PH231 Electromagnetism I (Spring 2024)
Lecture note 01: Introduction to Classical Electromagnetism
(This lecture note includes contents/topics that may be beyond the introductory level typically
encountered in undergraduate courses. It's perfectly normal for them to seem complex at first
glance. As you progress through the lectures in this semester and the next, you’ll find them more
familiar. I just aim to present goals/levels that a student in KAIST’s Physics program may aspire to.)
1.1 Preface
- Although Maxwell’s equations were “formulated” over one hundred years ago, the subject of
classical electromagnetism has not remained static. As one of the four fundamental forces of
nature, the electromagnetic (EM) force has recovered a lot of important subjects in Physics,
including electricity (origin of electrical engineering: EE), classical optics (high-frequency EM
effects), classical condensed matter physics (low-frequency EM effects in matter), and the
“special relativity”. Particularly, special relativity provides us deeper understanding on the nature
of Physics and the space and time.
- From the point of view of the standard model, classical electrodynamics (CEM) is a limit of
ሬԦ) and energy
quantum electrodynamics (QED). We will discuss the cases of small momentum (݇
transfers, large average numbers of virtual or real photons, and weak electric and magnetic
ሬሬԦ, mediating the light-matter interaction of matter.
dipole moments, ‫݌‬Ԧ and ݉
- QED is a consequence of a spontaneously broken symmetry in a theory called the electroweak
theory in which initially the weak and electromagnetic (EM) interactions are unified and the force
carriers of both are massless; at very high energies, the universe has four components of the
Higgs field whose interactions are carried by four massless gauge bosons. The symmetry
breaking leaves the EM force carrier (photon) massless with a Coulomb’s law of infinite range,
while the weak force carriers (W+, W–, and Z0) acquire masses of the order of a few tens of
G݁V/ܿ ଶ with a weak interaction at low energies of extremely short range (~2 × 10ିଵ଼ ݉).
- The purpose of this introduction lecture is therefore not to set the stage for a discussion of
Coulomb’s law and other basics of CEM, but rather to present a review and a survey of CEM.
1.2 Classical Electromagnetism: Maxwell Equations in Vacuum, Fields, and Sources
- The equations governing CEM phenomena in vacuum
1.
Partial differential equation (PDE): Maxwell’s equations
‫ܧ ڄ ׏‬ሬԦ =
ሬԦ െ
‫ܤ×׏‬
2.
ߩ
ߝ଴
ሬԦ = 0
and ‫ܤ ڄ ׏‬
ሬԦ
1 ߲‫ܧ‬ሬԦ
߲‫ܤ‬
Ԧ and ߘ × ‫ܧ‬ሬԦ +
=
ߤ
‫ܬ‬
=0
଴
߲‫ݐ‬
ܿ ଶ ߲‫ݐ‬
Charge conservation: the continuity equation for
߲ߩ
+ ‫ܬ ڄ ׏‬Ԧ = 0
߲‫ݐ‬
ሬԦ: electromagnetic forces on objects
- Concept of ‫ܧ‬ሬԦ and ‫ܤ‬
1.
The Lorentz force on a point charge ‫ ݍ‬is governed as
ሬԦ൯
‫ܨ‬