very slightly in frequency and wave number.
(4)
$1
=
*z
=
cos ( k z - W t ) ,
cos [(k 6k)z - (w
+
+ Fw)t].
The sum of these two is
(5) $ = $1
+
*z = 2
1
cos 3 (2 6k
- t 6w) cos
[(k + 3 ( + - 2 -
J
3 1 1
which is just, the familiar expression for the phenomenon of “beats.”
The field oscillates a t a frequmcy which differs negligibly from W, while
its effective amplitude
R
(6)
=2
cos +(z 6k - t
6w)
varies slowly between the sum of the amplitudes of the component waves
and zero. As a result of constructive and destructive interference the
field distribution along both time and space axes appears as a series of
I’’10.
59.-Beats.
periodically repeated “beats ’) or “groups I J in the manner portrayed by
Fig. 59. Now the surfaces over which the group amplitude A is constant
are defined by the equation
z 6k
(7)
--
t
6w = constant,
from which it follows that the groups themselves are propagated with
the velocity
6W
=
X‘
The group velocity is determined by the ratio of the difference of frequency to the difference of wave number. If the medium is nondispersive, 6k =
1
-
60 and, hence, the group velocity coincides with the
2)
phase velocity v. I n a dispersive medium their values are distinct.
I n the example just cited the superposition of two harmonic waves
leads to a periodic series of groups. A single group or pulse of any desired
shape may be constructed upon the theory of the Fourier integral by
332
P L A N E W A V E S I N UNBOUNDED, ISOTROPIC M E D I A
[CHAP.V
choosing the amplitude of the component harmonic waves as an appropriate funct,ion of the frequency or wave number and integrating.
The concept of group velocity applies only to such Fourier representations as are confined to a narrow barid of the spcctrum. If the amplitude
function A ( k ) is of negligible magnitude outside the region
ko - 6k 5 k I ko
we may replace (9) by
(10)
+=
Lfr',;
+ 6k,
A(k)ei("-"t) dk,
which represeIlts what is commonly called a "wave packet." For the
present we shall assume that k is real and that w is a known function of k .
Within a sufficiently small interval 26k, w ( k ) will deviate but slightly from
its value a t k o and may, therefore, be represented by the first two terms of
a Taylor series.
The condition that higher order terms in this expansion shall be negligible
imposes the necessary restriction on 6k.
A wave packet can now be represented by
where +o is a mean amplitude defined by
This amplitude is constant over surfaces defined by
from which it is evident that the wave packet is propagated with the
group velocity
In case the medium is nondispersive, u coincides with the phase velocity
v , but otherwise it is a function of the wave number ko.
SEC.5.181
333
W A V E FRONT A N D SIGNAL VELOCITIES
From the relations k = 2?r/X, w = kv, we may derive several equivalent expressions for the group velocity which arc occasionally more
convenient than (16).
It is apparent from the manner in which the group velocity was
defined that this concept is wholly precise only when the wave packet is
composed of elerncntary waves lying within a n infinitely narrow region
of the spectrum. As the interval 6k is increased, the spread in phase
velocity of the harmonic components in a dispersive medium becomes
more marked; the packet is deformed rapidly, and the group velocity
as a velocity of the whole loses its physical significance. Note that a
concentration of the field in space does not imply a corresponding concentration in the frequency or wave length spectrum, but rather the
contrary. Consider, for example, the case of a harmonic wave train of
finite length.
f(2) =
cos Jcoz,
when Iz[
L
< -.
2
The amplitude of t,he disturbance in thc neighborhood of any wave
number k is given by (40),page 292,
(ko - k )
sin ___L
2
A ( k ) := 3r
ko-k
'
and the form of this function is illustrated by Fig. 50. As the length L
increases, the wave number interval 6k within which A ( k ) has a n appreciable magnitude decreases. On the other hand a pulse produced by the
abrupt switching on and off of a generator is diffused over the spectrum
and cannot be represented even approximately by (10). This broad
distribution over the spectrum is a common grief t,o every laboratory
worker who must shicld sensitive electrical apparatus from atmospheric disturbances or the surges resulting from the switching of heavy
machinery.
5.18. Wave Front and Signal Velocities.-If
the dispersion of the
medium is normal and moderate, a pulse or wave packet may travel a
great distance without appreciable diffusion ; since the energy is presumed
t o be localized in the region occupied by the field, it is obvious that the
velocity of energy propagation must equal a t least approximately the
334
P L A N E W A V E S I N UNBOIJNDED, ISOTROPIC M E D I A
[CHAP.V
group velocity. I n a normally dispersive medium a n increase of wave
length results in a n increase of phase velocity and hence by (17) the
group velocity under these circumstances is always less than the phase
velocity. If on the contrary the dispersion is anomalous-as is the case
in conducting media-the derivative dv/dX is negative and the group
velocity is greater t'han the phase velocity. There is in fact no lack of
examples to show that u may exceed the velocity c. Since at one time
i t was generally believed thab the group velocity was nccessarily equivalent t o the velocity of energy propagation, examples of this sort were
proposed in the first years following Einstein's publication of t,he special
theory of relativity as definite contradict'ions t o the postulate that a
s i s a l can never be transmitted with a velocity greater than c. The
objection was answered and the entire problem clarified in 1914 by a
beautiful investigation conducted b y Sommerfeld and Brillouin,l which
may still be read with profit.
The particular problem considered by Sommerfeld is that of a signal
which arrives or originates a t the point x = 0 a.t the instant t = 0 and
continues indefinitely thereafter as a harmonic oscillation of frequcncy W.
The signal is propagat,ed t o the right, into a dispersive medium, and we
wish t,o know the time required to penetrate a given dist>ancc. We have
seen that the Fouricr integral of a nontjt:rminating wave train does not
converge, but this difficulty may be circumveiitcd by deforming thc path
of integration into the complex domain of the variable w. On the other
hand, the theory of the IJaplace t'ransform was introduced for the very
purpose of treating such functions; and indeed, i t may be shown by a
simple change of variable t'hat, t,ho complex Fourier int>egralemployed by
Sommerfeld is exactly the transform defined in Eq. (143) of Xec. 5.12.
It will facilitate the prescnt discussion t o modify the original analysis
accordingly.
At x = 0 the signal is defined by the function
The representation of this field a t any point z wit,hin the medium is
obtained by extending (20) to a solution of the wave equation, which
by (lo), page 323, is
1 SOMMERFELD,
Ann. Physik, 44, 177-202, 1914. BRILLOUIN,
ibid., 203-240. A
complete and more recent account of this work was published by Brillouin in the
reports of the Congrks International d'EZectricitd, Vol. 11, 1'" Sec., Paris, 1932.
S E ~5.181
.
335
W A V E FRONT A N D SIGNAL VELOCITIES
Equation (21) is satisfied by cilementary functions of the form
from which we construct our signal.
y
> 0.
I n a nondispersive medium, K' is constant and (22) is ident,ical with
(143), page 318. If, howevcr, the medium is dispersive, K' is a function
of s. We shall suppose that, the dispersion can be represented by a
formula such as (8), page 325, and upon replacing - i w by s we obtain
To include metallic conductors we need only put G i = 0. The wave
function which for t > 0 reduces to a harmonic oscillation a t x = 0 is,
therefore,
y
(24)
As s + l i m , p ( s ) + 1. If now
7
=
2
t -c
< 0,
> 0.
the contour may
be closed, according to Jordan's lemma, page 315, by a semicircle to thc
right of infinite radius. This path excludes all singularities of the
integrand and consequentlyfl:~,t ) = 0. Thus we have proved that at a
point z within the medium the field is zero as long as t < z / c , and hence
that the velocity of the wave front cannot exceed the constant c.
If
7 =
t -
5 > 0, the path may be closed only t o the left. The
C
singularities thus encircled occur a t the pole s = - i w and a t the branch
points of p. These last are located a t the points where /3 = 0 and
p = a . If P ( s ) is written in the form
wc see that
1
P=
00,
P
0,
(26)
=
i
when s = -- g l - d 4 G , 2 - g z ,
2
2
1
i
when s = --g -t 4 4 ( 4
a2) - g2.
2
2
-1
+
The disposition of these singularities in the complex s-plane is indicated
336
P L A N E W A V E S I N U N B O U N D E D , ISOTROPIC M E D I A [CHAP.V
in Fig. 60a, where
1
i
(27) u* = - 3 g L- 3d-,
b*
1
i
& -22 / 4 ( 4 + a2) - g.
2
= --g
Now if one encircles a branch point in the plane of s, one returns to the
initial value of 0 but with the opposite sign. This difficulty is obviated
by introducing a ( ( c u t ” or barrier along some line connecting a+ and b+,
and another between a- and b-. Over this rrc u tl’plane-which in the
function-theoretical sense represents one sheet of the Riemann surfacc
of b(s)-the function P(s) is single-valued; for, since it is forbidden to
traverse a barrier, any closed contour must encircle a n even number of
branch points. The path of integration in (24), which follows the line
drawn from y - i m to y
i 00 and is then closed b y an infinite semi-
+
--+
-_______ I
I
b
of integration in the s-plane.
circle to the left, may now be deformed in any manner on the “ c u t”
plane without altering the value of the integral, provided only that in
the process of deformation the contour does not sweep across the pole a t
s = -iw or either of the two barriers. In particular the path may be
shrunk to the form indicated in Fig. 60b. The contributions arising
from a passage back and forth along the straight lines connecting Co
and C1 and Co and Cz cancel one another and (24) reduces to three
integrals about the closed contours Co, C1, and Cz.
The first of these three can bc evaluated a t once.
whence by Cauchy’s theorem
SEC.5.181
W A V E FRONT AND SIGNAL VELOCITIES
33 7
BY (25)
wh
-_
fo(z, 1)
=
e
+
e
Thc remaining two integrals which are looped about the barriers cannot
be evaluated in any such simple fashion and they shall be designated by
Thus the resultant wave motion a t any point within the medium can
be represented by the sum of i.wo terms,
Physically these two components may be interpreted as forced and free
vibrations of the charges that constitute the medium. The forced
vibrations, defined by fo(z, t ) , arc undamped in t’ime and h a w thc same
frcqucncy as the impinging wave train. Thc free vibrations flz(z, t ) arc
damped in time as a result of the damping forces acting on the oscillating
ions and their frequency is determined by the elastic binding forces.
The course of the propagation into the medium can be traced as follows:
Up t o the instant t = z / c , all is quiet. Even when the phasc velocity
u is greater than c , no wave reaches z earlier than t = z/c. At t = z / c
the integral fI2(z, t ) first exhibits a value other than zero, indicating that
t,he ions have been set into oscillation. If by the term “wave front” we
understand the very first arrival of the disturbance, then the wave jront
velocity i s always equal to c, no matter what the medium. It may be shown,
however, that at t,his first instant t = z / c the forced or steady-state term
fo(z, t ) just cancels the free OK’transient term fI2(z, t ) , so that the process
st,arts always from zero amplitude. The steady st,ate is then gradually
built up as the transient dies out, quite in the same way that the sudden
application of an alternating o.m.f. to an electrical network results in a
transient surge which is eventually replaced by a harmonic oscillation.
The arrival of the wave front and the role of the velocity v in adjusting
the phasc are illust>ratedin Fig. 61, which however shows only the steadystat,e term. The axis x / c is drawn normal to the axis of t. The line at
45 deg. then determines the wave-front velocity, for i t passes through a
point z a t the instant z / c . The line whose slope is t a n 0 = c/v determines
the time t = z/v of arrival a t z of a wave whose velocity is v. Actually
the phase velocity has nothing to do with the propagation; i t gives
only t’he arrangement of phases and, strictly speaking, this only in the case
338
P L A N E W A V E S I N UNBOUNDED, ISOTROPIC ME’DIA [CHAP.V
of infinite wave trains. The phase of the forced oscillation is measured
from the intersection with the dotted line at t = x/v, and the phase at the
wave front t = 2/c is adjusted to fit. If v > c, 0 < 45 deg. The phase
of the steady state is again determined by the intersection with the line
z / c a t t = z/v, but the wave front itself arrives later.
The transition from vanishingly small amplitudes a t the wave front
to the relatively large values of the signal have been carefully examined
FIG 61.-Determination
of the phase in the steady state.
by Brillouin. This investigation is of a much more delicate nature and
we must content oiirsclves here with a statement of conclusions which
may be drawn after the evaluation of ( 3 2 ) . According to Brillouin a
signal is a train of oscillations starting a t a certain instant. In the course
of its journey the signal is deformed. The main body of the signal is
FIG. 62a.-Illustrating
the variation in
amplitude of the first precursor wave.
FIG. 62b.-Illustrating the arrival of
the first and second precursors and the
signal.
preceded by a first forerunner, or precursor, which in all media travels
with the velocity c. This first precursor arrives with zero amplitude,
and then grows slowly both in period and in amplitude, as indicated
in Fig. 62a; the amplitude subsequently decreases while the period
approaches the natural period of the electrons. There appears now a new
phase of the disturbance which may be called the second precursor,
GO
traveling with the velocity ____
c. This velocity is obtained by
4
-
assuming in the propagation factor P ( - i w ) of Eqs. (30) and (32) that the
impressed frequency w is small compared to the atomic resouance frequency Go. The period of the second precursor, at, first very large,
SEC.5.181
W A V E FRONT A N D SIGNAL VELOCITIES
339
decreases while the amplitude rises and then falls more or less in the
manner of the first precursor.
With a sudden rise of amplitude the main body, or principal part, of
the disturbance arrivcs, traveling with a velocity w which Brillouin
defines as the signal velocity. An explicit and simple expression for w
cannot be given and its definition is associated somewhat arbitrarily
with the method employed Lo evaluate the integral (32). Physically
its meaning is quite clear. In Fig. 62b are shown the two precursors
and the final sudden rise to Lhe steady state. When restricted to this
third and last phase of the disturbance the term “signal” represents that
portion of the wave which actuates a measuring device. Under the
assumed conditions a measurement should, in fact, indicate a velocity
.h. 1
42
x
....
\
\ /
.------
s
20
w
ido
-C / U , u = group velocity
--- C/U,
FIQ. 63.-Behavior
u = phase velocity
of the group, phase and signal velocities in thc neighborhood of a
resonance frequency.
of propagation approximately equal to w. It will bc noted, however,
that as the sensitivity of the detector is increased, the measured velocity
also increases, until in the limit of infinite sensitivity we should record
the arrival of the front of tho first precursor which travels always with
the velocity c. Qualitatively a t least we may imagine the medium as a
region of free space densely infested with electrons. An infinitesimal
amount of energy penetrat,cs the empty spaces, as through a sieve,
traveling of course with the velocity c. Each successive layer of charges
is excited into oscillation by the primary wave and reradiates energy
both forward and backward. By reason of the inertia of the charges
these secondary oscillations lag behind the primary wave in phase, and
this constant retardation as the process progresses through successive
layers results in a reduced velocity of the main body of the disturbance.
The wave-front velocity defined here is thus always equal to the constant c. The phase velocity v is associated only with steady states and
may be either greater or less than c. The group velocity u differs from
the phase velocity only in dispersive media. If the dispersion is normal,
340
P L A N E W A V E S I N UNBOUNDED, ISOTROPIC M E D I A
[CHAP.V
the group velocity is less than v; it is greater than the phase velocity
when the dispersion is anomalous. I n the neighborhood of an absorption
band i t may become infinite or even negative. The signal velocity w
coincides with the group velocity in the region of normal dispersion,
but deviates markedly from i t whenever u behaves anomalously. The
signal velocity is always less than c, but becomes somewhat difficult and
arbitrary to define in thc neighborhood of an absorption band. In
Fig. 63 the velocities which characterize the propagation of a wave train
in the neighborhood of a n absorption band are plotted as ratios of c.
Problems
1. Let E a he the amplitudt: of the electric vcctor in volts per nieter and 8 the mean
energy flow in watts per square nieter of a plane wave propagated in free space. Show
that
=
1.327 X
EO = 27.45 4
E; wntt/nietcr2,
3 volts/meter.
2. Let E Ohe the amplitude in volts per meter of the electric vector of a plane wave
propagated in free space. Show that the amplitudes of the magnetic vectors H n arid
B , are related numerically to E , by
+
E a ampere-turn/meter = X lo-* En oersted,
H , = 2.654 X
EOweber/meterz = X lop4E , gauss,
Ro = f X
+
where the oersted is the iinr:tt,iori:rlized c.g.s. electromagnetic unit.
in statvolts per centimeter, sliow~that
Ho
=
E , oersted,
B,
=
If E , is expressed
Eo gauss.
If a particle of charge q moves with a velocity v in the field of a planc electromagnetic wave, show that the ratio of the forccs exerted by the magnetic and electric
components is of the order v / c .
3. The theory of homogeneous, plane waves developed in Chap. V can be extended
to inhomogeneous waves. If II. is any rectangular component o f an electromagnetic
vector, we may take as a general definition of a plane wave the expression
$ = $, ei+,
4 = --wt
+ k . r,
in which $z0 is a complex amplit.ude : ~ n d4 the complex phase.
k is now a complex vector which wc shall writc as
k
=
kl
The propagation factor
+ ZIZ + ipnz,
= 0ln1
where n I and nz are unit vectors. Show that 01 is the real phase constant; that the
surfaces of constant, real phase are planes normal to the axis nl, and the surfaces of
constant amplitude are planes normal to n z ; and that B is the factor measuring attenuation in the direction of most rapid change of amplitude. The true phase velocity v
is in the direction nl.
Show that the operator V can be replaced by zk and the field equations are
k X E - - w B =O,
k X H i.wD = - iJ,
k * B = 0,
k D = 0.
-
341
PROBLEMS
Show also that, if the medium is isotropic,
-
Let kl kz = klkz cos 0 and find expressions for kl and kz in terms of p , E, U, and the
angle 0.
4. Continuing Problem 3, assume that the electric vector is linearly polarized.
Show that E is perpendicular to both kl and kz and that H lies in the plane of these
two vectors. What is the locus of the magnetic vector H? Inversely, show that if
H is linearly polarized i t is perpendicular to both kl and kz, while E lies in the kl,
kz-plane. These inhomogeneous plane waves are intimately related to the so-called
“ H waves” and “I3 waves,” Sec. 9.18.
6. The theory of plane waves can be extended to homogeneous, anisotropic
dielectrics. Let u = 0, p = pol and the properties of the dielectric be specified by a
tensor whose components are t j k . Choosing the principal axes as coordinate axes,
we have
Dj
=
0’
cOK~E~,
=
1, 2, 3).
The field equations relating E and D are then
k X (k X E)
+ ~*poD
=
0,
k * D = 0.
Let k = kn, where n is a unit real vector whose components along the principal axes
are nl, n2, n 3 . Let u = w / k and v j = c/&.
The v j arc the principal Velocities.
Show that the components of E satisfy the homogeneous system
and write down the equation which determines the phase velocity in any direction
fixed by the vector n. Show that this equation has three real roots, of which one is
infinite and must be discarded. There are, thcrefore, two distinct modes of propagation whose phase velocities are u‘ and u“ in the direction n. Correspondingly there
are two types of linear oscillation in distinct directions, characterized by the vectors
E’, D’ and E”, D”. Show that thme are related so that
.
= D’!. E’ = D’ . D” = 0,
E’ . E” # 0,
E * n # 0.
D’ E”
6. Show that in a homogmeou:, anisotropic dielectric the phase velocity satisfies
the Fresnel relation
$y&
50,
j=1
where the n1 are the direction cosines of the wave normal n with respect to the principal axes, and the vl are the phase velocities in t b directions of the principal axes.
(See Problem 5 . ) Show also that
342
P L A N E WAVES I N UNBOTINDED, ISOTROPIC M E D I A [CHAP.V
where the yi are the direction cosines of the vector D with respect to the principal
axes.
7. If in Problem 6 V I > 02 > 213, show that there are two directions in which v
has only a single value. Find these directions in terms of ul, v2, va, and show that in
each case the velocity is v 2 . The directions defined thus are the optic axes of the
medium which in this case is said to be biazial. What are the necessary conditions
in order that there be only one optic axis?
8. Show that for a plane wave in an anisotropic dielectric
where v is the phaso velocity defined in Problem 5.
A vectorial velocity of energy propagation u in the direction of S is defined by the
equation
S
=
hu.
The velocity u is related to the phase velocity v by
Show t h a t
and, hence, that the energy flow is equal t o the total energy density times the velocity
u, with
II=
E2n - (n E)E
E 2 - (n E)2 v,
or in magnitude
9. From the results of Problem 8 show that the magnitude of the velocity u of
energy propagation in a plane wave can be found in terms of the constants U I , u t , v3
from the equation
in which the lj are direction cosines of the vector u.
Show also that there are, in general, two finite values of u corresponding to each
direction. I n what directions are these two roots identical? What is the relation of
these preferred axes of u to the optic axes?
10. A nonconducting medium of infinite extent is isotropic but its specific inductive capacity K = t/en is a function of position. Show first t h a t the electric field
343
PROBLEMS
vector satisfies the equation
where k 2 = w2ep and the time enters only through the factor exp( --id).
Assume now that the spatial change of c per wave length is small, so that
IVK~
h
1. Show t h a t the right-hand term can be neglected and t h a t the
propagation is determined approximately by
<<
v2+ + k2+
= 0,
where 11. is a rectangular component of E or H and k is a slowly varying function of
position. Next let
11. = $ 0
pi+,
4 =
-wt
+ koS,
where 4 is the phase, k o = w / c , and S a function of position.
function S is determined by the equation
Show that the phase
where
Note that S is in general complex, even though K is real, and that the amplitude of the
wave is also a slowly varying function of position.
11. The transition from wave optics to geometrical optics can be based on the
results of Problem 10. Show that as the wave number k becomes very large, the
phase function is determined by the first-order, second-degree equation
while the amplitude satisfies
v 111 $0 * vs
=
-
:vzs.
Since S is now real, the wave fronts are represcnted by the family 1’1 = constant, and
the wave normals or rays are given a t each point by VS. The function S is called
the “eikonal” and is identical with Hamilton’s “characteristic function.” Let the
vector n represent the index of refraction
times a unit vector in the direction of
the wave normal. Then
dT
VS = n ,
n . v l n f i o = -4v-n.
Note that these relations hold only in regions where the change in K per wave length is
small. Hence they fail in the neighborhood of sharp edges or of bodics whose dimensions are of the order of a wave length. In such cases the complete wave equation
must be applied. (Sommerfeld and Runge, Ann. Physik, 36, 290, 1911.)
344
P L A N E W A V E S I N IJNBOUNDED, ISOTROPIC M E D I A
Let dr be an element of length in the direction of a ray.
[CHAP.V
Then
S is a function of the coordinates
50, yo, zu of a fixed initial point and the coordinates
z of the terminal point reached by the ray. Of all possible paths through these
two points, the ray actually follows that which makes S a minimum, or
z, y,
This is Fermat’s principle. If the disturbance originates at xO,yo, zo a t the instant t o ,
and arrives a t z, y, z a t t, then
S(Q, YO,zn;
Z, Y, Z ) = t - to,
and the path of the ray is such that the time of arrival is a mininium.
12. Plane waves are set up in an inhomogeneous medium whose inductive capacity
varies in the direction of propagation. Suppose this direction to be the z-axis and
assume the field to be independent of z and y. The wave equation is then
where k i = w2e~,u0and the time enters only in the factor exp( --id).
Suppose that K
K I as z + - m , and K + K Z as z 4 m , where K I and K Z are constants. It is possible to construct four particular solutions behaving asymptotically
as follows:
asz+ asz+ as z ---t
a s z j
a.
m.
a.
m.
Thus $ 1 and $ 8 reduce to plane waves tr:tveling in thc positive direction; $ 2 and G4,
t o waves in the negative direction. Only two of these solutions are linearly independent and, hence, there must exist analytic connections of the form
$3
=
al$1
+ azh,
$34
=
b h
+b$~.
Mathematically this rcpresents the anaiytic continualion of $ 3 from one region to
another, and physically the reflection coefficient is given by the ratio r = a,/u,.
Let
r
where
= ICnZ/S, K S is a constant and s an additional parameter. Replace z by
the new independent variable u = cxp(l) and show that thc wave equation reduces
t o hypergeometric form. Express $ 1 . . . $ 4 as hypergeometric functions and cnlculate the joining factors and the reflection coefficient. The form of K ( Z ) is such as
t o admit a very wide choice in strata. (Epstein, Proc. Nut. Acud. Sci., 16,627, 1930.)
13. Ten times the common logarithm of the ratio of the initial to the terminal
energy flow measures the attenuation of a plane wave in decibels. Show that for a
PROBLEMS
345
plane wave in a homogeneous, isotropic medium
power loss
=
8.6868
db/meter,
B is the attenuation factor defined in Eq. (49), page 276.
14. For the media whose constants are given below, plot thc attenuation of a
plane wave in decibels per meter against frequency from 0 to 1 0 7 cycles/seeond.
Use semilog paper or plot against the logarithm of the frequency.
where
1
i
Material
~
Seawater . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fresh water. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Wet earth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Dry e a r t h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I
Conductivity,
mhos/nieter
~-_
__~_
4
10-3
10-3
10-5
1
I
E/<O
1:
~
1
l
10
4
16. A function f(z) is defined h:y
f(x) = e--Bs,
f(z) =
z>o,p>o.
z < 0, B > 0.
e-Bz,
Show that
16. Obtain a Fourier integral representation of the function f ( t ) defined by
f ( t ) = 0,
f ( t ) = e--bl cos w t ,
t
t
<o,
> 0.
Plot the amplitudes of the harmonic components as a function of frequency (spectral
density) and discuss the relation between the breadth of the peak and the logarithmic
decrement.
17. The surges on transmission lines produced by lightning discharges are often
simulated in electrical engineering practice by impulse generators developing a unidirectional impulse voltage of the form
where 01 and p are constants determined by the circuit parameters. This function
represents a pulse or surge which rises sharply to a crest arid then drops off more
slowly in a long tail. The wave form can be characterized by two constants t i arid L 2 ,
where t l is the time required to reach the crest, and 1 2 is the interval from zero voltage
to that point on the tail a t which the voltage is equal to half the crest voltage. A
standard set of surges is represented by the constants
(a) t i
( b ) tl
(c) t i
=
=
0.5,
1.0,
t Z
=
1.5,
t z = 40,
tz =
=
5,
microseconds.
10,
From these values one may determine the constants
a
m(1 0
346
PL,ANE ILJAVES I N UNBOUNDED, ISOTILOPIC M E D I A
[CHAP.V
Make a spectral analysis of the wave form and determine the frequencies a t which
thc spectral density is a maximum for thc three standard cases.
18. The “telegraphic equation”
has the initial conditions
i)U
-
u = f(z),
at
=
gb),
when t
=
0.
Show t h a t
u =
arrcz
+ at) + f ( x
where
This solution was obtained independcntly by Hraviside and by Poincari..
19. An electromagnetic pulse is propagated in a homogmeous, isotropic medium
whose constants are c, p, u in the direction of the positive z-axis. At the instant
t = 0 the form of the pulse is givrn by
where 6 is a parameter. Find a n expression for t h r pulse f(z, 1) a t any subsequent
instant.
20. Wherever tho disp!:mcment current is negligible with respect to the conduc1), the field satisfirs approximately the equations
tion current (u/ew
>>
an
VXE-I-/.l--O,
at
Show that t o the same approxiriiation
nA
E=-.- J
H
at
aA
V’A - / . l ~- = 0,
at
and that the current density J satisfies
VzJ - pu
(3J
--
il1
=
0
’
347
PROBLEMS
What boundary conditions apply to the current density vector? These equations
govern the distribution of current in metallic conductors a t all frequencirs in the radio
spectrum or below
21. The equation obtained in Problcm 20 for the current distribution in a metallic
conductor is identical with that governing thc diffusion of heat. Consider the onedimensional case of a rectangular component J propagated along the z-axis
Show that
is a solution such that at t = 0, J .= f(z), and which is continuous with continuous
derivatives for d l values of z when t > 0.
22. In Problem 21 it was found that the equation
v-
av
a22
a$
=-
at
is satisfied by
--( a - z ) 2
f(a)e
4ut
dor.
From the theory of Fourier transforms show t h a t this can be written in the form
Obviously the equation is also satisfied by
Compare these solutions and from thc uniquencss theorem demonstrate the
identity
This relation has been applied by Zwald to the thcory of wave propagation in crystals.
23. Show that if radio waves are propagatcd in an electron atmosphere in the
earth's magnetic field, a resonxncr phcnonlenon may be expected near a wave length
of about 212 meters. Marked selrriive ahsorption is actually ohserved in this rrgion.
Assume e = 1.60 x 10-19 coulomb/(Llcctron, m = 9 X 10-31 kg /electron, and for the
density of the magnetic field of the cmth B , = 0.5 X lop4weber/mcter2.
24. A radio wave is propagated in an ionized atmosphere. Calculate and discuss
the group velocity as a function of frequency, assuming for the propagation constant
348
P L A N E W A V E S I N UNBOUNDED, ISOTROPIC M E D I A
CHAP.^
the expression given in Eq. (36), page 329. The concentration of electrons in certain
regions of the ionosphere is presumed to be of the order of 1OI2 electrons/meter3.
25. A plane, linearly polarized wave enters an electron atmosphere whose density
is 1012 electrons/meter3. A static magnetic field of intensity B O = 0.5 X lo-'
weber/meter2 is applied in the direction of propagation. Obtain a n expression
representing the change in the state of polarization per wave length in the direction
of propagation.
26. A radio wave is propagated in a n ionized atmosphere in the presence of a fixed
magnetic field. Find the propagation factor for the case in which the static field is
perpendicular t o the direction of propagation.
27. As a simple model of an atom one may assume a fixed, positive charge to
which there is bound by a quasi-elastic force a negative charge e of mass m. Neglecting frictional forces, the equation of motion of such a system is
m'i + f r = 0,
where r is the position vector of e and f is the binding constant. A static magnetic
field is now applied. Show that there are two distinct frequencies of oscillation and
t h a t the corresponding motions represent circular vibrations in opposite directions
in a plane perpendicular to the axis of t h e applied field. What is the frequency of
rotation? This is the elementary theory of the Zeeman effect.
28. T h e force per unit charge exerted on a particle moving with a velocity v in a
magnetic field B is E' = v X B. The motional electromotive force about a closed
circuit is
V'=$E'.as
=J[vX(vXB)].ndu.
At any fixed point in space the rate of change of B with the time is aB/at. If by E
we now understand the total force per unit charge in a moving body, then
aB
V X E = -- + V X
at
(V
XB).
Show that the right-hand side is thc negative total derivative
so that the Faraday law for a moving medium is
The displacement velocity is assumed to be small relative t o the velocity of light,
CHAPTER VI
CYLINDRICAL WAVES
An electromagnetic field cannot, in general, be derived from a purely
scalar function of space and time; as a consequence, the analysis of
electromagnetic fields is inherently more difficult than the study of heat
flow or the transmission of acoustic vibrations. The three-dimensional
scalar wave equation is separable in 11 distinct coordinate systems,'
but complete solutions of the vector wave equation in a form directly
applicable t o the solution of boundary-value problems are a t present
known only for certain separable systems of cylindrical coordinates and
for spherical coordinates. It will be shown that in such systems an
electromagnetic field can be resolved into two partial fields, each derivable
from a purely scalar function satisfying the wave equation.
EQUATIONS OF A CYLINDRICAL FIELD
6.1. Representation by Hertz Vectors.-We
shall suppose that one
set of coordinate surfaces is formed by a family of cylinders whose elements are all parallel to the z-axis. Unless specifically stated, these
cylindrical surfaces are not necessarily circular, or even closed. With
respect to any surface of the family the unit vectors il, iz, i3 are ordered
as shown in Fig. 64. il is, therefore, normal to the cylinder, i3 tangent
t o it and directed along its elements, and i, is tangent to the surface arid
perpendicular t o il and is. Position with respect to axes determined
by the three unit vectors is measured by the coordinates ul, u2, x, and
the infinitesimal line element is
(1)
ds
=
ilhl dul
+ i,h, du2 + i3dz.
Now let us calculate the components of the electromagnetic field
associated with a Hertz vector n directed along the z-axis, so that
II1 = 112 = 0, IIz # 0. We shall assume in the present chapter that the
medium is not only isotropic and homogeneous but also unbounded in
extent. Then by (63) and (64), page 32, the electric and magnetic
vectors of the field are given by
E(')= v x v x n,
H(l) =
EISENHART,
Annals of Math., 36, 284, 1934.
349
(c$
+ .) v x n;
350
[CHAP.VI
CYI,INDRICAI; TTAVES
since both 111and 112arc‘ zero, it is a simplc matter to calciilate from (81)
and (85), page 49, tlic componcnts of E(’) and H(I).
Thus we have derivcd from a scalar function IL = $, an electromagnetic
field characterized by the absencc of an axial or longitudinal component
X
V
lo
FIG.64.-Relation
Y
of unit vectors to a cylindriral suiface.
arc parallel t o the z-axis.
The generators of the surface
of the magnetic vector. Since n is the elect>ricpolarization potential, this
can bc called a field of electric type (pagc 30), but in the present instance
the term transverse magnetic field proposed recently by Schelkunoff seems
more apt.
Since nZis a rectangular component, it must satisfy the scalar wave
equation,
(5)
V’* -
21c.
a$
p € 7- pu - =
dt
at
0,
1 SCHELKUNOFF,
Transmission Tlicory of Plane Electromagnetic Waves, Proc.
Inst. Radio Enyrs., 25, 1457-1492, Novcrnbcr, 1037.
SEC.6.21
SCALAR AND VECTOR POTENTIALS
351
or by (82), page 49,
The elementary harmonic solutions of this equation are of the form
(7)
*
= j(u1, u2),kihz--iot,
whcre j ( u l , u Z )is a solution of
I n quite the same way a partial field can be derived from a second
Hertz vector II* by the operations
if
n*is directed along the x-axis, the components of these vectors are
The scalar function n,“is a solution of ( 5 ) , and the field of magnetic type,
or transverse electric, derived from it, is characterized by the absence of a
longitudinal component of E.
The electromagnetic field obtained by superposing the partial fields
derived from II, and 11; is of such generality that one can satisfy a prescribed set of boundary conditions on any cylindrical surface whose gcnerating elemcnts are parallel to the z-axis; i.e., on any Coordinate surface
defined by u1 = constant-or prescribed conditions on cither a surface
of the family u2 = constant or a plane z constant. The choice of these
orthogonal families, however, is limited in practice to coordinate systems
in which Eq. (8) is separable.
6.2. Scalar and Vector Potentials.-The transverse electric and transverse magnetic ficlds defined in the preceding paragraph have interesting
propertjies which can be made clear by a study of the scalar and vector
potentials. Consider first the transverse magnetic field in which
[CHAP.V I
In the present instance
+ = 11, and, hence,
The components of E(’) are then
for the components of BC1), we obtain
in which A without a subscript has been written for A,.
Now it will be noted that in the plane z = constant the vector E(’)
is irrotational and therefore in this transverse plane, the line integral of
E(‘)between any two points a and b is independent of the path joining
them; for an element of length in the plane z = constant is expressed by
ds = ilhl dul izhzdu2 and, consequently,
+
The difference of potential, or voltage, between a n y two points in a transverse
plane has a dejinite value at each instant, whatever the frequency and whatever
the nature of the cylindrical coordinates.
Next, one will observc that the scalar function A plays the role of a
stream or flow function for thc vector B(l). Let the curve joining the
points a and b in Fig. 65 represent the trace of a cylindrical surface intersecting the plane z = constant. We shall calculate the flux of the
vector B(‘) through the ribbonlike surface element bounded by the curve
ab whose width in the z-direction is unity. If n is the unit normal to this
surface and i3 a unit vector directed along the z-axis, we have
(18)
-
B(’) n d a
=
B(’)
-
(i3
x ds)
=
-
i3 (ds x B(l)),
SEC.6.21
353
SCALAR AND VECTOR POTENTIALS
where d s is again an element of length along the curve. Upon expanding
(18) we obtain
(19)
-
B(1) n d a
hence,
=
hlBhl) d u l - h2Bi1)d u 2 = - d A ;
lb
B(’J - n da
(20)
=
A(a) - A@).
T h e magnetic $ux through a n y unit strip of a cylindrical surface passing
through two points in a plane z = constant
i s independent o j the contour of the strip.
If the components of E(’) and B(I) are
expressed in terms of the scalar function $,
i t is easy t o show that
&l)B\l) + E‘1)B‘”
= 0
(21)
2
2
7
and consequently the projection of the vector
E(l) o n the plane z = constant i s everywhere
Y
FIG. 65-Thc
rurvc a-h renormal to B(IJ. Moreover, in the transpresents thc intcrscction of a cyverse plane the families of curves, = con- lindrical surface with the zy-plane
stant, A = c o n s t a n t , c o n s t i t u t e a n which contains also the vectors n
and B1.
orthogonal set of equipotentials and streamlines. When the time variation is harmonic, it is clear from (14) and (7)
that the equipotentials are the lines
+
f(ul, u2) = constant,
(22)
in whichf(ul, u2)satisfies Eq. (8).
The transverse electric field has identical properties but with the
roles of electric and magnetic vectors reversed. According t o ( 3 5 ) ,
page 27,
(23)
D(2J
=
If now we let $
-v x
=
I:,
II:
H(2) =
A*,
=
IIg
=
-v+* - dA*
~
0, then
(25)
The components of the field vectors are, therefore,
dt
- u- A * ,
6
354
CYLINDRICAL WAVES
[CHAP.V I
(27)
a 23’
H12’ =
a21C, - pu-.atC.
dz2 - pt-at2
at
The projection of H(2)on the transverse plane is irrotational and, consequently, the line integral representing the magnetomotive force
between two points in this plane is independent of the path of integration.
Likewise the flux of the vector D(Qcrossing a strip of unit width as shown
in Fig. 65 depends only on the tcrminal point,s.
(29)
lb
.n da = L b d A * = A * @ ) - A*(a).
D(2)
The projection of H(*)on the plane x = constant is everywhere normal
to D(2);hence, the families +* = constant,, A * = constant are ort>hogonal.
The field D(2),H(2)is in 1his sense conjugate to the field E(I),B(l).
6.3. Impedances of Harmonic Cylindrical Fields.-We shall suppose
that the time variable enters only in the harmonic factor ecWt. Then the
potentials and field components of a transverse magnetic field are
+
in which k 2 = pew2 ipuw. The upper sign applies to waves traveling in
the positive direction along the x-axis, the lower sign to a negative direction of propagation.
A set of impedances relating thc field vectors can now be defined on
the basis of Sec. 5.6. I t is apparcnt Ilmt the value of t,he impedance
depends upon the direction in which it is measured. By definition,
(33)
The intrinsic impedance of a homogeneous, isotropic medium for plane
waves is by (83), page 283,
SEC.6.41
ELEMENTARY WAVES
355
so that (33) can be expressed as
zp
(35)
=
h
t-2,.
-k
Likewise impedances in the dirrctions of the transverse axes can be
defincd by the rclat,ioris
(36)
&1’
= -Z(1)[j(1)
1
2
1
EL”
=
Z(l)H(l).
2
1
The correlation of the components of electric and magnetic vectors and
the algebraic sign is obvioiisly determined by the positive direction of the
Poynting vector. Sincc Hi1)is zero, there is no component of flow
represented by the terms EL1)HI1),
E$l)HL1),
and the associated impedances
conscquently are infinite. Upon introducing the appropriate expressions
for the field components into (%), we obtain
The determination of corresponding impedances for the harmonic
components of transverse clectric fields nceds no further explanation.
The potentials and field vectors are in this case given by
From these relat,ions are calcu1:rtcd the various impedances.
There emerges from these results the rather curious set of relations:
(44)
zpzi2’ = z&”z‘,2)= Z(l)Z(Z)
== 2 ‘ 2 ’
z
z
0 .
WAVE FUNCTIONS OF THE CIRCULAR CYLINDER
6.4. Elementary Waves.-By far the simplest of the separable cases
is that in which the family ,ul= conslant is represented by a set of
356
CYLINDRICAL WAVES
coaxial circular cylinders.
(1)
Then by 1, page 51,
u2 = 8,
u1 = r,
[CHAP.VI
hl = 1,
hz = T ,
and Eq. (8) of Sec. 6.1 reduces to
which is readily separated by writing f ( r , 8) as a product
f
(3)
= fl(T)fZ(@,
wherein fl(r) and f 2 ( 8 ) are arbitrary solutions of the ordinary equations
(4)
(5)
The parameter p is, like h, a separation constant; its choice is governed
by the physical requirement that a t a fixed point in space the field must
be single-valued. If, as in the present chapter, inhomogeneities and
discontinuities of the medium arc excluded, the field is necessarily
periodic with respect to 8 and the value of p is limited to the
integers n = 0, *I, &Z, . . . . When, on the other hand, a field is
reprcscnted by particular solutions of (4) and (5) in a sector of space
bounded by the planes 8 = 81, 8 = 82, it is clear that nonintegral values
must in general be assigned to p .
Equation (4) sahfied by the radial function f l ( r ) will be recognized
as Bessel's equation. Its solutions can properly be called Bessel functions but, since this term is usually reserved for that particular solution
J p ( d mr ) which is finite on the axis r = 0, we shall adopt the name
circular cylinder function to denote any particular solution of (4) and shall
designate it by the letters f l = Z , ( v " ~r). The order of the function whose argument is d k 2 - h2r is p . Particular solutions of the
wave equation ( 5 ) , page 350, which are periodic in both t and 8, can therefore be constructed from elementary waves of the form
(6)
=
ein6
z,(4-2
r ) ,+ihz--iot.
The propagation constant h is, in general, complex; consequently the field
is not necessarily periodic along the z-axis. An explicit expression for h
in terms of the frequency w and the constants of the medium can be
obtained only after the behavior of $ over a cylinder r = constant or on a
plane z = constant has been prescribed.
SEC.6.51
357
PROPERTIES OF THE FUNCTIONS Z&)
6.6. Properties of the Functions Z,(p).-Although it must be assumed
that the reader is familiar with a considerable part of the theory of Bessel’s
equation, a review of the essential properties of its solutions will prove
useful for reference.
If in Eq. (4) the independent variable be changed to p = d-2
T,
me find that Z p ( p ) satisfies
(7)
an equation characterized by a regular singularity at p = 0 and a n essential singularity a t p = m. The Bessel function J , ( p ) , or cylinder function
of the first kind, is a particular solution of (7) which is finite a t p = 0.
The Bessel function can, therefore, be expanded in a series of ascending
powers of p and, since there are no singularities in the complex plane of
p other than the points p = 0 and p = m , it is clear that this series must
converge for all finite values of the argument. For any value of p , real
or complex, and for both real and complex values of the argument p we
have
When p in (7) is replaced by - p , the equation is unaltered; consequently, when p is not an integer, a second fundamental solution can he
If, however, p = n is a n
obtained from (8) by replacing p by - p .
integer, J,(p) becomes a single-valued function of position. The gamma
function r(n m
1) is repheed by the factorial (n m ) ! ,so that
+ +
+
(9)
(n = 0, 1,2,
The function
it by
J,(p)
*
*
*
).
is now no longer independent of (9) but is related to
(10)
J-n(Pj
= (-l)nJn(P),
so that one must resort to some other method in order to find a second
solution.
The Bessel function of the second kind is defined by the relation
(11)
NAP)
1
=
[J,(P)
cos p n - J-*(P)l.
This solution of (7) is independent of J,(p) for all values of p , but the
right-hand side assumes the indeterminate form zero over zero when p
358
[CHAP.
CYLINDRICAL WAVES
VI
is a n integer. It can be evaluated, however, in the usual way by differentiating numerator and denominator with respect to p and then
passing to the limit as p --+ n. The resultant expansion is a complicated
affair1 of which we shall give only the first term valid in the neighborhood
of the origin.
(n = 1, 2,
*
-
. ),
where y = 1.78107 and [pi << 1. The characterizing property of the
functions of the second kind is their singularity at the origin. Since they
become infinite a t p = 0, they cannot be employed to represent fields that
are physically finite in this neighborhood.
Further light is cast on the nature of the functions J,(p) and N,(p)
when one examines their behavior for very large values of p. The expansions about the origin converge for all finite values of p, and J,(p) and
N,(p) are analytic everywhere with respect to both p and p, the points
p = 0 and p = 00 excepted. However, when p is very large the convergence is so slow as to rcndrr the series useless for practical calculation
and one seeks represcntations of these same functions in series of
inverse powers of p. It can, in fact, be shown that Bessel’s equation
is satisfied formally by the expansions
(15) P,(P)
=
1-
(4p2 - 1)(4$ - 9)
2 !(Bp)
(4p2 - 1)(4p’ - O)(4p2 - 25)(4p2 - 49)
4 !( 8 p )
+
-...
7
in which the phase angle 4 is given by
Now it turns out that these series diverge for a11 values of p , and
consequently do not possess the exact analytical properties of the func1See WATSON,“A Treatise on the Theory of Bessel Functions,” Chap. 111,
Cambridge University Press, 1922, and JAHNKE-EMDE,
“Tables of Functions,”
p. 198, Teubner, 1933.
SEC.6.51
PROPERTIES OF T H E F U N C T I O N S Z , ( e )
359
tions they are intcndcd to represent. On the other hand, if p is large,
the first few terms diminish rapidly in magnitude and in this sense t h e
series are “semiconvergent.” It can be shown that if the expansions
are broken off near or before the point a t which the successive terms begin
t o grow larger, they lead to an approximate value of the function and the
error incurred can be estimated. The larger the value of p the more
closely does the sum of the first few terms coincide with the true value of
the function; for this reason such representations are said to be asymptotic.
I n the present instance we note that when p is sufficiently large,
J,(p)
=
p (.
cos
TP
-r
+ .),
- 2p
At very great distances from the origin the cylinder functions of the first
and second kinds are related t o each other as the cosine and sine functions, hut are attenuated with increasing p due to the factor 1/z/p.
They are the proper functions for thc representation of standing cvlindrical
waves.
By analogy with the exponmtial functions one may construct a linear
combination of the solutions J , ( p ) and N,(p) to obtain functions associated with traveling waves. The Bessel functions of the third Bind, or
Nankel functions as they are commonly known, are defined by t h e
relations
(20)
(21)
f q ) ( P ) = J,(P)
+ iN,(P)l
H;)(P)= J, ( P ) - ~ N , ( P ) .
From the preceding formulas we find readily that for very large p
To the series expansions of the functions themsrlLw we shall add for
ready refcrence several of the more important reciirrence relations.
(24)
z,-1 .tz,+1
= 2P
- 8,.
P
360
CYLINDRICAL W A V E S
[CHAP.VI
6.6. The Field of Circularly Cylindrical Wave Functions.-Within a
homogeneous, isotropic domain every electromagnetic field can be represented by linear combinations of tbe elementary wave functions
Of these (28) alone applies to finite domains including the axis r = 0;
a t great distances from the source, (29) must be employed, since by (22)
it reduces asymptotically to a wave traveling radially outward. Each
elementary wave is identified by the parameter triplet n, h, k. Whcn
n = 0, the field is symmetric about the axis; when h = 0, the propagation
is purely radial and the field strictly two-dimensional. Functions such
as (28) and (29) can be said to represent inhomogeneous plane waves.
The planes of constant phase are propagated along the z-axis with a
velocity v = w / a , where a is the real part of h, but the amplitudes over
these planes are functions of r and 8. Such waves can be established
only by sources located a t finite distances from the origin of the reference
system, or in media marked by discontinuities. The plane waves studied
in the preceding chapter are in a strict sense homogeneous,since the planes
of constant phase are likewise planes of constant amplitude. They can
exist only in infinite, homogeneous media, generated by sources that are
infinitely remote.
From the formulas developed in See. 6.3 one may calculate the
impedances and the components of the field vectors in terms of a wave
function 4. We find
S E C . 6.71
CONSTRUCTION FROM P L A N E W A V E SOLUTIONS
361
Likewise for the transverse electric ficld,
azn(p>
(34)
When init,ial conditions arfl prescribed over given plane or cylindrical
surfaces, a solution is constructed by snpcq~ositionof clementary wave
functions. For fixed values of w (or k ) and h, one obtains for the resultant
field in cylindrical coordinates the equations
m
n=
-m
m
n=-
m
where a, and bn are coefficicnfs to be determined from initial conditions.
The direction of propagation is positive or negative according t o the
sign of h.
INTEGRAL REPRESENTATIONS OF WAVE FUNCTIONS
6.7. Construction from Plane Wave Solutions.-If { measiires distance along any axis whose direction with respect to a fixed reference
system ( 2 , y, z ) is determined by a unit vector n, then the most elementary
type of plane wave can be represented according to Secs. 5.1 to 5.6 by
362
CYLINDRICAL W AVES
[CHAP.V I
the function
9 = eikf-iut
(1)
?
in which the constants k and w are eithcr real or complex. Let R be the
radius vector drawn from thc origin to a point of observation whose
rectangular coordinates are 5 , y, z. The phase of the wave function a t a
given instant is then measured by
5
(2)
=
-
n R
=
n,z
+ n,y + n,z.
The direction cosines n,, nu,n, of the vector n are best expressed in terms
of the polar angles CY and p shown in Fig. 66.
(31
n, = sin a cos
p,
n,
=
sin a sin p,
n,
=
cos a ;
hence,
(4)
$=
As t,he paramekrs
CY
e i k ( z sin
a
cos j3+y sin a sin B f z 00s a)- i d
are varied, the axis of propagation can be
oriented a t will. With each direction of
propagation one associates an amplitudc
g(a, p) depending only on the angles a
and p ; since tho field equations are in the
prescrit case linear, a solution can be constructed b y mpcrposing plane waves, all
of the same frequency but traveling in
various dircct,ions, each with its appropriat,e amplitude.
and
Y
(5) +(x, 2 / ,
2,
t ) = e-'"'JdaJdPg(a,
&(z
sin
(I
coa B+y sin
(
I
P)
sin B f z cos (I).
FIG.tX-Thc
phase of an elementary plane wave is measured
along tlie r-itzis whose direction is
detcrmined by the unit vector n.
A fixed point of observation is located by the vcctor R.
If the angles are real, the limits of integration for a are obviously 0 and r ;p goes
from 0 t,o 2n. But such a solution is
mathernatically by no means the most
general, for (5) satisfies the wave equation for complex as well as real
values of the parameters a and /3 and we shall discover shortly that
complex angles must in fact be included if we are t o represent arbitrary
fields by such an integral.
When o is real, the wave funct'ion defined by (5) is harmonic in time.
To represent fields whose time variation over a specified coordinate
1 The general theory of such solutions of the wave equation has been discussed by
Whittaker. See WHITTAKERand WATSON,"Modern Analysis," 4th ed., Chap.
XVIII, Cambridge University Prcss, 1927.
PEC.6.71
CONSTRUCTION PROM PLANE W A V E SOLUTIONS
3 63
surface is more complex i t will be necessary t o sum or integrate (5) with
respect to the parameter w. Wc shall define a vector propagation constant
k = k n whose rectangular components are
(6)
kl
=
k sin a cos p,
kz
=
k sin a sin 0,
k 3 = k cos a,
so that the elementary plane wave function can be written
+ = ezk.R--zwt
(7)
It follows upon introduction of (7) into the wave equation
dz+
(8)
ax2
+ az+ + dz2 a2+
-@
!Je
az+
a+
z
2 - PJ d t = 0,
that the components of k must satisfy the single relation
(9)
k?
+ ki + k2, = pew2 + ipuw = k 2
and are otherwise completely arbitrary. Thus of the parameters kl, k 2 ,
k3,W , any three may be chosen arbitrarily whereupon thc fourth is fixed
b y (9).
Let 11s suppose then that ovcr the plane x = 0 thc function
is
prescribed. We shall say that +(z, y, 0, t ) = f(z,y, t ) . Thc desired
solution is
+
in which kl, k2 and w are real variables and
mined by
(11)
k;
= w2p€
k 3
is a romplex quantit,y detcr-
+ ipuw - k:: - k,2.
The amplitude function is t,o he such that
If f(z, y, t ) and its first derivatives arc pirccwise continuous and alxolutely integrable, then g(kl, k,, w) is its Fourior transform and is given by
When (r = 0, each harmonic component is propagatrd
along the x-axis
._
with a velocity v = w / k 3 , but since k a = d i & L - k2, - k ; is not a linear
combination of w, kl, and k,, it is apparent that the initial disturbance
~
~
364
[CHAP.V I
CYLINDRICAL WAVES
f(z, y, t ) does not propagate itself without change of form even in the
absence of a dissipative term.
More precisely, there exists no general
- t>.
(
Equally one might prescribe the function
solution of (8) of the typef
2, y,
1
-
#(s, y, z, t ) throughout all
space a t an initial instant t = 0. Let us suppose, for example, that
#(z, y, z, 0) = f(z,y, z). The field is to be represented by the multiple
integral
in which ?GI, kp, JC3 arc real variables and o is a complex quantity determined from (9).
+ Li + k:) - b2,
-
(15)
io = b
f i -\/a2(&
where a = l/dG, b = a/2e as defined on page 297.
weighting function g(k1, k z , JCS) is to be such that
The amplitude or
If f(z, y, z) has the necessary analytic properties, the Fourier transform
exists and is given by
Only positive or outward waves have been considered in the foregoing
and initial conditions have been imposed only upon the function #
itself. If both # and a derivative with respect to one of the four variables
are to satisfy specified conditions, it becomes necessary to include negative as well as positive waves according to the methods described for the
one-dimensional case in Secs. 5.8, 5.9, and 5.10.
6.8. Integral Representations of the Functions Z,(p).-In
any system
of cylindrical coordinates u1, u2, x , the wave equation (6), page 351, is
satisfied by
(18)
#
= f(u1,
u2) f + - ~ t ,
where h and o are real or complex constants. Following the notation of
~
_
_
the preceding section h = ks = k cos and, since L = d p c w 2 i p u w , it
(Y
+
SEC.6.81
365
INTEGRAL REPRESENTATIONS
follows that the angle a made by the directions of the component plane
waves in (5) with the z-axis is also constant. I n other words, the elementary cylindrical wave function (18) can be rcsolved into homogeneous plane waves whose directions form a circular cone ahout the
z-axis, but the aperture of the cone is in the general case measured by a
complex angle.
(19)
in which x and y are to be expressed in tcrms of the cylindrical coordinates
u',u2.
I n the coordinat,es of the circular cylinder we have x
=
r cos 0,
y = r sin 8, and, hence,
PO)
x cos p
+ y sin p = r cos ( 6 - p).
I n the notation of the preceding paragraphs we have also
(21)
kr sin a
=
r 4 F h Z = p,
so tha t
We now change the variable of integration in (22) from p to + = - 8,
and note that, since the equation for f(r, 0) is separable, it must he
possible to express g(+
0) as a product of two functions of one variable
each.
+
(23)
g(P) = g(+
+ 0)
= g1(+>g2(0);
hence,
The angle function gZ(0) must obviously be some linear combination
of the exponentials e i p O and eciPo, and the amplitude or weighting fact,or
gl(+) must be chosen so that the radial function fl(r) satisfies (4),
page 356, which in terms of p is written
Let us substitute into (25) the integral
366
CYLINDRICAL WAVES
[CHAP.
VI
Differentiating under the sign of integration, we have
(27)
hence from (25)
(28)
(p2
sin2 4
+
ip
cos 4
- p2)gl(+)e i p C o s
0
d+
=
0.
This equation is next transformed by an integration by parts.
(28) is evidently equivalent to
Equation
If P and Q are two functions of +, then
this applied t o (29) gives
The first of these two ntcgrals can be made to vanish if the contour of
integration C is so chosen t h a t the differential has the same value a t both
thc initial and terminal points; the sccond is zero if the intcgrand vanishes. Therefore (26) is a solution of the Bessel cquation if gl(+) satisfies
and if the path C of intcgrat,ion is such that
Equation (32) is evidently satisficd by
However, we shall find
that, if fl(p) is to be identical with the cylinder functions Z,(p) defined
1 -ip;
in Sec. 6.5, a constant factor - e
must be added. Then by choosing
el*+.
lr
SEC.6.81
367
I N T EGR A L REPRES E N T ATIO NS
we obtain Sommcrfeld's integral rr:prcsentat.ion of the cylinder functions,'
. a
P (P ) =
(35)
e
-apz
.I
e i b cos ++P+)
P
the contour C to be such that
&,
I
( p sin 4 + p ) e i ( p coa ++p+)
z 0.
(36)
C
The distinction between the various particular solutions J , ( p ) ,
N,(p), H,(p) is now reduced to a distinction in thc cont,ours of integration
i n the complex plane of 4. Consider first the most elementary case in
which p = n, an integer. Then clearly, if the path of integration is
extended along the real axis of 4 from - P t'o P, or any other segment, of
length 2 ~ the
, condition (36) is fulfilled and the definite integral
i-n
(37)
JJP)
=
7&-
eip
cos ++in+
d4
is a solution of the bessel equation. That the function defined in (37)
is actually identical with the particular solution (9), page 357, can be
verificd by expanding e i p C o s 9 in a series about the point p = 0 and
integrating term by t'erm.
I n the general case when p is not, an integer, or t o obtain the independent solution, one must choose complex values of 4 in order that (36)
shall vanish a t the terminal points. Let 4 = y
in. Then
+
+
+
+
i p 4 = p sin y sinh r] - pr] i p cos y cosh q i p y .
(38) ip cos 4
If p is complex we shall assiime that p = a
ib, where a is a n essentially
positive quantity. Then by an a'ppropriate choice of y and r] the real part
ip4) 4 0 .
of (38) can be made infinitely ncgative, while exp(ip cos 4
This condition will be satisfied, for example, if we let 7 + 00 and
choose for y either - ~ / 2or + 3 ~ / 2 ;but the exponential also vanishes
when 7 -+ - provided y = + ~ / 2 . I n order that (35) shall represent
a solution of the Bessel equation, it is only necessary that the contour C
connect any two of these points.
Sommerfeld has chosen as :L pair of fundamental solutions the two
int'egrals
+
+
SOMMERFELD,
Math. Ann., 47, 335, 1896.
+
3 68
CYLINDRICAL WAVES
[CHAP.VI
The contour C1 followed by the integral (39) starts a t q = 00, y = - ~ / 2 ,
crosses both real and imaginary axes a t C#I = 0, and terminates eventually
a t r ] = - - m , y = a/2. The contour C z followed by (40) starts a t the
terminal point of C1, crosses the real axis a t y = T , and comes to an end
at q = a, y = 3n/2. Both contours are illustrated in Fig. 67. The
point at which the crossing of the real axis occurs is, of course, not essential to the definition. The contours may be deformed a t will provided only
they begin a t an infinitely remote central point of one shaded area and
terminate a t a corresponding point in a second shaded area.
FIG.67.-Contours
of integration for the cylinder functions.
The advantages of carrying C1 and Cz across the real axis a t thc
particular points y = 0 and y = P become apparent when the integrals
(39) and (40) are evaluated for very large values of p. For if the real
part of p is very large, the factor cxp(ip cos 4 ) becomes vanishingly
small a; all points of the shaded domains in Fig. 67 with the exreption
of the immediate neighborhood of the points q = 0, y = 0, * T , * 2 ~ ,
. . . . At these points the real part of i p 30s 4, Eq. (38), is zero however
large p ; consequently, if C1 and Cz are drawn as in Fig. 67, the sole contribution to the contour integrals will be experienced in the neighborhood
of the origin and the point q = 0, y = a. Out of a shaded region in
which the values of the integrand are vanishingly small, the contour CI
leads over a steep “pass” or “saddle point” of high values a t the origin,
and then abruptly downwards into another shaded plane where the contributions to the integral are again of negligible amount. The contour
6 2 encounters a similar saddle point a t q = 0, y = P.
To confine the
integration to as short a segment of the contour as possible, one must
approach the pass by the line of steepest ascent, descending quickly from
SEC.6.91
FOURIER-BESSEL INTEGRALS
369
the top into the valley beyond by the line of steepest descent. This
means in the present case that the contours C1, Cz must cross the axis
at a n angle of 45 deg. The behavior of the integrals (39) and (40) in
the neighborhood of a saddle point has been uscd by Debye’ to calculate
the asymptotic expansions of the functions H Y ) ( p ) and H g ) ( p ) . When p
is very large, both with rcsprct to unity and to the order p , one obtains
the relations (22) and (23) of page 359, thereby identifying the integrals
(39) and (40) with the Hankel functions defined in See. 6.5. Debye
considered also the case in which p was larger than the argument p, but his
results have been improved and extended by more recent investigations.
A contour integral represcntation of J p ( p ) , when p is not an integer,
follows directly from the relation
The contour C S shown in Fig. 67 represents a permissible deformation
of c1
6.9. Fourier-Bessel Integrals.-It
has been shown how in cylindrical
coordinates the two fundamental types of electromagnetic field can be
derived from a scalar function y5. I n case the cylindrical coordinates arc
circular, y5 is in general a function obtained by superposition of elementary waves such as the particular solutions (28) and (29) of page 360.
Our problem is now the following: a t the instant t = 0 the value of
+(r, 0, x, t ) is prescribcd over the plane x = 0; to determine J, for all
other values of z and t.
Let us suppose then that when t = x = 0, $ = f ( r , 0). We shall
assume that f ( r , 0) is a bounded, singlr-valued function of the variables
and is, together with its first derivatives, piecewisc continuous. Then
f ( ~0), must be periodic in 0 aiid can be expanded in a Fourier series whose
coefficionts are functions of r alone.
+ cz.
If now f ( r , 0) vanishes as r
vergence of the integrals
so“
-+ m
in such a way as to ensure the con-
IfvL(r)/
& dr, then each coefficientf,(r) can
be represented by a modified Fourier integral.2
DEBYE,Math. Ann. 67, 535, 1909. See also WATSON,Zoc. cit., pp. 235$.
Tho Fourier-Bessel integral is established by more rigorous methods in Chap. XIV
of Watson’s “Bessrl Functions.”
3 70
C YLZNDRZCAL II'A VES
[CHAP.VI
Consider a function f(x, y) of two variables admitting the Fourier
integral representation
A transformation to polar coordinates is now made both in coordinate
space and in k space.
x
(45)
kl
=
y = r sin 8,
kz = X sin p,
r cos 8,
cos p,
= X
so that in the notation of earlier paragraphs X = k sin OL = 4-2.
Thcn klx k2y = Xr cos ( p - 8) and (44), as a volume integral in
k space, transforms to
+
(46)
f ( r , 8)
=
dh
27r
Lzr
do g(X, p)
eiiy
co3 ( 6 - 0 ) .
The physical significance of this representation is worth noting. The
function exp[iXr cos (0 - e) - id] represents a plane wave whose
propagation constant is X, traveling in a direction which is normal to the
z-axis ?nd which makes an angle p with the x-axis. Each plane wave is
multiplied by an amplitude factor g(X, p) and then summed first with
respect to p from 0 to 2a and then with respect to the propagation
constant, or space frequency A.
The transform of f(x, y) is
which when transformed t o the polar coordinates f
ieads to
(48)
g(x,
p)
p
=
Suppose finally that f(r,
e)
dp
so2'
d,u j ( p , p)
= fn(r) eins.
Then
= p cos p, 7 = p
e - i x p cos
(0-p).
sin p,
SEC.6.101
REPRESENTATION OF A PLANE W A V E
371
Likewise (46) becomes now
or upon placing 4
(52)
=
fi
jn(.)
-
0,
einQ =
einQ
Thus we obtain the pair of Fourier-Besscl transforms
(53)
(54)
The function
is a solution of the wave equation in circular cylindrical coordinates,
which a t t = 0 reduces tof(r, 0) on the plane x = 0. But this obviously
is not the most general solution satisfying these conditions, for we have
still at our disposal the two parameters w and k which are subject only to
the relation k 2 = pew2
i u p w . If w is real, harmonic wave functions
such as (55) may be supcrposcd to represent an arbitrary time variation
of
over the plane z = 0. If both positive and ncgative waves are
considered, i t is permissible also to assign values to both $ and d+/dz
at z = 0. The treatment of such a problem has been adequately illustrated in Chap. V.
6.10. Representation of a Plane Wave.-The Fourier-Bessel theorem
offers a very simple means of representing an elementary plane wave in
terms of cylindrical wave functions. Let the wavc whose propagation
constant is k travel in a dircction defined by the unit vector n p whose
spherical polar angles with respect to a fixed reference system are a and p
as in Fig. 66, page 362. Thcn
+
+
(56)
+
= ezk
sin a ( z cos @ + y sin @)
. ezka cos
and our problem is t o represcnt the function
f(z, y) = eik sin u ( z cos @+y 8in 8) =
(57)
ezkr sin a cos ( 8 - 0 )
as a series of the form
m
(58)
j ( r , 0) =
C
*=-
jn(r)cine.
00
cr-wt
372
CYLINDRICAL W A V E S
By (43)we have
(59)
l=
f&> = 1
2n
eikr sin a cos
[CHAP.VI
( p - 8 ) - in8
Upon replacing fl - 8 by 9, this becomes equal to
and we obtain thereby thc useful expansion
(61)
sin a cos ( p - 8 ) =
2
n=-
inJn(kr sin a ) ein(B-0).
cc
Several other well-known series expansions are a direct conscqiicnce
of this result. Thus, if we put
p =
kr sin a, 0 - p
=
9-
7r
--)
2
(61)
becomes
m
which upon separation into real and imaginary parts lcads t o
m
THE ADDITION THEOREM FOR CIRCULARLY CYLINDRICAL WAVES
b--ol,\
6.11. Several important relations pertaining to a translation of the
axis of propagation parallel to itself can be derived from the formulas of
the foregoing section. I n Fig. 68 0
and 01 denote the origins of two
rectangular r e f e r e n c e s y s t e m s .
The sheet of the figure coincides
-1_ _ _ _ _ _ XI with the zy-plane of both systems
01
and the axes 21,y ~ z1
, through O1 are
r0
e s:
respectively parallel to 2,y, 2. The
0
X
function Jn(Xr,) einel, when multiFIG. tX-Translation
of the referonce
plied by exp( +ihzl - iwt), represystem.
sents a n elementary cylindrical
wave referred to the zl-axis. We wish to exprcss t,his cylindrical wave
in terms of a sum of cylindrical wave functions referred to the parallel
z-axis through 0.
373
A D D I T I O N THEOREM
SEC. 6.111
We write first
From the figure it is clear that 81
(2)
Tl COS
rl sin
= 0
+ II. and that
II. = rl cos (el - e) = T - yo cos (e - e,),
II. = r1 sin (el - 0) = ro sin (e - e,).
Moreover, in virtue of the periodic properties of the integrand, (1) is
equivalent to
It follows from (2) that
(4)
r1 cos (+ -
4) =
r cos
4 - r0 cos
(+
+ 0 - eo);
hence,
By (61), page 372, we have
and in virtue of the uniformit,y of convergence one may interchange
the order of summation and integration.
m
m=-
Upon replacing
(8)
m
el by e + II. this result becomes
J,(XrI) ein* =
5
m=-
J,(xro)J,+,(xr)
eim(0-00).
m
An analogous expansion for the wave function Hil)(Xrl)einsl can be
obtained by writing
(9)
3 74
CYLINDRICAL WAVES
[CHAP.VI
where C1 is the contour described in Fig. 67, page 368, with a shift of
amount $ along the real axis to ensure the vanishing of the exponential
a t the terminals. Upon expanding the exponent by (2), we find
If now Irl > / r o c,os (0 is led to the expansion
&)I,
one can proceed exactly as above and
m
m= -
m
If ro = 0, the two centers coincide; J,(O) = 0 for all values of m except
zero, and J o ( 0 ) = 1. The angle II, is zero and the right and left sides
of '(11) are clearly identical. The expansion can be verified also in the
case when r and r1 are very large, for then the Hankel functions can be
replaced by their asymptotic representation, (22), page 359. The
angle
is approximately zero and rl N r - ro cos (6 - 0,). The
amplitude factor 1/2/arl can be replaced by m
r without appreciable
error, but the term ro cos (6 - 0,) must be retained in the phase. Then
(11) reduces to
+
and the right-hand side of (12) in turn is by (61), page 372, the correct
expansion of the plane wave on the left. Indeed as r1 becomes infinite,
the expanding cylindrical wave function designated by (11) must become
asymptotic to a plane wave.
When Irl < Ir0 cos (0 - @,)I the expansion (11) fails to converge and
is replaced by
(13)
H c ) ( l r l )ein*
=
3
m=-
H2)(Aro)Jn+m(Ar)
eim(e-eo),
m
which is finite a t r = 0. At this point Jn+,(Xr) is zero unless m = -n.
Moreover
N ?r
eo - e and, since IIYA(p) = enTiHk1)(p),the right
and left sides of (13) are clearly identical. I n the other limiting case of
r0 very large, we write rl = ro - r eos (6 - 0,) and find by means of the
asymptotic representations of the Hankel functions that (13) approaches
+
+
a plane-wave function propagated from 0
1 towards 0 along the line joining
these two centers.
SEC.
375
ELEMENTARY WAVES
6.121
WAVE FUNCTIONS O F THE ELLIPTIC CYLINDER
6.12. Elementary Waves.-Cirrular wave functions are in a sense a
degenerate form of elliptic wave functions obtained by placing the
eccentricity of the cylindcrs equal to zero. The analysis of the ficld and
the properties of the functions must inevitably prove more complex in
elliptic than in circular coordinates, but the results are also fundamentally
of greater interest.
From 3, page 52, we take
f(E,
upon introducing these into (8), page 351, we find that
satisfy the cquation
7) must
+ Co”(k2- h2)(f2 - q2)f
This in turn is readily separated by writing f
d2fi
(t2- 1) 7
+ E (if1dE f [c;(k2 - h2)E2- b]f l
dE
(5)
(1 - Vz)
d*fz
&T - 7
!fi
+ [b - C i ( k 2 - h2)72]
(ill
f2
0
and leads to
= fl(()fz(q)
(4)
=
=
0,
=
0,
where b is an arbit>rary constant of separation. Thus fI(E) and f Z ( 7 )
satisfy the same dzflerential equation. Equations (4) and ( 5 ) are in fact
special cases derived from the associated Mathieu equation1
(6)
( 1 - 9)w” - 2 ( a
+ 1)zw’ + ( b - czz2)w= 0
by putting the parameter a = -4. These equations are characterized
by an irregular singularity at infinity and regular singularities a t x = k 1.
When (>> 1, (4) goes over to a Bessel equation.
A certain simplification of (4) and (5) can be achieved by transformation of the independcnt variables. Let
E
(7)
=
cosh
U,
7 =
cos
V,
so that the transformation to rectangular coordinates is expressed by
(8)
x
=
cg cosh u cos v,
y
= co sinh
u sin v.
1 WHITTAKER
and WATSON,
Zoc. cit., Chaps. X and XIX; INCE,
“Ordinary Differential Equations,” Chap. XX, Longmans, 1927.
3 76
[CHAP.VI
CYLINDRICAL WAVES
Then in place of (4) and (5), we obtain
d”fz
+ (b - c;x2
dv2
COY2 V ) f i =
0,
where as in the preceding sections X = d k 2 - h2. The distance between
the focal points on the x-axis, Fig. 9, is 2c0 and the eccentricity of the
confocal cllipses is e = l/cosh u. The transition to the circular case
follows as the limit when co 4 0 and u 3 m . Then co cosh u + co sinh u
-+r , and v is clearly the angle made by the radius r with the x-axis.
For this reason we shall refer to fl(u) as a radial function and f2(v) as a n
angular function. When u = 0, the eccentricity is one and the ellipse
reduces to a line of length 2c joining the foci on the x-axis.
The Mathieu equations (9) and (10) have been studied by many
writers. We shall consider first the angular functions f2(v).
There are,
of course, solutions of (10) whatever the value of the separation constant
b. But the electromagnetic ficld is a single-valued function of position
and hence, if the properties of the medium are homogeneous with respcct
to the variable v, i t is necessary that fi(v) be a periodic function of the
angle v. Now Eq. (10) admits periodic solutions only for certain characteristic values of the parameter b. These characteristic values form a
denumerable set bl, bz, . . . , b,, . . . . Their determination is a
problem of some length, so we shall content ourselves here with a rcferencc
to tabulated results1 and proceed directly to the definition of the functions
satisfying (9) and (10) when b coincides with a characteristic value b,.
For the proper values of b, Eq. (10) admits both even and odd periodic
solutions. The denumerable sct of characteristic values leading to even
solutions may be designated as bg) and the associated characteristic
1 A very readable account of the theory of the Mathieu functions has been given
by Whittaker and Watson, Zoc. cit., Chap. XIX; further details with extensive references to the literature have been published by Strutt in a monograph entitled “Lam&
sche- Mathieusche- und verwandtr. Funktioncn in Physik und Technik,” in the collection ‘i Ergebnisse der Mathematik,” Springer, 1932. ‘rables of Mathieu functions
and characteristic values have been published by Goldstein, Trans. Cambridge Phil.
SOC., 23, 303-336, 1927. The functions Se, and So, defined in the text differ from
the CF, and se, of these authors only by a proportionality factor. Whereas Goldstein
chooses his coefficients such that the normalization fartor is
we have found i t
advantageous to normalize in such a way t h a t the even function and the derivative of
the odd function have unit value a t thc pole v = 0. See Stratton, Proc. Nut. Acad.
Sci. U . S., 21, 51-56, 316-321, 1935; and MORSE,ibid., pp. 56-62. Extensive tables
of the expansion coefficients I):, the chararteristic values b , and the normalization
factors have been computed by P. M. Morse and will appear shortly.
4,
SEC.6.121
377
ELEMENTARY W A V E S
functions can then be represented by the cosine series
(11)
Se,(coX,
cos v)
=
c' D:: cos nu,
(m = 1, 2, 3,
-
*
*
),
n
where the primed summation is to be extended over even values of n
if m is even and over odd values of n if m is odd. The recursion formula
connecting the coefficients Q(c0X) can be found by introducing (11) into
(10). All coefficients of the series are thus referred to an initial one which
is arbitrary. It is advantageous to choose this initial coefficient in such
a way that the function itself has unit value when v = 0, corresponding to
9 = cos v = 1. To this end we impose upon the D; the condition
n
The odd periodic solutions of (10) are associated with a second set of
characteristic values to be designated by b:). These functions can be
represented by a sine series
So,(coX,
cos v)
=
2' F;
sin nu
n
upon whose coefficients F;(coX) we impose the condition
c' nF;
=
1.
n
Conscquently the derivativc of So,(coX,
= 0.
cos v) will have unit value a t
2,
;[
So,(coX,
cos v)
= 1.
The characteristic functions Se, and So, constitute a complete
orthogonal set,. Let by) and by) be two characteristic values and Se,,
Sei the associated functions. They satisfy the equations
(15)
d2Sej
dv2
F
_
+ (6je) - c;h2 cos2 v)Sei = 0.
Multiply (15) by Sei, (16) by Sei, and subtract one from the other.
Upon integrating (17) from 0 to 27r and taking account of the periodicity
of the functions, we obtain
Sei(coX, cos v) Sei(coX, cos v) dv = 0,
Np,
i. # j .,
z = 3.
3 78
CYLINDRICAL WAVES
[(:HAP.
v1
The normalization factors N g ) can be computed from the series expansions of the functions.
In identical fashion one deduces that
su”” SOi(COX, cos v) S O i ( C O X , cos v) dv
and
(20)
I”’
Se,(coX, cos v) X o , ( c o X , cos
=
0,
Ay’,
i # j,
i = 3.
v) dv = 0,
the last for i = j as well as i # j .
With each characteristic value bg) there is associated one and only
one periodic solution Se,. For this same number bg) there must exist,
however, an independent solution of (10). Since the second solution is
nonperiodic it is unesscntial in physical problems so long as the medium
is homogeneous with respect to the angle v. When, on t8heother hand,
there are discontinuities in the properties of the medium across surfaces
v = constant, the boundary conditions may require use of functions of
the second kind.
We turn our attention now to the radial functions. It can be shown
without great difficulty that Eq. (9) is satisfied by an expansion in Bessel
functions whose coefficients differ only by a factor from those of (11)
and (13). Thus, when the parameter b assumes one of the characteristic
values bg), an associated radial function is
where [ = cosh u, im-n=
. Since all solutions of
Bessel’s equation satisfy the recurrence relations (24) and (25), page 359,
the even radial functions of the second kind are defined in terms of the
Bessel functions of the second kind by
The convergence of such series of functions is not easy to demonstrate
but appears to be satisfactory in the present instance. The great advantage of these representations is that they lead us at once to asymptotic
expressions for very large values of coXE. By (18) and (19), page 359,
we have
SEC.6.121
379
ELEMENTARY WAVES
By analogy with the Hankel functions, we are led to form the linear
combinations
The function Re; satisfies the same equation as the angle function
Se,; neither has singularities other than that at infinity. The two must,
therefore, be proportional to one another.
(29)
Se,(coX, 5)
=
6l g ) R e h ( c o X , E ) ,
2'
1
1:) =
in-m
__
D::
D$
nin-m
(30)
2
D"
( m odd).
Df
COAT
On the left, t replaces cos o.
(m even),
.Lz'
lr
Consequently when u = 0, 5
=
1, we have
Ret,(coX, 1) =
A corresponding set of relations can be obtained for the odd radial
functions. We define:
+
Rok(coX,[) = ROT,- iRo2,.
(33) Rof(coX, E) = Rot, iRoi,
The asymptotic expansions of the odd functions are identical with those
of the corresponding even functions. Finally,
(34) i Xo,,(coX,
.F)
=
6zg)Rot,(coX, t ) ,
4
__
12) =
7rCiX2
~
2
*COX
3'FFznE
n
7'
in-,
in-m
(meven) ,
(modd).
380
[CHAP.VI
CYLINDRICAL WAVES
At u = 0,
= 1, we have
With these functions a t our disposal we are now in a position to write
down elementary wave functions for the elliptic cylinder. The even and
odd wave functions, which are finite everywhere and in particular along
the axis joining the foci, are constructed from radial functions of the
first kind.
(36)
$t = Sem(coX,cos v) Rek(coX, E ) e+idkZzXi._iwt,
(37)
$& =
Xo,(cok, cos
v) Rot,(coX, l) e f i d k.~
2 - - X Zczot-.’
When it is known that at great distances from the axis of the cylinders the
field is traveling radially outward, the elementary wave functions will be
constructed from radial functions of the third kind.
(38)
J.,,, = Xe,(coX, cos v) Ref(coX, [) e*idEzLz-ht,
(39)
+m
:
= SO,(C~X,
cos
t)
v)
~ o k ( c ~ ~e *, i
d m z - i w t .
The components of the electric and magnetic vectors can now be
found by the rules set down in See. 6.3.
6.13. Integral Representations.--According to (19), page 365, we
have in any system of cylindrical coordinates
f(u1,
(40)
~
g(p)
=
)
2
eik sin 4 z cos 8+2/ sin 8)
dp.
If now the rectangular coordinates are replaced by elliptic coordinates
through the transformation x = co cosh u cos v, y = cu sinh u sin v, we can
write (40) in the form
f(u,v>
(41)
=
J g(P>
eihp
dp,
___
where as in the past X = d k 2 - h2 = k sin a, and where
+ y sin
co(cosh u cos v cos /3
sinh u sin v sin @..
This function j(u,v) is to satisfy the equation
(42)
p(u,v, 0)
(43)
but sincef
(44)
=z
cos /3
3
+
i3U2
3+
= ji(u)ji(v),
it is
=
ciX2(cosh2u - cos2 v)f
=
0,
clear t,hat it must also satisfy
azf + (ciX2 cosh2 u - b)f
aU2
+
= 0,
SEC.6.131
INTEGRAL REPRESENTATIONS
obtained by multiplying (9), page 376, by fz(v).
(41) with respect to u,we have
38 1
Upon differentiating
(45)
if, therefore, (41) is to satisfy (44), it is necessary that
[ 2 - ?$) +
2
(46)
g(p) i x
A2
cgX2 cosh2 u - b
Now p is a function of 0 as well as of u and it can be easily verified that
(47)
Consequently (46) is equivalent to
(49)
Finally by (30), page 366,
hcnce, (41) is a solution of (44) provided g(p) satisfies
3 + (b do2
c;x2
cos2 P)g
=
0,
and the path C of integration is so chosen that
It will be noted that the equation of the transform g(p) is identical with
that of the angular function j2(v),and in fact differs unessentially from
that for fl(u)with which we set out. This property is common to the
transforms of all solutions of equations belonging to the group defined by
(6), page 375.
The results just derived are valid whatever the value of the separation
If, however, we confine ourselves to the even and odd sets
constant b,.
of characteristic values bg) and b z ) , it will be advantageous to choose for
g ( p ) the periodic solutions of (51) which have been designated by Se,(coX,
cos p), Xo,(cGX, cos 0). Since p(u, v, p) is also periodic in p with period
27r,it is apparent that (52) will be fulfilled when the path of integration
382
CYLINDRICAL WAVES
[CHAP. VI
is any section of the real axis of length 2a. An integral representation
of f(u, v) which holds when b belongs to the set b z ) is then
(53)
From this an integral representing j l ( u ) can be found immediately by
placing v = 0.
jE(u) =
(54)
L2r
cash
Sc,(coX, cos p)
co8
B dj3.
There remains the task of identifying the particular solution defined
by (54). Upon introducing the cosine expansion of Se, and abbreviating
C O X cosh u = p, we obtain
(55) f E ( u ) =
2'D; Szr
eip
cos B
0
n
cos np dp
=
27r
2'inD:Jn(p).
n
Next we note that in = (-l)"i-",
and remember that the summation
extends over even values of n if m is even; over odd values only if m is
odd. Therefore (-l)n = ( - l ) m ,and in virtue of the definition (21))
page 378, it follows that
where = cosh u. Moreover, Sen,(coX, cos p) is proportional to ReA(coX,
cos a), in which [ has been replaced by cos 0,so that by (29)) page 379,
the function Rei(coX, t ) satisfies an integral equation,
Still another representation of RE: can be found by expanding the
integrand of (57). Thus
(58) Re:(coX, t ) =
4
Zz)
i-"D;
lzr
J,(coX cos p) eicaX[
B
dp.
n
But
(59)
whence,
la'
(60)
J,(coX cos p)
eicoXE cos
@
dp
=
i-"
cos n+ Jo[coX(E
+ cos +)I d4.
SEC.6.131
383
INTEGRAL REPRESENTATIONS
Remembering once more that ( - 1)” = (- l)m,
we obtain
(61) Rei(coX, E )
=
& lzw+
JO[COX(E cos +)] Sem(coX,cos 4) d+.
If’(
- l)m
Integral representations of the radial functions of the third and
fourth kinds can be obtained by a variation in thci choice of contour. To
simplify matters a little, let us take eosh ZL = .$,v = 0, cos /? = t . Then
the several radial functions are represented by integrals of the form
$$(E>
(62)
=
L Sem (coX , t ) ezcoAft(l- t2)-f
dt,
where the contour C is such as to ensure the vanishing of the so-called
“bilinear concomitant” (52), which now assumes the form
(63)
d
dt
(1 - t’)i[ icoX&!5‘e,(coX, t ) - - Sem(coX,t )
When t becomes very large, the asymptotic expansion of Xe, is
consequently, the vanishing of (63) at infinity is governed by a factor
exp icoX([ - 1)t. If, therefore, the real part of c(,X(E - 1) is greater than
zero, the contour can begin or end at t = i m , while for the other limit one
may choose either f l or -1. As a result we have
provided the real part of [cox(( - l ) ]
0-plane, we find (65) equivalent to
(67)
Ref(coX, t ) = i-m
> 0.
Reverting to the complex
* Se,(coX, cos
a)
eicohf ens
do,
with a corresponding integral for Rei(coX, [).
With the help of (67) one may deduce further integrals of the type
(61). Thus, in place of (58) we write
(68) Ref(coX,
6) = Z$
62’
n
f-i
J,(coX cos
0 ) eicohE
0 dp
384
[CHAP.VI
CYLINDRICAL WAVES
Now by (39), page 367, when p
provided the real part of
p
=
0, we have
> 0; consequently,
provided the real part of [coX(E - l)]
(70) leads to
> 0. A
combination of (61) and
A clue to the formulation of integral representations of the odd functions is given by
eiCoAi
CO8
B
cos np dp,
the result of a n integration by parts. The proof of the relations
is left to the reader.
6.14. Expansion of Plane and Circular Waves.-The periodic functions Xe,(coX, cos v) and SO,(COX, cos v) constitute a complete, orthogonal
set. Consequently, if a function f(u, v) is periodic in v with period 2~ and
together with the first derivative af/av is piecewise continuous with
respect to v, it can be expanded in a series of the form
m
(75) j ( u , v)
=
C jg)(u)Se,(:oX,
m=?
cos v)
+
oc
jz)(u)
SO,(C~X, cos 01,
m=O
SEC.6.141
385
E X P A N S I O N OF P L A N E A N D CIRCULAR W A V E S
whose coefficients are determined from
This expansion may be applied to the representation of a plane wave
whose propagation constant is k , traveling in a direction defined by the
unit vector n whose spherical angles with respect to a fixed reference
system are CY. and /3 as in Fig. 66, page 362.
$, =
(77)
ezk sin a(x
30s
0+y sin 0). ezke
cos or-iot
Upon expressing x and y in terms of u and v and putting X
have
(78) f(u,v, B)
=
ezhp,
p
=
co(rosh u cos v cos /3
=
k sin a , we
+ sinh u sin v sin p).
Since there is complcte symmetry in (78) between v and p, the expansion
(75) can be written
(79)
echp
=
2
a,(u> Se,,(ccX, cos
~ e , ( c o ~cos
, v)
m=O
m
-I-
2 b,(u) Xo,(coX,
cos p) Som(coX, cos u )
m=O
in which the coefficients a,(u) and b,(u) depend on u alone.
of (76), wc have
a,(u) NE)Se,(coX, cos p) =
(80)
I n virtue
Xe,(coX, cos v) e z h p dv.
is unaffected by the value of p, we may put P
1) = 1, p = co cosh u cos v. Then by (56),
since a,(u)
Se,(coX,
=
0,
Likewise
(82)
b,(u) N E ) S ~ , ( ~cos
~ Xp), =
~ o , , ( c ~cos
~ ,v)
eiAp
dv.
We now differentiate this last relation with respect to p and then put
6 = 0. I n virtue of (14) the coefficient b,,(u) proves to be
386
[CHAP.VI
CYLINDRICAL W A V E S
The complete expansion for a plane wave of arbitrary direction is,
therefore,
m
Sem(coX, cos
P)
1
+ NF
Ro;(coX,
I
cash U ) SO,(C~X,cos V ) XO,(coX, cos B) .
This last result enables us to write down a t once the integral representations of those elementary elliptic wave functions that are finite on
the axis. We need only multiply (84) b y Se,(cd, cos P) and integrate
over a period. From the orthogonal properties of the functions it follows
that
(85) Rei(cd,
similarly, for the odd functions,
With the help of (67) it can be shown also that the elementary functions
of elliptic waves traveling outward from the axis are represented by the
contour integral
The upper sign in the exponential is to be chosen when - ~ / 2 < v < ~ / 2 ,
the negative sign when r / 2 < v < 3 ~ 1 2 . A similar relation can be
deduced for the odd functions.
From the analysis of Sec. 6.8 it is easy to aee th a t the even circular
wave function cos n0 Jn(Xr),which remains finite on the axis, can be
represented by the integral
(88)
2 d cos n0 J,(Xr) =
where p = z cos j3
sozu
cos nB eiX* dp,
+ y sin /3 = r cos (j3 - 0).
D; and summation over even or odd values of
(89)
Rei(coX, cosh
U)
Se,(coX,
cos v )
=
Upon multiplication by
one obtains
n,
,&7'
in-mDzcos n0 J,(Xr),
which expresses the even elliptic wave function as a n expansion in circular
wave functions. A similar expression can be found for the odd functions.
387
PROBLEMS
An addition theorem relating circular and elliptic waves referred to
two parallel axes has been derived by Morse.’
Problems
1. Show t h a t the equation
is satisfied by wave functions of the type
t,b = Ce’(61+6P)cos (hlx - 61) cos (hzy
h:
hi
h: = k2.
-+ +
- 6z)eihsz-iwf,
Let il. = II,, II, = II, = 0, be the components of a Hertz vector oriented in the direction of propagation. Show that this leads t o a transverse magnetic field whoso
electric components are
E , = -ihlh3Cei(61+62)sin (hlx - 61) cos (hzy - 8z)eih3Z-imt,
E I -- -ihzhsCei(61+*2)cos (hlx - 61) sin (hzy - 62)eih3L-iwf,
E , = ( k Z - h:)Ceifb1+6z)cos (hlz - 61) cos (hzy - 6 z ) e i h r z - k f ,
arid whose impedances are
E.
z= = -H,
=
k e - hi
u p cotan (hlz ik’hi
&I),
Find the components of a corresponding transverse electric field by choosing
n:, n: = 11* = 0.
The factor eir61+62) is introduced to include traveling as well as standing waves.
Thus, if one chooses 6L = im , the field behaves as eihLz.
2. The equation of a two-dimensional wave motion in a nondissipative medium is
$b =
Show that
where
x‘ - x = r cos 8,
y’
-y
=T
sin 8,
and
+
=
dx, Y),
!?
at = G(x, y),
when t = 0.
+
This is the Poisson-Parseval formula. Note that a t time t the wave function is
determined by the initial values g and G a t all points on the circumference and the
interior of a circle of radius at. Thus even in nondissipative media the disturbance
1
MORSE,loc. cit.
388
CY L IN D R IC A L W A V E S
persists after the passage of the wave front.
wave motion in even-dimensional spaces.
3. Show that the wave equation
[CHAP.
VI
This behavior is characteristic of
r dr
has a particular solution of the form
iI.
=
1L 2 ” Y
+ ir cos a, t
(2
27r
:’)
--
sin a
dor,
which reduces to tl. = F ( z , t ) on the axis r = 0, where F is an arbitrary function.
Show that another solution of thc same equation is
+=
J-mm
F (z
-
)
1 -
r sinh a,
cosh a
da.
(Bateman)
4. A special case of the solution obtained in Problem 3 is the symmetric, twodimensional wave function.
$ =
F ( 1 - cosh
2x
a)
da,
which represents a wave expanding about a uniform line source of strength F ( t ) along
the z-axis.
Show that if F ( t ) = 0 when t < 0, then
which is zero everywhere as long a s t < r / c .
If the source acts only for a finite time T , so that F ( t )
0
=
0 except in the interval
< t < T, show that the wave leaves a residue or “tail” whose form, when t
>>
- I:
C
is determined by
* =
(Lamb, “Hydrodynamics,” 5th cd., p. 279, 1924.
6. Show t h a t the equation
is satisfied by
Cambridge University Press.)
+
sin k ( ~ a ) d [
r+ro d E 2
where
r2
6. Show that the equation
= x2
-
+ y2,
(5
-2
ri
2
- (Y
= 2:
- Yo12
+ yi.
(Bateman)
T,
389
PROBLEMS
is satisfied by
+
where r2 = 2 2
y2 and f and F are arbitrary functions,
T o what limitations is this
solution subjected?
Obtain in this way the particular solution
(Bateman)
7. The equation
is rcduccd by the substitution
J. = e-*tu(z, y, t )
to
The initial conditions are
1,et r he the radius of a sphere drawn about thc point of observation and a, p , y the
direction cosines of r with respect to the x, y, z axes. Show that a solution is represented by
u ( z , y, 2) =
--
t g(z
+ ~ t ay, -t uta) cos (ibtr) d Q
+2,
SS
1 G(x
+ ata, y + ato) cos (ibty) df2,
(Roussincsq)
where d Q is thc solid angle subtended by an clcment of surface on the sphcrc.
8. Show that
is an integral representation of the Bessel function of the first kind valid for all value:
of the order p provided E e ( p ) > - i.
9. Derive the integral representations
Imz
> 0.
for the Besscl function of the second kind and zero order, and
Imz <O.
390
C Y L I N D R I C A L WAVES
[CHAP.VI
10. Show that the modified Bessel equation
is satisfied by the function
where
pri
--
I,@)
=
e
when
~,(iz),
-T
< arg z < n--.
2
The most commonly used independent solution is the one defined by
A complete set of recurrence relations for these functions are given by Watson,
Bessel Functions,” page 79.
11. Demonstrate the following integral representations of the modified Bessel
functions.
‘(
> 0 and
It is assumed t h a t Re
/arg ZI
T
< -.
2
12. Show that for very large values of the argument the asymptotic behavior
of the modified Bessel functions is given by
m
n-
provided -2
3
< arg z < -r,
2
larg zI
m=O
where
(P’m,
r(p + m + +L- m + 3)
= m ! r(p
(4p2 - 1)(4p2 - 32)
..
. [4p2 - (2m - 1)2]
22mm!
3
< -2 r ,
391
PROBLEMS
Retaining only the first term,
-.
13. Prove that if R
= d r 2
+ z2 is the distance between two points, the inverse
distance is represented by
R
=
2
x
cos ,8z Ko(gr)d p ,
where Ko(pr) is the modified Bessel function defined in Problem 10.
14. As a special case of Problem 4,show that the equation
is satisfied by
where k = w / c and K Ois the modified Bessel function defined in Problem 10. This
solution plays the same role in two-dimensional wave propagation as does the function
eikH/R in three dimensions.
16. Show that the equation
is satisfied by
n
n
m=l
which has a period of 2nrr in 0 and hence is multivalued about the axis.
this solution can be represented by the integral
Show t h a t
iu
following a properly chosen contour C in the complex or-plane. This solution is
everywhere finite and continuous. $ behaves like a plane wave a t infinity provided
I 6 I < P , but vanishes (is regular) a t infinity if 0 lies between any pair of the limits
T
< B < 3rr,
3u
< e < h, . .
I
--& < e
< -",
which define the 2nd, 3rd, . . . nth sheets of a Riemann surface. Such multiform
wave functions have been applied by Sommerfeld to diffraction problems.
16. Derive explicit expressions for the radial and tangential impedances of elliptic
cylindrical electromagnetic waves.
17. Show that the integral
is a cylindrical wave function which is equal to e i k x / R , R2
=
r2
+ z2, when f(h) - 1.
CHAPTER VII
SPHERICAL WAVES
I n the preceding chapter it was shown how the analytic difficulties
involved in the treatment of vector differential equations in curvilinear
coordinates might be overcome in cylindrical systems by a resolution of
the field into two partial fields, each derivable from a purely scalar function satisfying the wave equation. Fortunately this method is applicable
also to spherical coordinates to which we now give our attention. The
peculiar advantages of cylindriral and spherical systems are a consequence of the very simple character of their geometrical properties. A
deeper insight into the nature of the problem and of the difficulties
offered by curvilinear coordinates in general will be gained from a brief
preliminary study of the vector wave equation.
THE VECTOR WAVE EQUATION
7.1. A Fundamental Set of Solutions.-Within any closed domain
of a homogeneous, isotropic medium from which sources have been
excluded, all vectors characterizing the electromagnetic field-the field
vectors E, B, D, and H, the vector potential and the Hertzian vectorssatisfy one and the same differential equation. If C denotes any such
vector, then
dC
= 0.
5 - pu v2c - p€ d2C
dt
Because of the linearity of this wave equation, fields of arbitrary time
variation can be constructed from harmonic solutions and there is no
loss of generality in the assumption that the vector C contains the time
only as a factor ctwt.
By the operator v2acting on a vector one must
understand V2C = V v C - V x v x C ; therefore, in place of (1)
we shall write
-
(2)
v v - c- v x v x
+
C+k?C
=
0,
where k 2 = epu2 iupw as usual.
Now the vector equation ( 2 ) can always be rcplaced by a simultaneous
system of three scalar equations, but thc solution of this system for any
component of C is in most cases impractical. [Cj. (85), page 50.1 It
is only when C is resolved into its rectangular components that three
392
SEC.7.11
A FUNDAMENTAL SET OF SOLUTIONS
393
independent, equations are obtained and in this case
V2Cj
(3)
+ kZCi = 0,
( j = r , Y,z).
The operator V2 can then be expressed in curvilinear as well as rectangular
coordinat'es. Very little attention has been paid to the determination of
independent vector solutions of ( Z ) , but the problem has been attacked
recently by Hansen' in a series of interesting papers dealing with the
radiation from antennas.
Let the scalar function $ bct a solution of the equation
v:'++ k2$
(4)
=
0,
and let a be any constant vectJor of unit lengt,h. We now construct three
independent vector solut'ions of (2) as follows:
(5)
L = V+,
M
=:
V x a+,
N
=
1
- V x M.
k
If C is placcd equal to L, M, or N, one will vcrify that ( 2 ) is indeed satisfied
identically by (5) subject to ( 4 ) . Since a is a constant vector, it is clear
t>hatM can be written also as
M
(6)
=
1; x a
=
1
- V x N.
k
For one and the same generating function $ thc vector M is perpendicular
t3 the vector L, or
(7)
L-M=0.
Thc vector fiinct,ions L, M, and N have ccrtain ridable properties
that follow directly from thcir definitions. Thus
vxL
(8)
=
0,
v-L
=
V2$
= -k2+,
whereas M and N are solenoidal.
(9)
V - M = 0,
V * N = 0.
The particular solutions of (4) which are finite, cont'inuous, and singlevalued in a given domain form a discrete set. For the moment wc shall
&note any one of these solutions by
Associated with each charactmistic function
are three vec:t)or solutions L,, M,, N, of (2), no two of
which are colinear. Prcsum:tbly any arbitrary wavc funct,ion can bc
rcpresented as a linear combinat,ion of the characteristic vect,or funct'ions;
since the L,, M,, N, possess ccrtain orthogonal properties which me shall
demonstrate in due course, t'he coefficients of the expansion can be deter-
+,.
$%
1 HANSEN,
Phys. Rev., 47, 139-143, January, 1935; Physics, 7,460-465. December,
1936; J . A p p l i e d Phys., 8, 284-286, April, 1937.
394
[CHAP.V I I
SPHERICAL WAVES
mined. When the given function is purely solenoidal, the expansion is
made in terms of M, and N, alone. If, however, the divergence of the
function does not vanish, terms in L, must be included.
The vectors M and N are obviously appropriate for the represent>ation
of the fields E and H, for each is proportional to the curl of the other.
Thus, if the time enters only as a factor e-i”t and if the free-charge density
is everywhere zero in a homogeneous, isotropic medium of conductivity
u jwe have
1
H=-VXE.
WJ
Suppose then that the vector potential can be represented by an expansion
in characteristic vector functions of the form
the coefficients a,, b,, c,, to be determined from the current distribution.
By ( 8 ) , (lo), and the relation pH = V X A, we find for the fields
(12)
E
=
-3
(a,M,
+ b,N,),
H=
(a,N,
+ b,M,).
n
The scalar potential
+ plays no part in the calculation, but can, of course,
be determined directly from (11). For by (27), page 27,
v -A
i
= - k2+;
w
hence, by (8)
n
Then V+ =
-2 cnLn and clearly the relation E = -V+ - aA
- leads
dt
n
again t o (12). Finally, if we recall that under the specified conditions
an electromagnetic field can be represented in terms of two Hertz vectors
by the equations
(14)
E
V X V X
+iwpV X
n*,
H
=
k2
--V X rI + V x V x II*.
ZWP
it is immediately apparent that
where a is a constant vector.
Before applying these results t o cylindrical and spherical systems, let
us consider the elementary example of waves in rectangular coordinates.
APPLICATION TO CYLINDRICAL COORDINATES
SEC.7.21
395
A plane wave whose propagation vector is k = kn is represented by
$ = eik * R - i d
(16)
J
where R is the radius vector drawn from a fixed origin. Then since
k R = k,x
k,y
k,z, it is easy to see that
(17)
+
+
.
L
=
i$k,
M
= i$k
x a,
.
N
=
1
-$(k
k
x a) x k.
-
I n this particular case L M := M N = N L = 0; all three vectors are
mutually perpendicular and L i s a purely longitudinal wave. Now the
polar angles dctermining the direction of k are a! and p, as defined in
Fig. 66, page 362, and general solutions of (2) can be constructed by
integrating these plane-wave functions over all possible directions. If
g(a, p) is a scalar amplitude or weighting factor, one may write for L
in any coordinate system, subject to convergence requirements,
L
= ie-iwt
da!
d p g ( a , @)k(a!,
8) eik.R;
da
d/3 g ( a , 8)k
likewise for M and N
M = ie-iwt
x a eikeR,
7.2. Application to Cylindrical Coordinates.'-The
scalar eharacteristic functions of the wave equation in cylindrical coordinates were set
down in Eq. (6), page 356. Therc are certain disadvantages in the use of
complex angle functions exp(in0) and it will prove simpler to deal with
the two real functions cos no and sin no which may be denoted simply
as even and odd. Cylindrical wave functions, constructed with Bessel
functions of the first kind and hence finitc on the axis, will be denoted by
$i'),
while the wave functions formed with N,(Xr) or HL')(hr) will be
called $i2)
and $3 respectively. Thus
where X = d k 2
1
- h2 = k sin a! as usual.
Functions of the first kind are
Compare the results of this section with Eqs. (36) and (37), p. 361.
396
SPHERICAL WAVES
[CHAP.VII
represented by the definite integrals
and representations of other kinds differ from these only in the choice
of the contour of integration.
From (23) we now construct integral representations of the vector
wave functions. Thus
(24)
Upon differentiating (23) with respect to r , 0, and z and noting that
k sin a cos (P - e), k sin a sin (P - O), and k cos CY are the components
of k respectively along the axes defined by the unit vectors ill i,, i, so that
(25) k = ilk sin
CY
cos (0 - 0)
+ izk sin a sin (p - 6) + iSk cos a,
we find for the even function
which is of the form (18). The corresponding odd function is obtained
by replacing cos np by sin np under t,he integral. When applying these
characteristic functions to the expansion of an arbitrary vector wave,
it proves convenient to split off the factor exp(ihz - i w t ) , and we shall
define the vector counterparts of th e scalar functionf(r, 0) of Chap. VI by
(27)
L, = lneihz-iiot
M, = mneiha-iiwt
N I t -- nneihz-iiwt.
From (26) we have
I n like fashion one finds integral representations of the independent
solutions. For the constant vector a we choose the u.nit vector is directed
along the x-axis. Then
whence it is easily deduced that
SEC.7.21
since
397
APPLICATION TO C Y L I N D R I C A L COORDINATES
v x M,
=
kN,, we havc
From these integrals one can easily calculate the rectangular components
of the wave functions. The vector functions of angle in the integrands
are resolved into rectangular components which can then be combined
with cos no or sin no. Thc resulting scalar integrals for the components
arc evaluatcd by comparison with (23).
To expand a n arbitrary vector function of r and 8 in terms of the
l:,LA,mznx,nEnh,
one must show that these functions are orthogonal. Let
il, i2,is be unit base vectors of a circularly cylindrical coordinate system,
Fig. 7, page 51, and Z,(Xr) a cylindrical Bessel function of any kind.
Then by (5) we obtain
a
z,(Xr)
(32) le,,A= dr
0
=
(33)
me,x
'Ps n8 il
SlIl
7;
n Z,(Xr)
f
zn(Xr)sin n8 i, 4- ih z,(Xr) cos no is,
cos
sin
n8 il - d z,(Xr) '0' n8 i,,
dr
sin
g3
'?'
sin
n8 is.
Now it is immediately evident that the scalar product of any two functions integrated over 8 from 0 to 2n must vanish if the two differ in the
index n.
(35)
if n # n'.
Let us understand by i the function 1 with the sign of ih reversed.'
the normalizing factor is to br found from the integral
Then
where 6 = 0 unless n = 0, in which case 6 = 1. The first two terms on
the right can be combined with the aid of the recurrence relations (24)
and (25), page 359, giving
1
This is not necessarily the conjugate, since X and h may be complex quantities.
(42)
J z W0
.I
2*
.
l 0e n i m;n,,
=
mE"h
";.A.
de
= J2*
lenA
0
*
fiZnxt
dB
=
0,
and by use of the recurrence relations once more
2a
-
do =
m:nA no, de = 0.
mp'
e"
There remains only one other combination, and this, unfortunately, leads
to a difficulty.
(43 )
(44)
':"A
JZr
lenA
*
0
0
*
tienx, dB = i(1
0
+ 6) ={
- hX'* Z,(Xr)
k
XX'h
- 2[Z,-,(Xr) Z-l(Xfr)
a quantity not identically zero.
Z,(X'r)
+ Zn+l(Xr) Zn+1(X'r)1
The set of vector functions lenh,menh,
0
0
nenh,therefore, just fail to be completely orthogonal with rcspect to e
0
because of (44). I n many cases this is of no importance, for if a vector
function is divergenceless-as are t,he electromagnetic field vectors in
the absence of free charge-it can be expanded in terms of menh,ne*x
0
0
alone. A complete expansion of the vector potential, on the other hand,
requires inclusion of the lenhel
0
The completeness of the set of vector functions has not been demonstrated but
presumably follows from the completeness of the scalar set
I&:.,.
SEC.7.31
E L E M E N T A R Y SPHERICAL W A V E S
399
I n the case of functions of the first kind the Fourier-Bessel theorem
has been applied by I-Iansen to complete the orthogonalit,y and to simplify
the normalization factors. From ( 5 3 ) and (54), page 371, after an
obvious interchange of variable, we have
(45)
f(h) =
J “ r dr
A’ dX’f(X’)J,(h’r)J,(Xr).
This is applied to (37), (39), (41), and (44) when Z,(Xr) is replaced by
,Jn(X~). For f(X’) we shall h:ive A’, h‘ = d m ,h’X’ and the like.
Now one must note that the validity of the Fourier-Bessel integral has
J
been demonstrated only when
If(X)\fi
dX exists, and that this con-
dition i s not fuljilled in the present instance. The artifices employed to
ensure convergence involve mathematical difficulties beyond the scope
of the prcsent work. With certain reservations one may introduce an
exponential convergence factor exactly as in the theory of the Laplace
transform on page 310. By 1;: let us undcrstand henceforth the limit
-
0
approached by l$~:e-’slAas s
0
0.
Then
Consequently the normalizing factors reduce to
(49)
while for the troublesome (44) we find’
THE SCALAR WAVE EQUATION IN SPHERICAL COORDINATES
7.3. Elementary Spherical Waves.-Since an arbitrary time variation
of the field can be represented by Fourier analysis in terms of harmonic
components, no essential loss of generality will be incurred hereafter by
the assumption that
(1)
+ = f ( R , 0, 4)e-ht.
1 The reader must note that the limit approached by (46) as s -+ 0 is not necessarily
equal t o the value of the integral at s = 0, for this would imply continuity of the function defined by the integral in the neighborhood of s = 0. This point underlies the
whole theory of the Laplace transform. Cf. Carslaw, “Introduction to the Theory
of Fourier’s Series and Integrals,” 2d ed., p. 293, Macmillan, 1921.
400
SPiiERICAL W A V E S
[CHAP.V I I
I n a homogeneous, isotropic medium the function f ( R , 0, +) must satisfy
(2)
v2f+ky-=0,
which in spherical coordinatcs is expanded by (95), pagc 52, to
The equation is separable, so that upon placing f
finds
= fl(r)fL(0)fS(+)
one
The parameters p and q arc separation constants whose choice is
governed by the physical requirement that a t any fixed point in space
the field must be single-valued. If the properties of the medium are
independent of equatorial angle +, it is necessary that f3(+) be a periodic
function with period 2n and (I is, thcrcfore, restricted to the integers
m = 0, i1, & 2, . . . . To determine p we first identify the solution f 2
as an associated Legendre function. Upon substitution of 7 = cos 8,
Eq. ( 5 ) transforms to
(7)
a n equation characterized by rcgular singularities at the points 7 = - 1,
9 = + I , 7 = to, and no others. I t s solutions arc, therefore, hypergeometric functions. Now when m = 0, ( 7 ) reduces to the Legendrc
equation. There are, of course, two independent solutions of thc
Legendre equation which may be expanded in ascending- power series
about the origin 7 = 0. These series do not, in gcncral, converge for
r] = _+1. If, however, we choose p 2 = n(n
l), where n = 0, 1, 2,
. . . , then one of the scries breaks off after a finite number of terms
and has a finite valuc a t the poles. Thcse polynomial solutions satisfying
the equation
d 2v - 2 7 -dU
(1 - 72) -n(n
l)v = 0
d 7.
d7
+
+
+
are known as Legendrc polynomials and are designated by P,(q).
now we differentiate (8) m times with respect to 7, we obtain
If
SEC.7.31
where w
ELEMENTARY SPHERICAL WAVES
= dmu/dqm,and
401
upon making a last substitution of the dependent
m
variable, w
=
(1 - qz)-5zlfp(v), we obtain (7) in the form
The solutions of (10) which arc’ finite a t the poles 7 = rt 1 and which are,
consequently, periodic in 0 arc’ the associated Legendre polynomials
fi(9) =
P;(,)
=
(1 -
”d-P ( )
-Ad 9-
92)Z
For each pair of integcrs there exists a n independent solution &;(q)
which becomes infinite a t 7 = I l , and which consequently does not
apply to physical fields in a complete spherical domain.
The definition of the associated Legcndre polynomial as stated in (11)
holds only when n and m are positive integers. The functions of negative
index are related in a rather simple fashion to thosc of positive index,
but we shall have no need of them here. To obviate any confusion on
this matter we choose for particular solutions of (6) thz real functions
cos m+, sin md, and restrict ?n and n to the positive integers and zero.
It is clear from (11) that P;(v) vanishes when m > n, for Pn(q)is a
polynomial of nth degree. We have in fact
m
The properties of the hypergeometric functions, of which the Legendre
functions are the best known c.xample, have been explored in great detail
and i t would be scarcely possible to give an adequate account of the
various series and integral representations within the space a t our disposal. We shall set down only the indispensable recurrence relations,
402
[CHAP.VII
SPHERICAL WAVES
T o these may be added the differential relations
1
2
= - [ ( n- m
+ l ) ( n+ m)P:-’
- Pm+l
n
I.
I t may be remarked in passing that these relations are satisfied also by
functions of the second kind Qz.
The functions cos rn4 P;(cos 6 ) and sin m4 P;(cos 6) are periodic on
the surface of a unit sphere and the indices m and n determine the number
of nodal lines. Thus when m = 0, the field is independent of the equatorial angle 4. If n is also zero, the value of the function is everywhere
constant on the surface of the sphere. When n = 1, there is a single
nodal line at the equator 6 = ~ / along
2
which the function is zero. When
-1
FIG.6g.-Nodes
of the function sin 3rp Pi(cos 0) on the developed surface of a sphere.
The function has negative values over the shaded areas.
n = 2, there are two nodal lines following the parallels of latitude at
approximately 6 = 55 deg. and 0 = 125 deg., so that the sphere is
divided into three zones; the function is positive in the polar zones and
negative over the equatorial zone. As we continue in this way, it is
apparent t ha t there are n nodal lines and n
1 zones within which the
function is alternately positive and negative. For this reason the
P,(cos 0) are often called zonal harmonics. If now m has a value other
than zero, it will be observed upon examination of Appendix IV that
+
m.
the function is zero at the poles due to th e factor (1 - q2)H, and that the
number of nodal lines parallel to the equator is n - m. Moreover, the
function vanishes along lines of longitude determined b y the roots of
ccs rnb and sin md. There are obviously m longitudinal nodes that
intersect the nodal parallels of latitude orthogonally, thus dividing the
403
ELEMENTARY SPHERICAL WAVES
SEC.7.31
surface of the sphere into rectangular domains, or tesserae, within which
the function is alternately positive and negative. For this reason the
functions cos m4Pz(cos 6) arid sin m+P:(cos 6) are sometimes called
tesseral harmonics of nth degree and mth order. There are obviously
2n
1 tesseral harmonics of nth degree. This division into positive and
negative domains is illustrated graphically in Fig. 69 for the function
sin 34 P,3(cos 6).
If the tesseral harmonics are multiplied by a set of arbitrary constants
and summed, one obtains the spherical surface harmonics of degree n
with which we have already had something to do in Chap. 111.
+
Yn(6,4)
(15)
(anr,, cos
=
+ b,,
m4
m=O
sin rnd)P;(cos 6).
The tesseral harmonics form a complete system of orthogonal functions on the surface of a sphere.l It is in fact easy to show through an
integration of (12) by parts that
From these relations follows the fundamental theorem on the expansion of
an arbitrary function in spherical surface harmonics: Let g ( 6 , 4) be a n
arbitrary function o n the surface of a sphere which together with all its first
and second derivatives i s continuous. Then g ( 6 , 4) oan be represented by
a n absolutely convergent series of surface harmonics,
(18)
g(6,
4)
=
m
n
n=O
m=l
2 [anoPn(cOs6) + 2 (anrncos m4 + b,,
PF(cos 611,
whose coeficients are determind by
ano = 2n
-__
+
Qr
(19) an,
b,,
+
izr
l g ( 6 , 4) P,(cos 19) sin 6 d6 d4,
2n
1 (n - m ) !
___2r
(n m ) !J zoU
+
2n + 1 (n - m ) !
2 r (n + m ) !
= ---
= -__
sin m$)
g (6, 4) P;(cos 0) cos rn4 sin 0 do d4,
Jg(o,4) P;(cos
6)
sin m+ sin e de ci4.
1 The proof of this statement and of the expansion theorem which follows will be
found in Courant-Hilbert, “ Methoden der mathematischen Physik,” 2d ed., Chap.
404
SPHERICAL JBAVES
[CHAP.
lrII
In the case of a function th at depends on 0 alone, the conditions determining the convergence of a n expansion in Legendre polynomials are
identical with those governing the convergence of a Fourier series.
Under such circumstances it is sufficient that g(0) and its first derivative
be piecewise continuous in the interval 0 5 0 I 27r. The convergence
theory of the expamion (18) in surface harmonics, on the other hand,
presents definitely greater difficulties and the extension of the theorem to
discontinuous functions involves considerations beyond the scope of the
present outline.
Tliere remains the identification of the radial function f l ( R ) satisfying
(4). If for f1 we write f l = (kR)-+v(R),it is readily shown that v(R)
satisfies
(20)
R2-d2v
dR2
+ R dR
dv f [k2R2- ( n
+3”]
v
= 0,
and hence, by Sec. 6.5, is a cylinder function of half order.
The characteristic, or elementary, wave functions which a t all points on
the surface of a sphere are finite and single-valued are, therefore,
(22)
f e
Om“
=-
1
dlcR
cos
Z,+,(kR) Pg(c0s 0) sin m4.
As in the cylindrical case, we choose for Z,++(kR) a Bessel function of the
first kind within domains which include the origin, a function of the third
kind wherever the field is to be represented as a traveling wave.
7.4. Properties of the Radial Functions.-Various notations have been
employed a t one time or another t o designate the radial functions
(kR)-*Z,++(kR)but none appears to have gained general acceptance.’
What seems t o be a logical proposal has been made recently by Morse2
and will be adopted here. Accordingly we define the spherical Bessel
functions by
,-
.-
,-
.-
VII, $5, 1931. For greater detail see Hobson, “ T h e Theory of Spherical and Ellipsoidal Harmonics,” Cambridge University Press, 1931.
1 See WATSON,
“Bessel Functions,” p. 55, Cambridge University Press, 1922.
MORSE:“Vibration and Sound,” p. 246, McGraw-Hill, 1936.
SEC.
7.43
PROPERTIES OF THE RADIAL FUNCTIONS
405
Series expansions of these functions about the point p = 0 can
be obtained directly from (8:) and (11), page 357. If we recall that
I'(x
1) = xr(z), I?(+) =
arid make use of the dupZicaEion formula
+
6,
whrnce it is apparent that j n ( p ) is an integral function. Likewise from
the rcllation N,+g(p) = (-l)n-lJ.-n-;(p),
we obtain for the function of
the second kind
We turn next t o representations when p is very large, and discover a t
once a notable property of the Bessel functions whose order is half an
odd integer. On page 358 expansions in descending powers of p were
given for the functions of arbitrary order p . Such series satisfy the
Bessel equation formally, but are only semiconvergent. We obscrve
now, however, that when p == n
1/2, the series P,+i(p) and &,+;(p)
break off, so that there is no longer question of convergence. Consequently (13) and (14) of page 358 are analytic representations of J,+;(p)
and Nn++(p); furthermore, i t is apparent that these functions of hulf order
can be expressed in finite terms. From (23) above and Eqs. (13) to (17)
of page 358, we obtain
+
where
(29) Pn++(p)
=
+
n(n2 - l)(n
2)
1- _ _ - ~
22. 2! p2
n(n2 - l)(n2 - 4)(n2 - 9)(n
24.4! ,,4
+-
+ 4) - . . .
For the functions of the third and the fourth kind this leads to
(-;)n+l
h'l'(P) = -- e"[Pn++(P)
iQn+:(~)li
P
(31)
;n+ 1
h;''(p) = __ e - i ~ [ P n + t ( ~ ) i&n+t(~)I.
+
P
1
406
SPHERICAL WAVES
[CHAP.V I I
These series converge very rapidly so that for large values of p the first
term alone gives the approximate value of the function. Thus, when
The recurrence relations satisfied by the spherical Bessel function
follow directly from Eqs. (24) to (27) of page 359.
Having defined the radial functions, we can a t last write down the
general solutions of (3) as sums of clementary spherical wave functions.
I n case f(R, 0, 4) is to be finite a t th e origin, we have
(36) f"'(Rj
$1
4) =
2 j n ( k R ) [asd'n(cos
n=O
0)
+2
cos m4
( ~ n m
m=l
+ b,,
sin m4)P;(cos O)],
while a field whose surfaces of constant phase travel outward is represented by f("(R, 0, $), obtained by replacing j,(kR) by hhl)(kR)in (36).
A spherically symmetric solution results when all coefficients except a00
are zero. Then apart from an arbitrary factor, we have
7.6. Addition Theorem for the Legendre Polynomials.-If g(0, 4) is
any function satisfying the conditions of the expansion theorem (18),
its value a t the pole, 0 = 0, must be
sin 0 d0 &,
since P,(l) = 1, P,m(l) = 0. In particular let us take for g(0, 4) any
surface harmonic Yn(6, +) of degree n expressed by (15). Then in
407
ADDITION THEOREM
SEC. 7.51
virtue of the orthogonality relations, the sum in (38) reduces to a single
term and we obtain the formula
We shall apply this result to the problem of expressing a zonal harmonic in terms of a new axis of reference. Let P in Fig. 70 be a point
on a sphere whose coordinates with
respect to a fixed rectangular reference system are 0 and 4. A second
point Q has the coordinates a and
8. The angle made by the axis OP
with the axis OQ is y. The zonal
Y
harmonics a t P referred to the new
polar axis OQ are of the form P,
(cos y ) and our problem, therefore,
is to expand P,(cos y ) in terms
of the coordinates 0, 4, and a,8.
It is apparent that on a unit
FIG. 70.-Rotation
of the axis of
reference from OP to OQ for a system of
sphere cos y is the projection of the
harmonica.
line OP on the axis OQ. If 2, y,
are the coordinates of P, x’, y’, z‘ those of Q, then
+ +
a) + cos 0 cos a.
(40) cos y = xx’ yy’ zz’ = sin 0 sin a cos (4 We now assume for P,(cos y) an expansion of the form
n
multiply both sides by P::(cos 0 ) cos m+, and integrate over the unit
sphere. With the help once more of the orthogonality relations, we
obtain
(42)
12=
0
J’P,,(eos r)~
( C O S0)
0
cos m+ sin e de d+
(n + m)!
+ I (n - m)!
-- 27r
2n
‘me
But by (39)
(43)
s,”
P,(cos y)~;(cos e> cos rn4 sin e do d 4
--
47r
2n
+ 1 [P;(cos
0) cos m d ~ ] , ,=~
4*
~ ( c o as) cos mp,
ml
488
SPHERICAL W A V E S
since when
y =
0, 8 = a , 4 = p.
2
Cm =
(44)
[CHAP.VII
Consequently,
(n - m ) !
P:(cos a ) cos m/3.
(n m)!
Likewise,
+
(n - m ) !
PF(cos a ) sin mp.
( n m)!
The desired expansion, or addition formula, is therefore
d,
(46) P,(COS
y) =
=
2
+
P,(COS a ) P,(COS
e)
+ 2 2 ~( n+-~mP) !~ ( c o as ) P:(COS
8) cos m(4 - p).
m=l
This result leads to an alternative formulation of the expansion
theorem stated on page 403. If g ( B , 4) and its derivatives possess the
necessary continuity on the surface of a sphere, its Laplace series (18) is
m
7.6. Expansion of Plane Waves.-It is now a relatively easy matter
to find a representation of a plane wave propagated in a n arbitrary
direction in terms of elementary spherical waves about a fixed center.
The direction and character of the plane wave will be determined by its
propagation vector k whose rectangular components are
(49) kl
=
k sin a cos p,
kz
=
k sin a sin 0,
k3
=
k cos a,
while the coordinates of any arbitrary point of observation are
(50) x = R sin 0 cos 4,
Then the phase is given by
(51)
-
k R
=
y
=
R sin 8 sin 4,
kR[sin a sin 0 cos (4 - p)
x = R cos 0.
+ cos a cos 61 = k R cos y
The function f ( l ) ( R ,8, 4) = exp(ikR cos y) is continuous and has continuous derivatives everywhere including the origin R = 0. It can, thcrefore,
be expanded according to (36), page 406. Consider first an axis that
coincides with the direction of the wave. Since the wave is symmetrical
about this axis, we write
m
SEC. '7.71
409
INTEGRAL REPRESENTATIONS
The coefficients are determined in the usual manner by multiplying
both sides by P,(cos y) sin y and integrating with respect to y from 0 to
T.
Then by (17)
To rid this relation of its dependence on R, differentiate both sides with
respect to p = kR and then place p = 0. From (25) it follows that
(54)
hence,
(55)
2n(n!)2 a,
(2.n
l)!
+
=
2n
+
in
2
1%
cosn y ~',(cos y> sin y dy.
Thc integral on the right of (55) is readily evaluated and we obtain
a, = (2% l)i", or
+
(56)
+
?:"(2n l)j,(kR) P,(cos 7).
eikR cos y =
n=O
I n case the z-axis of the coordinate system fails to coincide with the
direction of the planc wave, one may use the addition theorem (46) to
express (56) in terms of an arbitrary axis of reference. Then
+22
I
(n - m)!
~ ~ ( c a>
o ~s ~ ( c 0)o cos
s m(4 - p> .
m=l
7.7. Integral Representations.-We
have found it convenient on
various occasions to represent a wave function as a sum of planc waves
with appropriate weighting or amplitude factors. The direction of each
component wave is determined by the angles a and p. Integration is
extrndpd over all real directions in space, and may in certain cases include
imaginary values of a and /3 as well. As in Xec. 6.7, we wish to find
representations of the form
(58)
f(z, y, z) =
or, when x
(59)
=
dp g(&, p)
R sin 0 cos 4, y
f ( ~0,, 4)
=
=
&k(Z sin a cos
R sin 0 sin 4, z
B f u sin a slnB+z cos a)
=
J' dor 1cip g(a, a) eikn
R cos 0,
y,
where a and /3 are the angles shown in both Figs. 66 and 70. I n the
case of cylindrical waves a fixed frequency and a fixed wave length along
410
SPHERICAL WAVES
[CHAP.V I I
the z-axis lead to a constant value of a , so that the directions of the plane
waves representing a cylindrical function f(ul, u2) constitute a circular
cone. Since for spherical wnvcs there exists no such preferred axis, the
integration must be extcndcd t o both a and p .
An integral representation of tJlemcntary spherical wave functions
can be obtained directly from the expansion (57). If both sides of that
equation are niultiplicd by P:(cos a ) cos mp or by ~ ( C Oa)Ssin mp, one
obtains, thanks to the orthogonality relations (19), the result
sin a d a dp.
Upon multiplying both sides by the arbitrary constants anm,b,, and
summing over m, this can be written also as
(61)
j,(kZ2)%-,(8, 4)
=i-n
47r
Lz"l
eikn cos ?Yn(a,p) sin a d a do.
From these general formulas may be derived a number of useful special
cases. Thns by choosing m = 0, 0 = 0, we obtain a representation of the
spherical Bcssel function j,(kR).
j,(kR)
7.To'
=-
eikR
a
Pn(c0S a) sin a da,
It is in fact easy t o show that Eq. (4), page 400, is satisfied by the integral
provided the contour C is such that the bilinear concomitant
vanishes at, the limits. In place of r) = i 1, we may choose a value that
will cause the expoiiential t o vanish. Thus if lc is real, we take for the
upper limit r) = iw and obtain the rcprcsentations
SEC.7.81
41 1
A FOURIER-BESSEL INTEGRAT,
If on the other hand k is complex due to a conduct,ivity of the medium,
one may choose t h a t root of k 2 whose imaginary part is positive and
replace the limit 17 = i cc) by 17 = a .
Again from (60) and (54), placing 4 and R equal to zero, one obtains
a representation of the function Pz(cos 0) :
The integration with respect t o a cannot be carried out easily with only
the help of formulas from the present section at our disposal, but by olher
means i t may be shown that (67) reduces t o
J 2 r (cos 0
(68) P?(cos 0) = (n + nL)!iPm
2mz !
+ i sin 0 cos p)" cos mp do.
Finally, one will note that by making usc of the integral representation (37), page 367, for a Bessrl function, the right-hand side of (60) can
be reduced t o a single integral When 4 = 0,
(69) j,(kR) P,"(cos 0)
= -
l
eikR co8 a cos eJ,(kR sin 0 sin a )
P;(cos a ) sin a d a .
7.8. A Fourier-Bessel Integral.-It will be assumed that the arbitrary
function f(x, y, z) and its first dcrivativm are pitcewist. continuous and
that the integral of the absolute value of the function extended over all
space exists. Subject to these conditions the Fourier integral of f(x, y, z )
is
ilk1 d k , dks,
and its transform is
(71)
g ( h , k2, k3) =
(&)
Jm
:
y,
, .(m
f:J
m : J
~ ) e - ~ ( ~ ~ ~ + ~ ~ y f ~ ~ ~ )
dx dy dz.
We now transform these integrals to spherical coordinates under the
tacit assumption that a triple integral extended over an infinite cube can
bc replaced by an integral over a sphere of infinite radius. Proceeding
as in Sec. 6.9 and noting that klx
k2y
kzz = kR cos y , we obtain
+
(72) f ( R , 0, 4) =
(ky
(73) g ( a , p, k )
(&>'
=
0
+
f r J z r g ( a , p, k ) eikRcos y k 2 sin a dk d a d p ,
. 0
0
lml[f(R,
0,
4)e - i k R c o s y R2sin 0 d R d 0 d 4 .
412
SPHERICAL WAVES
[CHAP.V I I
Next we shall assume that f(R, 8, 4) = fn(R)Y,(6, 4 ) and impose
upon the otherwise arbitrary function f n ( R )the condition that it shall be
piecewise continuous and have R piecewise continuous first derivative,
and that the integral
LmIfn(R)[dR shall exist'.
Then
sin 6 d6 d4,
which in virtue of (61) and the fact th at j n ( - k B )
over into
=
(-l)"j,(kR) goes
or g(a, P, k ) = (-1),Yn(a, 0) g , ( k ) . Introducing this evaluation of the
transform back into (72) and again interchanging the order of integration,
we are led to
and hence to the symmetrical pair of transforms
(77)
(78)
from which the subscript n has been dropped as having no longer any
particular significance. I n the special case n = 0, Eqs. (77) and (78)
reduce to the ordinary Fourier integral (18), page 288. I n case the
chosen value of R coincides with a point of discontinuity in f ( R ) ,we write
(79)
:J
0
k 2 dk
1
- J ( P ) j , ( k p ) j,(kR)
1
p2 dp =
2 [f(R + 0) + f ( R - 011.
7.9. Expansion of a Cylindrical Wave Function.-When
calculating
the radiation from oscillating current distributions it is sometimes
necessary to express a cylindrical wave function in spherical coordinates.
The integral representation of wave functions that are finite on the axis is
Now
(81)
k R
= hr
cos (P - 4 )
+ ihz = kR[sin a sin e cos ( 4 - p)
+ cos a cos el,
SEC.7.101
413
A D D I T I O N THEOREM FOR z o ( k R )
and in ( 5 T ) , page 409, we have a n expansion in the appropriate spherical
wave functions of exp(ik. R). Upon integration with respect to p, we
obtain
e) j , ( k ~ ) .
P;(COS a ) P:(COS
Since P; vanishes if m > n the first m terms of (82) are zero and the
expansion can be writtcn
2
m
(83)
J,(XT) eihz =
4-2m
!
+ 1) (n n 2mp
P:+:,,(cos
a)
+
7k=O
P;+,(COS
When a = ~ / 2 h,
=
0) j , , + m ( ~ ~ ) .
0, X = k , and
(n
+ 2m - l)!
if n is odl
.*
.
The result is an expansion of a cylindrical Bessel function in a series of
spherical Bessel functions.
(41
(-1)l
z=o
+ 2m + 1) (21)!
+ 2m 1 ! ( +
~ m - I)!
21
2Zz+m-1
P ~ + , ( c o s6 ) j Z ~ + & W .
7.10. Addition Theorem for zo(kR).--Let P(R0, 19, $) be a point of
observation and &(Ill, el, $1) the source point of a spherical wave. The
coordinates Bo, 6, # and R1,61, (blare referred to a fixed coordinate system
whose origin is a t 0. The distance from & to P is, therefore,
R
=
dz++
122, - 2RoR1 cos y
where cos y = cos 6 cos 61 sin 6 sin 61 cos (4 - 41). Our problem is
to express a wave function referred to Q as a sum of spherical waves whose
source is located at 0. General treatment of this problem demands a
lengthy calculation and its practical importance is insufficient to warrant
discussion here. However, in the theory of radiation we shall have occasion to apply the special case in which the wave is spherically symmetric about the source &. One finds then without great difficulty the
expansions :
414
[CHAP.V I I
SPHERICAL W A V E S
(87)
The proof is left to the reader.
THE VECTOR WAVE EQUATION IN SPHERICAL COORDINATES
7.11. Spherical Vector Wave Functions.-As in the case of cylindrical
coordinates, one can deduce solut,ions of the vector wave equation in
spherical coordinates directly from the characteristic functions of the
corresponding scalar equation. Following the notation of the preceding
section we shall put 4, = f, e-wt, where f, is the characteristic
Om=
Om*
solution
(1)
fp
=
cos
sin m+P:(cos 0) x,(kZZ).
omn
According to See. 7.1, oiie solution of the vector wave equation
v v * c- v x v x
(2)
C+kT
=
0
can be found simply by taking the grsdiont of (1). We define L = v+,
and split off the time factor by writing L = k w tThen
.
by (95), page
52,
(3)
1,
a
= -z,(kR)
aR
cos
P:(cos 0) sill m+ il
cos
+ IZ1 z,(IcR) d-ae P,”(cos e) sin
rn4 i,
-
m
zn(kld)P;(cos
Tm
e) sin
cos m4 i3,
where il, iz, is are the unit vectors defined in Fig. 8, page 52, for a
spherical coordinate system.
To obtain the indeperident solutions M and N as described in Sec. 7.1,
one should introduce a jixcd vector a. Such a procedure in the present
instance is of course permissible, but then M and N will be neither normal
nor purely tangential over the entire surface of th e sphere. If in place
of a we usc a radial vector il, a vcctor function L X i, is obtained which is
tangent to the sphere; but il is not a constant vector and consequently i t
by no means follows B priori that it can be employed to generate a n
independent solution. However, we shall discover t h a t a tangential solution M can in fact be constructed from a radial vector. Unfortunately
the procedure fails in more general coordinate systems.
SPHERICAL VECTOR WAVE FTJNCTIONS
SEC.7.111
415
Let us try to find a solution of (2) in the form of a vector
V x (ilu(R)+) = L x ilu(R),
where u(R) is a n unknown scalar function of R. Then if hl,, M 2 , M S
are the R, 6, and cp components of M,we have
M
(4)
MI
=
=
0,
The divergence of M is wro and we now expand the equation
V x V x M - k2M = 0 into R, 8, and C#I components by (85), page 50.
The R-component is identically zero, whatever u(l2). The conditions
on the 0 and 4 components are satisfied provided u(R)is such that
(The peculiar advantagc of spherical coordinates over more general
systems is that the conditions on both tangential components are identical.) If, therefore, we choose u(R)= R, Eq. (5) reduces to the required
relation
V2$
k2+ = 0.
(6)
+
Hence in spherical coordinates, (2) is satisfied by
M = V X R# = L
(7)
xR
=
1
V x N,
k
whose components are
(8)
From the relation kN
be easily found.
=
V x M the components of a third solution can
+
Since must satisfy (4), page 400, the radial component reduces t o the
simpler form'
1 The first thorough investigations of the clcctroniagnetic pro1)lcin of the sphere
were made by Mie, Ann. Physik, 26, 377, 1908 and Dcxhycs, AWZ.I'hyszk, 30, 57, 1909.
Both writers made use of a pair of potential functions lcading directly to our vectors
M and N. The connection between these solutions and a radial Hertzian vector has
been pointed out by Sommerfeld in Ricmann-Webcr, " Diffcrentialgleichungen der
Physik," p. 497, 1927.
416
SPHERICAL WAVES
[CHAP.V I I
To obtain explicit expressions for the vector wave functions M and N,
we need only carry out the differentiation of (1) as required by (8) and
(9). The time factor is split off by writing M = me-&', N = n c i W t ,
and we find
l a
a
+kR dR [Rzn(kR)]ae P,m (cos 8) sin rn4 i,
'OS
7.12. Integral Representations.-The
wave functions 1, m, and n can
be represented as integrals of vector plane waves such as (18) t o (20),
page 395. For the functions of the first kind, which are finite at the
origin, we have by (60), page 410,
First, we calculate the components of the gradient; namely, af/aR,
'a
~ aa' e ~R' sin e a+'
differentiating under the sign of integration. Next we
observe from an inspection of Fig. 70, page 407, th a t if the vector k(a, p)
is directed along the line OQ and R(0, 4) along the line OP, then
+
= k cos y = k[sin a sin 0 cos (4 - p)
cos a cos 01,
k iz = k[sin a cos 0 cos (4 - p) - cos a sin 01,
k i3 = -k sin a sin (4 - p),
k. il
(14)
-
the unit spherical vectors i,, i,, il referring, of course, to the point of
observation at P. Then it follows without further calculation that
To find the corresponding representations of m(') and n(l) we note that
R
(16)
xk
=
?&{sin a sin (4 - B)i,
+ [sin a
COR
0 cos (4 -
- cos
(k x R) x k
=
LY
sin
a)
eli,),
k2R(sin2y il - cos y[sin N cos e cos (4 - p)
- cos a sin O]iz cos y sin a sin (4 - /3)i3],
+
and, upon differentiating according to (8) and (9), obtain
QEC.
417
ORTHOGONALITY
7-13]
e i k R cos
Y
~ ( C Oa)S‘OS m@sin a d a dfi.
sin
To obtain the corresponding integral representations of functions of the
third and fourth kinds the integration over a from 0 to T must be replaced
by a contour in the complex domain as described on page 410.
Rectangular components of the vectors 1, m, and n can be calculated
without great difficulty with the help of these integral representations,
although the resultant expressions are somewhat long and cumbersome.
The vector integrands are rcsolved into their rectangular components
and become explicit expressions of a, p, 0, 4. Thus, for example,
k , = k sin a cos p. The recurrence rtlations for the spherical harmonics
are then applied to eliminate these factors and to reduce the integrals to
a form that can be evaluated by (13).
7.13. 0rthogonality.-The scalar product of any even vector with any
odd vector, or with any vector differing in index m, is obviously orthogonal
and we need consider only such products as do not vanish when integrated
over 4 from 0 to 27r. Thus
(19)
s””
0
1,omn ’ 1,om* , d#$ = (1
a
+ 6)T
a
zn(kR) aiz z , ( k R )
z,(kR) z,(kR)
where 6
formula
=
0 if m
> 0,
1
i
6 = 1 if m = 0. To reduce (19) we apply the
- 01
+
2
(n m)!
n(n
2n+l(n-m)!
when n # n’,
+ l),when n
=
n’.
Integration of (19) over 0 gives, therefore, zero when n # n’, or, when
= n’,
n
A final reduction follows from the recurrence relations (33) and (34),
page 406, and we obtain the normalizing factor
418
[CHAP.VII
SPHERICAL WAVES
The same formulas lead directly to the integrals
while all products differing in the index n are zero.
Upon examining the cross products, we find first
(25)
J2m
1,
0'""
mom,,
e
&J
=
5 (1 3. 6)
?r
m d
- xn Zn' -- [P,"P z ] ;
sin
e ae
hence.
for arbitrary values of n and n'. Likewise
The complete orthogonality of the system is spoiled, as in the cylindrica,l
n, , whose integral over the sphere does not
case, by t,he product 1,
.
Omn
vanish if n
= n'.
Omn
I n that case we find
To complete the orthogonalization one might treat k' as a variahle
parameter and integrate over k' and R as in Sec. 7 . 2 ; such a procedure
usually proves unnecessary.
7.14. Expansion of a Vector Plane Wave.-To study the diffraction
of a plane wave of specified polarization by a spherical object one must
E X P A N S I O N OF A VECTOR PLANE N A V E
SEC.7.14j
419
first find a n expansion of the incident vector wave in terms of the spherical
wave functions 1, , me , n,
omn
onn
0""
.
Consider the vector function
f(z) = a e%kz = a &kR
(29)
cos 0
7
where a is a n amplitude vector oriented arbitrarily with respect to a
rectangular reference system. Let a be resolved into three unit vectors
directed along the z-, y-, z-axis respectively. Then
+
+
a, = sin 0 cos C#J il cos 0 cos cp i, - sin cp il,
a, = sin 0 sin cp il
cos 0 sin 4 i,
cos cp i,,
a2 = cos 0 il - sin 0 i,,
where il, i2, is are again the unit vectors defined in Fig. (8), page 52,
for a spherical coordinate system. Now the divergence of the vector
functions a, exp(ikz) and a, exp(ikx) is zero and consequently they may
be cxpanded in terms of the ch:tracteristic functions m and n alone. At
R = 0 the field is finite and we shall, tLerefore, require functions of the
first kind. It is apparent, moreover, that the dependence of (30) on
cp limits us to m = 1; whence upon consideration of the odd arid even
properties of (11) and (12), we set up the expansion
(30)
(31)
a,etkR
cos 0 =
2
+
(unrn::L
+ bnn::b).
n=O
To determine the coefficicntu
preceding scction.
(32)
s' J'*
a, m:;;
0
eskR
cofi
0
whence by (23),
(33)
which in virtue of (24) gives
(35)
hence,
W I apply
~
the orthogonality relations of the
sin 0 d0 d+ = 2rinn(n
+ 1)[j,(kR)]2,
420
[CHAP.V I I
SPHERICAL WAVES
The same process gives for the wave polarized in the y-direction the
expansion
(37)
Since the longitudinal wave function a, exp(ikz) has a nonvanishing
divergence, its expansion must involve thc 1 functions. It turns out, in
fact, that only this set is required and one finds without difficulty that
Problems
1. Show t h a t the spherical Bessel functions satisfy
0, when m # n,
r m
provided n and m are integers satisfying the conditions n 2 0, m > 0. For the application of such integrals to thc expansion of functions in Bessel series see Watson,
"Bessel Functions," page 533.
2. Show t h a t the cylindrical Besscl functions satisfy
and with the help of these formulas show that the spherical Bessel functions satisfy
j,(z) = ( - 1 ) " ( 2 2 ) " __
nn(x)
=
dn
cos z
(-1)"-'(22)" d(z2)"
~ -
3. Show that if R 2 = R:
(-;&)n &).
cos x
= -Z"
+ R: - 2 R o R 1cos 7,
sin kH
sin kRo sin kR1
d(c0s y ) = -- 9
kRo
sin
cos _
kR1
_ kRo
__
d(c0s y )
=
kRo
kRi
cos
kRo
sin
kR1
___ ___
kRo
kRi
7
kRi
Ro 5 Ri,
Ro 2 RI.
4. Show that in the prolate spheroidal coordinates defined on page 56 the equation
V2$
+ k2$
=
0
421
PROBLEMS
assumes the form
Let @
=
$,(.$)$2(v)+3(4), and show that the above separates into
where m and C arc separation constants.
wave functions can be constructed from
Next show that a set of prolate spheroidal
m
m
fimn= ( E * - 1)' (I .- +)'ScLn(c, v)Ee,,(c,
cos
sin
5 ) . m4
in which both the "angular functions" S c and the "radial functions" Re satisfy the
equation
(1 - 2')"'
- 2(m
1 ) ~ y ' (b - c222)y = 0,
+
+
with m an integer or zero, b = C - m, and c = kd = 2ad/h.
6. When the wave functions of Problcin 4 apply to a complete sphcroid one
must choose m = 0, 1, 2, . . . , and find the separation constant b such that the
field is finite at the poles TJ = 1. A discrete set of values can be found in the form
b, = n(n
+ 1) +fn(c),
n=0,1,2,
-
'
..
Show that the functions S e and Re can be represented by the expansions
m
__
Sc:n(c,
2) =
T,"(z)
Z'd,T,"(z),
=
(1 -
22)
">&(z)
s
where P:,,, is a n associated Legendrc: function and the priine over the summation sign
indicates that the sum is to be extended over even values of s if n is even and over
odd values if n is odd.
Re:,(c,
z)
' e-n -~
(am + s)!
=
2vYL !(C.)"
S!
d s j s + , (CZ)
where j s + m ( c ~is) a spherical Bessel function. The independent solution is obtained
Show that the coefficients satisfy the recursion
hy replacing j8+m(cz) by n,+,(cz).
formula
(s
(2s
+ 2m + 2)(s + 2m + 1)
+ 2m + 5)(2s + 2m + 3)
s(s - 1)
+ 2m - 1)(2s + 2m - 3) C2d,-z
2s2 + 2s(2m + 1) + 2m - 1
c2 + s(s + 2m + 1) - b
(2s + 2m - I ) ( % + 2m + 3)
C2ds+2
+
(2s
1
d, = 0
and from this relation give a method of determining the characteristic values b,.
422
SPHERICAL W A V E S
[CHAP.
VII
6. Prove t h a t the spheroidal functions defined in Problem 5 satisfy the relations
Se;, = i"2"+lrn! XnRe:,
etczt(1
Se:,(c, z ) = A,, S
'
l
- tz)%5'e~,(c,
1 ) dt
where X, is one of a discrete set of characteristic values.
7. Write the equation V2$ k2$ in oblate spheroidal coordinates and express
the solutions in terms of the functions Se and Re defined in Problem 5.
8. A system of coordinates E, 7, 9 with rotational symmetry is defined by
+
ds2 = hl d f 2
+ hz dq2 + r2 d @
where r is the perpendicular distance from the axis of rotation. If the field has the
same symmetry as the coordinate system, its components are independent of 9.
Show that the equations
V
)(
E - iwpH = 0,
V X
H
+
( b e
- u)E = 0
break up into two independent groups:
-i _a
rhz a?
i _
a
_
rhl aE
(rE+)- i w p H ~= 0,
(rEg)
+ iwgH,,
=
0,
Show that (11) is satisfied by a potcntial Q such that
(I) can be integrated in the same manner. (Abraham)
9. Apply the mcthod of Problem 8 to find a rotationally symmetric electromagnetic field in prolate spheroidal coordinates. I n the case of electric oscillations
directed along meridian lines show that Q satisfies
423
PROBLEMS
where Q
=
Q~(E)&~(V).
Note that both of these are a special case of the equation
(1 - 9 ) y ” - 2 ( a
when a = - 1 .
+ 1)zy’ + ( b -
c222)y =
0
Show that the field is given by
EE = imp
d2k2
dQz
-
&I
-
d(p-
1)(E2 -
?
72)
dq
10. Two types of spherical electromagnetic waves have been discussed i n this
chapter: the transverse electric for which E E = 0, and the transverse magnetic for
which H R = 0. Obtain expressions for the radial wave impedances as was done
previously for cylindrical waves.
11. Prove the expansions (86) and (87) of Sec. 7.10 and show t h a t when el = T ,
R 1+ m , Eq. (87) goes over asymptotically into the expansion of a plane wave.
12. Obtain expressions for the rectangular components of the vector spherical
wave functions 1, m, and n by the method suggested a t the end of Sec. 7.12.
CHAPTER VIII
RADIATION
I n the course of the past three chapters we have studied the propagation of electromagnetic fields with no concern for the manner in which
they are established. We consider now the sources, and the fundamental
problem of determining the intensity and structure of a field generated
by a given distribution of charge and current.
THE INHOMOGENEOUS SCALAR WAVE EQUATION
the problem
8.1. Kirchhoff Method of Integration.-Mathematically
of relating a field to its source is t h a t of integrating an inhomogeneous
differential equation. Let J/ represent a scalar potential or any rectangular component of a field vector, and let g(z, y, x , t ) be the density of
the source function. We shall assume that throughout a domain T’ the
medium containing the source is homogeneous and isotropic, and that
its conductivity i s zero. The presence of conductivity introduces serious
analytical difficulties which can be avoided in most practical applications
of the theory. The effect of conducting bodies in the neighborhood of
oscillating sources will be treated as a boundary-value problem in Chap.
IX. Subject to these restrictions, the scalar function $ satisfies the
equation
where v = ( ~ p ) - iis the phase velocity.
The Kirchhoff theory of integration is an extension to the wave
equation of a method applied in See. 3.3 t o Poisson’s equation. Let V
be a closed domain bounded by a regular surface S and let 4 and
be any two scalar functions which with their first and second derivatives
are continuous throughout V and on S. Then by (7), page 165,
+
where d / d n denotes differentiation in the direction of the positive, or
outward, normal. Let ( d ,y’, z’) be a fixed point of observation within
B and
+
+
r = d ( z ’ - x)’
(2’ - z ) ~
(y’ - y)2
(3)
the distance from a variable point (2,y, z ) within V or on X t o the fixed
424
SEC.8.11
KIRCHHOFF M E T H O D OF INTEGRATION
425
point. For 6 we shall choose the spherically symmetric solution
4
(4)
=
;r
+ ;)
(t
of the homogeneous equation
(5)
where f ( t
ment t
+ i)is a completely arbitrary analytic function of the argu-
+
As in the analogous problem of the stationary field, 4 has a singularity
at r = 0 ; this point must, therefore, be cxcludcd from the domain V
by a small sphere S1 of radius drawn about (x’,y’, 2 ’ ) as a center. The
volume V is now bounded externally by X and internally by X1. Furthermore, if we denote the left sidv of (2) by I and make use of (1) and ( 5 ) ,
we obtain
Or, since $-a2+ - J / -a2Q
at2
=
at2
a
-
at
at
(4 -;;
-
+--a+) ,
dt
(7)
at
Upon integrating over the ent,irc duration of time this givcs
+
(
To this end let t‘
The next task is to choose the function f t
the last term of (8) vanishes.
f ( t t’) the impulse function
+
(9)
f(t
+ t’)
1
= ---
e
- (t +t‘)2
____
2P
~
6
.L/%
=
S,(t
in such a way that
=
r/v and take for
+ t’),
defined by (35), page 291, with the property
(10)
J-*-f ( t + t’) d ( t + t’)
=
1.
+
To avoid any question as to the continuit,y of and its derivatives, wc
shall imagine 6 t o be exceedingly small but shall not pass a t once to thc
limit 6 = 0. Thus defined, S o ( t t’) vanishes everywhere outside an
infinitesimal interval in the neighborhood of t = - t’. With this in mind
+
426
RADIATION
[CHAP.V I I I
it is apparent that if F ( t ) is an arbitrary function of time, then
F(-t’)
(11)
=
J
-
-
S,(t
+ t’)F(t) dt,
and that a f l a t approaches zero with vanishing 6 for all values of t.
Since now 4 and a+/at vanish a t the limits t = - m and t = ~0 for
all finite values of T , it follows that the last term of (8) must likewise
vanish and, consequently, that
= -
sv
g(z, y, z, -t’)
dv.
When equated to the right-hand side of (2), this lcads to
Over t,he surface SI we know that d / a n = -a/ar, so that
Passing to the limit as r l
3 0,
this reduces to
and hence by (11),
(16)
J-wm
dt Jsl
(4
$ - J. %)
an da
= -4?nc/(z’, y’,
z’, 0).
Obviously these.limits may now be drawn in t o enclose a n infinitesimal interval
containing the instant t = - r / v .
SEC. 8.11
KIRCHHOFF M E T H O D OF I N T E G R A T I O N
427
Substituting this value into (13) gives
To reduce these integrals to a simpler form we notc that
and that
An integration by parts gives
and application of (11) leads to
(21)
I)(%’, Y’,z’, 0)
=
g(z,
Y, Z,
-t’)
dv
+
According to (21) the valuc of at the point d ,y‘, z’ and the instant
t = 0 is obtained by summing the contributions of elcments whose phases
have bcen retardcd by a n amount t’ = r/u. But the location of the
reference point on the time axis is purcly arbitrary; consequently, the
Iiirchhoff formula becomes
-
-
-
in which the symbol
will denote a function with retarded phase.
The volume integral in (22) is a particular solution of the inhomogeneous wave equation representing physically the contribution to
$(XI,
y’, z’, t ) of all sources containcd within V . To this part icu1:tr solution is added a general solution of the homogeneous equation ( 5 ) cxprcssed
428
[CHAP.V I I I
RADIATION
as a surface integral extended over S and accounting for all sources
located outside S. In case the values of II, and its derivatives are known
over S, the field is completely determined at all interior points, b u t it is
no more permissible to assign these values arbitrarily than in the static
case. We shall have occasion below to discuss this point further in
connection with the Huygens principle.
The behavior of
a t infinity is no longer obvious since r enters
+
explicitly into [g] through the retarded time variable t -
rU
The field,
however, is propagated with a finite velocity; consequently, if all sources
are located within a finite distance of some fixed point of reference and if
they have been established within some finite period in the past, one may
allow the surface S to recede beyond the first wave front. It lies then
entirely within a region unreached by the disturbance at time t and
within which II, and its derivatives are zero. I n t,his case
where V now represents the entire volume occupied by sources.
The reader has doubtless noted that the choice of the function
f (t
+ :) is highly arbitrary, since the homogeneous equation (5) admits
:>
also a solution - f t - - . This function leads obviously to an advanced
rl (
time, implying that the quantity II, can be observed before it has been
generated by the source. The familiar chain of cause and effect is thus
reversed and this alternative solution might be discarded as logically
inconceivable. However the application of “logical ” causality principles offers very insecure footing in matters such as these and we shall
do better to restrict the theory to retarded action solely on the grounds
that this solution alone conforms to the present physical data.
8.2. Retarded Potentials.-When
(24) is applied to the scalar potential and to the rectangular components of the vector potential we obtain
the formulas
I#+?,
t)
A(d, t)
=
:
:
4tie
Jv - p ( z , t*)
4”, Jv
= -
dv,
- J(z,t*) d ~ ,
which enable one to calculate the field from a given distribution of charge
and current. Here x’ represents the coordinate triplet x‘, y’, z’, and x
r
the triplet x, y, x, while the retarded time is expressed by t* = t - -.
2,
SEC.8.21
RETARDED POTENTIALS
429
This statement, however, is subject to one condition. We have in
fact only shown that (25) and (26) are solutions of the inhomogeneous
wave equation. I n order that they shall also represent potent,ials of an
electromagnetic field i t is neccwary that thcy satisfy the relation
The prime attached to the operator v denotcs differentiation with
PaSpc’ct to x’, y‘, z’ a t the point of observation.
We shall show that this condition is fulfilled by (25) and (26) provided
the densities of charge and current satisfy an equation of continuity.
Since a / a t = a/at* a t a fixed point in space we have
Now when the operator
V r n = -V’rn.
v
is applied to any power of r , onc may w r i k
r
The variables x’ occur in J(x, t * ) only through t* = t - -.
V’
By the divergence theorem
(34)
since the closed surface S hounding V can be taken so large that J is
zero everywhere over S. T h m on combining (28) and ( 3 3 ) , we find
430
[CHAP.V I I I
RADTATION
and this must vanish since thc conservation of charge requires that
(36)
(v . J)t*=ronstant -t -*ata P
=
0
at any point whose local time i s t*.
8.3. Retarded Hertz Vector.-By virtue of (36) it is sufficient to
prtscribe the current distribution in space and time. The charge density
can then be calculated and the potmtials det crinined by evaluating thc
integrals ( 2 5 ) and (26). However, in most cases it is advantageous to
derivc: the field from a single source function by means of a Hertz vector.
Let us express the current and charge densities in terms of a single
vector P defined by
ap
J = -p = -V.P.
(37)
at '
The equation of continuity is then satisfied identically and the field
equations are
dH
(111) V . H = 0,
(I) v x E -I- p Tf = 0,
(11) V x H - t -aE
-
at'
at
v .E
(IV)
=
1
--v.P.
As shown in Sec. 1.11, thcse cqiiations zre satisfied b y
where
(39)
n is any solution
of
vxvx
rI -
The rc,ctangular componcni 5 of
wave cquation
(40)
v211,
vv en
+
am
I*€--
at2
=
1
-P.
n, thcrc~fow,satisfy the inhomogencous
a2n,-
1
- I*€ 2 - - - P I ,
at
( j = 1, 2, 3).
It need scarcely bc said that the vector P defined by (37) is not
identical with the diclectric polarization dcfined first in See. 1.6. The
(x,,x,,x,)
vector P of the present section measures
thc polarization, or moment per unit vol( 1 1 , (2, (3)
ume, of the free-charge distribution and
R1
consequently is not equal to D - toE.
T o facilitatc the calculations that follow a small change in notation will be
0
made. The point of observation in Fig.
FIG. 71.--h
current element is
located at tl, t 2 , t 3 ,while xi, x 2 , x 3 71 whose radial distance from a fixed orilocate a fixed Pint of observation.
gin 0 is R will now be located by the rectangular coordinates ( 5 1 , z2, Q). The current element located a distance
y
SEC.8.41
43 1
D E F I N I T I O N OF T H E M O M E N T S
R 1 from 0 has the coordinates
of observation is r, so that
((1,
El,
[3)
and its distance from the point
We need consider only the harmonic components of the variation, from
which general transient or steady-state solutions may be constructed.
Let
1
J = Jo(E> cht, P =- P O ( $ ) c - ' ~ ~ , PO = .T V JdE),
(42)
aw
(43)
P
=
PO(l) c - ' ~ ' ,
i
P" = w- Jo.
Applying (24) to (40) and noting that in the present case k
we obtain the fundamental formula
=w
&= w/v,
(44)
A MULTIPOLE EXPANSION
8.4. Definition of the Moments.-To
evaluate the retarded Hertz
integral one must frequently resort t o some forin of series expansion.
The nature of this expansion will be governed by the frequency and the
geometry of the current distribution. In the present section we shall
consider an extension to variable fields of the theory o€ multipoles set
forth in Secs. 3.8 to 3.12 and Secs. 4.4 to 4.7 for clectrostatic and magnetostatic fields.
According to (87), page 414, when R > El, the integrarid of (44) above
can be expanded in the series
eik.
= ik 2 (Zn + 1)P,(cos y) jn(lcRl)hL1)(kR)
;
r
m
ri = 0
consequently, if R is greater than the radius of a sphere containing the
entire source distribution.
(an
n=O
+ l)hk1'(kR)
sv
Po(( ) j,(lcR1) P,(cos y) dv.
Suppose now that P,,([) is expressed as a function of 121, y, 4 , where 4
is the equatorial angle stbout an axis drawn from 0 through the point of
observation. Then t,he vector Pc(R1, y, 4) can be resolved into scalar
components and each component expanded in a series of sphtric:tl
432
[CHAP.V I I I
RADIATION
harmonics as in (36), page 406. I n virtue of the ortnogonality of the
functions P,(cos y ) , the integral on the right reduces essentially to an
expansion coefficient times [jn(kRl)l2.The success of this attack will be
determined by the nature of the current distribution and the difficulties
encountered in computing the expansion coefficients.
If the wave length of the oscillating field is many times greater than
the largest dimension of the region occupied by current, the series (2)
can be interpreted in terms of electric and magnctic multipoles located at
the origin. We shall assume that for all values of RI within V
where A is the wave length in the medium defined by e and p. Consequently the function j,(kRl) can be replaced by the first term of its
series expansion. By (25), page 405,
(4)
j,(kRl)
Znn!
= __-__
(kR1)".
(2n + l ) !
The total field will be represcnted as thc sun1 of partial fields by
m
(5)
ar(Z,
2) =
2
rI(*)(X, t ) ,
n=O
and, hence,
the first term
(7)
Let II. bc any scalar function of position.
By the divergence theorem
(8)
and, hence,
(9)
-
If X encloses thc entire source distribution, Po n will be zero over S
and the surface integral will vanish. Choose now for # a rectangular
component of the radius vector R1. Then
sVPoidv= -
S ijv.
Podv;
433
D E F I N I T I O N OF T H E MOII.IENP’S
SEC.8.41
conseauen tlv
The partial Jield n(O)i s generated by a n oscillating electric dipole whose
moment pC1j i s defined exactly as in the electrostatic case, Eq. (47), page 179.
To obtain the second term of (6) we note that Pl(cos y) = cos y ,
To interpret the integral we first expand the integrand
(13) PoRl cos y
=
-
P
(Ri R) -2
R
_ - 1 [(RI X PO)X R
2Et
+ P O @ R)~ + R1(R. Po)].
*
The magnetic dipole moment was defined in (28), page 235, as
m(1) =
:
(14)
a
-
R1 X
JO
dv-
Since R is independent of the variables 4 of integration, one has, therefore,
The electric quadrupole moments of a charge distribution were defined
in (48), page 179, as the components of a tensor,
Upon putting #
=
(17)
Furthermore,
tiffin
(9) this gives
pij
J (tipoi +
=
3
(18)
R1.R
=
R
x
j=1
Consequently,
aR
.$I%:
4201)
dv =
P0.R = R
pii.
s
dR
Poi--.
j=1
ax1
434
R A D I A TION
[CHAP.V I I I
The quantities p:z) arc the componc’nis of a w c t o r
(20)
p(2’ = 2 p . vn,
where 2p is the tensor whosc component., are pt3. [Cf. Eqs. (23), (24),
pagc 100.1
The partial ficlcl II(l) rcprrscnts the contrzbutions of a magnrtic dipole
and an electric puadrupolc.
Comparison of (21) with (7) shows that the two fields differ in the first
place by a factor 6 ,
or approximately 3 X lQs,
and consequently the
moments m(l) and pCz) must be very nzucli larger than p(I) if nc1)is to
be of appreciable magnitude. On the other hand, it must be remembered
that the electric moments of a current distribution arc functions of the
frequency a h wc,ll as of the gcometry
[Eq. (43), page 4311. If p(?) is expressed in terms of current rather
llian charge, wc scc that the magnitude of r P is indcpcndcnt of frequency, whcrcas the magnitude of
diminishes as 1/w with incrcasing frequency.
The itmaining tmms of the serics
in ncn)rrprescnt the contributions of
electric and magnetic multipolcs of
higher ordcr. These must be cxFIG.73.--Coniponerit,s of field about an
I)r(>ihedin terms of tensors and the
electiic dipole.
labor of computation incrcasm rapidly. The magnitudcs of the siiccessivc tcrms progrew csscritially
as ( o / c ) ~ . In c a s ~the rate of convcrgcncc is so slow that higher order
terms must hc taken into account, one is usually forced to adopt some
other method of calculating the ratliat ion.
8.5. Electric Dipole.-Let us suppcasc that, the charge distribution is
such that only the electric dipole momcnt is of importance. This is the
case, for example, of a current oscillating in a straight scction of wire
whose length is very small compared to that of the wave. The coordinate
system is oriented so that the positive z-axis coincides with the dipole
moment p(l). The vector p(l) can he resolved into its components in a
spherical system as illustrated in Fig. 7 2 .
(22)
p(I)
=
p(l) cos e il - p ( l ) sin e i,,
435
ELrTCTRIC DIPOLE
SEC.8.51
where i,, i,, is are the unit spherical vectors designated in Fig. 8, page 52.
According to (7) the Hertz vector II(O) is parallel to p(l) arid can be
resolved in the same manner. Thc field vcrtors arc now calculated from
the formulas (38), page 430, which in the case of a harmonic time factor
become
E =VV-n
(23)
+ k211,
H
=
-iW€V x n.
The differential opc~rntors,cxprtwed in sphcrical coordinates by (trj),
page 52, are applied t o thc v c c t o r
(24)
II(”) = r P cos 0 il - IT(o) sin 0 i,.
If Ro = V R is a unit vector dirccted from the dipole towards the observer
and t* = t -
R
-)
2,
then
The general structure of a dipole field is easily determined from these
expressions. Wc note in thc first place that the elect>ricvccLor lies in a
meridian plane through t’hc axis of the dipole and thc magnetic vector is
perpendicular to this plmc. The magnetic lines of force arc coaxial
circles about’ the dipole. At zcro frequency all terms vanish witb the
exception of the first in l / W in (25), which upon rcfcrcnce to (27),
page 175, is seen to be identical with the field of an electrostatic dipole.
Moreover, if we replace p(l) by the linear current element
i
-
w
I ds according
to (43), pagc 431, it is apparent fhat t’he first tcrm of (26) can he interprcted as a n extension to variable ficlds of the Biot,-Savart law, (14),
page 232. The terms in (25) and (26) have been arranged in invtrrsc
powers of R and the ratio of the magnitudes of successive terms is k R ,
R
or 2~ -.
x
I n the immediate neighborhood of t’he source the “static” and
“induction” fields in l / R 3 and l / R 2 predominate, while at distances such
that R >> X, or k R << 1, only the “radiation” field
need be taken into account. At great distances from the source, measured
in wave lengths, the Jield becomes transverse to the direction of propagation.
436
RADIATION
[CHAP.VIII
If we let R ---f 00 and at the same time increase p(l) so that the intensity
of the field remains constant, we obtain in the limit the plane waves
discussed in Chap. V.
This transverse property of the wave motion at great distances is
common t o the fields of all electromagnetic sources, but we must not
overlook the fact that in the vicinity of the .source there i s a longitudinal
component in the direction of propagation. The radiation field of an
clectric dipole, for example, vanishes along the axis of the dipole, but for
sufficiently small values of IZ there is an appreciable longitudinal component of E(O) of the form
This component, and not the radiation field, determines the distribution
of current in a vertical antenna above the earth.
Let us calculate next the energy flow within the field. For this
purpose we shall make use of thc complex Poynting vector derived in
Sec. 2.20. If is the mean value of the energy crossing unit area in the
direction of the positive normal per second, then
s
-
+E x 8.
To calculate the complex Poynting vector S*, we first write (25) and
(26) in component form.
(29)
S
=
Re(S*),
S*
=
From these we construct
(31)
and obtain
S* = +Eon+il - +ERR+i,,
Next the normal component of S* is integrated over the surface of a
sphere of radius R.
SEC.8.61
437
MAGNETIC DIPOLE
Now by (29), page 137, the real part of (33) gives the mean outward flow
of energy through a spherical surface enclosing the oscillating dipole.
Since this real part is independent of the radius of the sphere, it must be
the same through all concentric spheres and we discover that the system i s
losing energy at a constant rate
(34)
watts.
The quantity W measures the radiation loss from the dipole. I n the
expansion of an electromagnetic field, only those terms that vanish a t
infinity as R-' give rise to radiation. I n the present instance there is,
of course, a n instantaneous energy flow associated with the other terms
of (25) and (26), but the time average of this flow over a complete cycle is
zero. The terms in R-2 and I F account for the energy stored in the
field, energy which periodically flows out from the source and returns to
it, without ever being lost from the system.
I n free space po = 47r x lo-',
= Q x lo-*, and hence
W(a)= $ X 10-16 w 4 [ p ( 1 ) 1 2 = 173 X 10-16 y 4 / p ( 1 ) / 2 watts
(35)
where v is the frequency. The radiation increases very rapidly with
increasing frequency and decreasing wave length.
8.6. Magnetic Dipole.-The current distribution may be such t h a t
only the magnetic dipole moment m(') is of importance. At sufficiently
low frequencies this is certainly the case for any closed loop of wire. Then
The components of the field calculated from (23) are
(37)
The structure of this field is identical with that of the electric dipole but
the roles of E and H are reversed. The vector H(') lies in a meridian
plane through the dipole and has a radial as well as a transverse component. The lines of E(') are concentric circles about the dipole axis.
When k = 0, (37) reduces to the expressions derived in Sec. 4.5 for the
static field of a fixed magnetic dipole.
To calculate the radiation we construct the complex Poynting vector
as in (31) and integrate its radial component over a sphere of radius R.
438
[CHAP.VIII
RADTA TION
Only the terms of (37) in 1/12 cont,ributc to thc result and wc obtain
watts.
d&
= 376.6 ohms, in which case
I n free spacr
(39)
Tyc'j
10qn2(1q2
watts .
For the sake of example consider an alternating current in a single
circular loop of radius 12, meters. The maximum value of thc current is
l oand the wave length is assumed to be very much greatcr than R1.
The phase of the current is then essentially the same a t all points of the
circuit, wliencc i t follows from (43), page 431, and (11), pagc 433, that
the electric dipole momcnt is iicyjigible. The magnctic moment
Im(l)[= aR2,10;eonscquently, the cnergy radiatcd per sccond is
1
W
=
(2)4
16Qd
I,".
The radiation resistancc is defined as the equivalent ohmic resistance
dissipating energy a t a n equal mtc. Since I . rcprcsents the maximum
current amplitudr,
(41)
= +(fir;
and, hence, in the present instance
w
=
3 2 0 d 2:4
()
=
3.075 X l o s
@)
4
ohms.
This result is restricted in thc first place by t h condition
~
Idiat the sccond
term in the expansion of j1(kZil) shall b(. Iwgligible with rcycct to the
first.
(43)
and hmce jl(kR1) can be approximated by the first term if lcRl < 1.
It is in fact a gciieral rule that thew functions can be represented b y their
first term as long a$ their argiimcmt is not grcatcr than their order. If
then 1 = 10 Rl, wc find the radiation rciistance to be about 30 ohms.
As 1 decreases,
i1icrmsc.s very rapidly, but the clcctric dipole and
quadrupole moments now bcgin to play a part. On thc othcr hand if
the wave length increases, i t is clcnr that radiation soon becomes entirely
negligible. Currents whose wavc length is long compared to the largest
dimension of the metallic circuit give rise t o no appreciable radiation and
arc said to bc quasi-stationary.
RADIATION THEORY OF LINEAR ANTENMA SYSTEMS
8.7. Radiation Field of a Single Linear Oscillator.-The
rigorous
determination of t h e current distribution in a conductor is a boundary-
sE<’.
8.71
439
L I N E A R OiSCZLLrlTOR
value problem. A steady-state solution of the ficlcl cqrrationb n1rr.t bc
found, corrcsponding to a given imprcssrd c.rn.f., nhirh sat i h f i c h the
ncwssary conditions of finiteness at intclrior poiiiid of thc coiiciiictor and
continuity of the tangent ial components of fic,ld int c.n\ii?- itcros5 its
surface. This problem has nevcr becii completely solvcd for a firiittb
length of a cylindrical conductor, but a good idea of wlittt takes place may
be gained from a study of a conducting, prolntc sphoroid ~ ~ l i o s(wwir)
tricity diffrrs but little from unity. Such an investigation was madc first
by Rhraham,l and a numbcr of papers h:tvc dcalt morel rcccntly u it11
the sanic subject. All this work corifirnis what one IF oultl iiatui~:dly
txpc’ct: the current distribution along an isolated straight mire is essentially harmonic and
the ratio of antenna length to wave lcngth is 7
’
half an jntcxgcr, as long as I hc cross-sectional
dimensions arc very small with respect to
ihe length. A much grcntcr source of (’nor
than the finite diarnvtcr of the conductor is
thc damping due t o radiation. As the an- ,
tenna current flows outward from the feeding
point towards the cnds, energy is dissipatcd I
in heat and radiation. Consequently the
assumption of a conslant amplitude at thc
rurrcnt loops along the wire is iIlcorrc.(.t. lk+.
73.--Slnde h o a r osc111:Ltoi.
Superimposed on a standing antenna currcnt wave is a dainpcd traveling
wave supplying the losses. Thc effect of this modificd distribution on
the radialion pattern may bc considerable.
The current distribution is also affected by proximity to the ground
or other objects, The trend i n contemporary antenna design is towards
systems of linear antcmias supported some distance above the earth and
in this case the r.ffc.ct of thc rarth on current distribution can usually bc
neglected. When, on thc otlivr hand, a structuial steel towcr is used as
an antenna, the capacity t o ground may have an important influence on
the distribution.
I n thc prescnt section we shall consider the radiating properties of
lincar antennas completely isolated in free space. Thr. manner in which
the radiation is rcflected and rrfracted by the earth will be taken i i p in
the following chxptcr. We sliall assume that the current has been
measured a t a numbcr of points along each conductor and that this
measured distribution can be represented by one or more harmonic
standing or traveling waves.
Let us considcr first a single straight wire of length I as shonn in
Fig. 73. The current a t any point ( measured from the center of the
1
ABRAHAM,
Ann. Z’hysik, 66, 435, 1898.
440
RADIATION
wire is I = u(t)exp(-id).
of the distribution is
[CHAP.VIII
Then by (44), page 431, the Hertz vector
where d t is an element of length along the conductor.
jield of an element d t is by ( 2 7 ) )page 435,
The radiation
so t h a t upon integrat,ing the contributions of all elements along the length
of the wire we obtain
Now if r>> 1 one may consider sin e / r to be a slowly varying function of t. This means that the radius vectors from current elements to
the point of observation are essentially parallel, so t h a t sin e / r may be
replaced by sin O/R and removed from under the sign of integration.
This approximation is the more nearly exact the larger R, and since the
radiation is the same through all concentric spheres about the origin 0,
we may take R as large as we please. The phase, however, must be
considered with more rare. From the figure it is clear t h a t the time
required for a wave emitted by the element a t t to reach a distant point
is less than that required for a wave originating a t 0 b y the amount
t cos e / c , and consequently their phase eventually differs by k t cos 8,
since k = 21r/X. It follows that, as R -+ m , the field intensities approach
the limit
A sinusoidal current distribution is rcprescnted by
(5)
u([)= I , sin ( k t - a ) ,
and the condition that the current shall be zero a t the ends (
= +Z/2
is
The integer m is equal to the number of half wave lengths contained in
the length I, and 10 is the current amplitude a t a loop position. The
integral (4)is readily evaluated and leads to
SBC.8.71
44 1
LINEAR OSCILLATOR
T h c numerical factor is
=
Or
sin
Ea = 6oIo
59.92 N 60;
(7cos
8)
ezkR-Lzwl
7
when m is even.
sin 8
The expressions do not depend on the length of the radiator, but only
upon the mode of excitation. The j k l d zs that of a dipole located at the
center whose amplitudr i s modijird by a phase factor F(B).
The intensity of radiation in any given direction is again expressed in
terms of the complex Poynting vector which is radial and whose magnitude is S* =
(10)
~
cos2
S"
(11)
I2= 30
-2
~-
sin2
)
sin 2O
2n 1 2 2
30 2
I2
s* = 2n ELZ
@ cos 8)
(T cos
( m =1,3,5
- - . ),
0)
(m = 2, 4, 6, * . * ).
sin2 0
From the location of the zcros and maxima of these functions one ran
gain a fair idea of the radiation pattern. When m is odd, the zeros of
S* occur a t
m
1 3
cos Q =
.
.
.
-,
mrn
m
while the maxima are found from the relation
)
-)
-7
ma
tan e sin 8
= cotan
2
Likewise, when m is even, t'he zeros occur a t
2 4
cos 8 = 0) -> - 1 . . .
m
)
-1
m m
m
and the maxima correspond t o angles satisfying
mn
- tan Q sin 8 = tan
2
The radiation patterns of t,he first four modes are shown in Fig. 74.
In each case the current distribution is indicated by a dotted curve
442
[CHAP.VIII
RADIATION
along the radiator, which is broken up into oscillating current sections
alternately in phase and 180 deg. out of phase. The intcrfcrrnce action
of these current sections gives rise to lobes in the radiation pattern equal
in number to that of scctions m. Between lobes is a radiation node.
I
I
I
I
I
I
I
I
I
I
\
\
I
\
\
\
\
\
FIG.74a. m
FIG.74b. m
=
2.
F1~.74.-Radiation patterns of a linear oscillator excited in the modes m
=
=
1.
1, 2, 3, 4.
The radiat'ion intcnsity is greatest along the axes of the lobes lying
nearest the radiator and as m
t h e radiation is directed entirely
along the wire. Energy can be propagated along a n infinite conductor
but there i s no outward rwdial pow. We shall see later that a Joule heat
loss within the conductor gives rise to a compensating inward Bow.
These theoretical patterns are modified in practice by the damping
due to Joule heat and radiation losses. As a consequence, the successive
--f
SEC.8.71
L I N E A R OSCILLATOR
443
current loops diminish in amplitude as one proceeds along the antenna
outward from the point, of feeding. The radiation lobes in or near the
equatorial planc e = a/2 are strengthened a t the expense of the outer
ones. This effect becomes more pronounced as the resistance of the
antenna wire is increased. It is illustrated by Fig. 7 5 . The curves were
obtained by Bergmannl from mcasuremcnts on lincar oscillators excited
in the mode m = 3 and constriicted of bronze and two sizes of steel wire
respectively. The ratio of the current maximum a t the central loop to
that a t either of the outer loops was of the order of 1.4:l. When the
-___Bronze, 4.0 mm diameter
Steel, 4.0 mm diameter
diameter
- - _ _ _Steel, 0.5 mm
FIQ. 75.-Effect
of antenna resistance on the radiation according to measurements by
Bergmann.
current amplitudes were wcighted in this ratio, a calculation of the
radiation gave results in close agreement with the observed values.
The power dissipated in radiation is obtained b y integration of S*
over the surface of a sphere of very large radius R drawn about the center
0. When m is odd, we have
To evaluate this integral, we rcplace the variable 6' by u = cos 6' between
the limits + I and - 1. Then, since
(17)
(16) transforms to
(18)
1
1
+ cos m m du.
BERGMANN,
Ann. Physik, 82, 504, 1927.
444
Now let 1
[CHAP.VIII
RADIATION
+ u = v/mr, which leads to
W
(19)
or
=
151;
lmff
1 - cos 2,
V
dv,
W = 151,2[ln2mry - Ci 2m?r],
(20)
where y = 1.7811 and Ci 3; is the cosine integral
Ci z =
-91” -v
cos 2,
dv,
for which values have been tabulated in Jahnke-Emde, “Tables of
Functions. l 1
An identical result is obtained when m is even. In either case the
radiation resistance is
(R = 30(ln 2mry - Ci 2m?r), (m = 1, 2, 3, .
(22)
),
or, since In 2r-y = 2.4151,
6i = 72.45
30 In m - 3OCi 2rm
ohms.
(23)
-
+
21 , order of excitation
m =A
FIG.76.-Radiation
resistance of a linear antenna a t resonance.
If m > 2, the term in Ci 2rm is negligible. The relation of radiation
resistance to the order of excitation is shown graphically in Fig. 76.
Equation (23) is valid only a t resonance, when the quantity 21/X = m is
an integer. At intermediate wave lengths the radiation resistance
follows a curve which oscillates above and below the resonance values
indicated by Fig. 76. The exact form of this curve depends on the
nature of the excitation.‘
See, for example, an interesting paper by Labus, Hochfrequenztech. u. Elektroak.,
41, 17, 1933.
SEC.8.81
R A D I A T I O N DUE TO T R A V E L I N G W A V E S
445
8.8. Radiation Due to Traveling Waves.-The
current distribution
considered in the preceding paragraph constitutes a standing wave with
nodes located a t the end points. By a proper termination of the wire,
a part or all of the reflected wave may be suppressed with the result that
the current is propagated along the conductor as a traveling wave.
Such, for example, is the case of a transmission line, where in order to
secure the most efficient transfer of energy every effort is made to avoid
standing waves.
To calculate the radiation losses of a single-wire antenna of finite
length' with no reflected current component, let us assume for the current
distribution the function
(24)
u ( { )= IoeikE.
If, as in the previous case, the antenna is fed a t the central point, the
radiation field intensities are
which, when evaluated, gives
The radiant energy flow is determincd by the function
The power dissipated in radiation is, therefore, equal to
(28)
W
=
302;
sin 0 do.
(1 - cos 0)'
1 - cos 0, du = sin 0 dO, and reduce (28) to
~
To integrate we put u
=
klu
( 2 - u ) sin2--
W
=
301;
U
du.
Omitting the elementary intervening steps, one finds for W the expression
'
The problem of a line with parallel return was first treated correctly by
Manneback, J. Am. Inst. Elec. Engrs., 42, 95, 1923.
446
[CHAP.V I I I
RADIATION
and for the radiation resistance
I t appears that for equal wave lengths and the same antenna the radiation
resistance of a traveling wave is greater than that of a standing wave.
/
/
I
I
\
\
I
f
t\
is
/I
I
I
\
\
@
/
I
I
\
t:
\
\
Marconi-Franklin antennas
Idealized current distribution
FIG. 77.
8.9. Suppression of Alternate Phases.
The directional properties of a singlewire antenna can be accentuated by
suppression of alternate current loops.
This is accomplished in the Marconia
Franklin type antenna by proper loading
with inductance at half-wave-length in, tervals along the line. The alternate sections of opposite phase are shortened to
\
such an extent as to render them negligible
\
\ as radiators and the antenna is then equivFIG. 78.-Evaluation of the series
alcnt to a colinear set of half-wave oscilb e-%mb.
lators placed end to end and all in phase.
m=O
This is shown schematically in Fig. 77.
The contribution of each oscillating segment to the radiation field at
a distant point arrives with a phase advance of j3 = w d / c over its next
2
Consequently the resultant field is
lower neighbor.
(32)
447
SUPPRESSION OF ALTERNATE P H A S E S
SEC.8.91
.
E0 -OOiIo
-
sin 19
_
~
-
m=O
where n is the number of current loops. Now the quantity b e-i4 is a
vector in a complex plane rotated through an angle -mP from the real
axis. The sum indicated in (32) can, therefore, be evaluated by simple
geometry. The vcctor diagram is drawn in Fig. 78, and the sum is
clearly the chord of a regular polygon. Following the notation of the
figure WE have
’
2
b = 2a sin -2’
2 a s i n -n7P
2
=
b sin nP
whence
= ___
sin P
a t an angle a
Since in the present instancc b
=:
= - (n -
P with the real axis.
1) 2
1, we obtain
(33)
For P we write w d l c = ?r cos 8, since d
x
2 cos 8, and so obtain
= -
The radiation intensity is
(T cos
sin
(35)
2a
sin 8
sin
1’’ 6
6
and the total power radiated
cos2
(36)
W
=
301;
8)
cos 6) sin2
cos 0)
(5
cos 8)
______ -
sin 0
sin2
d8.
cos 8)
448
RADIATION
[CHAP.VIII
This integral has been evaluated by Bontsch-Bruewitsch,' who obtains
ns
W
(37)
= (-1)n-1301;
(n-1)r
X - 4X
0
+ 8s
(n--21% ( n - 3 ) r
0
0
- 128
0
+. . .
A
4(n
- l)i],
0
where
mr
sin2 u
0
U
du
=
600 -
500 T
400 -
2
300 -
200 100 -
0
1
(In 2may - Ci 2m7r).
2
--
ing segments; Fig. 79b is a plot of
radiation resistance which shows
t h a t a marked increase of radiating
efficiency is achieved b y the suppression of alternate interfering
phases.
8.10. Directional Arrays.-The
single-wire antenna is directive
only i n the sense that the radiation
is concentrated in cones of revolution about the antenna as an axis.
A colinear set of half-wave oscillators excited i n phase confines the
radiant energy to a thin, circular
disklike region such as that of Fig.
duce a preferred direction in the
equatorial plane itself, or any other
form of axial asymmetry, the
single-wire radiator must be replaced by a group or array of
half-wave antennas suitably spaced and excited in the proper phase.
In practicc the spacing between centers is usually regular, so that
the array constitutes a lattice structure. There is in fact a certain
parallel between the problems of x-ray crystallography and the design
of short-wave antenna systems. From the x-ray diffraction pattern the
physicist endeavors to locate the centers of the diffracting atomic dipoles
and hence determine the structure of the crystal; the radio engineer must
choose the lattice spacing and phases of the current dipoles so as t o
FIG.79a.
FIG.79b.
FIG.79a.-Radiation pattern about antenna whose current distribution is indicated
by the dotted curve.
FIG.796.-Radiation resistance of n halfwave segments oscillating in phase.
I
1
BONTSCH-BRUEWITSCH,
Ann. Physik, 81, 437, 1926.
SEC.
8.101
DI REC T I ON AL A R R A Y S
449
obtain a prescribed radiation pattern. I n both cases the distribution of
radiation is characterized by a phase or form factor dependent upon the
polar angles 0 and 4, and t’he methods which havc been developed for
the analysis of crystal structure can be applied in large part to antenna
design.
To simplify matters we shall assume that all radiators of the array
are parallel. The antennas are excited a t the central point by means of
FIQ.80.--A
system of half-wave oscillators arrayed in a regular lattice structure defined by
the base vectors 8 1 , 8 2 , 8 3 .
transmission lines which can be designed to radiate a negligible amount of
energy. Moreover the phase velocity of propagation along these lines
can be adjusted to values either less or greater than c ; consequently, the
phase relations between the members of the array can be prescribed
practically a t wil1.l
The phase factor of a single half-wave antenna will be denoted by
PO,
cos
(40)
Fo(e)
=
6
cos
sin
e)
The centers of the oscillators are spaced regularly in a lattice structure
whose constants are fixed by the three “base vectors” al, a2,a3,as shown
‘The technical aspects of this problem have been discussed by many recent
writers. One may consult: Carter, Hansel1 and Lindenblad, Proc. Inst. Radio
Engrs., 19, 1773, 1931; Wilmotte and McPetrie, J . Inst. Elec. Eng. (London),66, 949,
1928; Hund, “Phenomena in High-frequency Systems,” McGraw-Hill, 1936; Hollman, “Physik und Technik der ultrakurzen Wellen,” Vol. XI, Springer, 1936.
450
RADIATION
in Fig. 80.
[CHAP.
VIII
The j t h oscillator is located in the network by a vector
rj
(41)
=
+ jzaz + j3a3,
j 2 , j , are three whole numbers not excluding zero. The phase
where jl,
of thc j t h oscillator with respect t o the phase of the oscillator a t the
origin is pi,
Pi
(42)
=j
1,l
+ jz, +
j3a3.
is the phase of any oscillat'or relative to it,s nearest neighbor on
Thus
the axis defined by the base vector al, and so on for the others. Thc
radiation field of the j t h oscillator is, therefore,
(43)
Ej
=
-iGOIojFo(0)
e i k R - iwt - ikRo.r, - ipj
R
where Ro is a unit vector in the direction of the radius vector R. The
resultant field int'ensity is obtained by summing over the entire array.
(44)
By a proper adjustment of amplitude, spacing, and phase in (44) a
radiation pattern of almost any dcsired form can be obtained. We shall
consider only the most common arrangement in which all current amplitudes are the same and the base vectors al, a2, a3 define a rectangular
structure whose sides are parallel t o the x-, y-, and z-axes. I n each row
parallel to the x-axis there are nl radiators; in each row parallel to the
y-axis there are 722; and in each column parallel t o the z-axis there are n3,
so that the lattice is completely filled.
(45)
RO ri = ~ I U sin
I 0 cos 9 + jzu2sin 0 sin 4
+ j3u3cos 8
in which UI, u2, a3 are the spacings along the 2-, y-, z-axes respectively.
The field intensity a t any point in the radiation field is
whose complex phase factors arc
m
(47)
DIRECTIONAL AREAI.N
SEC.8.101
451
If we let
(48)
71
=
kal sin
e cos 4 + a l l
7 3
=
yz =
kas cos 0
kaz sin
+ a3,
e sin
4
+ a2,
then (47) can be written
by (33). The complex factors 3,re eliminated from the radiation intensity
and we get
(50)
where
. nsYs
sin
2
~
F,
=
(s =
-7
sin
1, 2, 3 ) .
7s
All the radiation formulas derived above are special cases of this
result. For example, the single wire of Sec. 8.7 excited in an upper mode
constitutcs a linear distribution of half-wave oscillators with a1 = a2 = 0,
a3 = X/2, a3 = a. Then
(52)
F1= F 2 = 1,
n 3a
sin -- (cos 8
2
F 3 -cos
6
+ 1)
cos 0)
which leads directly to (11) on page 441. If the colinear oscillators are
in phase, as describcd in Sec. 8.9, wc take a1 = a2 = 0 , a3 = X/2, a p = 0,
and obtain ( 3 5 ) .
Considcr next a row of vertical half-wave oscillators spaced along the
horizontal x-axis. We shall assume them t o be excited in phase and
spaced a half wave length apart. a1 = X/2, a2 = a3 = 0, a1 = 0. Then
n o w depends o n the equatorial angle 4. The zeros occur a t those
angles for which the quantity nla/2 sin e cos 4 has any one of the valucs
T , 27r, . . . .
In the equatorial plane e = ~ / and
2 there are radiation
which
452
RADIATION
[CHAP.
VIII
nodes along the lines determined by
cos 4
(54)
or
cos 4
=
=
2 2-4 . . . i-J
nl
nl
n1
nl
i - 7
2
i - 7
nl
4
in1
FIG. 8la.-Radiation pattern in the
equatorial plane of
two half-wave oscillators operating in
phase and separated
by a half wave length.
pattern in the equatorial
plane of four half-wave
oscillators operating in
phase and with half-wavelength separation.
as nl is even or odd. The principal maximum occurs a t 4 = r / 2 , so that
this array gives rise to a broadside emission of energy. Subsidiary
maxima are determined by the condition aFl/a+ = 0, which in the
present instance leads to the relation
(55)
nl tan
6
sin
e cos cp .
(Y
)
nl, so that the radiation intensity in the
)
e cos 4
=
tan - sin
When 0 = cp = r / 2 , FoFl =
direction of the principal m a x i m u m i s ni times the m a x i m u m intensity of a
single oscillator. On the other hand, each oscillator receives only the
nlth part of the total power delivered by the source (assuming equipartition), and hence the net gain in radiation intensity in the preferred direction
i s given by the factor 71.1, the number of oscillators.
DIRECTIONAL ARRAYS
SEC.8.101
453
The neighboring zeros lying on either side of the principal maximum
in the equatorial plane are fixed by the relation cos 4 = f2/nl.
Hence
the larger nl, the narrower the heam. The beam is evidently confined to
a region bounded by the angles $1 and &, where cos 42 - cos q h = 4/nl,
I
FIG.82.-Radiation
pattern in the equatorial plane of two half-wave oscillators 90 deg. out
of phase and with quarter-wave spacing.
lying on either side of 4 = 7r/2. The radiation patterns in an equatorial
plane have been plotted in Figs. 81u and 81b for arrays containing two
and four oscillat<ors.
Lastly, we shall apply (50) to two antennas spaced a quarter wave
length apart and 90 deg. out of phase. Let ul = X/4, u2 = u3 = 0,
*
0
cos
6
J,
8
(sin e cos 4 - 1)
*
J
The radiation pattern in the equatorial plane is shown in Fig. 82.
FIG. 83.-Radiation
patterns in the
The energy is thrown forward in a equatorial
plane for a horizontal row of
broad beam in the direction of the vertical half-wave oscillators backed by a
of reflectors, illustrating the concenpositive x-axis. Radiation to the row
tration of radiation in a forward beam by
rear is reduced to a very small increasing the number of oscillators.
amount. I n many commercial in- (Hollmann.)
stallations the rearward antenna is not connected directly t o the sourcc.
Its excitation is then due to inductive coupling with the forward oscillator
and its action is that of a reflector. In this case, however, the spacing is
not quite X/4 and a small adjustment must also be made in the length of
the reflector.
454
RADIATION
[CEAP.
VIII
A nwdlcliltr beam in space can be obtained by a combination of the
simple arrays described above. Concentration of the radiation in a
meridian plane is accomplished by means of a horizontal row of vertical
oscillatcrs. Extending the length of these vertical lines into a series
of half-wave radiators excited in
,,
p =r/2
phase results in a concentration in
the equatorial plane. Finally, this
two-dimensional array is backed up
8 by a corresponding set of reflectors
a t a n approximate distance of X/4.
In Fig. 83 the radiation diagram in
the plane 0 = a / 2 of a horizontal
row of vertical oscillators backed by
FIG. ~4.--1tadiation patterns in the reflectors is drawn for several values
emidg
p ----*
I I
II
I 1
49L-
meridian plane for a vertical set of halfwave oscillators backed by reflectors.
(Hollmann.)
'% the number in a roW. I n Fig.
84 the same diagram is drawn for a
vertical set in t'he meridian plane 4 = ~ / 2 . I n both cases the effectiveness of an increase in n is apparent. The greatest improvement naturally
occurs when n is small, in which case the addition of another oscillator
makes a very considerable difference. The relative gain decreases with
increasing numbers and a point is soon reached a t which any further
addition to the array is economically unOf
warranted.
8.11. Exact Calculation of the Field
~r
j: ,
1
of a Linear Oscillator.-The method of j
Poynting, which has thus far been emj j
cp, 9,2)
ploycd to calculat,e the radiation resist'
r1
ance, is based entirely on a determination
of the outward flow of energy at great j
distances from the centcr of the radiating
$1
system. Now if cnergy is const,antly lost + + + 4 0
from the system i t milst also be supplied FIG.85.--Coordinates of the linear
oscillator discussed in Sec. 8.11.
by the source; consequently, work must
be done on the antenna currents a t a constant rate. There must be a component of the e.m.f. parallel to the wire a t its surface which is 180 deg. out
of phasc with thc antenna current. A knowledge of the phase relations
between current and e.m.f. along the conductor must lead to the same value
of radiation resistance. I n fact it can give us more; for this resistance is
but the real part of a complex radiation impedance which must eventually
be determined if the radiator is t o be t,reated as a circuit element.
It is apparent t h a t the result's of the preceding paragraphs cannot be
applied to this alternative method, for they are valid only a t great distances from the source. It is essent)ial that we have a n exact, expression
I
1
i
A
+!
I
i
E X A C T CALCIJLATION OF T H E FIELD
SEC.8.111
455
for the field in the immediate neighborhood of each conductor. The
demonstrations by Brillouin, I Pistalkors, and Bechmann3 that a n
exact expression in closed form can in fact be found were an extremely
important contribution to antenna theory.
We shall consider again a single linear conductor coinciding with a
section of the z-axis, but thc center of the conductor need no longer be
located a t the origin. A point on the conductor is fixcd by its coordinate
[, and its length is I = f z - [I, as shown in Fig. 85. The currcnt distribution is sinusoidal, but the current need not be zero a t the elids if
terminal loading is taken into account. Such loading can be accomplished
through the capacitative effect of mctal plates, by grounding through a
lumped inductance or resistance, or by any of a variety of other devices.
The current is, therefore, again
1 = I. sin (kf - a)e-iwt
(57)
and the Hertz vector of the distribution is
which can be written also in the form
p,
The cylindrical coordinates of a n arbitrary point in the field are
4 , z, and
the distance froni any current element to this point is
____
= dpz ( z - t)2. Now let
~
T
+
u=r
(60)
Then
ir
+ [ - z,
eik(r+E
U
du = - d [ .
r
Ll
u 2 eaku
d l = elkz
- du,
(61)
r
u
where u1 = r 1
- z, u2 = r2
l2- z, and T I , r2 are the distances
from the ends of the wire to the point of observation. Likewise, if
+
+
v =r -[
(62)
where
forc.
1
v1
= rl
-
41
+ z,
dv
+ z, v 2 = r2 - EZ + z.
=
V
-;
d[,
The Hertz vector is, Iherc-
BRILLOUIN,
Radioblectricitk, April, 1922.
PISTALKORS,
Proc. Znst. Radio Engrs., 17, 562, 1929.
BECIIMANN,
ibid., 19, 461, 1471, 1931.
Next Page
456
RADIATION
[CHAP.VIII
A simple transformation reduces these exponential integrals to a form
for which tabulated values are available. We note first that
By definition the cosine and sine integrals are
(66)
Cikul
since Si.o
=
=
-I:
cos ku
211
Sikul
du,
sin ku
---du;
U
=
7r/2, we have
(67)
or
l,
u2
(68)
pku
UZ
UZ
-du=
u
CikuIul+iSiku( .
u1
Tables of these functions will be found in Jahnke-Emde.
To calculate the field we apply the formulas
+
E = V V - I I k2n, H
(69)
which in cylindrical coordinates reduce to
=
-i W E V x n,
Since ull uzl etc. in (64) depend upon both z and p , the integrals must be
differentiated with respect to their limits. For example,
1:'
&,
duz
eikui
-du = eiku2
__ - - __ u 1 dz'
U
uz a2
eik*
(71)
eikz
2 l:'.i""
eik(ri+
U
Similarly,
€1)
du = ___ rl
eik(rz +Cz)
~.
rz
-
P
t-1
-
(2 - .$I)
&Tl+
$1)
~.
TI
After differentiating in this manner and combining terms one obtains
for the components of field intensity
Previous Page
SEO.8.121
R A D I A T I O N RESISTANCE
457
I n the case of an isolated wire whose ends are free the current will he
zero at the points 5 = f l , .$ = &. Then k = ?rm/l,where m is again the
number of half wave lengths along the wire. Since
.
The field intensities then reduce
to the comparatively simple expressions
8.12. Radiation Resistance by the E.M.F. Method.-The expressions
for the field derived in the preceding paragraph are valid and exact a t
all points of space, including the immediate neighborhood of the current.
The occurrence of singularities at the end points r1 = 0, r2 = 0, results
from the assumption of a line distribution of current. In practice the
finite cross section of the wire ensures finite values of the field intensity,
although relatively large intensities must be expected near these terminals. At any other point along the conductor the phase relation
between current and the parallel component of E can be determined from
(73) or (76); consequently, the work necessary to maintain the current
against the reactive forces of the field can be calculated.
I n Sec. 2.19 i t was shown that for real vectors E, H, and J and a current distribution in free space the divergence of the Poynting vector
satisfies the relation
(77)
458
[CHAP.VIII
RADIATION
If the field is periodic, the average of the time rate of change of stored
energy is zero, so that
(78)
V . S = V * S = -E*J.
If now in place of real vectors we deal with complex field intensities and
currents, we have by See. 2.20
-
S
(79)
=
+&(E X H) = ReS*,
-
12eE lZeJ
=
&Re(E
-
- J),
the sign indicating conjugate values. The total average power radiated
from a systcm of currents is, therefore,
W
=
L s - n d a = - - 1R e J v E - $ d v .
2
The radiation can be calculated either by integrating the normal compone,nt
of the Yoynting vector over a closed surface including all the sources, or by
integrating the power expended per unit volume over the current distribution.
Suppose next t h a t there are n linear conductors carrying current in
otherwise empty space. Without any great loss of generality it may be
assumed that the conductors of the system are parallel to one another
and to the x-axis. The current i n the j t h conductor is I f , a complex
quantity. The z-component of the electric field at the jth conductor
due t,o the i t h current is Eii Then by (80) the total radiation from the
system is
the integrations to be extended the length of each conductor. This is the
rate a t which the induced e.m.f.'s do work on the currents. Let us write
(82)
Ii
=
I o i sin ( k & - cyi)e-iwt,
E..
1 % = IoiU..eciUt.
2%
The amplitudes lo?
and I o i are in general complex, and the function
U j iis determined by (73). Then
where
The quantit'ies Zii are coupling coefficients or, in a certain sense, transfer
impedances. The real part of the coefficient Zii represents the radiation
resistance of the ith conductor when all other conductors are removed.
SEC.8.121
459
R A D I A T I O N RESISTANCE
A very neat method of calculating the radiation has been suggested
by Bechmann.' If the current dish-ibution is sinusoidal as in (82), one
may write
(85)
The field intensity Eji satisfies the relation
and hence
W
(58)
=
$Re
ti
n
]=I
%=I
32
[TI,
From (64) and (68) we have
+
+
5%- E l , u = rtl - tt t,. The
where, in the present case u = rll
coordinate refers to points 011 the antenna whose current is 1%)while
E, is a point on the j t h conductor. The subscripts 1 and 2 denote as
I d o r e the coordinates of the lower and upper ends respectively of an
antenna. No further integrutzon i s necessary to $nd the radiution f r o m a
system vf linear conductors. Siricc thc current amplitudes l o and
, lo,are
factors in (88), an integrated expression for the impedance Z,, can be
found.
In tlw case of an unloadcd antcrina the current 1,is zero at the terminals El, and
Thcn k t ~ , a? = 0, k[Z1 - a j = m ? ~and
, we obtain
,$$
(90)
w=
a
n
RC
n
33
k f o j [ ( -l ) m l n , ( t 2 , > - r ~ ( t ~ ~ ) i .
3=1
% = I
A difficulty ariscs whcn thiJ formula is applied to colincar conductors.
Consider, for cxamplc, the radiation from a single linear oscillator as in
SCC.8.7. This case is the limit of two parallel wires whosc separation
approaches zero. If we take the terminals of the two conductors to be
coincident, t h m
1
B E L H M A Zoc.
~ N ,czt , p. 1471.
But k
=
am/l
=
1 27r/X, - d p o / e 0 = 60; hence b y (90)
2a
W
= -151;Ci-
amu
2z
1
0
-
This result fails, however, since CiO is infinite. The difficulty can be
avoided by calculating first the radiation of two parallel wires separated
by a small but finite distance p . Upon passing to the limit p = 0, the
infinite terms drop out‘ and one obtains, as in See. 8.7,
W
(91)
=
1512,(ln2may - Ci2ma).
THE KIRCHHOFF-HUYGENS PRINCIPLE
8.13. Scalar Wave Functions.-Lct V be any region within a homogeneous, isotropic medium bounded by a closed, regular surface s, and
let +(x, y, z ) be any solution of
(1)
v2*
+
k2+
=
0
which is continuous and has continuous first derivatives within V and on
S. Then i t follows directly from Green’s theorem, ( 7 ) , pagc 165, that
the value of a t any interior point x‘, y’, z’ can be expressed as an integral
of and its normal derivative over S.
+
+
where as usual
r = d(z’ - 212
(3)
+ (y’ - g>2 + (z’
- z)2
is the distance from the variable point x, y, z on S to the fixed interior
point x’,y’, z’. Equation (2) is in fact the special case of (22), pagc 427,
for which all sources are located outside S and the time factor exp (-id)
has been split off.
1
Details are given by Bechmann, Zoc. cit., p. 1477.
SEC.8.131
461
SCALAR WAVE FUNCTIONS
The function defined by (2) is continuous and has cont,inuous derivatives a t all interior points, but exhibits discontinuities as the point
x’, y’, z’ traverses the surface S . The transition of the function
f(s’,y’,
(4)
2’) = -
across S was shown to be continuous in See. 3.15; consequently, the discontinuities of (2) are identical with t,hose discussed in Scc. 3.17 in
connection with the static potential. If d, y’, z‘ is a n y point, either
interior or exterior, the function defined by the integral
(5)
\
L-’
FIQ.%.--The
/
surface SI may be closed a t infinity.
is a t all interior points identical with the solution J. whosc values have
been specified over S , and a t all exterior points is zero.’
As in the static case, too, the values of J. and its derivatives at all
alone on S
interior points are uniquely determined by the values of
(Dirichlet problem), or by d J . / d n alone (Neumann problem). The values
of both J. and dJ./an, therefore, cannot be specified arbitrarily over S if
u and are to be identical a t interior points. The function u defined by
(5) satisfies (1) and is regular within V whatever the choice of
and
over S , but the values assumed by u and &/an on X will in general
differ from those assigned to and d J . / d n .
All elements of the surface S which are infinitely remote from both
the source and the point of observation contribute nothing to the value
+
+
+
The analytic properties of the Kirchhoff wave functions are discussed in detail
by PoincarB, “ThBorie math6matique de la lumii?re,” Vol. 11, Chap. VII. See also
the very useful treatise by Pockels, “ a e r die partielle Differentialgleiehung
Au
k*u = 0,”Teubner, 1891, and “Lectures on Cauchy’s Problem” by Hadamard,
Yale University Press, 1923.
+
462
R A D I A TION
[CHAP.
VIII
of the integral (2). I n Fig. 86 all elements of the source are located
within a finite distance 121 of a fixed origin &. The surface X is represented as an infinite surface 81 closed by a spherical surface X, of very
large radius r about P as a center. Thc point of obscrvation P(z’, y’, 2’)
lies within S and all elcrnents of the source are exterior t o S. If R , the
distance from Q to an clement of area on Xt,is very much larger than
the largest value of R1, then by (87), page 414, and ( 3 2 ) , page 406,
eikRz
(6)
-
~ ‘v
-
I22
-e?!!
R
2
(-i)%(2n
+ 1)P,(cos TI) j,(kR1)
n=O
n-0
(7)
hence,
Over Sz
The terms in r -I, r P 2 ,and r - 3 in the expanded integrands of ( 2 ) )therefore,
cancel one another, while the remaining terms are of the order r-”,
n > 3, and consequently contribute nothing t o the integral over Sz
as r + m .
Let us suppose now that the surface X represents an opaque screen
separating the source from the observer. I n virtue of the theorem just
proved it can be assumed that open surfaces of infinite extent are closed
a t infinity. If a small opening 81 is made in the scrcen, the field will
penetrate to some extent into the region occupied by the ohserwr. Thc
problem is to determine the intensity and distribution of this diffracted
field. Clearly if the values of and a + / d n were known over thc opening
XI and over the obscrver’s side of the screen, the diffracted field could
be calculated by evaluating (2). Thcse values are not known, but t o
obtain an approximate solution one may assume tentatively with Kirchhoff that:
( a ) On the inner surface of the scrcen
+
SEC.8.131
SCALAR W A V E F U N C T I O N S
463
( b ) Over tjhe surface S1 of the slit or opening the field is identical with
that of the unpc:rturbetl incident wave.
Nearly all calculations of the diffraction patterns of slit>sand gratings
made since Kirchhoff’s time have been based on these assumptions.’
It is easy t o point out fundamental analytic errors involved in this
procedure. I n the first place, the assumption that $ and a$/& are zero
over t’heinside of the screen implies a discontinuity about the contour C1
which bounds the opening S1,whereas Green’s theorem is valid only for
functions which are continuous everywhere on the complete surface X.
This difficulty cannot be obviated by the simple expedient of replacing
the contour of discontinuity by a t’hin region of rapid but continuous
t,ransition. If $ and a$/an are zero over any finite part of S , they arc
zero a t all points of the space enclosed by AS.^ I n the eecond place, an
electromagnetic field cannot in general be represent.ed by a single scalar
wave function. It is characterized by a set of scalar functions which
represent rectangular componcnts of the electric and magnetic vectors.
Each of these scalar functions satisfies (1) and its value at an interior
point z’, y’, z’ is, therefore, expressed by ( 2 ) in terms of its values over the
houndary X. But these components a t a n interior poirit must not only
satisfy the wave equation, they must also be solutions of the Maxwell
field equations. The real problem is not the integration of a wave
cquation, either scalar or vector, but of a simultaneous system of firstorder vector equations relating t’he vectors E and H.
I n spite of such objections the classical Kirchhoff theory leads to
satisfactory solutions of many of the diffraction problems of physical
optics. This success is due primarily to thc fact that the ratio of wave
length to the largest dimension of the opening in optical problems is
small. As a consequence the diffracted radiation is thrown largely
forward in the direction of the incident ray and the assumption of zero
intcnsity on the shadow side of the screen is approximately justified.
Measurcments are usually of intensity and do not take account of the
polarization. As the wave length is increased, the diffraction pattern
hroadens. A computation of $ from ( 2 ) now leads to intensities directly
h h i n d the screen which are by no means zero, contrary to the initial
assumption. Various writers have suggest’ed that this might be conOn the classical theory of diffraction see, for example, Plmck, “ I ~ h f i i h r u n gin
die theorctische Optik,” Chap. IV, Hirzel, Lcipzig, 1927, or Born, “Optik,” Chap.
IV, Springer, Berlin, 1933. To obtain a clearer understanding of the physical significance as well as the shortcomings of Kirchhoff’s met,hod the readcr should consult
thc papcrs by Rubinowicz, Ann. I’hysik, 63, 257, 1917, and Kottler, ibid., 70, 105,
1923. Since the writing of this and the following sections a treatment of the subject
Ins a p p m r d in book form: “The Mathematical Theory of Huygerrs’ Principle,” by
Baker and Copson, Oxford, 1939.
A direct consequence of Green’s tlieorern. See POCKELS,
loc. cit., p. 212.
464
RADIATION
[CHAP.
VIII
sidered as the first of a series of successive approximations, the values of
$ and a$/& over S obtained from the first calculation to be used as the
boundary conditions in the second. There is, however, no proof that
the process converges and the difficulties of integration make it impractical.
Recent advances in the technique of generating ultrahigh-frequency
radio waves have stimulated interest in a number of problems previously
of little practical importance. A natural consequence of this trend
towards short waves is the application of methods of physical optics to
the calculation of intensity and distribution of electromagnetic radiation
from hollow tubes, horns, or small openings in cavity resonators. I n
radio practice, however, the length of the wave is commonly of the same
order as the dimensions of the opening, and the polarization of the diffracted radiation is easily observed. It is hardly to be expected that
the Kirchhoff formula (2) can be relied upon under these circumstances.
8.14. Direct Integration of the Field Equations.*-The problem of
expressing the vectors E and H at an interior point in terms of the
values of E and H over an enclosing surface has been discussed by a
number of writers2 A simple and direct proof of the desired result can
be obtained by applying the vector analogue of Green’s theorem to the
field equations. In Sec. 4.14 it was shown that if P and Q are two vector functions of position with the proper continuity, then
(10) ~ ( Q . V X V X P - P - V X V X Q ) d v
=$s (P x v x Q - Q x B x P) encia,
where S is as usual a regular surface bounding the volume V .
Let us assume that the field vectors contain the time only as a factor
exp (-id) and write the field equations in the form
(1)
V x E - i ~ p H= -J*,
(111) V * H
=
1
- p*,
P
The medium is assumed to be homogeneous and isotropic, and of zero
conductivity. The quantities J* and p* are fictitious densities of
“magnetic current’’ and ‘(magnetic charge,” which to the best of our
8.14 and 8.15 were written in collaboration with Dr. L. J. Chu.
z L o ~Phil.
,
Trans., (A) 197, 1901; LARMOR,
Lond. Math. SOC.Proc., 1, 1, 1903;
IGNATOWSKY,
Ann. Physik, 23, 875, 1907, and 26,99, 1908; TONOLO,
Annali d i Mat.,
3, 17, 1910; MACDONALD,
Proc. Lond. Math. SOC.,10, 91, 1911 and Phil. Trans., (A)
212, 295, 1912; TEDONE,Linc. Rendi., (5) 1, 286, 1917; KOTTLER,Ann. Physik, 71,
457, 1923; SCHELKUNOFF,
Bell. System Tech. J . , 16, 92, 1936; BAKER
and COPSON,
Zoc. cit., Chap. 111.
1 Sections
SEC.8.141
465
INTEGRATION OF FIELD EQUATIONS
knowledgct have no physical existence. However, we shall have occasion
shortly to assume arbitrary discontinuities in both E and H, discontinuities which are in fact physically impossible, but which would be
generated by surface distributions of magnetic current or charge were
they to exist. Currents and charges of both types are related by the
equations of continuity,
V * J - imp
(V)
=
V J* - i m p *
0,
=
0.
The vectors E and H satisfy
V x V x E - k2E = iwpJ - V X J*,
(11)
V x V x H - k2H = iweJ*
V X J,
(12)
with k2 = w2ep.
I n (10) let P = E, Q = 4a, where a is a unit vector in an arbitrary
direction and 4 = eikr/r. Distance r is measured from the element a t
x, y, z to the point of observation a t x’, y’, z’ and is defined by (3). We
have identically
+
(13)
V x Q
+
-
V 4 x a,
V X V x Q = ak24 V(a V+),
V X V X P = k2E iwpJ - V X J*.
=
+
Following the procedure of See. 4.15, i t is easily shown that a is a factor
common to all the terms of (10) and, since a is arbitrary, it follows that
=
l [ i m p ( n x H)4
+ (n x E) x v@+ (n - E)v4 - n x J*+] (la.
The identity
reduces this to
(16)
s
V
(iwJ4
-
J* X V 4
=
+
[iwp(n x
H)
+ (n x E) x V 4 + (n- E)V+] du.
The exclusion of the singularity a t r = 0 was described in Sec. 4.15.
A sphere of radius r1 is circumscribed about the point z’, y’, z’, its nornial
directed out of V and consequently radially toward the center.
466
[CHAP.V I I I
R A D IA T IO N
and on the sphere n = ro. The area of the sphere vanishes with the
radius as 4ar2,and, since
+ -
(n x E) x n
(n E)n = E,
(18)
t)he contribution of the sphcrical surface to the right-hand side (16)
reduces to 4nE(z’, y’, 2 ’ ) . The value of E a t any interior point of V is,
therefore,
(19) E ( d , y’, z‘)
-
sv
4‘,
-
=
&
(iwrJm
-
[iwp(n x H ) +
J* x v+
+
+ (n x E) x V + + (n - E)V+] da.
An obvious interchange of vectors leads to the corresponding expression
for H,
(20) H(d, y’,
2‘) = 4a
Ss
+ 45,
-
sv
(iwrJ*m
[k(n
x E)$
+ J x v+ + P
- (n x Ip)
x
V+ -
-
(n H)V+] da.
This last is the extension of ( 2 3 ) , page 254, to the dynamic field.
If all currents and charges can be enclosed within a sphere of finite
radius, the field is regular a t infinity and either side of X may be choscn
as its “interior.”
I n the earlier sections of this chapter the calculation of the fields of
specified distributions of charge was discussed in terms of vector and
scalar potentials arid of Hertzian wetors. We have now shown that
E and H may be calciilated directly without the intervention of potcntials. The surface intvgrals of (19) and (20) rcymscnt the contributions
of sources located outside S. If S rccedcs to infinity, it may be assumed
that these contributions vanish. Discarding the densities of magnetic
charges and currents, one obtains the uwf ul formulas
Since the current distribution is assumed t>obe known, the charge density
can be determined from the equation of continuity.
I n Sec. 9.2 it will be shown that an electromagnetic field within a
bounded domain is completely determined by specification of the tangential componcnts of E or H on the surface and the initial distribution
of the field throughout the enclosed volume. It follows that when
a x E and n x H in (19) and (20) have been fixed, the choice of n E
-
467
INTEGRATION OF FIELD EQUATIONS
S w . 8.141
-
and n H is no longer arbitrary. The selection must be consistent with
the conditions imposed on a field satisfying Maxwell’s equations. The
same limitations on the choice of 1c, and d1c,/an were pointed out in Sec.
8.13. The dependence of the normal component of E upon the tangential
component of H is equivalent to that of p upon J.
Let i i s suppose for the moment that the charge arid current distributions in (19) are confined to a thin layer a t the surface S. As the depth
of the layer diminishts, the densities niay be increased so that in the
limit the volume densities arc rcplaccd in the usual way by surface
densities. If the region V contains no charge or current within its
interior or on its boundary S , the field a t an intcrior point is
(22)
E(z’,y’,x’) = -G s [ i o p ( n X H ) + + ( n X E ) x V + + ( n - E ) V + ] d a .
l.I
Itj is now clear that this is cxactly the field that would be produced by a
distribution of clertric currcni over S with surface density K, a distribution of magnetic current of density K*, and a surface electrir charge of
density 7, where
(23)
K
=
-n
x
K* = n x E,
H,
-6n.E.
?1 =
The values of E and H in (23) arc those ,just inside the surface S.
The function E(z’, y’, z ’ ) defined by (22) is discontinuous across A!.
It can be shown as above that discontinuities associated with = e c h T / r
arc idcntical with those of the stationary regime,
= l / r . Then by
See. 3.15 the integral
+
+
(24)
suffers a discontinuity on tr:tnsilion through S equal to n * AE3 = q / c ,
where AE, is the diffcrencc oi‘ the values outsitlr and inside. The third
term of (22)’ thercforc, docs not affect the transition of the tangential
component but reduces the normal component of E t o zero. Likewisc by
Sec. 4.13 the discontinuity of
Ez(z’, y’,
L
4‘,
2’) = -
K* X v + d a
is specified by n x AEz = K*, so that the srcond term in (22) reduces the
tangential component of E t o zero without affecting thc normal coinponent. The first term in (22) is continuous arross 8, but has discontinuous derivatives. The curl of E a t z’, y’, x’ is
V’ x E(d, y’,
2’)
=
-2
Jy
(n
x H) x V+ da
468
RADIATION
[CHAP.V I I I
which is of the type ( 2 5 ) . The vector E and the tangential component of
its curl are zero o n the positive side of S; E i s , thwefore, zero at all external
points. The same analysis applies to H.
8.16. Discontinuous Surface Distributions.-The results of the preceding section hold only if the vectors E and H are continuous and have
continuous first derivatives at all points of S . They cannot, therefore,
be applied directly to the problem of diffraction a t a slit. T o obtain
the required extension of (22) to such cases, consider the closed surface X
(surfaces closed a t infinity are included) to be divided into two zones
S1 and S , by a closed contour C lying on S. as in Fig. 87. The vectors
E and H and their first derivatives
are continuous over S1 and satisfy
the field equations. The same is
true over Xz. However, the components of E and H which are tangential to the surface are n o w
FIG.87.--C is a contour on the closed subject to a discontinuous change
surface S dividing it into the two parts Si in
across c from
to
and S2.
the other. The occurrence of such
discontinuities can be reconciled with the field equations only by the
further assumption of a line distribution of charges or currents about the
contour C. This line distribution of sources contributes to the field,
and only when it is taken into account do the resultant expressions
for E and H satisfy Maxwell's equations.
A method of determining a contour distribution consistent with the
requirements of the problem was proposed by Kottler.' It has been
shown that the field a t an interior point is identical with that produced
by the surface currents and charges specified in (23). A discontinuity
in the tangential components of E and H in passing on the surface from
zone S1 t o zone S z implies therefore an abrupt change in the surface
current density. The termination of a line of current, in turn, can be
accounted for according to the equation of continuity b y an accumulation
of charge on the contour. Let ds bc an element of length along the
contour in the positive direction as determined b y the positive normal
n, Fig. 87. Let n l be a unit vector lying in thc surface, normal to both n
and the contour element ds, and directed into zone (1). The line densities of electric and magnetic charge will be designated by u and u*.
Then Eqs. (V), when applied to surface currents, become
UOTTLER,
Ann. Physik, 71, 457, 1923.
SEC.8.151
DISCONTINUOUS SURFACE D I S T R I B U T I O N S
469
and hence by (23)
(28)
i w g = nl
- (n X
-
H2
-n
iwa* = nl (n X El - n
-
X HI) = (H2 - Hl) (nl x n),
x E2) = - (Ez - El) (nl x n).
-
The vector nl x n is in the direction of ds. If S2represents an opaque
screen over which E2 = H2 = 0, the field a t any point on the shadow side
is
which can be shown to be identical with
It remains to be shown that the fields expresscd by these integrals are
in fact divergenceless and satisfy (I) and (11). Consider first the divergence of (29) a t a point z’, y’, 2’.
n
taking into account the relation V’ = -V when applied to 4 or its
derivatives. Now
(33)
-
(n x H) v + d a =
4v x H Snda - f C 4 H ds.
470
RADIA TION
[CHAP.V I I I
The line integral resulting from t.his transformation is zero when S is
closed, hut otherwise just cancels the contour integral in (32). Inversely,
only the presence of th,e contour integral in (29) leads to a zero divergence and
waves which are transverse at great distances f r o m the opening S1. From
(11) and the relation Ic2/ioe = - i w p , follows immediately the result
V’ * E(x’, y’, 2’) = 0.
(34)
An identical proof holds for H(z’, y’, 2’).
Finally, i t will be shown that (29) and (31) satisfy (I) and (11).
(35)
V’ x H(z’, y’, z’)
= -4lT
L*
-
[(n x H1) V V +
- iwc(n
since the curl of the gradient is idcntically zero.
(n x HI) . vv+ da
(36)
=
J,(n
= -$c
+
=
V x H J v + da -
-
X El) X
n (Vvm
V+]da,
Furthermore,
-1,- x
L x v - (Hlv+)
A1
V+ HI ds - i w c
+ k2+(n x HI)
H l ) v + da
n
da
hl(n - El)v + da,
where the operator vv+acts on v+ only. The last integral takes account
of t,he fact that t h e field equations arc by hypothesis sahfied on X1.
Then
(37)
V‘ X H(z’, y’,
2’) =
lTpCV + H1 - + 2 kl[iwdn x H1)+
ds
-
+ (n X El) X V + + (n - El) V+] da
= -iwc
E ( d , y’,
2’).
The validity of (I) is established in the same manner.’
FOUR-DIMENSIONAL FORMULATION OF THE RADIATION PROBLEM
8.16. Integration of the Wave Equation.-In SCC.1.21 i t was shown
how by means of tensors t,he field equat’ions could be written in an exceedingly concise form. If the imaginary distance x4 = i c t be introduced as a
coordinate in a four-dimensional manifold, the equations of a variable
field are in fact formally identical with those which govern the static
regime, and the methods which were applied to the integrat,ion of Poisson’s
equation can be extended directly t o thc more general case. The fourThese formulas have bccn applied by L. J. Chu and the author to thc c:tlciilntion
T h e results
compare favorably with those obtained by Morse and Rubenstein, Phys. Rev., 64, 895,
1938, who solved a two-dimensional slit problem rigorously by introducing coordinates
of a hyperbolic cylinder. Phys. Rev., 66, 9% 1939. See also the treatment of such
problems by Schelkunoff, ibid., p. 308.
of diffraction by a rectangular slit in a perfectly conducting screen.
471
I N T E G R A T I O N OF T H E W A V E EQUATION
SEC.8.161
dimensional theory is more abstract than the methods described earlicr
in this chapter, and consequently has not been applied to the practical
problem of calculating antenna radiation. Quite apart from its formal
elegance, however, the four-dimensional treatment sometimes leads in
the most direct manner to very useful results. This is particularly true
of the rather difficult problem of calculat>ing the field of an isolated
charge moving in an arbitrary way.
The discussion will be confined to the case of charges and currents in
f r w space. AH in (82), page 73, the vector and scalar potentials ran be
represented by a single four-vector whose rectangular components are
(1)
@I
=
A,,
=
fly.
+3
A,,
=
@4
=
i
C
4.
Likewise the current and chargc dcnsitics are represented by a four-vector
whose romponents are
(2)
J1 =
J,,
Jz
=
J,,
JX
J,,
=
Jq
= icp.
The four-potential satisfies thr relations
(3)
According to the notation of ( 3 5 ) , page 64,
is a symbolic four-vector
whose components are d / a m T h c x n ( 3 ) can k)e c.xyresscd concisely as
(4)
-
nzQ,=-poJ,
Q,
=
0.
Let V be a region of a four-dimcnsional spacc hounded by a threedimensional “surface” S. n is the unit outward normal to S. Then in
four as in three dimensions
(5)
If u and w are two scalar functions of the four coordinates zl, xz, za, z4
which together with their first and second derivatives are continuous
throughout V and on S , then
(6)
-
Jv 17 ( u n w ) dv
=
(7)
Thus, one obtains a four-dimensional analogue of Grecn’s theorem,
472
[CHAP.V I I I
RADIATlON
Let xi be the coordinates of a fixed point of observation within S,
and xi those of any variable point in V or on 8. The distance from
xi to xi is
(9)
It can be verified by direct differentiation that the equation
(10)
02w
=
0
is satisfied by
w = -1
R'
(11)
at all points exclusive of the singularity R = 0, which will be excised in
the usual fashion by a sphere S1 of radius R1. If u is identified with any
component @j of the four-potential, there follows
Over SI we have
To determine the area of the hyperspherical surface S1 polar coordinates
are introduced,
21 =
(14)
R cos
x2 = R sin 81 cos 02,
5 4 = R sin O1 sin O2 sin 4,
$1,
x3 = R sin O1 sin O2
COB
4,
which satisfy
x;
(15)
+ x; + xi + xi = R2.
The scale factors hi are calculated as in (70), page 48, and are found to be
(16) hl
=
1,
h2 = R,
h~ = R sin
$1,
hd = R sin
el sin 02.
The element of volume is
(17)
dv
=
hlh2h3h4dR do1 do2 d 4
=
R3 sin2 el sin O 2 d R do1 d o z d+;
hence, the area of the hypersphere is
(18)
Therefore,
R3
izT
sin2 O1 sin B2 do1 do, d+
=
2dR3.
473
F I E L D OF A MOVING P O I N T C l I A l i G B
SEC.8.171
Upon replacing
by - p o J , and recombining the components
a four-vector, one obtains for @ a t the point x: the expression
into
To find the field vcctors oiic must compute the components of the
tensor 2F a t the point 2;. By (79), page 72,
( j , k = 1, 2, 3, 4).
Let us suppose for the moment, that all sources are contained within V .
Then the integral over S contributes nothing in (20) to the value of
@ and in this case
where (J x R)?k = J,Rk - . J ~ R 3and
) where R is the radius vector drawn
from 2 to 2’. The cross product J x R is a six-vector or antisymmetric
tensor whose components are defined as in (62), page 69. In concise
form,
8.17. Field of a Moving Point Charge.-An isolated charge q movcs
with an arbitrary velocity v in frcc space. It will be assumed that the
distance from the charge t o thr observer is such that the charge can hc
represented by a geometrical point. Thc motion of the charge along its
trajectory is specified by cxprcwing its coordinatrs as functions of t ,
(24)
21 =
fi(t),
x z = f2(t).
2 3 = f3(t),
2 4
= f4(t)
=
id.
Onc must note that here 1 is riot the observer’s time but time as measurcd
on the charge. Then
and by (20)
(26)
The origin on the tirnc axis can be shifted a t will without loss of
generality. I t will he convcnicritl t o assume that the observer’s time
im
x,-Plane
chargo to the ohservcr. The polcs of the
irit'cgrand in (26) are locatcd a t
(28)
::
-w
x4
=
-ir,
2 4
=
+ir.
x., = ir
Now the data of the problcm specify the
jcocoordinat~csof the trajectory for real values
t which are less than t', and in the present
'*
x,=-ir
instancc less, thcrefore, than zero. Thus
the fj(t) arc given only for values of 5 4
on the negativc imaginary axis. Howcver,
-102
where vr is the componcnt of chargt; yelocity in the direction of thc radius
vector r drawn from chargc to oljscrrer.
Thc integral (26) now has thc forin
The contour is traced in the rlockwisc direction ; consequently, the
integral in (32) is by Cauchy's theorem equal to - 2 ~2.
(ifi
til
SEC.8.171
F I E L D OF A M O V I N G POINT CHARGE
475
or, in terms of vector and scalar potentials,
These arc the formulas of T i h a r d mid Wiechwt. Thc values of T and v
are those associated with y at an instant preceding the observation by the
interval r/c.
Rather than differentiate thescb expressions to obtain the field vectors,
i t is most direct to apply the s a m ’ procedure to ( 2 3 ) . This has been done
by Sommerfeld‘ who obtains formulas, the derivation of which by older
methods led to great complications. It is convenient to rcsolve the
expressions for the field into two parts: a velocaty field which contains no
terms involving the acceleration v, and an acceleration field which vanishes
as v goes to zero. For the velocity field one finds
(35)
where
and ro is a unit vector in the direction of r from charge to observer at the
r
instant t’ - -. These formulas can be obtained also by applying the
c
Lorentz transformation (111),page 79, t o the field of a static charge and
noting that (111) refers to the ol3server’s time.
The acceleration field is
1 SOMMERFELD,
in Riemann-\T.c~licr, “ Partiellen Diffcrcntialgleichungcn der
1935. See also, ABRAHAM,
“Theorie der
matheiriatischen Physik,” p. 786, 8th d.,
Elektrizitat,” Vol. 11, pp. 74J., 5th ed., 1923, and FRENKEL,
“Lehrbuch der Elektrodynamik,” Vol. I, Chap. T’I.
476
RADIATION
[CHAP.VIII
Note first that both velocity and acceleration fields satisfy the condition
-
H = 42.'. x E.
(38)
PO
The magnetic vector i s always perpendicular to the radius vector drawn from
the eflective position of the charge, by which one means the position a t the
r
instant t' - -. The electric vector of the velocity field is not transverse,
C
but in the acceleration field
(39)
E = &H
x ro,
~
.
=
~
r
~. = 0.
r
~
The velocity field decreases as I/?, while the acceleration field diminishes
only as l / r . At great distances the acceleration Jield predominates; it i s
purely transverse and i t alone gives rise to radiation.
I n case the velocity of the charge is very much less than that of light,
the equations (37) for the field at large distances become approximately
which remind one of (27), page 435, for the radiation field of a dipole.
The results of this section have been based on the assumption that
the motion and trajectory of thc charge zre known, just as earlier the
assumption was made that the currcnt distribution can be specified.
In neither case is i t exact t o treat the problem of the motion or distribution apart from t h a t of the radiation. Let us suppose, for example, that
a charge is projectled into a known magnetic field. If the radiation is
ignored, the mechanical force exerted on the charge is gv x B, from which
the motion can be calculated by the methods of classical mechanics.
I n case the velocities are large, a correction can also be made for the
relativistic change in mass. Now the effect of this force is to accelerate
the charge in a direction transverse t o its motion and, consequently, to
introduce an energy loss through radiation. The dissipation of energy
through radiation is not accounted for by the force qv x B. An additional force, the radiation reaction, which can be compared roughly to
friction, must be included. The radiation reaction in turn affects the
trajectory, whence it is obvious t h a t the exact solution can be found only
by introducing the total effective force from the outset-a very much
more difficult problem. Fortunately the radiation reaction is in most
cases exceedingly small, so that a satisfactory approximation for the
motion can be obtained by ignoring i t entirely.
PROBLEMS
477
Problems
1. Show that the r.m.s. field intensity a t large distances from a half-wave linear
oscillator is given by the formula
where W is the radiated power in watts, R the distance from the oscillator in meters,
and e the angle made by the radius vector R with the axis of the oscillator.
Show that the r.m.s. field intensity st large distances from an electric dipole is
voIts/meter,
provided the wave length is very large relativc to the length of the dipole.
2. Let $ be a solution of the equation
which together with its first and second partial derivatives is continuous within and
on a closed contour C in the zy-plane, 3.nd let
d(d
+
where r =
- xl2 (y’ - yI2, denotes the outward normal to C, and H;*)(kr)
is a Hankel function. Show that, if the fixed point x‘, y’ lies outside the contour C,
then u = 0, while u = $ if it lies within.
The theorem holds also when H i ’ ) ( k r ) is replaced by the function N ( k r ) . This
is a two-dimensional analogue of the Helmholtz formula (5), page 461, derived by
Weber, Math. Ann., 1, 1-36, 1869.
3. Discuss :he analogue of the Kirchhoff formula in two dimensions. A proof
based on the Weber formula of Problem 2 has been given by Volterra, Acta Aiath.,
18, 161, 1894. (Very interesting work on the propagation of waves i n a two-dimensional space has recently been done by M. Iliesz. See the discussion by Baker and
Copson, “ Huygens’ Principle,” Cambridge University Press, p. 54, 1939.)
4. Let z’, y’ be any fixed point within a plane two-dimensional domain bounded
by a closed curve C , and let \L be a solution of
which is continuous and has continuous first derivatives on C and within the enclosed
area S. Show that
+
(y‘ - y)2 and K O ( i k r ) is a modified Bessel function diswhere r2 = (I’- 2)’
cussed in Problem 10, Chap. VI. The normal n is drawn outward from the contour.
478
[CHAP.VIII
RADIATION
6. Lct t,he ciirrerit distribution throughout a volume V be spcclfied by thc function J(z)ePiwt,
where z stands for t,hc three coordinates z, y, z. Show that the electric
field intensity due to the distribution can be represented by the integral
eihr
&iwt
E(z’, t )
=
__
4rwe
Jv [ ( J .V)V
+ kzJl ;dv,
-
where
2’ stands
for the three coordinates of thc point of observation, k
r2
=
+
(z‘ - z ) ~ (y’ - y)z
+ (z’
= w d p e ,
and
-z)~.
6. When the theorem of Problcrn 5 is applied to a perfect conductor one obtains
eik r
[(K- V)V 4 k2K] - d~
r
where K is the surface current density a t a point z, y, z on the conductor S. Let S
now be the surface of a linear conductor. Rssurnc that the curvature of the wire is
continuous and that a t all points the cross scction is small in comparison with the
radius of curvature and with the wave length. Show that the field of such a linear
conductor is given by
where V’ is applied a t the point of observation and the integrals arc extended along
t h e contour C of the wire between points SI and s2.
7. Apply the formula of Problem 6 t o obtain the field of a linear oscillator of
length 1 and compare with the results of Sec. 8.11.
8. In case the linear circuit of I’rohlern 6 is closed, S I = sz and the integrated
term is zero. On the surface of the conductor the tangential component of E is
approximately
whence the current I must be of the form
I
=
A,,
e--zkms,
k,
=
2 ~ m
-9
1
where 1 is the length of the circuit and m is an integer.
a closed, oscillating loop is given by
Show
.
a t the field of such
The expression is exact in the limit of vanishing cross section and is approximately
correct if the cross section is srnall in comparison with the wavc length and the radius
of curvature at, each point of the circuit. Its application to Hertzian oscillators,
such as wwe used in the c:trly days of wireless telegraphy, was discussed in an Adanis
Prize essay by Mxcdonald in 1902 entitled ‘‘ Elcctric Waves.”
9. A scmi-infinite linear conductor carrying a current of frequency w / 2 ~coincides
with the negative z-axis of a coordinate system. Find expressions for the components
479
PROBLEMS
of electric and magnetic field intensity and calculate the total energy crossing a plane
transverse to the conductor.
10. Compute the radiation resistance of a linear half-wave oscillator by the
e.m.f. method described in Sec. 8.12.
11. An isolatcd, linear, half-wave antenna radiates 50 kw. at a wave length of
15 meters. Plot the pnrallel and perpendicular components of clcctric fioltl intensity
along the antenna on the assumption of a harmonic current distribution and a No. 4
wire.
12. The odd and cven functions
occur frequently in the theory of linear oscillators. Let
these function are solutions of
Put F =
(B), page 375,
q =
cos 0 and show that
G, and show that G satisfies the associntctf Mathieu equation
(I - q2)w”
- 2 ( a + I jqw‘
+ ( b - c2q2)w
=
0
for the particular case a = 1, b = a2 - 1, c = a.
Although F(’) and Fc0) are periodic in 8 for all values of a,notc t h a t they remain
finite a t the poles only for certain characteristic vnlurs. For thc csen function, these
27n
I
values belong to the discrete set a:) = --- T ;for thc odd [unction, they form the
2
set a$’ = mr, where m is an intrgrr.
Demonstrate the orthogonality of tlic functions exprcsscd by
+
m=n
m z n’
the subscripts referring to characteristic values of a,and show that the functions are
normal. Discuss the relation of these functions to thc radiation field of an arbitrary
current distrihution on a linear antenna.
13. An electromagnetic source is located a t a point Pi, and another operating
at the same frequency is located a t P 2 . Thc intrrvening mcdiurn is isotropic h u t
not necessarily homogeneous. All relations betwcen field vectors nrn linear, and the
time variation is harmonic. Let the field vectors due to the sourrc at PI be El and
HI,
t,hose due to the source a t I’2 be EParid Ha. Show that wherever these vectors
are continuous and finite, they satisfy thc symmetrical relation
V
*
(ELX Hz - Ez X Hi)
= 0.
This result is due to Lorentz and has been developed into a number of reciprocity
theorems of fundamental importance for radio communication.
480
[CHAP.VIII
RADIATION
14. On page 434 it was shown t h a t the Hertz vector of a n electric quadrupole is
+ x:,
R 2 = x2 +xi
, pq, be the spherical components
where po(&, &, f 3 ) is the density of charge. Let p ~ pe,
of the vector pCz). Show that the radiation field of the quadrupole, in the region
lcR
1, is given by
>>
,-
r
while the radial components vanish as 1/R2. Show that the mean radiation intensity is
in the preceding prob16. Show that the components ps and p g of the vector
lem are related to the components pij of the quadrupole tensor by
pe
=
p+ =
+ PIZsin 26)
+ cos 20 (Pl3 cos 4 + sin +I,
- pll) + p12 sin e cos 24 + pZ3 cos e cos 4 - palcos B sin e.
4 sin 28 ( p l l cos2 + + PZZsin2 4 - p33
+, sin 8 sin 24
P23
(p22
Rotate the coordinate system to coincide with the principal axes of the quadrupole
tensor. Then p 1 2 = pZ3 = p31 = 0 and the above reduces to
ps = sin 20
(&I
where
Qi
=
$(pi1
- Qz
+
pzz
pg
cos 24),
-2
~ 3 3 ) ~
=
2 sin e sin 2+ Q 2 ,
&z = t(pzz
-pd.
Show now t h a t t h e total quadrupole radiation is
-
If the medium about the quadrupole is air,
+
(9: 3Q3
pe
watts.
= l/c2, and
16. A charge - e is located a t each end of a line rotating with constant angular
velocity about a perpendicular axis through its center. A charge +2e is fixed at the
center. The dipole moment of the configuration is zero. Calculate the components
of the quadrupole moment and find the total radiation. (Van Vleck.)
17. Two fmed dipoles are located in a plane, their axes parallel but their moments
directed in opposite sense. The dipoles rotate with constant angular velocity about
PROBLEMS
48 1
a parallel axis located halfway between them. Calculate the components of the
quadrupole moment and find the total radiation. (Van Vleck.)
18. A positive and negative charge are bound together by quasielastic forces to
form a harmonic oscillator. Show that the radiation losses can be accounted for by a
frictional force which, however, is proportional to the rate of change of acceleration
rather than t o the rate of change of displacement.
19. A positive point charge oscillates with very small amplitude about a fixed negative charge of equal magnitude. Show that the formulas for the field of an oscillating
dipole can be obtained directly from Eqs. (35) and (37) on page 475 which apply to
an accelerated point charge.
CHAPTER IX
BOUNDARY -VALUE PROBLEMS
It can now be assumed that the reader is familiar with the principles
that govern the generation of electromagnetic waves and the manner in
which they are propagated in a limitless region of homogeneous, isotropic
space. The “boundlcss region” is, of course, simply an abstraction.
I n point of fact the most interesting electromagnetic phenomena are
those induced by surfaces of discontinuity or rapid change in the physical
properties of the medium. These boundary phenomena are roughly
of three types. Suppose that a wave, propagated in one medium, is
incident upon a surface of discontinuity marking t h e boundary of
another. I n the first case the linear dimensions of the surface measured
in wave lengths are very large. A fraction of the incident energy is
reJlected a t the surface and t h e remainder transmitted into the second
medium. The direction of propagation is in general modified and this
bending of the transmitted rays is referred to as rpfraction. The laws
that govern the reflection and refraction of electromagnetic waves at
surfaces of infinite extent are relatively simple. If, however, any or all
the dimensions of the surface of discontinuity are of the order of the
wave length, the difficulties of a mathematical discussion are vastly
increased. The perturbation of t h e primary field under these circumstances is referred to as dij’hction.
I n both cases t h e secondary field of
induced charges and polarization is excited by a primary wave of independent origin. Both are inhomogeneous boundary-value problems as
defined first for t h e static field on page 195.
The third case referred t o above is the homogeneous problem. A
conducting body is embcdded in a dielectric medium. Charge is displaced from the equilibrium distribution on the surface and then released.
The resulting oscillations of charge are accompanied by oscillations in
thc surrounding field. This field can in every case be represented as a
supcrposition of characteristic wave functions whose form is determined by
the configuration of the body and whose relative amplitudes are fixed by
the initial conditions. Associated with each characteristic function is a
characteristic number that determines thc frequency of that particular
oscillation. The oscillations are damped, partly due to the finite conductivity of the body and partly as a rcsult of energy dissipat,ed in radiation. However, the positions of conductor and dielectric relat,ive to the
surface of separation can be interchanged. Electromagnetic oscillations
482
SEC 9.11
483
B O U N D A R Y CONDITIONS
then take nlace within a dielectric cavity bounded by conducting walls.
If the conductivity is infinitr, tlierc is neither heat loss nor radiation and
the oscillations arc undamped a1 all frcquencies.
GENERAL THEOREMS
9.1. Boundary Conditions.- A convmtional proof of the conditions
satisfied a t a boundary by norrnal and tangential components of the
field vectors was presented in S ~ T .1.13. In Chaps. I11 arid I V the relation of these boundary conditiom to the discontinuous properties of
certain integrals was cstablishcd for the stationary rcgime; on the basis
of S c ~ s 8.13
.
and 5.14, the same procedure c:in be clxtendzd to the variable
field. We shall forego this aIial1 but s l d l give some further attention
to thc important casc in which the conductil ity of one of the two media
approaches infinity.
At the intrrfacc. of two m d i : t thc transition of the tangential component of E and the riorrnal component of D is expresscd by
(1)
n x (Ez - El)
= 0,
-
n (Dz - D,)
=
6,
where by previous convention n is: the unit normal directed from tho
medium (1) into ( 2 ) and 6 dr1iott.s the surfacv charge l o avoid confusion
with w = 2av. The flow of chcirge across or to the boundary must also
satisfy the equation of continuity in caw eithci or both the conductivities
are finite and not zero.
a6
JI) = - at
Suppose now that the time enters only as a common factor c x p ( - i ~ t )
and that apart from the boundary thc t w o media are homogeiieous and
isotropic. Then (1) and (2) togither givc
12et11s s r e first uncler what circimstaiiccs thc surface charge can be zero.
If 6 = 0, the determinant of (3) must vanish and h n c e
U1EZ - U > € I = 0.
(4)
If 6 is not zero (and this is tEw iisual case), it may be vlirninated from (3)
and wc obtain as: a n alternattue OoiLndary condztion o n the normal component
of E :
/.~cllkiE,,- /J&EI, = 0.
(5)
If either conductivity is infinite, (5) becomes indcterminatci; but from (3)
484
BOUNDARY-VALUE PROBLEMS
and in the limit as
u1-+
[CHAP.IX
m
(7)
From the field equations, moreover,
so that, if by hypothesis the field intensities are bounded, both Eland
H1 approach zero a t all interior points of (1) as u1 + 00. The transition
of the tangential component of E is continuous across the boundary and,
consequently, in the case of an infinitely conducting medium
n x Ez = n x El = 0.
(9)
Consider now the behavior of the magnetic field a t the boundary.
I n general
-
n (B, - B1) = 0,
n x (H, - H1) = I(,
(10)
where the surface current density K is zero unless the conductivity of
medium a t the boundary is infinite. It has just been shown that H1
vanishes if u1 becomes infinite and in this case
(11)
n - B 2 = 0,
n x H, = K.
A further useful boundary condition on the magnetic field in the case
of perfect conductivity can be derived as follows.' Let the boundary
surface S defined by the equation,
[ = fl(z, y, z ) = constant,
(12)
coincide with a coordinate surface in an orthogonal system of curvilinear
,
tangential component
coordinates E, 7, { as in Sec. 1.16. If ul-+ ~ i the
of E2 and the tangential component of V X H,, which is proportional to
it, approach zero. Therefore by (SO), page 49,
The normal coordinate is {. Since u1 is infinite, I-11 is zero; hence, since
the coefIicients of the unit vectors must vanish independently, we have
just outside the conductor
d
(htH6) = 0.
The quantities h,H, and h:HE are covariant components (page 48) of
the vector Hztangent to the surface. The boundary condition is, therefore: the normal derivatives of the covariant components of magnetic jield
The author is indebted to E. H. Smith for the proof.
9.11
SEC.
485
B O U N D A R Y CONDITIONS
tangent to the boundary vanish as the conductivity o n one side becomes
infinite.
Under the head of “boundary conditions” we ~ h s l lconsider also
the behavior of the jield at injinity. I n this respect the variable field
differs notably from the static. If in the stationary regime all sources
are located within a finite distance of the origin, the field intensities
vanish a t infinity such that lirn R2E and lirn R2H are bounded as R -+ m ,
I..rhere R is the radial distance from the origin. The scalar potential
satisfies V2# = 0 and lim R# is bounded as R + a. Every function C$
which is regular a t infinity and which satisfies Laplace’s equation a t all
points of space (no sources) is necessarily zero. On the other hand, if
variable sources are located within a finite distance of the origin, the field
intensities vanish such that lim RE and lim RH are bounded as R -?. a.
The scalar potential and rectangular components of the vector potential
and the field intensities all satisfy llhe equation,
a t points where the source density is zero. Moreover there exist functions $ satisfying (15) throughout all space and vanishing at infinity
which are not everywhere zero.
A solution of Laplace’s equation is uniquely determined by the sources
of the potential and the condition that it shall be regular a t infinity.
These two conditions are not sufficient to determine uniquely the wave
function J,, for (15) admits the possibility of convergent as well as divergent
waves. This question has been investigated by Sommerfeld’ in connection with the Green’s function of (15) for spaces of infinite extent.
To conditions that are analogous t o those of the static problem must be
added a “radiation condition.” The problem is formulated as follows:
The density g(x, y, z ) of the sourcc distribution is specified, and these
sources are assumed to lie entirely within a domain of finite extent.
Then is uniquely determined if:
( a ) a t all points exterior to a closed surface S (which can, if necessary,
be resolved into a number of separate closed surfaces), $ satisfies
+
vz+ + k2+
(16)
(b)
CY*
=
-9;
+ satisfies homogeneous boundary conditions over S of the type
+ /3 an
- = 0, where
01
and 0 are constants or specified functions of
position;
+
( c ) vanishes in such a way t,hat lim R+ is bounded as R
dition we shall again refer to as regularity a t infinity;
SOMMERFELD,
Jahresber. deut. math, Ver., 21, 326, 1912.
3 00,
a con-
486
BOUNDARY-VALUE PROBLEMS
(d)
[CHAP.IX
9 satisfies the radiation condition
(17)
R--*
-
which ensures that a t great distances from the source the field represents
a divergent traveling wave. It has been tacitly assumed that the time
enters explicitly in the factor exp( -id).
The significance of (17) may be made clear by applying Green's
theorem to (16) within a region bounded internally by X and externally
by a surface So. Then, as in Sacs. 8.1 and 8.13,
The first two terms to the right of (18) reprercnt traveling waves diverging
from the source. The third term, ho~z'cwr,expresses the sum of all waves
traveling inwards from the elements of So and mnst, therefore, vanish as
Xu recedes to infinity. If this third term be designated by U , we have
where d B is an clement of solid angle. The second integral in (19) is
extended over the finite domain 47r and vanishes as R + rn provided
is regular a t infinity as prescribed in (c) above. I n order that U shall
vanish, i t is sufficient that
+
If a t grrak distanccs So is replac~cdby a sphere of radius R, (20) is identical
with (17).l
9.2. Uniqueness of Solution.-Lct V be a region of space bounded
intcrnally by the surface S and cxternally by SO. The surface X can bc
resolved, if the case demands, into a number of distinct closed surfaces
Si as in Fig. 18, page 108. Thcn V is multiply connected (page 226)
clnd t'he surfaces Si represent the boundaries of various foreign objects in
the field. It will be assumed for t'he moment that the propert,ics of V
are isotropic but the parameters p , E, and rn can be arbitrary functions
of position. Now let El, HI, and E,, H2 be two solutions of t,he field
equations which a t the instant t = 0 are ident'ical a t all points of V .
We wish to find t'he minimum numhcr of conditions t o be imposed on the
* An alternative proof
was given in Sec. 8.13, p. 461.
SEC.
U N I Q U E N ESS OF SOL U TION
9.21
487
components of t’he field vectors a t .the boundaries S and Soin order that
the two solutions shall remain idontical a t all times t > 0.
In virtue of the linearity of the field eqiiations (we exclude ferromagnetic materials) the differmca jicld E = EP - El and H = H2 - HI
is also a solution. One may atstsimc wit,hout, loss of generality that the
sources of the fields lie cntircly outside the region V , since it was shown
in the preceding chapter that thti field is uniquely determined whcn the
charges and currents are prescribed. Or, if sources appear within V ,
one must specify that the dist,rihut,ion and rate of working of the electromotive forces are in both cases the same. Then within V , by Poynting’s
theorem, page 132, the difference field satisfies
In ordrr that the right-hand mcmbclr of (21) shall vanish, it is only
necessary that either the tangential components of El and E2,or the
tangcntial components of HI and H2 11c identical for all values of t > 0;
for thcn either n x E = 0 or n x H = 0 and E x H has no normal
component over the boundaries. In that case we have
2
dt
.I
(5 E2 +
v 2
5
The right-hand member of (22) is always equal to or less than zero.
The energy integral on the left is essentially positive or zero and vanishes
a t t = 0. Hence (22) can only be satisfied by E = E2 - El = 0,
H = H2 - HI = 0 for all values of t > 0, as was to be shown. A n
electromagnetic field i s uniquely determined within a bounded region V
at all times t > 0 by the initial valuc:; of electric and magnetic vectors throughout V , and the values of the tangcntial componmt of the electric vector (or
of the magnetic vector) over the boundaries for t 2 0.
If So recedes to infinity, V is cxtcmally unboundcd. To ensure t h r
vanishing of the integral of a Poynting vector over an infinitely remote
surface, it is only necessary to assume that the medium has a conductivity,
however slight. If the field was initially established in the finite past,
the difficulty may also be circumvented by the assumption that Solies
beyond the zone reached a t time t b y a field propagated with a finite
velocity c. The theorem just proved does not take full account of this
finiteness of propagation. We have established that the valurs of
E and H are uniquely determincd throughout V a t time t by a tangcntial
boundary condition and the initial values everywhere in V . Physically
i t is obvious, however, that this is more information than should be
necessary. The field is propagated with a finite vclocity and, consequently, only those elenients of V whose disiance from the point of
488
BOUNDARY- VALUE PROBLEMS
[CHAP.IX
observation is r ( t - to)c need be taken into account. The classical uniqueness proof given above has been extended in this sense by
Rubinowicz.'
The theorem applies also to anisotropic bodies. Electric and magnetic energies are then positive definite forms (cf. page 141), which is to
say that they are positive or zero for all values of the variables. The
proof remains unmodified.
There is one aspect of the theorem which at first thought may be
puzzling. We have seen that the field is determined by either the values
of n x E or n x H on the boundary, yet in the problems to be discussed
shortly we find it necessary to apply both boundary conditions,
n x (E, - El) = 0,
n x (H, - HI)
= 0,
(23)
where here E, and El denote values of field intensity on either side of the
boundary. The reason for the apparent discrepancy is, of course, that
n x E and n x H refer to the tangential components of the resultant
field on one side of the surface. These are in general unknown and it is
the object of a boundary-value problem to find them. Equations (23)
simply specify the transition of the field across a surface of discontinuity
and the two together enable us to continue analytically a given primary
field from one region into another. Having thus determined the total
field, the uniqueness theorem shows that there is no other possible
solution. If, however, one side of a boundary is infinitely conducting,
we do know the tangential component of the total field, for in this case
n x E = 0, and a single-boundary condition is sufficient for the solution
of the problem.
9.3. Electrodynamic Similitude.-Since Newton's time the principle
of similitude and the theory of models have had a most important influence on the development of applied mechanics. This is particularly true
of ship and airplane design which is governed very largely by the data
obtained from small models in towing tanks and wind tunnels. It is
customary to express the conditions of an experiment in terms of certain
dimensionless quantities such as the Reynolds number. Thus the
results of a single measurement of a given model can be applied to a series
of objects of identical form and differing only in scale, provided the
viscosity of the fluid and the velocity of flow are varied in such a way
as to keep the Reynolds number constant.
Similar principles prove very helpful in the design of electromagnetic
apparatus. We shall write first the field equations in a dimensionless
form. In a homogeneous, isotropic conductor
l
R
~
~Physik.
~ Z.,
~ 27,
~ 707,
v 1926.
~ ~
~
,
SEC.9.31
ELECTRODYNAMIC SIMILITUDE
489
Now let
E
(25)
E
=
H
= eE,
I.c =
EoKe,
length
=
=
hH,
U = UoS,
/JoKm1
time
ZOL,
=
toT,
where E, H, K,, K,, s, L, and T are the dimensionless measure numbers of
the field variables in a system for which the unit quantities arc e, h,
The measure numbers satisfy the equations
€0, PO, lo, uo, and to.
V X E+
L
Y
l3E
V X H - PK, -aT
f3H
K
-
~= ~
0,
~ s=
E 0,
where
are dimensionless constants. Upon eliminating the common ratio e / h ,
one obtains
From the first of these it follows a t once that the product poco must have
the dimensions of an inverse velocity squared. This is, no doubt, the
most fundamental approach to the problem of units and dimcnsions.'
I n order that two electromagnetic boundary-value problems be
similar, it is necessary and sufficient that the coefficients LYK,,,, OK,, and ys
be identical in both. For t o let, us take for example the period 'T of the
field, and for lo any length that characterizes one of a family of bodies
which differ only in scale. Thus lo may be the radius of a set of concentric spheres, or the major axis of a set of ellipsoids. The condition of
similitude requires th at the two characteristic parameters C1 and 6 2 in
be invariant to a change of scale. Suppose that the characteristic length
lo is halved. Then both C1 and Czremain unchanged if the permeability
This is an awkward remedy
I.( a t every point of the field is quadrupled.
from a practical standpoint, but it is the only way in which the initial
state can be simulated through the adjustment of a single parameter. If
p and e are left as they were, constancy of C1 results also from halving
the period, or doubling the frequency, but this alone does not take care
of C,. I n order th at the half-scale model shall exactly reproduce the
1
C j . Secs. 1.8.4.8,and 4.9.
490
BOUNDARY-VALUE PROBLEMS
[ C H A PI. X
iull-scale conditions, i t is necessary also that the conductivity be doubled
a t every point.
This principle can be illustratcd by rcfercnce to the type of highfrequency radio generator that maintains standing electromagnetic waves
in a cavity resonator bounded by metal walls. The frcquency of oscillation is determined essentially by the dimensions of the cavity, while
the losses depend largely on the conductivity of the walls. If the dirnensions are halved, the frcqncncy will approximatcly be doubled; if the
conductivity of thc walls is unchanged, it is entirely possible that the
resulting increase of the rplative loss will pass the critical value, so that
the half-scale apparatus fails to oscillate.
REFLECTION AND REFRACTION AT A PLANE SURFACE
9.4. Snell's Laws.-Two
homogeneous, isotropic media have as a
common boundary the plane 8, and are ot,herwisc infinite in cxtent. The
unit vector n is normal t o the plane X and directed from the region
FIG.Sg.-Reflertion
and refraction a t a plane surface S.
(€1, 11, a,) into thc region (€2, 12, u2).
Let 0 be a fixed origin, which for
convenience we locate on S. Then, if r is the position vector drawn from
0 to any point in either (1) or (2), the interfacc Xis defined by thc equation
(1)
n . r = 0.
A plane wave, traveling in medium (2) is incident upon 8. By (27),
page 272,
where EOis the complex amplitude of the incident wave and no a unit
vector which fixes its direction of propagation,' as in Fig. 89. The plane
defined by the pair of vectors n and n, is called the plane of incidence.
1
It is apparciit that so far :is tlic present problem is concerned it would be neater
49 1
SNELL’S LAWS
SEC.9.41
The continuation of the primary field into medium (1) is determined
b y the boundary conditions a t S. To satisfy these boundary conditions,
a reflected or secondary field must bc postulated in ( 2 ) . Physically i t is
clear that the primary field induces an oscillatory motion of free and
bound charge in the neighborhood of S , which in turn radiates a secondary
field back into ( 2 ) as well as forward into (1). We shall make the
tcritative assumption that both transniitted arid reflected waves are
plane, and write
where i.he unit vectors nl and n, are in the directions of propagation of
transmitted and reflect’ed waves respectively, and El arid E, are complex
amplitudes, all as yet undetermined. By hypot’hesis, El, E,, and Eo
are independent of the coordinates and, conscquent,ly, if the tangential
components of the resultant field vect,ors are to he continuous across S ,
it is necessary that the arguments of t,hc exponential factors in (2) and
(3) be identical over the surface n r = 0. But
.
r = ( n - r ) n - n x (n
(4)
hence, a t any point on the interface r = -n
x r);
x (n x r). Therefore,
kznu* n x (n x r) = k2n2 n x (n x r),
k2no n x (n x r) = klnl. n x (n x r),
-
(5)
-
x n) (n X r),
(n, x n - nz x n) . n x r
(kznox n - klnl x n) n x r
or, since no n X (n X r) = (n,
(6)
-
=
=
0,
0.
From these two relations it follows that n, no, n,, and n2 are all coplanar.
The planes of constant pkasc of both transmitted and rejlectcd waves are
normal to th,e planc of in.cirlcncc. From t,hc first of (6) also
(7)
sin
02
=
sin
(T -
6,;) = sin Oo,
whence the a,ngle of inciclemc B0 is equal to the angle of reJEection
From thc second of (6)
(8)
k 2 sin 60 =
Equations (7) and (8) expr
k 1
e2.
sin 81.
h d 1 ’ s ~niusof reflection and refraction.
__
from a notational point of view to let t,he incident wave travel from (1) into (2).
Shortly, however, the plane S will bc rcplnced by :I closed surf:ioe S bounding a cornplete body. By previous convention n is tlircct.c:d outwnrd frnin a closcd surface and
from medium (1) into ( 2 ) . The choice of n o :is above will facilitate the comparison
of formulas from the present with thosc o f latcr sections.
492
[CHAP.IX
BOUNDARY- 'VALU E PROBLEMS
9.6 FresneI's Equations.-The boundary conditions will now be
applied to determine the relation between the amplitudes Eo, El and E2.
At all points on 8
+
+
n x (Ho HJ = n x HI.
n x (Eo E2) = n X El,
(9)
By virtue of (2) and (3) the second of these two can be expressed in terms
of the electric vectors.
(10)
n x (no x Eo
+ nz X E2) -kz
=
k
n x (nl x El)-2.
Ccl
P2
Expansion of (10) leads to such terms as
-
.
n x (no X Eo) = (n Eo)no- (n no)Ec.
(11)
The orientation of the primary vector Eois quite arbitrary but can always
be resolved into a component normal to the plane of incidence and consequently tangent to S, and a second component lying in the plane of
incidence. (Cf. Sec. 5.4, page 279.) The analysis is greatly simplified
by a separate treatment of these two components of the incident wave.
Case I. Eo Normal to the Plane of Incidence.-Then
-
-
n Eo = no Eo = 0.
Since the media are isotropic, the induced electric vectors of the transmitted and reflected waves must be parallel to Eo and hence also normal
to the plane of incidence, so that n El = n Ez = 0. From Fig. 89
n no = cos (T - Oo) = - cos eo,
(12)
n nl = cos (T - 0,) = - cos el,
n n2 = cos O2.
Upon multiplying the first of Eqs. (9) vectorially by n and making use of
(11) and (12), we find that the amplitudes must satisfy
-
-
.
The relative directions of electric and magnetic vectors for this case are
shown in Fig. 90. Solving (13) for El and E2 in terms of the primary
amplitude Eo leads to
El
=
Ez
=
(14)
(COS
e2 + cos e,)
P l ~ zcos
+
P1kz cos
e2 +
e2
cos el EO,
(when n Eo = 0 ) ,
Plkzcos eo - P2kl cos el
-
P 2 ~ COS
l
el Eo.
These relations are not quite so simple as they appear a t first sight,
for 81 is complex if either (1) or (2) is conducting and may be complex
even if both media are dielectric. By (7) and (8)
SEC.
(15)
4 93
FRESNEL'S EQUATIONS
9.51
kl cos Ol = .\/k:
cos B2 = cos Bo,
- k; sin2 Bo.
The angles of reflection and refraction can be eliminated from (14) and
we obtain as a n alternative form the relations
El
=
2plk2 cos eo
-
p l k z cos
00
+
p2
- Eo,
d k i - k i sin' Bo
-
(when n Eo = 0),
(16)
E2 =
p l k 2 cos Bo - pz
plkz cos Bo
m-k i sin2 Bo Eo.
_-_
-
+ p 2 d k 2 , - k ; sin2 Bo
Complex values of the coefficients of Eo imply that the amplitudes El
and E2 themselves are complcx and that the transmitted and reflected
waves differ in phase from tho incident wave.
/S
FIG.90.-Polarization
normal to the plane of incidence.
Case 11. Eo in the Plane of Incidence.-The magnetic vectors arc
then normal to the plane of incidence and parallel to S.
-
.
.
n Ho == n HI = n H,.
From (2) and (3) we have
which when substituted into (9) give
cos eo H~ - c~~ e2 H,
=
+ Hz
=
Ho
PlkZ
~
P'kl
HI,
cos
el H ~ ,
494
[CHAP.IX
BOUNDARY-VALUE PIZORLEMS
as the conditions at the boundary. The relative directions of electric
and magnetic vectors for this case are shown in Fig. '31. Solution of
(18) leads to
-
(when n H,
or, upon eliminat,ion of 81 and
(20)
82
= 0),
by (15),
-
(whcn n Ho = 0),
fiClpkT
cos 0,, - p l k z d k : - h$ sill2 B0
H 2 = - ____-- ~- -~ Ho.
p & cos Oo f p l k 2 2/:--kg sin2 B0
-
When thc iricidcnce is normal, $0 = 0, the two caws cannot be distinguished and thc amplitudes of t,ransmitted and rcflected waves reduce
to
The relations expresscd by Eqs. (14) and (19) werc first derived in a
slightly less general form by Fresricl in 1823 from the dynamical properties of a hypothetical elastic ether.
/S
FIG.Sl.-Polariaation
parallel t o the planc of incidence.
9.6. Dielectric Media.-We sh:tll study first the case in which the
two conductivities u1 arid u2 arc zcro, so that both media are perfectly
transparent. The permeabilities will differ by a negligible amount from
po and Snell's law can now be written
Snc. 9.61
495
DIELECTRIC MEDIA
whcre v 1 and 212 are the phase velocities and n12is the relative index of
refraction of the two media. If t l > € 2 it follows that n12 < 1 and there
corresponds to every angle of incidence 8 0 a real angle of refraction
81. If, however, €1 < cz, as is the case when a wave emerges from
a liquid or solid dielectric into air, then 81 is real only in the range for
which nl2 sin eo I 1. The phenomena of total reflection occur when
n12sin 0 0 > 1. We shall cxcludc this possibility for the moment and
considcr the Fresnel laws for real angles between the limits 0 and a/2.
When E is normal to the plane of incidence we now obtain from (14)
El
2 cos 0” sin
=
7-
Slll
(23)
(0,
el
+ Bo)
EO,
(n.E
=
0),
sin (01 - 8,)
E 2 = sin
. (0,
e,) EO,
+
and for the components of E lying in the plane of iiicidence from (19)
nl x E l
= __
sin (0,
2 cos 0, sin 0,
el) cos (0, -
+
el) no x
Eo,
-
(n Ho = 0),
(24)
Since the coefficients of EOin (23) and (24) are real, thc reflected and
transmitted waves are either in phahe with the incident wave, or out of
phase by 180 dcg. It is apparent that the phase of the transmit,ted wave
is in both casos idcntical with that of the incident wave. The phase of thc
reflected wave, however, will depend on the relative magnitudes of
8 0 and 61.
Thus if €1 > c2, thcn O1 < eo, so that ELis opposed in direction
to E” in (23) and therefore differs from i t in phase by 180 dcg. Under
the same circumstances tan ( 0 0 - 0,) is positive, but the dcriomiriator
tan (8,
01) becomes negative if Oo
el > a/2 and the phase shifts
accordingly.
The mean energy flow is g i w n by the real part of the complex Poynting
vector. I n optics this quantity is usually rclfcrrcd to as the “intensity”
of light, but the term is ambiguous since i t is also applied to the amplitudcs
of the fields. I n the present case
+
+
496
[CHAP.IX
B O U N D A R Y - V A L U E PROBLEMS
Now the primary encrgy which is incident per second on a unit area of
the dielectric interface is not
but the normal component of this flow
si
6
vector, or n Si = -~ E,2 cos 80. Likewise the energies leaving a
2
unit area of the boundary by reflection and transmission are
(26)
.-
n S,
=
4%
--
2
E2, cos
n St =
80,
-*
2
E,2 cos
el.
According to the energy principle the normal component of energy flow
across the interface must be continuous,
n
(27)
- + 3,)
(Si
=
n S,,
or
(28)
6E,2 cos
80
=
6Et cos 81 + G
E ; cos
80.
It can be easily verified that the Fresnel formulas (23) and (24) satisfy
this condition. The reJlection and transmission coeficients are defined by
the ratios
R+T=l.
I n case
E, is normal t o the planc of incidence, these coefficients are
sin2 ( 8 1 - 80)
sin2 (81 ooj'
RI= .
+
sin
sin 281
sin2 (81
8,) '
TI= .
+
and in the case of En lying in the plane of incidence
If the incidence is normal, 80
(22) that
= 81 = 0,
and i t follows from (21) and
R =
(32)
T =
-
-
There is only one condition under which a reflection coefficient is
zero. As 8 0
81
a/2, the t a n ( 8 0
81)
co and in this case 211-+ 0.
The reflected and transmitted rays are then normal to one another
+
(nl* nz = 0) and sin 81 = sin
+
6
- 00) = cos 0"; it follows from (22)
SEC.9.71
TOTAL REFLECTION
that
tan
(33)
eo =
497
.$
= n21.
The angle that satisfies (33) is known as the polarizing, or Brewster angle.
A wave incident upon a plane surface can be resolved, as we have seen,
into two components, one polarized in a direction normal and the other
parallel to the plane of incidence. The reflection coefficients of the two
types differ and, consequently, the polarization of the reflected wave
depends upon the angle of incidence. I n particular, if incidence occurs
a t the polarizing angle, the reflected wave is polarized entirely in the
direction normal to the plane of incidence. Use is sometimes made of
this principle in optics to polarize natural light, although in practice it is
less efficient than methods based on double-refracting prisms.
The optical indices relative to air are usually of the order of n21= 1.5;
a t radio frequencies they may be very much larger with a corresponding
increases
increase in the polarizing angle. Thus in the case of water, nZ1
from about 1.33 to 9 a t radio frequencies and the polarizing angle from
53 to 84.6 deg., which is not far removed from grazing incidence. Irregularities of radio transmission over water can doubtless be attributed 011
certain occasions to this cause. One will note also that at the polarizing
angle Ti1 is unity and th at this is the only condition under which all the
energy of the primary wave can enter the second medium without loss by
reflection a t the surface.
9.7. Total Reflection.--We return now t o the case excluded from
Sec. 9.6 of transmission from medium (2) into a medium (1) whose index
of refraction is less than that of (2). The formulas of See. 9.6 are valid
whatever the relative values of el and e2, but if eo is such that
n12sin
eo =
.$sin eo > 1,
they can be satisfied only by complex values of el. Physically a complex
angle of refraction implies a shift of phase and the appearance of a n
attenuation factor.
Let us suppose, then, that sin 81 > 1. The cosine is a pure imaginary.
(34)
cos
_____
i
01
= __
6
deZsin2 eo - €1 = inlzl/sin2 e0 - nil.
1
The radical has two roots whose choice will be governed always b y the
condition that the field shall never become infinite. T o simplify matters
a bit, the reflecting surface S will be made to coincide with the plane
2 = 0, as in Fig. 92. All points of medium (1) correspond to negative
values of 2. The phase of the transmitted wave is, therefore,
[CHAP.IX
BO U N D A K Y -V AL Uh’ PROBLEMS
498
-
klnl r
(35)
= w
&I*,
=
6(-iz
W
(--z
where it is assumed that p l
transmitted wave is
Et
(36)
cos fI1 ~4z sin 6,)
______
6
1
2
= pz = po,
=
e0 - ng,
+ z sin
BO),
and the field intensity of the
El e / 3 i z + < a z - i w t
(x < O),
where
The field defined by (36) vanishes exponentially as x 3
confirms the choice of the positive root in (34).
which
I
Medium (1)
FIQ.92.-Total
-m,
reflection from a surface coinciding with the yz-plane.
The amplitudes of reflected and transmitted fields are next determined from the Fresnel laws after elimination of el. From (14) we obtain
2 cos
=
cos
60
__-
eo + i \/sin2 eo - nzl
EOL,
and from (19) and (3), for polarization in t,he plane of incidence,
Since the coefficients of E, are complsx, i t is apparent that the transmitted and reflected waves are no longcr in phase a t the surface with
the incident wave. Thc reflection coefficient according to (29) is
R = Ez @E:, and i t follows a t once from (38) and (39) that
-
(40)
RL
=
Rli
=
1,
TI
=
T I I= 0.
The intensity of energy flow in the reflected wave is exactly equal t o the
SEC. 9.71
TOTAL REFLECTION
499
intensity in the incident wave; there i s no average flow into the m e d i u m o j
lesser refractive index. The field: intensity in medium (I), however, is by
no means zero. There is, in f:Lct, an instantaneous normal component of
energy flow across the surface whose time average is zero; the time average of the flow wit,hin (1) parallel to the surface (i.~.,in the direction of
z) does not vanish. This lattcr component is unattenuated in the direction of propagation but falls off very rapidly as the distance from X
increases due to the factor exp pix. The surfaces of constant phase in
the transmitted wave are the planes z = constant, which are normal to
surfaces of constant amplitude, x = constant. It is clear that Etll and
Htll have components along the z-axis, which is the direction of propagation within (1). The excitation within mcdium (1) in the case of total
rejlection i s a nontransverse wave (it m a y be either transverse electric or
transverse magnetic, page 350) confined to the immediate neighborhood of the
surface.
This analysis gives no clue as to how the energy initially entered (l),
for i t is based on assumptions of a steady state and of surfaces and wave
fronts of infinite extent. Actua'lly, thc incident wave is boundcd in both
time and space. The total reflection of a beam of light of finite cross
section has been treated by Picht' who showed that the average flow
normal to the surface is in this case not strictly zero. Any change
causing a fluctuation in the energy flow of the transmitted wave destroys
the totality of reflect.ion.
Figures showing the course of the magnetic lines of force and t h e
lines of energy flow in total reflection have been published by Eichenwald
and by Schaefer and Gross2
Finally, let us examine the relative phases of the reflected waves. I n
(38) and (39) write
E,II = e--iaI! E 011,
Ezl = e-"'l Eel,
(41)
[c--iSd.J = le-i6ll/ = 1.
Then since
we have
Suppose that the incident wave is linearly polarized in a direction that is
neither parallel nor normal to the plane of incidence. W7e then resolve
i t into components and discover that the resultant reflected wave is
PICHT,
Ann. Physik, 3, 433, 1929. See also NOETHER,
ibid., 11, 141, 1931.
SCHAEFER
a n d GROSS,
Untersuchungen uber die Totalreflexion, Ann. Physik,
32, 648,1910.
2
BOU N D A R Y- V A L U E PROBLEMS
500
[CHAP.IX
formed by the superposition of two harmonic oscillations a t right angles
to one another and differing in phase by an amount
6 = 611 - 61.
(44)
T h e rejlected wave is elliptically polarized, since by (43) the relative phase
difference 6 i s not, in general, zero.
6
tan 2
6 II - tan 61
tan 2
2
_ _ ~
6
61
1 tan 1tan 2
2
= _
-
+
cos eol/sin2
eo - nZl.
sin2 O0
The phase shifts 6 1 and 611 both vanish a t the polarizing angle
Oo = sin-' n21, and their difference 6 is also zero a t grazing incidence,
Bo = ~ / 2 .The relative phase 6 attains a maximum between these limits
a t an angle found by differentiating (45) with respect to Oo and equating
to zero. The maximum occurs when
which, when substituted into (45), gives
(47)
This property of the totally reflected wave was used by Fresnel to
produce circularly polarized light. It is first necessary that the incident
wave be linearly polarized in a direction making an angle of 45 deg. with
the normal to the plane of incidence. The amplitudes E z l and Ezll
are then equal in magnitude. The relative index n21 and the angle of
incidence Oo are next adjusted so that 6 = n/2, or tan 6/2 = 1. According to (47) this condition can be satisfied only if 1 - ngl 2 2n21, or
nzl < 0.414, or n12 > 2.41. I n the visible spectrum such a minimum
value of the index of refraction is larger than occurs in any common
transparent substance. To overcome this difficulty, Fresnel caused the
ray of light to be totally reflected twice between the inner surfaces of a
glass parallelepiped of proper angle. I n the radio spectrum, on the other
hand, the index of refraction may assume very much larger values. Thus,
in the case of a surface formed by water and air, n12= 9, nzl = 0.11.
The condition tan 6/2 = 1 is then satisfied by either of the angles
00 = 6.5 deg. or 00 = 44.6 deg. The latter figure has been confirmed
by measurements at a wave length of 250 cm.1
9.8. Refraction in a Conducting Medium.-The phenomena of reflection and refraction are modified to a striking degree by the presence of a
BERGMANN,
Die Erzeugung zirkular polarisierter elektrischer Wellen durch
einmalige Totalreflexion, Physik. Z., 33, 582, 1932.
501
R E F R A C T I O N I N A CONDUCTING M E D I U M
SEC. 9.81
conductivity in either medium. The laws of Snell and Fresnel are still
valid in a purely formal way but, as in the case of total reflection, the complex range of the angle 81 leads to a very different physical interpretation.
Let us suppose that medium ( 2 ) is still a perfect dielectric but that
the refracting medium (1) is now conducting. The propagation constants are defined b y
where a 1 and 8, are cxpreswd in terms of
page 276. By Snell’s law we have
61,
p ~ and
,
UI
by (48) and (49),
(49)
which it is convenient to writc as
sin
(50)
01
=
( a - i b ) sin 80.
The complex cosine is then
___---__
cos 81 = 4 1 - (u2 - b2 - 2abi) sin2 B0
(51)
= peir.
The magnitude p and phase y are found by squaring (51) and equating
real and imaginary parts on either side.
p2
(52)
p2
cos 2y = p2(2 cos2 y - 1) = 1 - (a2 - b2) sin2 Bo,
sin 2y = 2p2 sin y cos y = 2ab sin2 BO.
The phase of the refracted wave is, as in (35) and Fig. 92,
+ i@l)( 8, + z sin 8,)
=
cos y - p1 sin y) - i z p ( p 1 cos y + sin y)
+ ~ ( c t a+l sin B0 + iz(ap1 bal) sin BO.
From (49) and (50) it is readily seen that (aal + bP1) sin O0 = a2 sin o,,
(53)
klnl.r
= (a1
--J
CQS
-zp(aI
a1
bP1)
-
and (up1 - bad = 0. Within the conducting medium the transmitted
wave is represented, therefore, by
(54)
Et = El e p r + i ( - g z + n z z s i n B o - ~ o t )
where
(55)
p
q
= p(P1
= p(a1
7
(x < 01,
+
cos y
a1 sin y)
cos y - P 1 sin y).
Note that the surfaces of constant ampZitude are the planes p z = constant,
the surfaces of constant phase are the planes --gz
a2 s i n t10 z = constant,
and these two families do not, in general, coincide. Within (1) the field
is represented by a system of inhomogeneous plane waves, as in the case of
+
502
[CHAP.IX
BOUNDARY-VALUE PROBLEMS
total reflection. The planes of const)ant amplitude are parallel to the
reflecting surface S; the direction of propagation is determined by the
normal to thr plancs of const,ant phase. Thc angle II, made by this wave
normal with the normal to the boundary plane (in the present instance
the negative x-axis) is the true angle of refraction and is defined by
(56)
or
(57)
cos +b =
-x cos J/
+ x sin # = constant,
Q
sin +b
-~
fl+CY;Sin2e,’
=
CQ
sin Bo
ayit sin2
+
eo
The relation of the planes of constant amplitude to the planes of constant
phase is illustrated in Fig. 93.
I X
Medium
////,,?
/
Medium (1)
FIG.93.-Refraction
a t a plane, conducting surface.
A modified Sncll’s law for real angles is expressed by (57).
The quantity n(&)is a real index of refraction which now depends on the
angle of incidencc, a notable deviation from the law of refraction in
nonabsorbing media. The phase vclocity, defined as the velocity of
propagation of the planes of constant phase, is
(59)
Not only does this velocity depend on the angle of incidence, but as in the
case of total reflection there are also components of field intensity in the
direction of propagation. The field within the conductor is not strictly
transverse.
SEC.
9.81
REFRACTION I N A CONDUCTING
nmnirAu
503
The calculation of p , q, and n in terms of the constants of the media
and the angle of incidence is a tedious but elementary task. From ( 5 2 )
and (55), together with the definition of a and b implicit in (49), one
obtains the following rclations :
p2(eo) = +[-a;
(60)
+ + a; sin2 Oo
+ 4 4 0 3 3 + (a! - ~9
- a; sin2
00)2],
P:
- a; sin2
@I,
+ a; sin’
+ f i a $ j _+_ _ _-~p:
- a; sin2
0,)2].
qz(Oo) = +[a;- /3; - a; sin2 Bo
+ 64zE-+-cLy’;--
a;nZ(Bo)= +[a; -
p:
00
(01;
From these a subsidiary set of relations, known as Kettclcr’s equations,
can easily be derived.
The junctional relation of n(B0) to thc angle of incidence and the
constants a1 and 01 expressed by (60) has hecn confirmed by mcasurements in the visible region of the light spectrum.’ No direct connection
exists, however, bctwecn the observed values of a1 and 81 at optical frequencies and the static or quasi-static values of the parameters €1, p1,
and ul. I n fact a1 and can assume values a t optical frequencies which
at radio frequencics are possible only in densely ionized media. Thus Shea
found for copper aI/az = 0.48, a value less than u n i t y , and P l / a z = 2.61,
instead of the exceedingly large value one might anticipatc for such a
good conductor. I n this case the apparent phase velocity within the
metal is greater than that of light. The anomalous behavior of these
parameters a t optical frequencies gives rise to some very interesting
phenomena in the domain of metal optics, which lie beyond the limits we
have imposed upon the prcsent t heory.2
Although there appear to be no experimental data available in
support of Eqs. (60) a t radio frequencies, there is every reason t o believe
them exact. We shall discuss only the case in which the conduction
current in the medium is very much greater than the displacement
current. Let 7 = u / w e . Then the assumption is that q:>> 1, and
under these circumstances it will be recalled (page 277) t h a t
~SHEA
U’ied.
,
Ann., 47, 271, 1892; W~LSEY,
Phys. Rev., 8, 391, 1916.
2 An excellent account of the optical problems of reflection and refraction is given
by Konig in his chapter on the electrornagnetic theory of light in the “Handbuch der
Physik,” Vol. XX, pp. 197-253, Sprlngcr, 1928.
504
BOUNDARY-VALUE PROBLEMS
[CHAP.
JX
but: since 7; is assumed to be very much larger than the maximum value
of sin2 B0, we obtain the approximate formulas
hence, n 4
w
as a 1 4
00
or o -+ 0. ,4t the same time
and $ 4 0 . As the conductivity increases or the frequency decreases, the
planes of constant phase align themselves parallel to the planes of constant
amplitude and the propagation i s into the conductor in a direction normal to
the surface.
In the case of copper, u1 = 5.82 X lo' mhos/meter, and it is obvious
that $ differs from zero by an imperceptible amount. Whatever the
angle of incidence, the transmitted wave travels in the direction of the
normal. The factor
measures the depth of penetration. It is characteristic of all skin-effect
phenomena and gives the distance within the conductor of a point at
which the amplitude of the electric vector is equal to l / e = 0.3679 of its
value at the surface. In this distance the phase lags 180 deg. Since the
value of 6 measures the effectiveness of a material for shielding purposes,
it is interesting to know its order of magnitude. The table below gives
the values of 6 in the case of copper for several frequencies. They are
obtained from the approximate formula 6 = 6 . 6 ~ cm.
4
Y
cycles/sec.
60
103
106
6,
cm.
0.85
0.21
0.007
The depth of penetration is decreased by an increase in permeability,
but this is usually offset by the poor conductivity of many highly permeable magnetic materials.
SEC.9.91
REFLECTION AT A CONDUCTING SURFACE
505
The angle 1c. may be approximately zero for materials of much lower
conductivity than the metals. In the case of sea water,
u1
= 3 mhos/meter,
el =
81co,
and we find
At v = lo6 we see that n = 164, J. < 0.35 deg., and 6 = 29 cm. The
approximation is just valid a t v = los, for then vf = 45 >> 1, and in this
case n = 16.4, J. < 3.5 deg., 6 = 2.9 cm.
9.9. Reflection at a Conducting Surface.-We shall examine next the
phase and amplitude of the wave reflected a t the plane interface of a
dielectric and a conductor. By Snell’s law
kl cos el
(68)
=
l/lcq - k%sin2 oO,
and from (60) and (61) it follows that
pp =
(69)
WP1,
q2 - p 2 = a: - p: - a; sin2 O0,
q2 p 2 = [4a4p4
(a? - p: - a: sin2 &)2]+.
+
+
It can easily be verified from these relations that
where
Upon substituting (70) into the Fresnel equation (14) for the reflected
component of the electric vector polarized normal to the plane of incidence, one obtains
(72)
Ezl =
pla2
cos
60
pla2
cos
00
- pa 4
-
+
pLp
ia
eT
4 EOl.
dFW e
The fraction must be rationalized to give amplitude and phase.
(73)
Ezi = p l e - i 6 J - E,,l,
Then
and after a relatively simple calculation we find
(74)
The other component of polarization is found in the same way but
the computation is considerably more tedious. According to (20),
506
[CHAP.IX
BOUNDARY-VALUE PROBLEMS
the second Fresnel equation is
pz
(75) H ~ L
=
cos
00
+
(a?
_ _ P?~
-
v'FW
+ p l a 2 d=i
2a1~1i)-
+
/*la2
c
e2
Hal.
t!D
p2 cos 00 (a: - py
2a1pli)
e3
The lengthy process of rationalizing this expression may be skipped over
and the result set down at once. We abbreviate
n2 x E ~ I=I pile-26iI no x EGII,
(76)
and find
[ 2 p 2 a l ~cos
1 eo - pla2p12
= [ p 2 ( a ? - PT) cos eo - P1a2q12
[pLz(a? - E ) cos 8 0
p1azq12
[2p2a1P1 cos 8 0
p1azp1"
+
+
+
+
If the conductor is nonmagnetic, so that p 1 = p2, the expression for the
amplitude pi1 factors into the form deduced by Pfeifferl in the course of
his optical studies :
- (q - a2 cos
&J2
p 2 (q - a2 sin 0 0 t a n Bo)2
p2
_____
(78)
1'1
- ( q a 2 cos eo)2 p2 ( q a2 sin o0 t a n o 0 ) 2 p f
As in the case of total reflection, the two components of polarization
are reflected from an absorbing surfacc according to different laws.
Consequently a n inciderit uiaw which zs linearly polarized, but whose direction of polarization i s neither normal nor parallel to the plane of incidence,
will be rejected with elliptic polarization. The polarization is determined
by the ratio
+
+
+
+
+
+
(79)
Only in the case of nonmagnetic materials do the expressions for p and 6
reduce to a relatively simple form. If p 1 = p 2 ) one finds after another
laborious calculation that
p2
O,, t,an B o ) 2 + p 2
+ a2 sin eo tan eo)2 +
~ - _2a2
_ 11_sin- Oo
- _t.an_ 80_ ~ .
a: sin2 Bo tan2 Oo - (q2 + p ' )
- a2 sin
= (q
___
~
(q
(80)
tan 6
=
p2'
The reflection coefficients are again defined as the ratio of the energy
flows in the incident and reflected waves. Thus
(81)
RL = p i 2 ,
I n the case of normal incidence R ,
(82)
a:= k
(pza1
-
+
RII =
PII'.
= ail =
a,
Fla2)2
Pla2)2
+PX.
+
P X
PFEIFFER,Diss. Giessen, 1912; K ~ N I GZoc.
, cit., p. 242. Cf. also WILSEY,
zoc. cit., p. 393.
SEC.9.91
REFLECTION A T A CONDUCTING SURFACE
507
The degree of polarization is commonly measured by the ratio
where g2 = E2,II/E:l, the ratio of the incident intensities.
When one considers that the quantities q and p are functions of the
angle of incidence as well as of a11 the parameters of the medium [Eqs.
(60)], the complexity of what appeared a t first to be the simplest of
problems-the reflection of a plane wave from a plane, absorbing surfaceis truly amazing. The formulas nre far too involved to be understood
from a casual inspection, and the nature of the reflection phenomena
becomes apparcnt only whcn one treats certain limiting cases. Of
these, one of the most important for electrical theory is that in which
Case
I.
$
>> 1.
*
7;"=
lhen
€1
as was shown on page 504, and
(I -
2
cos 0")?
p 1 2 'u --
( + p1ff9 cos
1
(85)
--:
pza1
0")'
+1
+1
Since p1/p2 = pJp0 = ~,,,l i s the magnetic permeability of the conductor
and e z / e o = ~~1 is the specific inductive capacity of the dielectric, we may
write
(86)
5 =
E! =
.Pzff 1
Egi
=
2.11 X 10-4
where v is the frequency. I n the case of all metallic conductors
<< 1, and (85) reduccs furthcr t o
p~a2/p~za1
This approximation is 0btainc.d by applying the binomial theorem to
(85), and i t is valid as long as the square of p l a z / p 2 a l can be neglected
with respect t o its first power. Likewise in the case of polarization
parallel to the plane of incidence
508
B O U N D A R Y - V A L U E PROBLEMS
"I2
(88)
=
tan 611
=
2 cos2
00
&Gs2
80
[CHAP.I X
- 22 cos
+ x2
+ 22 cos 00 f x2'
00
22 cos 6 0
- cos2 0 0
5 2
Here the square term in x has been retained since a t a certain value of
Oo the numerator of ,oil2 becomes very small. This critical angle is obviously the analogue of the Brewster angle discussed above. The minimum
value of p112 corresponds to an angle of incidence satisfying the relation
ficos O0 = x. Consequently if the minimum is sharply defined the
ratio pl/ul can be determined directly from the observed reflecting
properties of the material.
It is clear from these results that the reflection coefficients of the
metals are practically independent of the angle of incidence and differ
by a negligible amount from unity even at the highest radio frequencies. Thus a t an air-copper surface, K,I = 1, ~~2 = 1, u1 = 5.82 X 10'
mhos/meter, so that 5 = 2.77 X
In the case of iron the conductivity is about one-tenth as large and the permeability' may be of the
order of lo3, but x is still exceedingly small, even a t wave lengths of a few
centimeters. The formula
-
R
(89)
= 1 -
4.22
x
10-4
YKm*Ke2
J - i y
to which (87) and (88) reduce a t normal incidence, was verified in the
infrared region of the spectrum by Hagen and R u b e m 2
The results of their measurements are shown in the following table
,
1p = l W 4 em. The theoretical values
a t X = 12p and X = 2 5 . 5 ~ 1where
with which they are compared were calculated from the electrostatic
values of the conductivities. At wave lengths less than 12p the devia1 - R, % a t X
=
1 2 ~
Observed
Ag
cu
A1
All
Pt
Ni
Sn
Hg
1
-
R, % a t
A = 25.5~
~-
Metal
1.15
1.6
-
2.1
3.5
4.1
Calculated
1.3
1.4
1.6
3.5
3.6
-
-
-
-
Observe(
Calculated
1.13
1.17
1.97
1.56
2.82
3.20
3.27
7.66
1.15
1.27
1.60
1.39
2.96
3.16
3.23
7.55
At ultrahigh frequencies the permeability may be very much smaller.
pare the measurements by Glathart, Phys. Rev., 66, 833, 1939.
* HAGENand RWENS, Ann. Physik, 11, 873, 1903.
Com-
SEC.9.91
REFLECTION AT A CONDUCTING SURFACE
509
tions become more marked and a t optical wave lengths there is no
correspondence. These figures are extremely significant. They i m p l y
that a s long as we are concerned only with metals, the macroscopic electromagnetic theory i s valid for all wave lengths greater than about
cm., or
frequencies less than about 3 x 1013cycleslsec.
The reflection of radio waves from the surface of the sea or land gives
results of another order. Consider first the case of an air-sea surface,
fi.
for which ~,,,l= K~~ = 1, a1 = 3 mhos/meter, and x = 1.22 X
Since the approximations (87) and (88) are valid only when z << 1, we
are limited now to frequencies less than 106 cycles/sec. Let us take, for
example, a 600-kc. broadcast, frequency, X = 500 meters. Thenx =
0.094. In Fig. 94 are plotted the
reflection coefficients R l and RIIof
waves polarized normal and parallel
to the plane of incidence as functions of the angle of incidence. One
will note that RI > Rll, and that a t
86.2 deg. the coefficient R I Ireaches
its minimum. Hencc?,,if the electric
vector of the incident wave is linearly polarized in a direction that is
neither normal nor parallel to the
plane of incidence, the wave rejlected
70 80 90
8,,degrees
from the surface of the sea will be
94.-Reflection coefficients of plane
elliptically polarized. Moreover, if FIG.
radio waves from the surface of the sea.
the transmitting antenna is vertical,
the wave will be polarized in the plane of incidence; thus, if it meets
the sea a t a grazing angle, it may be almost completely absorbed.
If now the frequency is increased, or if the wave is incident upon
fresh water or earth for which the conductivity is usually of the order of
or less, the approximations (87) and (88) no longer hold but the
character of the reflection phenomena is not greatly modified. We have
then to consider
Case 11. 7;
a
'
= 2
<< 1.
By (52) and (53), page 277,
&I%::
If the frequency is sufficiently high al>> pl, Eqs. (60) above reduce to
510
BOUNDARY-VALUE PROBLEMS
Then
P I
(92)
(tiel
- K e 2 sin2 00)+
(tiel
-
-
tieZ
[CHAP.IX
dLl
+ dii
- -___
cos On
sin2 O,J+
cos On’
cos O0 (tiel - ~~2 sin2 Oo)*
PiI N
Kel cos 06
JK,2 ( K e l - K e 2 sin2 On)+
To this approximation, only the dielectric properties of the medium
enter and (92) is identical with the results obtained in See. 9.6. The
next order would introduce a small correction in PI.’
Let us take for example a fresh-water surface. tie2 = 1, ~~1 = 81,
u1 ‘v 2 x
mho/meter. Then
4.45 x 104
, L\I1 ‘V 1 . ~ x8 1 0 - 7 ~ , p1 N 4.18 x 10-3.
171 =
K~~
+
V
The approximation (92) holds, therefore, if v > lo6. The constants for
earth and rock vary widely, but we may take as typical the case tiel = 6,
u1 ‘v
Then
3 x 104
o1~lrl5.ix i 0 - 8 ~ ,
p1 N 7.7 x 10-4,
171 = ___ ,
so that again (92) is valid for v > lo6. At frequencies greater than lo6
cycleslsec., fresh water and dry earth or rock act as dielectrics in so f a r as
their reflecting properties are concerned. The transmitted wave is of course
rapidly attenuated due t o the factor P I . T h e reflection coefficients
for these two cases have been plotted
in Fig. 95 against angle of incidence.
Such curves have been verified experimentally b y Pfannenberg.2
Although in the case of metals
the formulas hold for frequencies
extending into the infrared, this is
not true for dielectric or poorly conducting m a t e r i a l s . At w a v e
lengths of t h e order of a few centimeters most dielectrics exhibit a
marked increase in absorption ow%, degreee
ing to causes other than electrical
FIQ.95.-Reflection coefficients of plane
radio wave8 from earth and fresh-water c o n d u c t i v i t y * Thus a t a wave
surfaces a t frequencies greater than one length of 2.8 cm., the value of 0 for
megacycle per second.
distilled water computed on the basis
of a n electrical conductivity u = 2 X
mho/meter is 4.2 x
while
1
An account of various approximation formulas useful in optics is given by Konig,
loc. cit., p. 246.
2
PFANNENBERG,
2.Physak, 37, 758, 1926.
SEC.9.101 REFLECTION A N D T R A N S M I S S I O N COEFFICIENTS
511
the mcasured value' is 494 (m.k.s. units, distance in meters!). This is a n
extrcmc case corresponding to dipole resonance of the water molecule, but
it is obvious that under such circumstances the approximations (92) are
invalid. The general formulas may always be applied, even in the optical
spectrum, provided the valuw of a and /I are those determined b y
measurements a t a specific frequency.
PLANE SHEETS
9.10. Reflection and Transmission Coefficients.-We shall consider
only the case of normal incidence, which is of considcrable practical
FIQ.96.-Reflection
and transmission of plane waves by a plane sheet a t normal incidence.
interest. Three arbitrary homogeneous media characterized by the
propagation factors k l , k,, kS are separated by plane boundaries as shown
in Fig. 96. We need deal only with the magnitudes of the vectors and
write for the incident and reflected waves in medium (1):
The field within the sheet (2) must be expressed in terms of both positive
and negative waves.
(3)
while the transmitted wave is
E t -- E , eikjz--iot
(4)
,
Ht
=
k3
-Et.
WP3
It will be convenient to employ here the intrinsic impedance concept
introduccd in Sec. 5.6, page 282. For a planc wave in a homogeneous,
isotropic medium
Ei
(5)
1
=
&Z,Hj,
BAz, Physik. Z.,40, 394, 1939.
Z.
= UP;.
-
ki
512
[CHAP.IX
BOUNDARY-VALUE PROBLEMS
We shall now define t,he impedance ratios
(6)
and note that
(7)
ZjkZkj
= 1,
zijzjk
=
Zik.
I n terms of the impedance ratios the boundary conditions lead to
four relations between the five amplitudes.
+
(8)
F+
ikd
J2 e
E$ eikzd
+
Eo Ei = E$
EF,
Eo - El = Zi,(E,+ - E r ) ,
+ E , e-ikzd = E3 eikad
- E- e-ikzd = z 2 3 E 3&kad.
Upon solving for the amplitudes of the reflected and transmitted waves,
one finds
El
(9)
=
+
+
(1 - Z I Z ) ( ~ 2 2 3 )
(1 . Z 1 2 ) ( 1
22,)
+
E a = (1 - zI2)(1
-
+ (1 + zlz)(1
+ (1 - Z,2)(l -
-
_ _ _ _Z.2_3_
)e2ik2d
Z23)e2ikzd
4e-ik3dEo
zz3)eik2d
+ (1 +
~12>(1
+
zZ3)e-ik2d’
We shall now define the complex ratios
The physical significance of these ratios is apparent when one notes that
The Rik are the reflection coefficients a t normal incidence for the plane
interface dividing two semi-infinite media [Eq. (82), page 5061; the
quantity r l k is the complex ratio of the amplitudes of reflected and incident
waves. The reflection coefficient a t an interface is zero when the intrinsic
impedances of the adjoining media are equal.
I n terms of these r j k , (9) reduces to
The reflection and transmission coefficients of the sheet are equal to the
SEC.9.111
APPLICATION TO DIELECTRIC M E D I A
squares of the absolute values of these ratios.
513
We write
One will note that yjr~= - Y k j , but 8 j k = +&j.
Upon introducing these
phase angles into (13), and then multiplying each ratio by its conjugate
value, w e obtain
(16)
9.11. Application to Dielectric Media.-The intcrference phenomena
represented by (16) and (18) have frequently been applied to the measurement of the optical constants of materials in the infrared and electrical
as well as the visible regions of the spectrum.’ To make clear the
1 See, for example, Drude, “Lehrbuch der Optik,” Chap. 11, 1900. The method
has been used in the infrared by Czerny and his students. Czerny, 2. Physik, 66,
600, 1930; Barnes and Czerny, Phys. Rev., 38, 338, 1931; Cartwright and Czerny,
2. Physik, 86, 269, 1933; Woltersdorff, &id., 91, 230, 1934. Recently the same
514
BOUNDARY-VALUE PROBLEMS
[CHAP.IX
physical behavior of the reflection coefficient as the thickness of the sheet
is varied, we shall treat the simple case of a thin dielectric sheet separating
two semi-infinite dielectric media as in Fig. 96. The inductive capacities
of the three media are €1, € 2 , €3. 01 = pz = pa = 0, and consequently
Then
612 = 823 = 0.
and arz = 21r/X~,where hz i s the wave length within the sheet, Now if €2
2 3positive and R oscillates
lies between el and in value, the product ~ 1 2 ~ is
80
70
60
Y
850
r
P
.B 30
10
0
FIQ. 97.-Reflection
d
coeEcients of plane dielectric sheets as functions of thickness and
dielectric constant.
as a function of d. The minimum value of R occurs at d = &/4, and is
zero when r12 = r 2 3 ; i e . , when ez is the geometric mean of €1 and €3, or
6.
Under these conditions all the energy is transmitted from
medium (1) into medium (2), none is reflected a t the surface. T h e introduction of a quarter-wave length sheet of the proper dielectric constant
accomplishes the same purpose as the matching of impedance at the junction
of two electrical transmission lines. I n Fig. 97 curves are plotted which
show the effect of a dielectric sheet at the interface of air and water.
This principle has been applied recent,ly to the manufacture of
optically “invisible” glass. The reflection of light from a glass surface
is reduced almost to zero by coating with a film of the proper index of
refraction.
e2 =
method has been applied to ultrahigh-frequency radio waves. See for example
Bas, Phys. Z., 40, 394, 1939, and Pfister and Roth, Hochfreq. und Elektroak., 61, 156,
1938, who have extended the formula for the reflection coefficient to arbitrary angles
of incidence.
SEC.
ABSOEBING LAYERS
9.121
515
9.12. Absorbing Layers.-If
the medium on either side of the sheet
is the same, Eqs. (16) and (18) reduce to
10
I I 1 I I I I I I I
516
BOUNDARY-VALUE PROBLEMS
[CHAP.IX
oscillations of the reflection coefficient. The numerical values of the
minima are determined by the losses, and a curve such as that of Fig. 98
provides one of the best means of finding pz at ultrahigh frequencies.
In the case of a metal sheet the attenuation is so great, and the reflection coefficient so nearly unity, that transmission, through even the
thinnest sheets, is wholly negligible. The transmission coefficient
through a copper sheet 1p thick
cm.), at lo8 cycles/sec. for example,
is less than 10P. However, this calculation gives a wholly erroneous
impression of the effectiveness of shielding, for it applies only to plane
surfaces of infinite extent. The reflection losses from surfaces whose
radius of curvature is small compared to the wave length is by no means
always large,l and the only significant general criterion is the value of
the attenuation factor 8 2 , or its reciprocal, the skin depth, defined on
page 504.
SURFACE WAVES
9.13. Complex Angles of Incidence.-The possibilities of the Fresnel
equations are far from exhausted by the discussion of the preceding
sections, for it has been confined exclusively to real angles of incidence.
These equations represent the formal solution of a boundary-value problem which analytically is valid for complex as well as real angles. Complex angles of refraction already have occurred in connection with total
reflection and reflection from conducting surfaces. There the associated
plane waves proved to be inhomogeneous: the planes of constant amplitude fail to coincide with the planes of constant phase. Thus one may
reasonably anticipate that a complex angle of incidence will be associated
with an inhomogeneous primary wave. We shall not consider a t this
point how such waves are generated, but recall that in Secs. 6.8 and 7.7
it was shown that cylindrical and spherical waves of the most general
type can be represented as a superposition of plane waves whose directional cosines include complex as well as real values.
If the angle of incidence is real, the primary wave gives rise in general
to a reflected wave at the surface of discontinuity. Only in the case of
a plane wave polarized in the plane of incidence and meeting the interface of two semi-infinite dielectrics at the Brewster angle (page 497) does
there occur an exception to this rule. There is always a reflected wave
if one of the media is conducting. If, however, complex angles of incidence are admitted, the reflection coefficients can be made to vanish
whatever the nature of the media.
We shall consider in some detail the case of a wave whose electric
vector is parallel to the plane of incidence. Letting rII represent as in
' A point wcll illustrated in the case of cylindrical shields by Schelkunoff, Bell
System Tech. J., 13, 532, 1934.
COMPLEX ANGLES OF INCIDENCE
SEC. 9.131
517
(15), page 513, the ratio of the reflected to the incident magnetic vector,
the Fresnel equation (19), page 494, gives for this case
cos
TI1 = __
00
cos oo
-
212
cos 01
+ z12cos 81)
as defined on page 512. I n order that there shall be no reflected wave
the numerator of (1) must vanish. This condition, together with Snell's
law, leads t o the two relations
(3)
cos 81
=
Z X cos Oo,
sin
el
P2
= - Z,,
PI
sin eo,
which determine the angles of incidence and refraction.
nating one obtains
Upon elimi-
To simplify matters we shall assume that the interface of the two
media coincides with the surface z = 0. The z-axis is normal to the
I \
Wave normal
FIG.99.-Refraction
of the planes of constant phase characteristic of a surface wave.
surface and is positive in the direction leading from medium (I) into
medium (Z), as shown in Fig. 99. The plane of incidence is parallel to
the xz-plane; consequently, the field has only the one magnetic component Hy. The electric vector lies in the plane of incidence and has in
518
BOUNDARY-VALUE PROBLEMS
[CHAP.IX
general both the components E , and E,. T h e jield is transverse magnetic.’
Since the magnetic vector is tangent to the surface of discontinuity, its
transition from one medium to the other is continuous; hence, by Sec. 9.4
(5)
where C is an arbitrary constant.
Next we shall let
h = kz sin Bo = k l sin el,
(6)
and define the real and imaginary parts of the factors appearing in the
exponents of (5) as follows:
h =a
ik, cos e,,
ik, cos el
(7)
=
=
+ ip,
-i
+i
4
4
-
=
=
+ iq2,
+ iq,.
p2
-p,
The signs of the radicals have been so chosen that the field will vanish for
infinite values of k x . Then in terms of (7),
H,
H,
(8)
=
=
where
\El = exp[+plz - pz
q 2 = exp[-pzz - pz
(9)
(x > 01,
Csz,
cs,,
(5
< 01,
+ i(-qlz + az) - id],
+ i(-gzx + az) - id].
Let us consider first the nature of the incident wave.
the planes of constant phase are defined by
We note that
+
-92s
az = constant,
(10)
while the planes of constant amplitude satisfy
-p2x - pz = constant.
(11)
T h e planes of constant phase and constant amplitude are not coincident.
The real angle of incidence #Q is the angle made by the normal to the
planes of constant phase with the positive x-axis. It satisfies the relation
(12)
whence
(13)
z cos
cos $0 =
rC.o
+ z sin q0
Q1
=
w’ sin
constant,
J.O
=
Vzi3i
The planes of constant phase are propagated in the direction of their
positive normal with a phase velocity
W
(14)
v2
See Sec. 6.1, pp. 349ff.
=
&T2’
COMPLEX ANGLES OF I N C I D E N C E
SIC. 9.131
519
a velocity which is less than the normal velocity of a homogeneous plane
wave in an infinite medium of the same character. On the other hand,
the velocity parallel to the z-axis of a given point on a phase plane may be
greater than the velocity in free space. The reason is clear from Fig. 100.
The wave length Xz is the distance between two planes of corresponding
phase measured in the direction of the normal.
27r
xz = _____.
GTz
(15)
The distances between the same two planes measured along the x- and
z-axes, respectively, are
The phase point must travel these distances in the course of one period
so that the apparent velocities are
(17)
and these quantities have values which lie between
The normal to the planes of constant
X
amplitude makes an angle
with the
z-axis, satisfying the relation
02
and infinity.
+A
(18) --z cos
+; + z sin +;
=
constant,
whence
(19)
/
t a n fi;
=
--.PBz
/
/
2
Planes of constant amplitude, which in
component of phase
the case of large conductivity are ap- velocity parallel to t h e z-axis may be
greater than the velocity of light.
proximately a t right angles to planes of
constant phase, have not been drawn in Fig. 99.
Turning next to the transmitted wave in the region z < 0, we see
from (9) that the planes of constant phase and constant amplitude are
defined respectively by
(20)
and
(21)
+ az
=
constant
plz - pz
=
constant,
-q1x
whose normals make angles
and $[ with the negative x-axis defined by
520
BOUNDARY-VALUE PROBLEMS
[CHAP.
Ix
The phase velocity is
The components of the electric vector are best calculated from the
field equation
iWjl
E=-fVXH,
k
(24)
which resolved into rectangular components gives
(25)
E z --
-!&?d%!,
k2
E z -- i___.
m p a H#
1c2 ax
az
From (8) and (9), it follows that
E,
=
(a
Ez
=
iWP2
E,=F(
(27)
E,
+ ip)CS,,
-=
k2
(26)
iwp1
a
(pz
+ iP)C*l,
=7
(pi -
kl
+ iqz)C*Z,
(x > 01,
(x < 0))
iqi)C*i.
Physically, the field described by (8))(26), and (27) is associated with
a current in the x-direction whose distribution with respect to y is uniform.
Such a field is transverse magnetic, and the longitudinal component of
electric field accounts for the losses in either medium, since E,H, is the
component of energy flow normal to the interface. The transverse component E,, on the other hand, gives rise to a n energy flow parallel to the
plane. The whole problem is comparable to the limiting case of a
current propagated along a wire whose radius becomes infinite. The
other symmetry, corresponding to the condition r l = 0, leads to transverse electric waves and may be interpreted as th e limiting problem of
propagation along a solenoid of very large radius. I n this case the current is transverse to the elements of the cylindrical surface.
9.14. Skin Effect.-If
both media are perfect dielectrics, the angle
00 is real in the case of T I [ = 0 and is identical with the Brewster polarizing
angle. Of much greater practical interest is the case in which medium
(1) is a conductor and medium (2) a dielectric. One may then assume
that 1Zl2I2<< 1. According to (4) and (6)
Next Page
SEC.
521
S K I N EFFECT
9.141
which can now be expanded in powers of
Z12.
of which we shall retain in the present approximation only the square
term. Since by hypothesis the conductivity of (2) is zero, we may write
where as above 71 =
Then
UI/EM.
Note that a, the real part of this expression, is less than kz and that,
consequently, the velocity in the z-direction of a point on a plane of
constant phase is greater than that of a free wave in the dielectric. We
saw above that this is only an apparent velocity and that the true phase
velocity is in fact less than that fixed by k 2 .
We now make the further assumption that $>> 1, and that l/oT
can, therefore, be neglected with respect to 1/71. To this approximation
Thus the velocity in the z-direction is essentially the characteristic phase
velocity of the medium and the attenuation approaches zero with
increasing conductivity.
To the same approximation
(33)
which by (7) leads to
92
(34)
N p 2 N WE2
Likewise,
(35)
(36)
k!
-
h2 N W2(elPl
q1 ‘v P
lN
- E2P2)
+
6.
WPlUl
From these results one will note, first, that
(37)
iwp1u1,
Previous Page
522
BOUNDARY- VALUE PROBLEMS
[CHAP.
Ix
whence it appears from (13) and (19) that the planes of constant amplitude
in the incident wave arc normal to the planes of constant phase. As u1 -+ ,
tan #o -+ 00 and +o -+ 7r/2. A s the conductivity increases, the planes of
:onstant phase in the dielectric become very nearly normal to the surface of the
conductor.
Within the conductor, the corresponding relations are determined
by (22).
+:
, both +I and approach zero. The planes of constant ampliAs CQ-+
tude and constant phase within the conductor are very nearly parallel to the
interface. At the same time the wave length becomes very small, for
Q,
(39)
I t will he recalled that in the case of metals (TI is of the order of 107
mhosj’mcter. The deviation of 90 from 7r/2 and of $1 from zero is then
complctely negligible, and the wave length in meters within the metal is
FIG.101.-The
solid lines represent planes of constant phase, the dotted lines planes of
constant amplitude.
in magnitude of the order v-i. Even in the case of sea water, at a
frequency of 1 megacycle/sec., the forward tilt of the phase planes above
the surface amounts to only some 10’ of arc. The forward tilt of a radio
wave propagated over dry earth may be more marked. The relation of
the planes of constant phase and constant amplitude is then somewhat
as illustrated in Fig. 101. It must be remembered, however, that in such
a case the assumption TJ; >> 1 is not necessarily just,ified.
Upon introducing the approximations (32), (34), and (36) into (26)
and (27), we obtain for the field components the expressions
Note that the tangential component E , is continuous a t the boundary
SEC.9.141
523
S K I N EFFECT
= 0 and equal in magnitude to (pLo/ul)f times the magnitude of the
tangential component of magnetic field. As u1+ 00, the longitudinal
electric field vanishes.
The normal and tangential components of E differ in phase by 45 deg.
(A more exact calculation shows this angle to be less than 45 deg. in
the case of poor conductors.) Consequently the locus of the resultant
electric vector at a n y point i s an ellipse. At a point within the dielectric
just above the surface, the ellipse is inscribed within a rectangle whose
sides are 2a2 = 2 ( p 2 / 4 f , 2b2 = 2 ( p l o / a l ) f , b, << a2, as shown in
x
I Ex
IEx
-4
L- 2 b ,
FIG. 102a.Locus of the electric
vector a t a point in
the dielectric.
FIG. 102b.-Locus
of the
electric vector a t a point in the
conductor.
Fig. 102a. Since the particular value of z is of no importance, the
coordinates of the ellipse are
(42)
E,
=
a2 cos wt,
E,
=
b2 cos
(
wt
2
+- .
T h e electric vector in the dielectric rotates in a counterclockwise direction.
The ellipse is identical in form a t othcr points of the dielectric but
reduced in size by reason of the attenuat,ion factor.
Within t,he conductor the eltxtric vector describes a n ellipse bounded
by a rectangle whose sidcs are 2al = 2 w ( e Z p Z ) + / u 1 , 2bl = 2 ( p l w / a l ) i ,
bl >> al. The coordinates of the ellipse are
E, is now retardcd with respcct t o E , and, consequently, within the
conductor the electric vector rotatrs in a clockwise dircetion.
The drpth of penctration within the conductor is determincd by the
damping factor pl, whose reciprocal is exactly the factor 6 dcfined on
pagc. 504. The current density in the x-direction is J , = alE., and the
total current passing through an infinite strip 1 meter in width parallcl
to the xy-plane in the conductor is
524
(44)
BOUNDARY- VALUE PROBLEMS
I
=
(T1
so1$:
m
E, dx dy
=
[CHAP.IX
Ce-Bz+i(az-ot).
On the other hand, the current through such a strip of depth d is
(45)
z
Z(1 - , - p d + i q d ) .
Since p1 is very large in the case of good conduct,ors, the current is confined almost entirely to a thin “skin” in the neighborhood of the surface.
One will note, moreover, that the phase of a current filament varies
with its depth. This effect may be expressed in terms of an internal
self-inductance.
PROPAGATION ALONG A CIRCULAR C n I N D E R
9.16. Natural Modes.-A circular cylinder of radius a and infinite
length is embedded in an infinite homogeneous medium. The propagation constant of the cylinder is k l , that of the external medium kz. No
restriction is imposed as yet upon the conductivity of either. The
components of the field in circular coordinates were derived in Sec. 6.6.
This field must be finite at the center and, consequently, the wave functions within the wire will be constructed from Bessel functions of the
first kind. Outside the cylinder the Hankel functions HA1) ensure the
proper behavior at infinity. According to (36) and (37), page 361, we
have at all interior points, r < a,
* See the treatment by Sommerfeld in Riemann-Weber, “Differentialgleichungen
der Physik,” Vol. 11, pp. 507-511, 7th ed., 1927.
SEC.9.151
NATURAL MODES
At all exterior points, r
n=-
H,”
=
525
> a,
m
2- [Hil)(Xzr)RIF,.
n=
41
I n these relations
A: = Ic:
F,
(5)
=
- h2,
exp(in8
A: = k,2 - h2,
+ ihz - id),
and the prime above a cylinder function denotes differentiation with
respect to the argument Xr.
The coefficients of the expansions and the propagation factor h are
as yet undetermined. However, a t the boundary r = a, the tangential
components of the field are continuous, and this condition imposes a
relation between the coefficients. From the continuity of the tangential
components of E, we obtain
nh
- J,(u)
u2
(6)
+ y-J,!,(u) bi = nh
3H ~ ” ( V ) +
iPlW
U:
U:
J,(u) a;
=
iP*W
Hil)’(v) b;,
H p ( v ) a:,
while from the tangential components of H
\ ,
.-
J,(u)
q
u=
XlU,
=
H p ( v ) Q,
where
(8)
v
= Xza.
Equations (7) and (8) const,itute a homogeneous system of linear
relations satisfied by the four coefficients a;, Pi, a;, b;. The system
526
BO U N D AR Y-VAL UE PROBLEMS
[CHAP.
IX
admits a nontrivial solution only in case its determinant is zero. The
propagation factor h is determined by the condition that the determinant of
(6) and ( 7 ) shall vanish. Expansion of this determinant leads to the
transcendental equation
whose roots are the allowed values of the propagation factor h and so
determine the characteristic or natural modes of propagation. These
roots are discrete and form a twofold infinity. The Bessel functions are
transcendental; thus, for each value of n there i.; a driiumcrable infinity
of roots, any one of which can be dvnoted by the subscript m. Any root
of (9) can then be designated by hnm.
No general method can be stated for finding the roots of (9), but
fortunately i t is possible t o make approximations in practical problems
involving metallic conductors which greatly simplify the solution. One
will observe from (6) that if the conductivity is finite, a superposition of
electric and magnetic type waves i s necessary to satisfy the boundary conditions, except in the symmetrical case n = 0. The field is not transverse,
with respect to either the electric or the magnetic vector. If, however,
the conductivity of either medium is infinite, the boundary conditions
can be fulfilled by either a transverse magnetic or a transverse electric
field. I n this case the problem is completely determined by either the
conditions
Es
(10)
= 0,
E, = 0,
(r
=
a>,
or, alternatively, by
a
- (THO) = 0,
ar
d
H , = 0,
taken from (14), page 484, noting that the tangential components of H
do not vanish. I n case the cylinder is a perfect conductor embedded in a
dielectric, the allowed transverse magnetic and transverse electric modes
are determined by roots respectively of the equations
The amplitudes a: and b; are now quite independent and are determined
solely by the nature of the excitation. I n the inverse case of a dielectric
cylinder bounded externally by a perfect conductor, the transverse
magnetic and transverse electric modes arc determined by the roots
respectively of
SEC.9.161
CONDUCTOR E M R E D D E D I N A D I E L E C T R I C
527
In practice the conductivity of a metal, while not infinite, is yet so very
large that the cylinder functions in such a region can be replaced by their
asymptotic representations. It, is then found that in each possible mode,
either a transverse eleci ric or transverse magnetic wave predominates.
A small component of opposite type which spoils the transversality may
bc considered a correction term taking account of the finiteness of the
conductivity.
If only the symmetrical mode n = 0 is excited, the field is independent
of 0 and the coefficients ao, bo arc independent of one another, whatever
the conductivity. From (6) it is apparent that in this case the transverse
magnetic wave satisfies
a t the boundary, while for the transverse electric wave
The allowed transverse magnetic modes are then determined by the
roots hornof the transcendental equation
while the transverse electric modes are found from
9.16. Conductor Embedded in a Dielectric.-We shall discuss only
the case of waves that are predominantly transverse magnetic, corresponding to the practically important problem of a n axial current in a
long, straight wire.’ The axially symmetric solution, n = 0, shall be
considered first. It will then be shown that the asymmetric modes,
if excited, are immediately damped out and, consequently, never play
a part in the propagation of current along a solid conductor.
Suppose, first, that the conductivity of the cylinder is infinite.
According to (12) the propagation factor is then determined by
1 I n the case of transverse electric waves the lines of current are circles concentric
with the axis of the conductor. This solution may, therefore, be applied to the study
of propagation along an idealized solenoid. Cf. Sommerfeld, Ann. Physik, 16, 673,
1904. If the conductor is solid thc transverse electric modes are damped out.
528
BOUNDARY-VALUE PROBLEMS
[CHAP.IX
If v is very small, Hk1)(v)can be represented approximately by
2i y v
dH&”(V) ‘V -27 i
H f ) ( v )N - In - 9
(19)
7r
22
dv
3-2,
where y = 1.781. These expressions are obtained by evaluating the
integral (39), page 367, in the neighborhood of the origin. The Hankel
function is evidently multivalued, and as in the case of the logarithm we
use only its principal value. This principal value can be defined most
easily in terms of the asymptotic expression
which holds when v >> 1, p << v. If one puts v = lvle+, then the
principal value of H&’)(v)lies in the domain --a/2 < 4 < 3 ~ / 2 . The
principal value or branch of Hkl)(v) vanishes at all in$nite points of the
positive-imaginary half plane. It is only with the roots of the principal
branch that we are concerned, and it can be shown that the principal
branches of Hhl)(v) and Hil’(v) have none.’ Consequently, the ratio
has only t,he solution v = 0, and hence h = k2. I f the cylinder i s infinitely
conducting, the field i s propagated in the axial direction with a velocity
determined solely by the external medium. If this is free space, the waves
travel along the cylinder without attenuation and with the velocity of
light.
In case the conductivity of the cylinder, while not infinite, is very
large, as of a metal, h differs from k 2 but there will be a t least one mode
for which the difference is small, so that v = a 4
is a small quantity. The approximation (21) still applies to the right side of (16).
On the left side we note that lkll >> k2; since kz N lhl, u = k l a >> 1.
The Bessel functions Jo(u) and J l ( u ) can, therefore, be replaced by their
asymptotic representations. The imaginary part of u is positive. One
will remember that kl = a1 $1 and that, if u1 is very large, a1 1~ 01.
Then in virtue of (18)) page 359,
+
Equation (16) now assumes the form
(23)
whose right-hand member is independent of h.
1
SOMMERFELD,
in Riemann-Weber, loc. cit., p. 461.
~ E C 9.161
.
CONDUCTOR EA4BEDDED I N A DIELECTRIC
529
This equation for h can be solved easily by successive approximations.
The method proposed by Sommerfeld' goes as follows. Let
f In f = 7.
The quantity 1 is known and f is to be determined. Now the logarithm
of E is a slowly varying function with respect to f itself. Hence, if the
nth approximation has been found, the (n 1)th is obtained from
+
(25)
or
En+l
In
En
= 1,
The convergence can be demonstrated for complex as well as real values
of 1. Roots of (24) which lie near = 1 or which are associated with
other branches than the principal of the logarithm are rejected. It turns
out that In { is a negative number of the order -20, and the convergence
is hastened by assuming this value for In to. From (24), with suitable
approximations,
(27)
N
(1
+ i)i.75 x 10-13av
>-&
K I ~ K ~ ~ V
where v is the frequency and a is expressed in mcters.
To illustrate, the following numerical example is quoted from Sommerfeld.2 The wire is of copper, 1 mm. in radius. The frequency is
lo9 cycles/sec. and the outside medium is air. Then (27) gives
rl =
Assuming In
to=
-20, we obtain for the first approximation
=
In El = -16.8
1
+ q 7 . 2 5 x 10-7.
-(I
(1
+ i)3.6 X lo-',
+ i -4 = - (16.8 - 0.78i).
%-
Gottinger Naclir., 1899. Iksrrihed in Iticmann-Weber, Zoc. cit., p. 529.
LOC. d.,
p. 530.
' RIEMANN-WEBEH,
530
BO U N D A R Y -V AL U E PROBLEMS
Similarly
In
+
17 =
= __
lz
[CHAP.IX
(4.1 4.5i)10-',
In t1
= -(16.7 - 0.83i),
(2
'
17 = (4.2
- __
- In
3
t2
+ 4.5i)10-8
& differs but slightly from t2 and the series may bc broken off here.
The srnallness of 4 justifies the initial approximations for the Bessel and
Harikel functions. Itet,iirning now to (24), we calcuhte the value of D,
and hence of the propagation factor h, for a = 1 mm.
h 'V k2[1
+ (6.0 + 6.4i)10-5]
CY
+ ip.
The phase velocity along the wire i s
V 'V ~ ( -l 6.0 X lO-j),
while the distance the wave must' travcl to he roduced to l / e of its initial
amplitude is x = 770 meters. Alt>houghf.he velocity differs an inappreciable amount from that of free space, the damping at this high frequency
and for this size of wire (about No. 12, Brown and Sharpe gauge) is relatively high from the point of vicw of a communication line.
If the radius a is extremely small, the quantity u 'V akl may also be
small in spite of a large conduct'ivity. To illustrate this point, Sommerfeld has worked out the case of a platinum wire, conductsivityone-eighth
that of copper, a = 2p = 2 X
meter and v = 3 X lo8 cycles/sec.
One then finds t'hat u N (1 QO.19, and consequently the Bessel
functions J o ( u )and J l ( u ) cannot he replaecd b y thcir asymptotic exprcssions. I n the case citcd the value of u is so small that one can put
approximately Jo(u) = 1, J l ( u ) = u/2, and procecd according t'o the
same method outlined above. One obtains
+
h N 0.083
+ 0.061 i.
The phase velocity proves to be only 76 per cent of its free-space value
and the attenuat'ion i s so large that t,he amplitude is reduced to l / e of its
initial value in a distance z = 16 cm.
Thus far we have considered only one root of the dcterminantal
equation (16), t'he root which in the case of high conductivity makes
h 'V k 2 and leads to what will be tc:rmed t,hc principal wave. There are
othcr roots, howcver, for which v = a dk,Z - h2 is very la,rge. With
the help of the asymptotic expressions for the Hankel functions, (22),
pagc 359, we may now approximate (16) by
If the lcft side of (28) is very largo, the roots are given to the first approxi-
SEC.9.171 FURTHER DISCUSSION OF T H E PRINCIPAL WAVE
53 1
mation by Jl(u) = 0. But the roots of Jl(u) lie on the real axis of the
u-plane. Consequently the imaginary part of h2 must be equal to the
imaginary part of k!. This means that in the case of metallic conductors
dKm,
the attenuation factor P of h = a ip is of the order 2 X 1 0 - ~
and hence the complementary waves corresponding to the other roots of (16)
are damped out at once.
The roots of (17), which give the modes of axiallysymmetric transverse
electric waves can be investigated according to the same procedure.
However, since there is no “magnetic conductivity,” there is also no
principal wave characterized by low attenuation. All axially symmetric
transverse electric waves in a solid metal cylinder are damped out too
quickly to be observable.
I n the asymmetric case wc have secn that the field is a linear combination of transverse magnetic and transverse electric modes. The
existence of a principal wave with small attenuation implies that h ‘v k 2 ,
v << 1. Since n is now different from zero, this would lead to extremely
large values of the right-hand member of (9)) a condition that could be
fulfilled only by nearly infinite values of the conductivity gl. It turns
out that even with a conductivity as high as that of copper, the imaginary
part of the propagation factor h is still very large and the attenuation
such as to preclude the existence of asymmetric waves at any finite
distance from the source.
The results of this study may be summed up as follows. There are an
infinite number of modes of propagation along a solid conducting Pylinder.
The amplitudes of these modes are t h r coefficients a, and b,, whose values
are determined by the initial excitation-the nature of the source. Of
all these modes, however, only one, the principal mode, a symmetric,
transverse magnetic wave, possesses a relatively low attenuation. All
others are damped out within a centimeter or two from the source, even a t
~ the nrxt section that
frequencies as low as 60 cycles/sec. We shall s c in
the field of the principal wave within the conductor tends t o conccntratc
in a thin skin near the surface. At high frequencies the crnter of the
cylinder is essentially free of field and current, with a correspondingly
small conversion of energy into heat. On the other hand the field and
current distribution within the cylinder in the case of complementary
waves is nearly uniform. The “skin effect” is then external, and the
conversion into heat correspondingly large. As t h e conductivity of the
cylinder is decreased, the at Ienuation of the principal wave increases,
while that of the comp1ement:iry waves is diminished until finally, in the
limit of a perfect dielectric cylinder, we find the principal wave to have
vanished and the complementary waves propagated without attenuation.
9.17. Further Discussion of the Principal Wave.-We have seen that
in the case of a solid conduetirig cylinder only the principal wave plays an
+
532
BOUNDARY-VALUE PROBLEMS
[CHAP.
Ix
important role. Our next task is to put the results of the preceding
section in the form best adapted to numerical discussion. The first step
is to express the coefficients a$ and a; in terms of the total conduction
current in the cylinder, since this is usually the quantity most easily
measured. The current density parallel to the axis at any point within
the cylinder is J , = alEj, and the total current, therefore,
(29)
I
=
J , r de dr
= 27ra&1
Jo(Alr)r dr
eihz-iiot.
BY (261, page 36%
The amplitude of the total current is defined by
I = loeihz--iut 1
(32)
and hence
(33)
Likewise by (14),
(34)
The longitudinal and transverse impedances were defined on page 354
by the relations
E, = Z,Hs,
Ez = -Z,He.
(35)
Then from (30), page 360, or directly from Eqs. (1) to (4), in See. 9.15,
one obtains
(36)
(37)
It will be noted in passing that by (16),
(38)
zf
=
2;
at
r = a.
T h e continuity of the tangential components of E and H at the boundary i s
ensured by matching the radial impedances at r = a.
The surface impedance Z , is defined as the ratio of longitudinal e.m.f.
measured a t the surface of the conductor t o the total current.
(39)
E, = Z,I
at
r = a,
SEC.9.171 F U R T H E R DISCUSSION OF T H E P R I N C I P A L W A V E
533
whence by (1) and (33),
Finally, upon introducing the expressions for af and at into (2) and
(4), one obtains
He;
(41)
=z
Hee =
___
J1(X1r),
27raw~mJl(X1a)
- ik;l Hil)(Xzr)
__--.
27rawp1u1 Hp(X2a)
- ik:I
From these relations and the various impedancc functions the components
of the field can be dctermined at any point.
The approximation U ~ / E ~ >W> 1 is valid for all metallic conductors.
In all practical cases it can be assumed, therefore, that
k2, N iW/.llUl,
(42)
Iki[ >> kz
N
h,
k1=
d G ,
XI
kl.
N
Thus one obtains for the longitudinal impedances
,-
(43)
Outside the cylinder the longitudinal impedance approaches the intrinsic
impedance of the dielectric modium. Inside the cylinder it is a pure
imaginary; the mean axial flow of energy within the cylinder is, therefore,
zero. For
(44)
8:
=
-+Re(e.R+)
=
-+Re(Zj
l ~ O i j 2 )N
0.
The transport of energy along the c:ylinder takes place entirely in the external
dielectric. The internal energy surges back and forth and supplies the
Joule heat losses.
The transverse impedance a t an internal point reduces to
(45)
If r 6
-+ w , the ratio Jo(Xlr)/Jl(Xlr)
3 -i, as in (22) above.
this limit
In
or exactly the re1at;ion obtained in (40), page 522, for a surface wave
propagated over an infinite plane. The conclusions to be drawn from
this result are of the greatest practical importance. If by r we under-
534
BO TJNDAR Y-VALU E PROBLEMS
[CHAP.
Ix
stand a radius of curvature of any highly conducting surface, it is plausible
>> 1, one can approximate
to assume that when r
(47)
where Et and H z are components tangent to the surface and orthogonal
t o each other. To the same approximation it can be assumed that the
intensity of the magnetic field a t the surface of a metallic conductor
differs negligibly from the intensity a t the same point were the conductivity infinite. Consequently a problem involving metallic guides
5
4
3
P
2
1
0
X
FIQ.103.-Plot
J O ( X4)
of ___ = pe-’O
JI (X
4)
can be solved first on the assumption of perfect conductivity. An
approximation to the proper value of tangential component of E is next
found from (47), and the losses of the system then determined by calculating the energy flow into the metal and assuming that this is converted entirely into heat. Examples of this procedure will be given later.
In Fig. 103 are plotted the magnitude and phase angle of the ratio
against 5 = T 6.
The values are taken from the tables of Jahnke
and Emde.’ It is apparent that even for values of x as low as z = 3,
the deviation from the asymptotic value --i is very small. In m.k.s.
units
(49)
&jGG
=
2.81
x
10-3
dZZ,
&
“Tables of Functions,” 2d ed., pp. 316-317, 1933.
=
2-81 x 10-3$$
SEC.9.171 FURTHER DISCUSSION OF THE PRINCIPAL W A V E
535
and the conductivity of metals is of the order of lo7 mhos/meter. The
radius of the wire must be expressed in meters. Consequently a t low
frequencies x may be very much less than one unless the radius is relatively large. Take, for example, the case of a No. 12-gauge copper wire,
A t v = 60 cycles/sec., x = 0.17.
r = 1.026 mm. Then z = 0.022 6.
At v = lo5 cycles/sec., x = 6.9, and from the curves of Fig. 103 it is
clear that for this and all higher frequencies the field near the surface of
the wire is very nearly the same as that in the neighborhood of an infinite
plane.
The external radial impedance (37), with the help of (21), reduces to
Inside the magnetic field is given approximately by
while outside, if r is not too large,
,
Dielectric
Conductor
FIG.104.-Structure
of the field near the surface separating conductor and dielectric.
The structure of the field can be visualized by noting that lZ:l << lZj,
IZ:I >> IZI. Consequently IE:I .<<]El,
>> IEI. Inside the cylinder
the lines of E run almost parallel to the axis, trailing behind thc
wave; outside they leave the surface almost radially, with a very slight
forward slant. The charge on the surface of the cylinder is propagated
as a wave in the axial direction. Thus there are alternate bands of
positive and negative surface charge moving with the field. Lines of E
emerge from a positive band and return to a neighboring negative band.
A few terminate a t infinity. The structure is pictured roughly in
Fig. 104.
536
BOUNDARY-VALUE PROBLEMS
[CIIAP.
Yx
The current distribution within the cylinder is expressed by the
quantity
(53)
which is the ratio of current density at any internal point r to the current
density at the surface. Here x = r z/wplul,as before, and xo = a 2/wplul.
If x << 1, U may be calculated from the power series representations of
the Bessel function; if x>> 1, the asymptotic expansions can be used.
Tables of this ratio are given in Jahnke and Emde and also appear in
many handbooks and texts on electrical engineering. I n Fig. 105 the
ratio of the current densities is plotted
1.0
as a function of the radius for several
values of the parameter 2 0 . For suffi0.8
ciently large values of xo and x the
asymptotic expression for U becomes
0.6
0.4
4
I
1
2’ !0
‘ T d 6
I
&=8
I
IA
-U
t
r
l
0.8
1.0
FIG.105.-Current distribution in a
cylinder of circular cross section.
U =
8
(<-U(a-d
(54)
Jo(x
di),
4)
=
U s -e
6
,
where 6 = 2/2/wppl is the depth of penetration first defined on page 504. The
magnitude of the current density ratio
is, therefore,
(55)
[ u(‘v
,$r
e-y
6,
The current density decreases exponen-
tially from the surface inwards and, to
this approximation, in the same manner as the field near an infinite, plane conductor. If a>> 6, the center of
the cylinder is practically free of current; if a < 6, the current layers interfere with one another. One m a y assume f o r conductors of arbitrary cross
section that the field and current distributions near the surface difler negligibly f r o m those near the surface of an infinite plane provided the radius of curvature i s very much greater than the s k i n depth.
In conclusion we shall derive the classical expression for the alternating-current resistance of the cylinder. According to (39) the surface
impedance 2, is the ratio of the longitudinal component of electric field
at a point on the surface of the wire to the total current. The circuit
equation for a unit length of line is
JO(Z0
5 0
(56)
=adzz.
E,
=
RI
+ L d1
- = (€2 - iwL)I,
dt
where R is the a.c. resistance and L the internal inductance per unit
W A V E S I N HOLLOW P I P E S
SEC. 9.181
length.
537
Then
(57)
and hence
(58)
where again xo = ad=,
and RO= l / m 2 u l is the direct-current
resistance per unit length. The magnitude and phase of the Bessel
function ratio can be taken directly from Fig. 103. If 20 << 1, we
obtain
or
If 5 0 >> 1, as in the case of a wrll-developed skin effect, the asymptotic
expansion of (58) leads to the Raylcigh formula
9.18. Waves in Hollow Pipes.-An
interesting application of the
general theory developed in Sec. 9.15 has been made to the propagation
of waves along dielectric cylinders bounded externally by a medium of
different inductive capacity or by a metallic sheath. The propagation
of elect'romagnetic waves through tubes was discussed first in detail by
Itayleigh, while the theory of transmission along dielectric wires was
investigated some years later by Hondros and Dcbye2 and confirmed
experimcntally by S ~ h r i e v c r . ~Until recently no great practical importance was attached to this work because of thc difficulty in generating
waves of sufficiently high frequency. Progress in the development of
short-wave oscillators has largely overcome this obstacle and the possibilities of such a mode of guiding waves were examined independently
Since
in 1936 by workers at the Bcll Laboratories* and by B a r r ~ w . ~
then numerous contributions 1.0 the subject have appeared.
At extremely high frcquencies the losses in the best available commercial dielectrics are considerable; for this reason there is obvious
RAYLEIGH,
Phil. Mag., 43, 125, 1897.
HONDROS
and DEBYE,Ann. P h y s i k , 32, 465, 1910.
3 ZAHN,
Ann. Physik, 49, 907, 1916; SCHRIEVEH,
Ann. Physik, 63, 645, 1920.
4 SOUTHWORTH,
Bell System Tech. J . , 16, 284, 1936; CARSON,MEAD,and SCHELKUNOFF, ibid., p. 310.
6 BARROW,
Proc. Inst. Radio Enyr::., 24, 1298, 1936.
1
2
538
BOUNDARY- VALUE PROBLEMS
[CHAP.IX
advantage to a wave guide from which all diclectric supports within
the field have been eliminatcd. This can be accomplished by propagating
the wave within a hollow cylinder whose walls are highly conducting.
A further advantage is due t o thc ahscncc in such a system of stray ficld
outside the metal cylinder. At the extreme frequencies that prove
necessary, even a very thin sheath acts as a perfect shield. We may
assume, thcrefore, that the interior of the cylinder is a perfect dielcctric,
while the exterior is an infinite, conducting medium. The allowed
modes of propagation are in general given by (9). If the conductivity
of the external cylinder is finite, there will always be some small degree
of coupling between the transverse magnetic and transverse electric
modes except in the axially symmc%ric case. Practically, however, the
conductivity of a metal is so high t h a t this effect can usually be neglected
and each mode treated as belonging to one type or the other. A wave
that is essentially transverse magnetic is now commonly called a n E
wave (longitudinal component of E) ; a wave that is rsscntially transverse
electric is called a n 11 wave (longitudinal component of H).
We shall consider first the nondissipative case in which the conductivity of the extcminl medium is infinite. The nllowcd modrs of E and
11 waves are given b y the roots of (13) and at, once we comc upon a fundamental diffcrcnce with respect t o t h e earlier problcin of a solid metal
cylinder. It was noted above that the principal branches of the Hankel
functions H t ) ( u ) and H:l)(u) have no zeros and consequently thcir ratio
vanishes only a t v = 0. This lcd to the principal wave, for which
h N kp. When the conductivity of the wire mas finite, crrtnin complementary waves could be cxcitcd but these disappeared immediately
because of their high attenuation. Now, on the other hand, the situation
is just reversed. The func*tioiisJ T L ( da) i d JA(u) have an infinity of real
zeros so long as n is real, arid the zeros of Jn(u)arid Jn+l(u)can be shown
to be interlaced (to separate one anothcr).’ Therefore Jn(u) and
Jh(u) cannot vanish simultaneously and the modes of E’ waves are
determined by
J,(u)
(62)
=
0,
while the modes of H waves arc found from the roots of
where u
Xla = a l / k : - h2. Kow u = 0 is not a root of J o ( u ) = 0.
= 0 for n > 0, but in this case the field itsclf vanishes, as is apparent from Eqs. (I), page 524. Consequently there i s n o
principal wave in hollow pipes. If such guides are to be useful for com=
It does satisfy J,(u)
1
WmsoN, “Theory of Bessel Functions,” Chap. XV, Cambridge University
Press. 1922.
539
WAVES I N HOLLOW PIPES
SEC.9.181
miinication purposes, i t must be shown that the attenuation of t h e
modcs allowed by (62) and (63)) corresponding to the complementary
modes of the prcvious analysic. is relatively small.
For each order n of (62) there is a denumerable set of roots, any one
The propagation factor h,, correof which will be designated ti7*,.
sponding to the indicated mode of E wave is, therefore,
If k l is real and greater than u,,/a, then h,, is real and the wave is
propagated without attcnuation. If k l < UJU,
h,,, is a pure imaginary
and the attenuation is such that the mode is immediately suppressed.
I n this sense the tube acts as a high-pass wave filter. Cutoff takes place
a t the critical frequency v,.
I n a dielectric kl = a 6 1 ; hence,
The same condition can bc expressed in terms of a critical wave length
Anm.
with the understanding that A,, is the length of a wave of frequency
Y,,, propagated in an unbounded medium k l . If the frequency is less
than vnm, or the free wavc length greater than A,, t'he mode is suppressed.
The first few roots of (62) are found t o be
U O=
~
(67)
~
1
~
2
=
1
=
1
2.405,
3.832,
5.136,
utj? = 5.520,
~ 1 ;=
: 7.016,
..........
',
0 =
3 8.654,
.........'
~
.,
............
The longest possible free wave length corresponds t o the mode u . ~
and
~ ,is
(68)
X o l == 2 . 6 1 ~ .
Thus the limit is very roughly the diameter of the tube, so tha.t a hollowpipe wave guide can be of practical utility only in the centimeter wave
band.
The allowed modes of the transverse electric, or H waves, are dctermined in the same manner. Designating the roots of (63) by u;,, wc find
(69)
ui1 = 3.832,
u:, = 1.84,
= 7.016,
u:, = 5.33,
*
*
*
...
,
7
Thc formulas for the propagation constant and critical frequency are
identical with (64) and (65). It appears that of all possible modes, the
540
BOUNDARY- V A L U E PROBLEMS
[CHAP.IX
HI1component of the H wave has the longest critical free wave length,
but in practice this advantage is outweighed by the fact that it also
suffers a more rapid attenuation than some of the others.
The character of the propagation in thc neighborhood of the critical
frequencies can be made clear by graphs of phase and group velocities.
We write the propagation factor of any allowed mode in the form
Then the phase velocity is
where V I = l/&
is the phase velocity in an unbounded medium whose
propagation constant is k l , while the group velocity, according to (17),
page 333, is
(72)
. ,
Phase and group velocities are
related by
(73)
Vnmvmm = ~ 7 .
The relation of phase and group
velocities to the impressed frequency is shown in Fig. 106. One
will note that the phase velocity in
the tube is always greater than that
in the free medium. It increases
with decreasing frequency and
approaches infinity near the cutoff
frequency vnm. The group velocity
decreases with decreasing frequency
Vnrn
v
FIG. 106.-Phase and group velocities in and is zero at v = vnm. The dispersion is normal and becomes very
a hollow pipe. 31 = l / G p ; .
large in the neighborhood of vnm.
The components of the field within the tube can be written down
directly from (1) and (2), page 524. In virtue of (70) and (65), we have
(74)
nm
XI = kl V= Unm
-.
V
0
541
W A V E S I N HOLLOW PIPES
SEC.9.181
For any mode of E-wave type, the field is
For any mode of H-wave tJype,one finds
The arbitrary amplitudes a,,,,,, and bn, will be determined by the
initial excitation. There are theorems analogous to those of Fourier
analysis which govern the representation of a function f ( r / u ) in terms of
Bessel functions. Suppose that f(t) is a function defined arbitrarily in
It2j((t)j dt exists.
the interval 0 It I 1, and that,
Let
(79)
r / u be any internal point of an interval (xl, z2) such that
has limited total fluctuation in (zl,
Q).
Then the Fourier-Bessel series
Let
0
II: =
< 2 1 < x 2 < 1 and such that f(t)
(80)
2
m =1
1
a n m Jln(2Unm)
:= 2
[A.
+ 0) + f(x
-
O)]
542
[CHAP.IX
BOUNDA R Y-VALI'E PRO BLBMS
converges and its sum is the right-hand side of (80).l Of the modes
which are excited, only those will be propagated whose propagation
factors hnmare real.
It may be noted in passing that at, the critical frequency v = vnm, the
energy flow takes place entirely in thc transverse plane. There is no
flow parallel to thc axis of the cylinder. The longitudinal impedance of
the E waves is
which approaches zero as v
--+
vnm, while for the
H waves
Let us calculate next, the total energy flow in the axial direction.
time average of this quantity is
".=so"J-
The
8, r ds dr,
where
8, = ;R~(E,& - E&J.
(84)
For the E waves
(85)
8, =
'"p" ,&$-(y)'(L)*{[J:,
~.
Vnm
u . ~ ) ] sin2 no
cos2
The sin2 n0 and cos2 n0 factors reduce l o T nftcr intcgration. Ilct r = ax.
Then in virtue of the reciirrence relations (24) arid (25) on page 359 and
the condition Jn(unn,)= 0, we obtain
and, therefore,
WATSON,loc. cit., Chap. XVIII
SEC.9.181
WAVES I N HOLLOW PIPES
543
Likewise for t'he H waves,
By using t,he same recurrence formulas and the rc.lation JA(ub,)
this integrates t o give for the total energy flow
=
0,
There remains the question of dissipation. If the conductivity of the
walls is finite, the propagation factors hn, have a small imaginary part
which gives rise to an attenuation of the field as it travels along the axis
of the tube. These complex propagation constants must now be determined from Eq. (9). The calculation is simplified to some extent by
the fact that thc Hanlicl functions may be rcplaccd by their asymptotic
representations, owing to the 1:zrgc conductivity of the outer metal
cylinder; thc roots of (9) will differ, therefore, by a very small amount
from the values obtained from (62) and (63) for the case of infinite
conductivity. They can bc determined without grcat difficulty b y
expanding the functions J,(u) and J',(u) about the points u,, and ukm.
It will be morc instructive a t this point to calculate thc attenuation
by the approximate method outlined on page 534. If the conductivity
is infinite, the tangential components E , and EQare zero a t r = a ; if the
cylinder is metal, the skin depth 6 will certainly be very much less than
the radius a. Then to a very good approximation, a t r = a
Thus we can compute the radial flow of energy into the metal; since this
outflow is a t tho expense of the flow down the tube, we can find th e
attenuation. The propagation factor is written in the form
(91)
h,,
+
= anm
The mean radial flow per unit length at the surface of the cylinder is
where
544
BOUNDAR Y - V A L U E PROBLEMS
[CHAP.IX
For the E’ waves
while for the transvcrse electric, or €1 waves, one obtains
To the same approximation the axial flow is still given by (87) and
(89) when thesc expressions are multiplied by an attenuation factor
exp(-22pn,z) or exp(-2&,,z).
The rate at which
decreases along
the axis is
vz
Consequently the attenuation factors of the E waves are given by
These results are valid only when v is greater than the critical frequency.
If either a or v approaches infinity, the attenuation factors of the
E waves approach the limit Po, where
(99)
which increases in the normal manner as the square root of the frequency.
In Fig. 107 the attenuation ratios Pnm/poand 2pArn/Po are plotted as functions of vO1/v for the two lowest modes of E and H waves. yo1 is the
critical frequency of the fundamental Eol wave. In the neighborhood of
the critical frequencies the attenuation increases with great rapidity.
Most notable is the behavior of the attenuat,ion curve of the Hol wave,
which decreases with increasing frequency and approaches zero as v -+ 00.
A further investigation shows, however, that this anomalous behavior is
not common to all hollow tubes, but is peculiar to certain symmetrical
SEC.9.191
PROPAGATION C O N S T A N T OF C O A X I A L L I N E
545
cross sections, such as the circle and square. The stability of the lower
modes with respect to small changes in cross section has been discussed
FIG.107.-Attenuation
ratios of E and N waves.
v01 =
critical frequency of Em wave.
by Brillouin’ and by Chu2in connection with the propagation of waves in
hollow tubes of elliptic and rectangular cross section.
COAXIAL LINES
9.19. Propagation Constant.--The most serious disadvantage of the
hollow-tube transmission line is the absence of a principal mode of
oscillation and the consequent limitation to wave lengths which are of
the order of the diameter of the tube or less. This difficulty can be
overcome by introducing an axial conductor. A principal mode then
exists which allows propagation at all frequencies in the space bounded
by two concentric cylinders. The coaxial line shares with the hollow
tube the advantage of confining the field to its interior and thus eliminating interference with external circuits a t high frequencies. Its losses,
on the other hand, tend t o be larger due t o skin effect in the central
conductor and imperfect insulation a t th e points of support.
In practice the outer cylinder is,, of course, of finite thickness. At low
frequencies the skin depth may exceed this thickness, in which case the
field penetrates to the outside. The technical problem of “cross talk”
BRILLOUIN,
Electrical Communication,16, 350, 1935.
Proc. Znst.
CHU,J. Applied Phys., 9, 583, 1938. See also CHU and BARROW,
Radio Engrs., 26, 1520, 1938.
1
546
BO U N D A R Y-VALU E PROBLEMS
[CHAP.
IX
between telephone cables and of interference caused by currents induced
in communication lines through shielded power cables is a very serious
one. However, a n exhaustive discussion of such problems has no place
here, for i t is our purpose merely t o illustrate the application of electromagnetic theory. Coaxial lines now play a very important part in radio
and telephone communication a t frequencies
greater than 100 kc./scc. and the analysis of
this section will be restricted to frequencies
of this order.
(3)
We assume, therefore, that the skin depth
is very much less than t h e thickness of the
outer conducting cylinder, which can hence be
assumed t o have a n infinite outer radius. I n
Fig. 108 t h e concentric cylinders (1) and (3) are
FIG. 108.-Cioss
section of
conductors, while the intermediate region (2)
a coaxlal line.
is a dielectric to which, however, it may be
necessary to ascribe a small conductivity. The inner and outer radii
are a1 and u2. Only the symmetric, transverse magnetic modes ( E waves)
need be considered. Asymmetric modes can be excited, but unless the
wave length is of the same order as the difference a2 - a ] , their propagation constants will be pure imaginaries and they will be immediately
damped out. The components of the field are taken directly from (36)
and (37), page 361, for the case n = 0. Let I be the total current in the
central conductor and 1 0 its amplitude.
I
(1)
= IOCzhz--iot
Then exactly as in See. 9.17, we have for the central conductor, T
< al,
The longitudinal and radial impedances are defined by the relations
(3)
E,
=
Z,Ns,
E,
=
-Z,Hs.
For the region (1) we have, as in Scc. 9.17,
(4)
To construct a solution appropriate to the intermediate region (2)
which can be fitted a t both internal and external boundaries, two independent Bessel functions must be used. Which two is not important,
since every solution can be represented as a linear combination of any
pair of independent solutions. If the radii al and u2 are very large, the
region (2) is bounded effectively by two plane surfaces. To represent a
SEC.9.191
547
PROPAGATION C O N S T A N T OF C O A X I A L L I N E
solution under these circumstances one would naturally use sine and
cosine functions. Now it will be recalled (page 359) that if the argument
p
is very large, t h e function
6J o ( p ) -+
.$
$
cos
function of the second kind .\rp N o ( p ) --+
quently we shall write for the region al
(p -
sin
(p
i))while the
-
Conse-
< r < a2,
in which A and B are undetermined coefficients. The components of
E(2)are then obtained from (3) and t,he relations
I n the outer conductor the proper behavior of the field a t infinity is
ensured by choosing a Hankel function, as in See. 9.15. If r > a2,
The coefficients A , B , and C are next determined from the boundary
conditions. At both r = a1 and r = a , the x-components of E and the
@-componentsof H are continuous. These two conditions are equivalent
to the continuity of the radial impedance functions. Hence
(9)
or
at r
at r
=
=
al,
a,,
548
BOUNDARY-VALUE PROBLEMS
[CHAP.IX
The roots of this determinantal equation are the characteristic values
h h , (m = 1, 2, 3 . . . ), which fix the allowed symmetric modes of
propagation.
9.20. Infinite Conductivity.-The complexity of (11) is such that one
must resort to approximate methods of determining the roots. Let us
consider first the relatively simple case in which the inner and outer cylinders are perfect conductors. We recall that 1; = 12; - h21 X23 -- k23 - h2.
The imaginary part of h must remain finite, otherwise the wave would not
be propagated. Hence, as the conductivity approaches infinity, XI -+ kl,
X3 3 k 3 ; at the same time the absolute values of both lcl and k, approach
infinity. Then (11) reduces to
If X2 is very small, Jo(X2al)= Jo(h2a2)
N 1; for the functions No we
use the approximation (12), page 358. It is then apparent that the
principal root of (12) corresponds to A 2 = 0, or
h
(13)
= k2.
There is a principal wave whose propagation constant i s that of the medium
bounded by the two infinitely conducting cylinders.
There are also complementary waves. If Xz is very large, the
asymptotic representations (18) and (19), page 359, are valid and (12)
takes the form
(14)
sin X2(a2 - al) N 0,
whose roots are
If k 2 is real and greater than
mr
---J
a2
propagated without attenuation.
- a1
then horn is real and the wave is
If, however, k 2 is less than
-7
mr
- a1
then hornis a pure imaginary and the attenuat,ion is such that the wave is
immediately suppressed. All t,his is comparable to the hollow tube discussed in Sec. 9.18. The critical frequencies occur at
a2
(16)
vom
=
and critical wave lengths a t
-
mr
2r(a2 - all
4;~'
SEC. 9.201
I N F I N I T E CONDUCTIVITY
549
I n communication practice the distance u2 - ul usually amounts to a
few centimeters at the most and complementary waves are completely
ruled out. The same considerations apply to the asymmetric modes,
so that henceforth we shall dcal only with the principal wave.
I n (5) put A = 0 and note that as XZ -+0, N1(XZr) -+ -22/?rX2r. Then
for sufficiently small values of XZ the magnetic field between the conductors is equal t o
The major part of the conduction current follows the surface of the inner
and outer cylinders. If the intervening dielectric has some conductivity,
there will also be a transverse component of current from one cylinder to
the other, with a resultant decrease in current amplitude along the line.
The longitudinal current carried by the inner cylinder can be found
directly from Amphre’s law. It is 2 a ~ l H s ( ~if ) ,He(2) is measured a t
r = al. Hence
(19)
where l ois the conduction-current amplitude along the inner conductor.
The transverse voltage a t any point along the line is by definition
(23)
.while the ratio of the transversr volt,age to the current in a n infinite line
is called its characteristic impeda,nce 2,.
where 2 0 = o p z / k z is the intrinsic impedance of a medium whose propagation constant is kz, as derived in (83), page 283.
Consider for a moment the distributed parametcr line represented
schematically in Fig. 109. I, is the series inductance; R , the series
resistance; C, the shunt capacity; and G, t)he shunt conductance; all per
unit length of line and uniformly distributzd, The equations governing
550
[CHAP.IX
BOUNDARY- VALUE PROBLEMS
t h e distribution of current and transverse voltage along the line are
easily shown to he
The quantity Z
=
Iz
FIQ.109.--Schernatic
-
i w L is the series impedance; Y
=
G - i d is the
representation of a distributed parameter transmission line.
shunt admittance of the line. Upon solving ( 2 5 ) and ( 2 6 )simultaneously,
one obtains the so-called “ telegrapher’s cquation,”
zv - (LG + RC) 8V - RGV
a2v
- LC adxat2
at
= 0,
with an identical cquation for thc current,. The propagation constant
and the characteristic impedance < Z T as in Sec. 5.6.
is i
If now we differentiate ( 2 3 ) with respect to x , the result is
av
a2
- -- iwp2
In - I.
27r
al
I n the prcsent instance thr scries rcsistance is zero; consequently, i t
follows that the serics inductance per unit length of the coaxial line is
dx
L
(30)
p2
a2
= -111-
2a
henrys/meter.
al
The propagation constant is k 2 , from which the shunt admittance is
found to be
The shunt conductance and shunt capacity are, therefore,
2auz
G = mhos/metcr,
a2
In a1
C
2TE2
= __
a2
In a1
farads/metcr.
SEC.9.211
F I N I T E C ON DUC T I VITY
551
It will be observed that L and C satisfy the fundamental relation
(33)
LC
=
€@2.
9.21. Finite Conductivity.--In practice the cylindrical guides are of
metal whose conductivity, alhhough riot infinite, is very large. To a
very good approximation the distribution of the field between the coaxial
cylinders is the same as in thc case of perfect conductors, but the propagation factor must be modified to take into account the additional attenuation. To simplify the problem it will be assumed that both inner and
outer cylinders are of the same material, as is ordinarily the case. Then
(34)
Since X2, although no longer zero, must still bc very small, the approximations (12), page 358, are valid.
y = 1.781.
Then with reference to (11) we havc
(37)
We have admittcd the possibiIit,y of a small conductivity in the intermediate dielectric k z , liut this may be neglected in estimating orders of
magnitude. Thus assuming for the moment that k 2 ili w / c , we have for
copper
Remembering that a1 and a2 are measured in meters and hence are also
small quantities, it is apparent that the dcnominnt,ors of the determinantal
equation (11) can be rcplaced b y unily and, therefore,
whence we obtain
552
But A;
BOUNDARY-VALUE PROBLEMS
=
[CHAP.
rx
ki - h2 and so finally
=
k,(l - Ak).
The approximate value of the ratio A / B as expressed in (39) can now
be introduced into (5) to determine the magnetic field intensity a t any
point in the dielectric. One obtains
A logarithmic term has been dropped as obviously negligible. We assume
that the conductivity u2 of the dielectric is very small, if not zero. Then
the second term of (43) increases essentially with the three-halves power
of the frequency. An easy computation will show as in (38) that unless
a1 is very small, or r very large, this second term is entirely negligible with
respect to the first, even a t frequencies as high as 10’0 cycles/sec. The
= ~I ) a t the surface
constant B is determined by the condition 2 7 r ~ ~ H &
of the inner conductor, so that very nearly
(44)
as in the previous case of perfect conductivity. Physically this mean4
that the conduction and displacement currents in the dielectric ar”
negligible wit>hrespect to the conduction current carried by the inner
conductor.
The radial electric field intensity is
(45)
and the transverse voltage
This gives for the characteristic impedance
(47)
SEC.9.211
553
FINITE CONDUCTIVITY
the series impedance Z and shunt admittance Y of the line follow directly
from the relatims
h2 = - Y Z ,
(48)
2
-ihZ,,
=
z:
=
z
7’
y = -i-.h
zc
The approximate value of the shunt admitt,ance-and
it is an exceedingly good approximation-is
in practical cases
(49)
a1
a1
or exactly the same expression as was obtained in t,he case of infinitcl.:
conducting guides. For the series impedanw, on thc other hand, we
obtain
(50)
Z
N
-?
’ ?!??
2T
2
(1 -- A k ) z I n aa1
=
R - iwL.
The behavior of the line from an engineering standpoint is now completely determined by the Eqs. (25) and (26).
If x1 >> 1, as is usually the case, one can replace the cylinder functions
by their asymptotic values
(51)
Then
This correction term is small, so that (Ak)2 can be discarded.
series resistance and inductance are now
The
For copper guides, ~1 = 5.8 x lo7 mhos/meter, and we obtain
R N 0.416 X 10-7
(54)
LN2
x
ohms/meter ,
6
[In - + 30.2 .$(;
“
__
+
k)]
henrys/meter.
554
30 U N D A R Y-VALUE PROBLEMS
[CHAP.IX
The atteiiuation is calculated from the propagation factor
h
=
kZ(l - Ak)
= a
+ ip.
At very high frequencies the losscs in the dielectric spacers rise rapidly,
but when Y is less than a mcgacycle the shunt conductance of a wellconstructed line can often be neglected. If we assume that k, = w
the attenuation factor reduces to
6,
I n engineering practke it is cust,omxry t o define the at'tenuation in terms
of the logarithm of t'lic power rat,io. T e n times the logarithm to the base
10 of the ratio of output power to i n p u t power expresses the attenuation in
decibels. Since the power ratio is exp(-22px), the attenuation per unit
length of line is
(56)
Ndb
=
-2Op log,,
E
= -8.686
6 decibels/meter.
OSCILLATIONS OF A SPHERE
9.22. Natural Modes.-Thc problems of the plane and the cylinder
treated prrviously in this chapter concern infinite surfaces which serve
as guides to traveling waves. The sphere, on the other hand, is an
example of a body bounded by a closed surface within which there can
be set up a system of standing waves.
We shall suppose t,hat the sphere, of radius a and characterized
electrically by t.lie constant k l , is embedded in a n infinite homogeneous
medium k?. The ficld will bc represented in terms of the spherical,
vector wave fnnct'ions discussed in Secs. 7.11 t o 7.14. According t'o
(121, Page 394,
Here the subscript n stands for all indices, of which there are in the present
case three. The functions may be odd or evcn, and the indices m and ?z
will determine the number of nodes with respect t o the spherical angles
4 and 8. It will be recalled that the radial component of every function
M e is zero. Hence, if the coefficients a , are all zero, only the b, being
Omn
excited, the field has a radial component of E but the magnetic vector
is always perpendicular to the radius vector. There must be, therefore,
a distribution of electric charge on the surface of the sphere. T h e
oscillations whose amglitudes are represented by the coeficients b, are of
SEC.9.221
555
NATIIRAL MODES
clcctric type. They have in the past been referred t o also as transverse
magnetic, or E waves. If, on ihc other hand, only the a, are excited, the
field is such as would bc produced by oscillating magnetic charges on the
surface of the sphcre and the field is said to be of magnetic type. These
oscillations may also be said to be transverse electric, or H waves.
Let us consider first a n oscillation of magnetic type.
E
H
(2)
+ AomnMomn),
+
= - (Aem&Lmn
=
k
--
(AemnNemn
ZWP
The coefficients A ,
AOm,Nm,)-
are as yet arbitrary and the frequcncy
w
will shortly
Om”
be determined by the boundary conditions.
The functions M e
and
omn
N,
must now be chosen such that the field is finite a t the origin and is
omn
regular at infinity.
By (11) and (12), page 416, we have when R
< a:
E”R = 0)
(3)
(4)
where Pin is the tesseral harmonic
(5)
YLn
=
(A:,, cos m+
+ A:,,
sin m+)Pr(cos 0))
and the prime in (4)denotes differentiation with rcspcct to the argument
klR.
The external field in the region R
arid (4) Yhn by Yh,,
(6)
Y;,
=
(A;,, cos md
> a is obtained by replacing in
(3)
+ AZmnsin m+)Pp(cos 4 ) ,
kl and p 1 by lcz and p z , and j,(klR) by hil’(knR).
At the surface of the sphwe the continuity of the tangential components leads t o the relations
(7)
Esi
Hsi
=
=
Elie,
Hoe,
E+i= E+e,
HBi= H4e,
(r
=
a),
556
BOUNDARY-VALUE PROBLEMS
[CEfAP. IX
all four reducing to the conditions
Akmnjn(k la)
(8)
ha' ( h a ) ,
=
1
1
Pl
Pz
- A;,,[kla jn(kla)]' = - A;,,[k2a hil'(kza)]',
with an identical pair relating the odd coefficients Ad,, and A&,,. Let
now
(9)
p =
kza,
kl = Nkz,
kla = Np.
Then the boundary conditions (8) are satisfied only by a discrete set of
characteristic values pns which are the roots of the transcendental equation
Since the allowed values of p belong to a discrete set, it follows that
there is a corresponding set of natural frequencies, or modes of oscillation.
According to ( 9 ) ,
(11)
P t = (w:,€zPcCz
+ iwn,Pzaz)a2;
hence, the natural frequencies of the magnetic modes are
As in every characteristic-value problem, the amplitudes of the allowed
modes are determined by the initial distribution of the field. Suppose
that within the sphere a t the instant t = 0 the radial component of Hi is
a specified function of the form fl(O, 4 ) j 2 ( R ) . We note first that thc
natural frequencies wnS are independent of the distribution in 4. The
coefficients Ad are uniquely determined by f(0, 4 ) as in (18), page 403.
Om-
The series
2 Pi,,then
, depends in the desired manner on 0 and 4
00
where
=
m=O
a t t = 0. But there are an infinite number of such series, one for each
root s of (10). Each of t,hese series can be multiplied by another coefficient A2 and the summation extended over s.
SEC.9.221
NATURAL MODES
557
in which the Af are to be determined so that (14) matches f2(R) at
t = 0. The representation of arbitrary functions in Fourier-Bessel
series is treated by Watson,‘ and the theory can be adapted to the spherical Bessel functions jn(klB). Finally, we note that before summing one
can multiply the particular solutions by klR and then differentiate with
respect to the same variable. No change is necessary in the coefficients
Ad , but the summation over s is modified to fit the conditions imposed
Omn
on either Hoi or H,;. A fioltl has now been determined which fulfills the specified conditions within the sphere at t = 0. The factor
exp(--iiw,,t) fixes the behavior a t all subsequent times, and in virtue
of the uniqueness theorem of Sec. 9.2 the field is also determined a t all
external points.
The oscillations of electric type are quite indepcndcnt of the magnetic
modes, but the fields and natural frequencies arc determined in exactly
the same manner. I n place of (2) we have in this case
E
(15)
H
=
=
-(BemnNemn
k
+
BomnNomn),
-- ( B , d L ,
ZWP
and the field a t any internal point R
+ BomnMomn),
< a is
The amplitudes of the internal and external fields are relatcd by thc
boundary conditions
which lead to the transcendental equation
1
WATSON,loc. cit., Chap. XVIII.
558
BOUNDARY-VALUE PROBLEMS
[CHAP.IX
for the characteristic values pns. The natural frequencies wnSare obtained
from (12).
9.23. Oscillations of a Conducting Sphere.-We
shall suppose the
sphere to be conducting, but the medium in which it is embedded to be a
perfect dielectric.’ Consider first the limiting case in which
-+ 0 0 .
The two determinantal equations (10) and (19) now reduce to
( T ~
(20)
hp(p) =
0,
[ p @)(p)]’
=
0,
whose roots are respcctivcly thc chnractcristic values of the magnetic
and electric modes. However, caFh function h i l ) ( p ) can be represented
as a n exponential times a polynomial, See. 7.4, page 405. The roots in
question are the roots of these polynomial* and a r e , therefore, finite in
n u m b e r for each f u n c t i o n of finite order 1%. Take, for example, the first
electric mode for which n = 1. By (31), page 405,
[ p h\l)(p)]’ =
i
-->
e”P(p2
P
+ ip - l),
+
and the desired roots are those of p 2
i p - 1 = 0, or p I 1 = 0.86 - i0.5,
p1z = -0.86 - i0.5. Since (TZ = 0, the natural frequencies are given by
(23)
ans
=Pns
U
6;
hence, in the casc of the lowcst electric mode
The amplitude is reduced to the eth part of its initial value in the time
required for a wave to travel a distance equal to the diameter of the
27r
sphere. The wave length in the external medium is X = o.86 a = 7 . 3 ~ .
Since the conductivity is infinite, the damping must be attributed entirely
t o loss of energy by radiation. This loss is very rapid, even in the case
of the lowest mode, and in the example just cited the amplitude is reduced
to the eth part of its initial value in a time equal to the 0.27th part of a
complete period.
The first few charactcristic values for the problem of the infinitely
conducting sphere are tabulated below. The minimum damping factor
is that of the lowest electric mode.
DEBY YE, Ann. Physik, 30, 57, 1909.
SEC.
9.231
O S C I L L A T I O N S OF A CONDUCTING ASPHERE
553
Electric rriotlcs
-1
+ i0.86
-0.50
-0.50
+
-
i0.86
-1 .50
i0.86
-1.50 - i0.86
-1 . G O
-0.70 f i 1 . 8 1
-0.70 - Zl .81
-2.26
-1.87
-1.87
-2.17 + i 0 . 8 7
-2.17 - i0.87
-0.83
i2.77
- 0 . 8 3 - i2.77
+ i1.75
-
il.75
+
When the sphere is of metal, the conductivity is finite, but N is very
large none the less. We may expect, therefore, to find roots of (10)
and (19) in the neighborhood of those of (20). If l N p l >> 1, the
asymptotic representation of (10) is
Let the deviation from any root pns of the equation h k l ) ( p ) = 0 be Apns.
The right sidr of (25) is expanded in a Taylor series about p = pns.
Retaining only the lincar term, we obtain
which determines the corrections to be applied to the natural frequencies
of the magnetic modes. Likewise for the electric modcs, we replace (19)
by
Let pns be a root of
conductivity.
(28)
[ p hkl)(p)]’ =
0 and Apns the deviation due t o a finite
4-
[ P hA1)(p)I’ = [ P h L 1 ) ( p ) ] ~ = = p n s A ~ n a *
*
.
*
From (4),page 400,
(29)
P[P
hil’(P)l”
Apn8 N
~
+
[P2 -
4%+ l)lhA1’(P) = 0;
hence,
2
(30)
Pns
_
PLZNPZ
- n(n
+ 1) tan
_
_
(Np,,
-n
*T). 2
560
BOUNDARY- VALUE PROBLEMS
[CHAP.IX
Thus, in the case of the fundamental electric mode, one obtains
since the tangent of a very large complex number is +i.
metal sphere
In the case of a
It will be recalled finally that in our study of the natural modes of
propagation along a cylinder, the principal wave corresponded to the
condition: Xla very large inside the cylinder and Xza nearly zero outside.
A complementary set of waves was shown t o exist, however, for which
Xla was small. The same circumstances arise in the case of the sphere.
Debye has pointed out that sets of roots can be found for (10) and (19)
such that N p is a finite number even when N itself is very large. If the
conductivity of the sphere is finite, the damping of the oscillations is
extremely rapid, as was the case of the complementary cylindrical waves.
On the other hand, if the sphere is a perfect dielectric, the characteristic
values are real: there i s n o damping, whence it appears that these modes do
not radiate.
9.24. Oscillations in a Spherical Cavity.-So far as it is possible to
foresee a t the present time, ultrahigh-frequency radio circuits will be
reduced eventually to hollow-tube transmission lines and cavity resonators. The latter are electromagnetic analogues of the Helmholtz
resonators used in acoustics. Such systems do not radiate; one may,
therefore, ascribe to them definite values of electric and magnetic energy,
and consequently effective inductance, capacitance, and resistance
parameters which depend solely on the configuration. The damping
results from a penetration of the field into the metal walls a distance
equal approximately to the skin depth.
The character of the oscillations in a resonator can be illustrated by
the example of a spherical cavity. Let us suppose now that the interior
of the sphere is a perfect dielectric, so that k l = w 6,
while theexternal
region has an infinite conductivity. Then N = 0, and the characteristic
values of the magnetic and electric modes are determined respectively by
(33)
j,(kla)
=
0,
llclajn(kla)]’ = 0.
Let unsbe the roots of j,(kla) = 0. The lower of these mow are
(34)
Since kla
= w
u11
= 4.50,
~
=
1
2
6a, the
5.8,
U I ~=
7.64,
...
characteristic frequencies of the magnetic
SEC.9.241
OSCILLATIONS I N A SPHERICAL CAVITY
56 1
modes are given by
(35)
Formally this is identical with Eq. (65), page 539, for the characteristic
frequencies of a hollow pipe. The free wave length, or wave length a t
the same frequency in an unbounded medium, is
The longest allowable wave length among the magnetic modes is
= 1 . 4 ~ . In this case the field has only the components
E,, H E , and Ho, and the charge flows along parallels of latitude. The
All = 2?ra/4.5
H
E
FIG.110.-The
lowest magnetic mode of oscillation i n a spherical resonator, showing the
field in a meridian plane.
distribution of the electric and magnetic lines of force in a meridian plane
is illustrated by Fig. 110.
The roots of the second of Eqs. 33 will be denoted by ut and the
characteristic frequencies of the electric modes by vL8. Apart from a
prime, the formulas (35) and (36) remain valid and a calculation shows
that the free wave length of the fundamental electric mode is
Xil --
2ra/2.75
=
2.28a.
This is the lowest oi all possible modes, both electric and magnetic.
The axially symmetric case corresponds t,o an oscillating flow of charge
along meridian lines from pole to pole; the magnetic lines follow parallels
of latitude. The lines of force in a meridian plane are shown in Fig. 111.
At any internal point of tht: sphere, when m = 0, n = 1, we have from
(16) and (17):
562
[CHAP.IX
BOUNDARY-VALUE PROBLEMS
(37)
where w:l = 2.75/a 4G.
The surface current density K a t any point on the wall of the cavity
is K = n x H, where n is a unit vector directed radially inwards and H is
measured on the cavity side of the
....... surface. Since there is only the single
...
.. . . ..
component H+, the current density is
......
... .
.
... ... . . .......
directed along meridian lines. The
.
total charge crossing the equator per
I
second is
[1.
(38) I = 2aaK
I-
E
H
FIG.111.--The lowest electric mode
of oscillation in a spherical resonator,
showing the field in a meridian plane.
An equivalent inductance and resistance for the oscillating system can
be defined in terms of this current.'
The average magnetic energy within
the cavity is
The integral can be evaluated without difficulty either by applying
general formulas for the integration of cylindrical Bessel functions, or
directly by noting that
(40)
We find
j2(x)x2 dx
(41)
whence
(42)
1
=
x3
2 [31(x)
"
- jo(~>j2(,)1,
- 2n
T = - e1A2a3X 0.054.
3
HANSEN,J . Applied l'hys., 9, 654, 1938.
Next Page
SEC.9>25]
E X P A N S I O N OF T H E DIFFRACTED FIELD
563
Next we define an inductance L by the formula
F = +LP = $LIT.
(43)
If the current I in (43) is arbitrarily taken to be that which crosses the
equator, the equivalent inductance is by (38), (42), and (43)
L =0 . 0 7 7 ~ ~ ~
henrys.
Thus far it has been assumed that the conductivity of the walls is
infinite, in which case EO vanishes a t R = a and the attenuation is zero.
If, however, the walls are metallic, the tangential component of E a t the
surface may be obtained from the approximation (47), page 534. A t
R=a
(44)
There is now a radial flow of energy into the metal whose mean density is
The energy lost per unit time is obtained by integrating (46) over the
surface of the sphere.
where 6
=
d+
is the skin depth defined on page 504. An equiva-
wlIP2cz
lent series resistance 6i is expressed by
(48)
whence
w = a72 = +@IT,
(49)
The parameter Q , defined as the ratio of inductive reacta,nctt to series
resistance, is often used as a measure of circuit efficiency. Tho Q of the
cavity oscillating a t its fundamental frequency is
DIFFRACTION OF A PLANE WAVE BY A SPHERE
9.26. Expansion of the Diffracted Field.-A periodic wave incident
upon a material body of any description gives rise to a forced oscillation
of free and bound charges synchronous with the applied field. These
constrained movements of charge set up in turn a secondary field both
Previous Page
564
BOUNDARY-VALUE PROBLEMS
[CHAP.I X
inside and outside the body. The resultant field at any point is then
the vector sum of the primary and secondary fields. In general the
forced oscillations fail to match the conditions prevailing at the instant
the primary field was first established. To ensure fulfillment of the
boundary conditions at all times, a transient term must be added, constructed from the natural modes with suitable amplitudes. Such transient oscillations, however, are quickly damped by absorption and
radiation losses, leaving only the steady-state, synchronous term.
The simplest problem of this class and at the same time one of great
practical interest is that of a plane wave falling upon a sphere. As in the
preceding section, we shall suppose the sphere of radius a and propagation constant kl to be embedded in an infinite, homogeneous medium k2.
A plane wave, whose electric vector is linearly polarized in the x-direction,
is propagated in the direction of the positive z-axis. The expansion of
this incident field in vector spherical wave functions has been given in
(36), page 419.
where Eo is the amplitude and
the prime denoting differentiation with respect to the argument k2R.
The induced secondary field must be constructed in two parts, the one
applying to the interior of the sphere and the other valid at all external
points with the necessary regularity at infinity. By analogy with the
problem of plane boundaries discussed in Secs. 9.4 to 9.9, these two parts
will be referred to as transmitted and reflected waves, although such
terms are strictly appropriate only when the wave length is very much
smaller than the radius of the sphere. Let us write
SEZ.9.251
EXPANSION OF THE DIFFRACTED FIELD
valid when R
> a, and
which hold when R
< a.
565
The functions mL3) and ni3) are obtained by
*I”
replacing j,(k2R) by hA1)(k2R)in (2) and (3). kl replaces k , in (5).
The boundary conditions at h! = a are
il
+
E,) = i, X E,,
x (Hi t HT)= il x Ht.
il X (Ei
(6)
These lead to two pairs of inhomogeneous equations for the expansion
coefficients.
a5n(NP) - a3L1’(p) = j n b ) ,
(7)
IL2ah“pjnWP)l’
-- P l a 3 P h?(P)l’ = P l b jn(PIJ’7
PZNb;jn(Np) - P ~ V
K ( P=)~ i j n ( ~ ) ,
(8)
b f i N p j , ( N ~ ) l ’ - N b i b h2’(~)1’= NLPjn(p)I’,
where again
kl = Nk2,
p = kza,
kla = N p .
This system is now solved for the coefficients of the external field
(9)
If the conductivity or inductive capacity of the sphere is large relative
to the ambient medium, and at the same time the radius a is not too small,
then lNpl >> 1 and (10) and (11) can be greatly simplified by use of the
asymptotic expressions
In this case
(13)
Since h;l)(p)
form
(1 4)
= jn(p)
a:
+ i n,(p),
N
these coefficients can also be put in the
ieiy- sin yn,
b,’
N
i e i ~ hsin y’
n,
566
BOUNDARY- VALU E PROBLEMS
[CHAP.IX
where
To sum up, we have found that the primary wave excites certain
partial oscillations in the sphere. These are not the natural modes
discussed in Sec. 9.22, for all are synchronous with the applied ficld.
These partial oscillations and their associated fields, however, are of
electric and magnetic type for the same reasons that were set forth in
Sec. 9.22. The coefficients a, are the amplitudes of the oscillations of
magnetic type, while the 6 , are the amplitudes of electric oscillations.
Whenever the impressed frequency w approaches a characteristic frequency wns of the free oscillations, resonance phenomena will occur.
Now the characteristic frequencies of the magnetic oscillations have been
shown to satisfy (lo), Sec. 9.22. But this is just the condition which
makes the denominator of a;, in (10) above, vanish. Likewise the
natural frequencies of the electric oscillations satisfy (19) in Sec. 0.22,
which is the condition that the denominator of b; shall vanish. Note,
however, that the wns are always complex, whereas the frequency w of the
forcing field is real. Consequently the denominators of a; and b; can
be reduced to minimum values, but never quite to zero, so that the
catastrophe of infinite amplitudes is safely avoided.
In Fig. 112 the electric and magnetic lines of force are shown for the
first, four partial waves of electric type. The drawings are reproduced
from the original paper by Mie.l
The incident wave is linearly polarized with its electric vector parallel
to the z-axis. A t very great distances from the sphere the radial component of the secondary field vanishes as 1/R2, while the tangential
components Ere and E,, diminish as 1/R. In this radiation zone the
field is transverse to the direction of propagation. Moreover, the components EO
, and E , are perpendicular to each other and differ, in general,
in phase. T h e secondary radiation f r o m the sphere i s elliptically polarized.
There are two exceptional directions. When (b = 0, we note that
E,, = 0, and when (b = 9/2 we have E,o = 0. Consequently, when
viewed along the x- or the y-axis, the secondary radiation is linearly
polarized. Inversely, if the primary wave is unpolarized, as in the case
of natural light, the secondary radiation exhibits partial polarization
dependent upon the direction of observation. This effect has been
studied in connection with the scattering of light by suspensions of
colloidal particles. The most complete numerical investigation of the
problem thus far in the visible spectrum has been made by Blumer.2
MIE, Ann. Physik, 26, 377, 1908.
2 B ~ n2.~Physik,
~ ~ 32,
, 119, 1925; 38, 304, 1926; 38, 920, 1926; 39, 195, 1926.
SEC.9.251
EXPANSION OF T H E DIFFRACTED FIELD
First Mode
-
S e c o n d Mode
T h i r d Mode
F o u r t h Mode
Magnetic Lines of Force
Electric Lines of Force
of force corresponding to the first four modes of electric type.
FIG.ll2.-Lines
567
568
BOUNDARY-VALUE PROBLEMS
[CHAP.IX
9.26. Total Radiation.-The resultant field at any point outside the
sphere is the sum of the primary and secondary fields.
+
+
E = Ei E,, H = Hi H,.
(16)
The radial component of the total complex flow vector is
SE = +(EBB+
- EJJe),
(17)
which can be resolved by virtue of (16) into three groups:
(18) 8;
=
a(EieAi+- EiJie) f * ( E r Z r + - Ev&frO)
+(EtBgr+ EreHi, - EiJ:,s - EJTie).
+
+
Let us draw about the diffracting sphere a concentric spherical surface of
radius R. The real part of 8; integrated over this sphere is equal to the
net flow of energy across its surface. To simplify matters we shall
assume the external medium (2) to be a perfect dielectric, so that
k , = a&..
If the diffracting sphere is also nonconducting, the net
flow across any surface enclosing the sphere must be zero. If, however,
energy is converted into heat within the sphere, the net flow is equal to
the amount absorbed and is directed inwards. We shall call the total
energy absorbed by the sphere W,,
W , = -Re
(19)
cr
SER2 sin 0 d$ do.
The first term on the right-hand side of (18) measures the flow of energy
in the incident wave. When integrated over a closed surface this gives
zero so long as uz = 0. The second term obviously measures the outward
flow of the secondary or ficattered energy from the diffracting sphere, and
the tot,al scattered energy is
To maintain the energy balance the third term of (18) must be equal in
magnitude to the sum of the absorbed and scattered energies.
(21)
Wt = W,
+ W,
=
1
--Re
2
Liz'
+ Er&+
(Eisg,.+
- E+Brs
- Er+Rie)R2sin 0 d8 d+.
W t measures, therefore, the total energy derived from the primary wave
and dissipated as heat and scattered radiation.
To calculate W , and Wt we allow R to grow very large and introduce
the asymptotic values
SEC.
TOTAL RADIATION
9.261
569
into the field functions (2) and (3). The integrals can then be evaluated
with the help of Eq. (20), page 417,
ifn #m,
and the relation
One finds for the scattered cnorgy
for the sum of absorbed and scattered energies,
The mean energy flow of the incident wave per unit area is
The scattering cross section of the sphere is defined as the ratio of the total
scattered energy per second to the energy density of the incident wave.
m
Likewise one may define the cross section Qt by
(29)
If the conductivity or inductive capacity of the sphere is so large t h a t
lNp\ >> 1, then thc approximations (14) and ( 1 5 ) can be introduced and
the cross sections reduce to
3
m
(30)
Q8 =
(2n
n=l
m
+ l)(sin2 Y~ + sin2d),
570
BOUNDARY-VALUE PROBLEMS
[CHAP.IX
If the sphere is nonabsorbing, y n and y: are real, in which case it is
obvious that Q3 = - Q t .
9.27. Limiting Cases.-Although
in appearance, the preceding formulas are exceedingly simple, the numerical calculation of the coefficients
usually presents a serious task. There are two limiting cases which can
be handled with ease. If IpI = (klal >> 1, the functions jn(p) and
hA1)(p)may be replaced by their asymptotic values. The quantity p is
essentially the ratio of radius to wave length, so that this is the case of a
sphere whose radius is very much larger than the wave length. Results
obtained by this method should approach those deduced from the
Kirchhoff-Huygens principle of Secs. 8.13 to 8.15. If, on the other hand,
the wave length is very much larger than the radius of the sphere, so that
IpI << 1, the radial functions can be expressed by the first term or two
of their expansions in powers of p .
Case I. IpI >> 1. If the asymptotic representations (12) and (22)
are substituted into (10) and ( l l ) , one obtains for the coefficients
a;:-% .ne
p1
sin x - p 2 N cos x tan y '
PI - i p 2 N tan y
What is apparent here is also true in the general case: the expansion
coefficients are oscillating functions of p and the order n. Small changes
in either p or n may give rise to wide variations in the values of the
coefficients. The absolute magnitudes of the coefficients oscillate
between the limits zero and one.
If in (32) we replace n by n 1, it will be observed that a',+l 'v
T h e amplitude of an electric oscillation of order n i s approximately of the same
magnitude as the amplitude of the magnetic oscillation of next higher order.
The asymptotic expressions used in the derivations of (32) and (33)
are valid only so long as the order n is very much less than the arguments
Ip[ and ( N p l . Asymptotic formulas for the Bessel functions which hold
for all orders have been found by Debye, who has also shown that the
number of terms to be retained in the series expansions is just equal to
the number 1pI.l Methods of summing the coefficients have been discussed by Jobst.2
Case 11. Ipl << 1. From See. 7.4, page 405, we take the expansions
of the radial functions in powers of p or Np.
+
2
DEBYE,
loc. cit.
JOBST,Ann. Physik, 76, 863, 1925; 78, 158, 1925.
x.
571
LIMITING CASES
SEC. 9.271
(35)
(36)
The coefficients can now be expanded in powers of
(38)
+
+
+ +
+ + +
p.
p2n+l
22"n!(n 1)!
- {PI - P ~ N ~
(272) !(2n l ) !
1)
P2N%
- pi[(n
1) (n
__- 3)N2] - p2N2[(n
3)
(n 1)N2]
P2f.
(2n
2)(2n 3)
b i N i __
+
+
+ + +
If (PI is so small that all powers above the fifth can be neglccted, only tjhe
first four coefficients need be considered.
Furthermore, if
p1
= p2,
these expressions reduce to
So long as the conductivity remains finite, the rule still holds that the
magnetic oscillation of order n
1 is of the same magnitude as the
electric oscillation of order n.
When the ratio of radius to wave length is so small that p 5 can be
neglected with respect to p3, only the first-order electric oscillation need
be taken into account. This case was first investigated by Lord Ray-
+
572
BOUNDARY-VALUE PROBLEMS
[CHAP.IX
leigh. The amplitude of the oscillation is then
b',
'v
2i ___
N2- 1
3 N Z 2p3.
+
If this value is int,roduced in the expressions ( 2 ) , (3), and (4), it will be
noted on comparison with (30), page 436, that the field of the fundamental
mode i s that of a n electric dipole, oriented along the x-axis and with a dipole
moment equal to
NZ - . I
p = 4T€2-a3Eo.
N2 2
+
Reference to (32), page 206, shows this to be identical with the result
obtained in the electrostatic case. The amplitude of the scattered or
diffracted radiation at great distances from a small sphere increases,
therefore, as the square of the frequency; and the energy, as the fourth
power. This is the celebrated Rayleigh law of scattering which was
applied by its discoverer to explain the blue of the sky. The light which
reaches us from the heavens, unless the observer is looking directly at the
sun, has been scattered by dust particles or the molecules of the atmosphere. Since the wave length of visible light is much longer than the
radius of these submicroscopic particles, the fourth-power law is valid and
the short wave lengths in the natural sunlight are strongly scattered relative to those near the red end of the spectrum. If one looks directly a t
the sun, the opposite effect is observed, for as the blue suffers the
greatest scattering, its intensity in the direct beam is weakened relative
to the red, which now predominates. This effect is accentuated when
the sun is near the horizon, so that the path lies in a region heavy with
dust particles.
The varied shades of mist and smoke can be accounted for in the same
fashion, and it is well known that the colors of colloidal suspensions of
metallic particles in liquids or gases often differ completely from the
true color of the metal itself. However, as the radius of the particle
increases, terms of higher order play an increasingly important part.
The extremely selective character of the Rayleigh scattering disappears
and it becomes impossible to predict the pattern of the diffracted radiation on any basis other than the results of a difficult computation. The
higher order terms can be interpreted as electric and magnetic multipoles.'
An experimental test of the theory for radio waves has been made by
Schaefer.2
The diffraction of a plane wave by an ellipsoidal body can be treated
in the same manner as the sphere, but the analytical properties of the
GANSand HAPPEL,Ann. Physik, 29, 277, 1909.
SCHAEFER
and WILMSEN,2.Physik, 24, 345, 1924.
SEC.9.281
SOMMERFELD SOLUTION
wave functions are very much more complex.
573
An approximate solution
has been given by Gans' and a more rigorous analysis by Herzfeld.2 T h e
most thorough investigation to date is that of M o g l i ~ h . ~
I n no case,
however, have the numerical calculations been carried far enough to be
particularly useful, and the many practical applications of the theory
make it desirable that further work be done along this line.
EFFECT O F T H E EARTH ON THE PROPAGATION OF RADIO WAVES
9.28. Sommerfeld Solution.--The classical investigation of the effect
of a finitely conducting plane upon the radiation of a n oscillating dipole
was published by Arnold Somrnerfeld in 1909.4 Sincc that time a n
enormous amount of work has appeared on the subject and it may be
fairly said that no aspect of the problem of radio wave propagation has
received more careful attention. It is impossible within the confines
of this chapter to give a complete account of the theory, but it should
prove instructive to see how the problem is set up. As in all but t h e
simplest diffraction problems, the real difficulties arise with the necessity of reducing the formal solution to a state that permits numerical
computation.
The first problem attacked by Sommerfeld was that of a vertical
dipole located a t the surface of a plane, finitely conducting earth. The
axis of the dipole coincides with the z-axis of a rectangular coordinate
system and the plane z = 0 represents the earth's surface. All points for
which z < 0 lie within the earth, whose propagation constant is kl. The
propagation constant of the air in ihe region z > 0 is k z = w / c . A point
of observation is located by the cylindrical coordinates r, 4, x, and its
radial distance from the origin is
These relations are illustrated in Fig. 113.
The field is cylindrically synimetrical about the z-axis, and in the
neighborhood of the origin it must grow infinite like that of a dipole.
Now the field of a dipole, we recall from (7), page 432, can be represented
by an axial Hertzian vector which behaves as l / R as R --+ 0. We shall
write, therefore, the z-component of the Hertzian vector of the resultant
field as the sum of two parts:
GANG,Ann. Physik, 37, 881, 1912; 47, 270, 1915; 61,465, 1920.
HERZFELD,
Wiener Rer., 120, 1587, 1911.
Ann. Physik, 83,609, 1927.
MOGLICH,
4 SOMMERFELD,
Ann. Physik, 28, 665, 1909; Juhrbuch d. druhtl. Telegruphie, 4,
1
2
157, 1911. These earlier papers contain an error in algebraic sign which was corrected in the Ann. Physik, 81,1135, 1926. Also in Riemann-Weber, loc. cit., p. 542.
Some of the consequences of this error appear, however, to have been overlooked.
574
[CHAP.IX
BOUNDARY-VALUE PROBLEMS
+ Fl(kl;
1 .
I12(r,z ) = - eZkzR+ Fz(kz;
Iz
1 .
IIl(r, z ) = - etklR
Iz
(2)
T,
z),
(2
011
T,
z),
(2
> 0).
The first term accounts for the singularity at the origin. The dipole
moments have arbitrarily been chosen such that the coefficient of this
term is unity, and a common time factor exp( -id)has been suppressed.
The functions F1and F 2 are finite everywhere, including the point R = 0,
and satisfy the wave equation and the radiation condition a t infinity.
They represent the contribution of the diffracted wave and are to be
I
FIG. 113.-Vertical
electric dipole located at the surface of a plane, finitely conducting
earth.
chosen such as t o ensure the fulfillment of the boundary conditions a t
z = 0. The field vectors are derived from the Hertzian vectors through
(3)
E
= VV-II
+ k2n,
H
=
k2
-V
tWP
x KI,
whence the tangential components are
(4)
It may be assumed for the earth that p1 = 1.12 = p,,. At any point on
the plane z = 0 the functions ( 2 ) must satisfy the condition
(5)
(2 =
0).
These relations hold for all values of r and can, therefore, be integrated
with respect t o r. The functions and their derivatives vanish as r -+ 00 ,
so that the constant of integration must be zero. Thus the boundary
conditions reduce t o
(2 =
0).
To find functions PI and Fz t h a t satisfy these conditions, we shall
expand both the ((primaryexcitation” eiEr/Rand the “secondary excita-
SEC.9.281
575
SOMMERFELD SOLUTION
t,ion” F in terms of cylindrical wave functions. I n Sec. 6.6 it was shown
that every electromagnetic field within a homogeneous, isotropic medium
might be represented by the superposition of elementary cylindrical wave
functions of the type
(7)
$ ( r , 4, z) = ein+J , , ( d m i r ) eihz.
In the present instance the fields are symmetric and consequently n = 0.
As a parameter we shall choose X2 = k 2 - h2 rather than h, and multiply
each wave function by an amplitude f(X) dependrnt upon A. This
parameter can assume any complex value. The sign of the root
ih = -t d m i must be chosen such that the fields vanish as z 4 A m .
We do this by making the real part of 4always positive. Then
the conditions a t the origin and a t infinity are fulfilled by a discontinuous
Hertz vector whose radial and azimuthal components are zero and whose
z-component is
1
IT1 = - &iR
.
(8)
R
IT2 = - eskzR
’
R
+
+J
f l ( x > J o ( A r > e + m dA,
z
f2(X>Jo(Xr)e-d\/X‘-z d l ,
(z
< o>,
(2
> 0).
The next step is to represent the known primary excitation in the
same manner. To this end we make use of the Fourier-Bessel theorem
(53), page 371. If f ( r ) vanishes as r -+ m such that the integral of
f ( r ) 4;converges absolutely, then
(9)
fb-1 =
C h
1O
so-
P
dPf(P)JO(XP)JO(h-).
For f(r) we take first ezkr/r.
.“‘
=
r
X dX Jo(Ar)
d p eikPJo(Xp).
Integration over p can be effected if the Bessel function J&p) is replaced
by its integral representation (37), page 367.
It is assumed here that k has a positive imaginary part so that the
integrand vanishes a t the upper limit. k , is in fact real, but for purposes
of integration the atmosphere can be assigned a very small conductivity
which may be reduced to zero after the integral has been evaluated. The
last integral is equivalent to an integration about a closed contour. Let
(12)
u = e*,
du = i u d p .
576
BOUNDARY-VALUE PROBLEMS
[CHAP.IX
Then
where the path of integration is the unit circle about the origin in the
complex u-plane and where u1 and u2 are the roots of
u2+
x + 1 = 0.
2ku
~
The product of the roots is unity, so that u1 lies inside the circle and
outside.
u2
by Cauchy’s theorem (133), page 315. Consequently
This represents the primary excitation eikR/Rat all points on the
p l u e z = 0 as a sum of elementary cylindrical waves whose amplitudes
are expressed by the function X(kz - k2)-+. A representation a t any
other point follows directly from (7).
For the Hertzian vector of the resultant field, we now have
The functions fl(1) and fz(h) are next determined from the boundary
conditions (6). A slight difficulty is encountered with the first of these
relations, for if (17) is differentiated under the sign of integration with
respect t o z and z then placed equal to zero, the integrals diverge. But
SEC.9.291
W E Y L SOLUTION
577
Sommerfeld points out that
and hence in this case the primary excitation in (18) can be ignored.
simple calculation leads us then to the result:
h
fl(V
=
f2(X)
=
(20)
4-f
A
k ; d r n - k : d F q
Lj f i z z : + k ; d m ; '
x
Ic; 4FTGi Ic; 4
m;
___
d m i k: 4m;+ k ; 4A-C
-
Upon substitution of these expressions into (18), we find a t last
A formal solution has been obtained in terms of infinite integrals expressing the Hertzian vector of the resultant field a t all points both above and
below the surface.
9.29. Weyl Solution.-It was recognized from the outset by Sommerfeld that his solution could be interpreted as a bundle of plane waves
reflected and refracted from the earth's surface a t various angles of
incidence. This point of view was developed later by Wey1.l Its basis
is an integral representation of the function hbl)(kR) demonstrated in
Sec. 7.7. According to (66), page 410,
As in Fig. 70, page 407, the wave normal of each elementary wave makes
an angle a with the z-axis and an angle /3 with the x-axis. The point of
observation a t P is located by the spherical coordinates ( R , 6, +), while
the angle between the radius vector R and the wave normal k is y. Thus
k R = k R cos 7,
(24)
7 = cos y = sin 6 sin a cos (4 = x sin a cos p
y sin a sin p
R cos y
+
WEYL,Ann. P h y s i k , 60, 481, 1919.
p) + cos 6 cos a ,
+ z cos
Ly.
578
B O U N D A R Y - V A L U E PROULEMS
[CHAP.IX
'If the integration in (23) be carried out with respect to y instead of 1,
we have
where rl. is an equatorial angle in a system of polar coordinates whose
axis coincides with the radius vector R. But sin y dr d$ is an element of
area on a unit sphere concentric with the origin and the value of the
integral is invariant to a rotation of the reference axes. Hence,
This is the desired representation of the spherical wave function in terms
of plane waves. Sommerfeld's representation (17) is obtained from (26)
by carrying out the integration with respect to p. Weyl points out,
however, that in so doing one loses the freedom to rotate the coordinate
system into a n orientation that facilitates evaluation of the integrals.
For this purpose the axis of the radius vector R is more appropriate than
the z-axis normal to the surface of the earth.
Let us suppose that the dipole is located a distance zo above the surface
of the earth. The resultant field llz at any point for which z > 0 can be
expressed as the sum of the direct or primary field no, and the reflected
field II,. In the region 0 Iz Izo between the dipole and the earth the
primary radiation is represented by
eikZR
(27)
no = __
R
where
d 0 2 = sin a2dag do2.
(28)
In this region a2 is the angle made by the wave normal with the negative
z-axis, as shown in Fig. 114. It is thus also the angle of incidence at
which an elementary plane wave meets the earth's surface.
The Hertz potential II, of the reflected field can be constructed from
plane waves in the same manner. Each elementary wave of the bundle
will have the form
j r &kz[z
sin a2 cos Bz+v sin az sinBt+(zo+r)
cos
4.
SEC.9.291
579
W E Y L SOLUTION
1,ikewisc the refracted, or transmitted, field in the domain z
structed from the elementary waves
f t e i k z z ~cos as
. eiki(z sin
a1
cos B i + y sin a1 sin
81-2 cos
< 0 is con-
ai).
The undetermined functions fiand f t are reflection and refraction coefi-
If
Air
I
I
I
I
k,
+
,,,/,,////,////////?
/
waves
z=o
////,,//
Earth h ,
FIG.114.-The radiation from a dipole can be represented by superposition of plane wave?
which are reflected and refracted at tho surface of the earth.
cients. To simplify the notation wc write
-
k2 r = k 2 ( x sin a2 cos PZ
k l - r= k l ( x sin a1 cos P I
(29)
+ y sin a 2 sin p2),
+ y sin sin Pl).
a1
Then a t all points in the domain 0 I z 5 zo the resultant field is
At all points for which z I0,
At z
=
0, the boundary conditions lead to
Imposing these relations on (30) and (31), one obtains first
The elementary waves form a circular cone about the z-axis and the angles of
resection and refraction satisfy Snell's law (8), page 491. A further
calculation gives for the coefficients:
580
[CHAP.IX
BOUNDARY-VALUE PROBLEMS
jr
=
ft
=
(34)
where ZZl = p2kl/plk2. Now the lines of magnetic field intensity are
circles whose centers lie on the z-axis. The meridian planes in Fig. 114
are also planes of incidence, whence it is apparent that the magnetic
vector of each elementary wave is normal to its plane of incidence, while
its electric vector is parallel to this plane although not transverse to the
direction of propagation. Rcference to (19),page 494, shows that apart
from a factor in f t , the junctions f r and f t are the Fresnel reflection and
refraction coeficients for a plane wave.
When z > ZO,Eq. (30) must be modified such as t o vanish properly at
infinity. We now measure a from the positive x-axis and write
If finally zo = 0, this reduces to
while beneath the surface
(37)
It is now a simple matter ,to effect the transition to Sommerfeld’s
solution. By (80), page 412,we have
Jo(Xr) e i h z =
(38)
1
2n
71-
eiXr cos
(@-8B)+ihzdp.
Let
(39)
X = k z sin a z ,
When a2 = 0, X
=
h
0 , and when 01
relations, (36) transforms to
=
k z cos a2 = i 4-i.
7r
= - - i.o
2
X
=
00.
In virtue of these
SEC.9.291
WEYL SOLUTION
58 1
while the same substitutions transform (37) to
where N is the function defined by ( 2 2 ) . Thus the formal solution of
Weyl is identical with that of Sommerfeld in the special case zo = 0.
FIG.115.-An
image of the dipole is located in the earth a t the point z =
-20.
In the general case where zo # 0, we replace f r in (35) by
Then (35) can bc resolved into three distinct parts, reducing in virtue of
(27) and (40) to
where R I and R2 arc the distances shown in Fig. 115. This expreesion
holds for all values of z > 0. The resultant field at the point of observation is thus composed of a direct contribution from the vertical electric
dipole at z = +zo, a contribution from its image at x = -20, and zt
supplementary term which takes into account the effect of a finite conductivity of the earth. If k l 3 00, the last term of (43) approaches
and
(45)
by virtue of Eq. (17).
582
[CHAP.IX
BOUNDARY-VALUE PROBLEMS
Identical results have been obtained by Niessen’ through a n application of the Kirchhoff-Huygcns method discussed in Sec. 8.13. The surface
integrals are extended over the plane earth and over small spheres
which exclude the singularities occurring a t the source and the point of
observation.
9.30. Van der Pol Solution.-An interesting and useful transformation
of the Sommerfeld-Weyl expressions has been discovered by van der PoL2
Starting from the integral for the reflected field
*
(46) II,
=
k2, 4
q
kf 4 x
4 p T @+ 1;
qp
:1
/
x 2
- k %-
___
2
- k i J,,(Xr)e-Z/X’-”’”(z+Zd
1/T---Q
X dX,
obtained from (18) and (20), it is shown that the total field-primary and
reflected-at any point above the earth’s surface can be expressed as a
volume integral extended over a certain domain in real space. The
successive steps in the demonstration involve properties of Bessel functions which have been discussed in the earlier pages of this volume.
Details will be omitted here. For all values of z > 0, van der Pol obtains
the expression
--
The volume element dv = p d p d l d p , and the domain of integration is
____
0 I p I 0 0 , 0 4 { 4 0 0 , 0 Ip I; 23r. The distance p = 4 p 2 + r 2 ,
and the quantity Ri is a “complex distance” defined by
The sign of the root is chosen such that (48) has a positive real part, and
it is assumed that k z is real. The domain of integration can be interpreted as the half space below the gcometxical image of the source. This
is the shaded part of Fig. 116. The variables p, p are thus cylindrical
coordinates with the image point a t x = -20 taken as a n origin. 1 is
measured positive downwards.
When the conductivity of the earth is large, the propagation constant
r,
kl
has a large imaginary part.
Consequently the wave
2c:
t?)
is
rapidly attenuated and contributes an appreciable amount to the integral
only when p is small. I n other words, only that part of the lower medium
2
NIESSEN,Ann. Physik, 18, 893, 1933.
VhN pEfl POL, Physica, 2, 843, 1935.
SEC. 9.311
APPROXIMATION OF T H E INTEGRALS
583
(earth) in the immediate neighborhood, but below, the geometrical image
sends secondary waves to the observer. I n the limit of infinite conductivity
the region containing the effective sources of secondary radiation shrinks
to a point coincident with the geometrical image itself.
An analogous, somewhat simpler, solution is given also by van der Pol
for the case of a magnetic dipole, and the method can be extended to
include horizontal dipoles. Since any source of electromagnetic radiation
can be represented in terms of a distribution of electric and magnetic
dipoles, the bearing of these calculations on many other problems of
physical optics and electromagnetic theory is apparent. One might
mention, in passing, the rather complicated phenomena associated with
FIG.116.-In the van der Pol solution the domain of integration is the half-space below the
plane z =
-20.
total reflection, which we have treated only for the idealized case of a n
infinite plane wave.
9.31. Approximation of the 1ntegrals.-The
several methods of
approach t o the problem of reflcction from a plane surface all lead to
integral representations of the field, or more precisely of its Hertzian
vector, that arc easily shown to be identical. A much more difficult
task, which has occupied the attention of many investigators, is the
reduction of these integrals to practical formulas for the field intensity.
Sommerfeld’s attack was based on a deformation of the path of integration in the complex X plane. The function l,” defined in (22) has
branch points a t X = & kl and X = kz,and a pole at
(49)
The corresponding Riemann surfacc has four sheets; on only one of these
are the conditions fulfilled necessary for the convergence of the integral
at infinity. According to Sommerfeld the path of integration can be
resolved on this sheet into three parts: the first a loop from infinity about
584
BOUiVDA R Y-VALUE PROBLEMS
[CHAP.IX
the branch point X = +kI, the second a similar loop about X = +kz,
and the third any small circle about the pole a t X = Xo. The contributions of the loops about the branch points are Q1 and Q2, while the residue
at the pole gives a term P , so that 112 = Q1 Q 2
P. The terms &I
and Q2 were developed in inverse powers of the distance from the dipole,
the dominant terms of the series being proportional respectively to
eiklR/Rand eikzR/R. Consequently Q1 and Q2 were interpreted as “space
waves.” On the other hand, for P Sommerfeld obtained a function of
the form
+ +
which was interpreted as a ‘(surface wave.”
Historically these surface waves are of considerable interest. The
guiding of a plane electromagnetic wave along the plane interface
separating a dielectric and a good conductor seemG to have been first
investigated by Cohnl and shortly thereafter by Uller.2 Zenneck3 recognized the bearing of these researches on the propagation of radio waves
and showed that the field equations admit a solution that can be interpreted as a surface wave guided b y a plane interface separating any
two media. This solution is identical with the surface wave derived in
another manner in Sec. 9.13. Zenneck was quite clear as to the limitations of his results. He showed that a wave with a forward tilt, following
a plane earth and attenuated in the vertical as well as the horizontal
direction is compatible with Maxwell’s equations, and that such a wave
would explain many of the observed phenomena of radio transmission.
There was no proof as yet, on the other hand, t h a t a radio antenna docs
in fact generate a wave of this type. Sommerfeld’s 1909 paper undertook
to complete the demonstration, and the term P of Eq. (50) was interpreted as the surface wave component (in cylindrical coordinates) of the
total field.
Doubt as to the validity of Sommerfeld’s resolution was first raised
by Weyl in his 1919 paper. Weyl obtains an asymptotic series representation of the diffracted field by applying the method of “steepest descent”
(page 368) to the integral (36). I n case Iklj/k2>> 1, Weyl’s solution
also reduces to a form which can be interpreted as the superposition of a
space and a surface wavc, but the Weyl surface wave is not identical with
that of Sommerfcld and Zenneck.
COHN,“Das elektromagnetische Feld,” Leipzig, 1900.
ULLER.
Rostock Diss., 1903.
3 ZENNECK, Ann. Physzk, 23, 846, 1907.
1
SEC. 9.311
APPROXIMATION OF THE INTEGRALS
585
The discrepancy between the two forms of solution has been examined
more closely in recent years, although it can hardly be said that the
problem is closed. Burrows1 has pointed out that numerically the
transmission formulas based on Sommerfeld’s results differ from those
of Weyl by just the surface wave term P , and has made careful measurements which support the results of Weyl. The discrepancy is large only
when the displacement current in the ground is comparable to the conduction current. Asymptotic expansions have been given by Wise2
and by Rice3 which show that the term P in Sommerfeld’s solution is
canceled when all terms of the series for &I, Q 2 , and P are taken into
account. Finally, attention was called by Norton4 to an error in sign
occurring in Sommerfeld’s 1909 paper. This error does not appear in
the 1926 paper, but the numerical results given there are accurate only
when the conductivity of the earth is large.
From a practical standpoint the situation can be summed up as
follows: The results of Sommerfeld, and the charts of Rolfs which are
based upon them, cannot be relied upon when the displacement current
is appreciable, as is the case at ultrahigh frequencies. The formulas of
van der Pol, Niessen, Burrows and Norton6 are for the most part extensions of Weyl’s results. Field intensities calculated by these formulas are
in accord with each other and apparently in good agreement with experiment. The same is true of the results obtained by Strutt,? who applies
an alternative method to obtain an asymptotic solution.
To give some idea of the character of the formulas to be used for
practical calculations, we present a summary of the results obtained by
Norton for the vertical electric dipole. The Hertz vector of the resultant
field a t any point above the earth’s surface is
where V is the infinite integral expressed by the-third term of (43).
The approximate formula proposed for V is
(52)
v
=
[(l -
-
Z,Zl cos
where
(53)
Rv
-
el - 4.~22
-~
sin2 el
.z;~cos el + 4222, - sin2 el’
BURROWS,
Nature, Aug. 15, 1936; PTOC.
Inst. Radio Engrs., 26, 219, 1937.
WISE, Bell System Tech. J., 16, 35, 1937.
RICE,Bell System Tech. J . , 16, 101, 1937.
NORTON,
Nature, 136, 954, June 8, 1935; Proc. Inst. Radio Engrs., 26, 1102, 1937.
I, ROLF,
Proc. Inst. Radio Engrs., 18, 391, 1930.
6 NORTON,
Proc. Znst. Radio Engrs., 24, 1367, 1936; 26, 1203, 1937.
STRUTT,
Ann. Physik, 1, 721, 1929; 4, 1, 1930; 9, 67, 1931.
1
2
’
586
BOUNDARY-VALUE PROBLEMS
cos 01
(54)
[CHAP.IX
ZZl = k-.l
= ___
+ 20,
R1
k2
R, is the Fresnel reflection coeffcient for a plane wave incident a t an
angle 01. The apparent source is in this case a t a distance zo below the
surface. (See Fig. 115.) The function F is defined by
F
(55)
where
=
[I
+ i 2/?rw
e-w
erjc( -i
431,
(57)
iLzR1
PI = 7
(Z& - sin2 el)
(58)
222,
=
pe”.
Since
(59)
one may also write
where A is the free-space wave length. A detailed numerical discussion
of the function F is given by Norton.
In order that the final formula shall express the field intensity of a
current element, the Hertzian vector is multiplied by an appropriate
amplitude factor. According to (44), page 431, the Hertz vector of a
linear current element I dz is
Upon multiplication of (51) by il dz/4awe2 and applying (3), Norton
finds for the z-component of electric field intensity in volts per meter the
expression
eikzR1
+ Iz, sin2 el __
4*
R1
.
+ - Rv) (Z& - Z,Z, + sin2 el)F Iz + higher order terms in R,
iWP0
(62) dE, = _ _ I dz
eik2Ri
--
2
2
4
1
1
‘I
The terms in l / R i and 1/Rf referred to represent the contributions of the
“static” and “induction” fields and may be neglected a t distances of
ten wave lengths or so from the source. The angle O2 is defined by
SEC.9.311
APPROXIMATION OF T H E INTEGRALS
587
An analogous expression can be obtained for the component dE,, which
together with (62) determines the tilt or polarization of the field.
I n the special case of a current element on or very near the earth
zo = 0 and R1 = Rz = R, 01 = 0 2 = 0. The total electric field intensity
a t any point such that R >> X can then be expressed as the vector sum
of two components,
i W 0
dE.,,,,
(65)
iWP0
= -
4P
I dz sin 0 (1
+ R,)
eiktR
---
R
ae,
eikiR
dEau,fsce
=
I dz (1 - R,)P R
4n
~
~
where ae is a unit vector tangent to the meridian circle through the point
of observation in the direction of increasing 0 and, consequently, perpendicular to the radius vector R, while a, and a,. are the unit vectors in a
cylindrical coordinate system parallel t o the z- and r-directions. As the
conductivity of the ground approaches infinity, R, -+ 1. The component
(65) then vanishes and thtx total field (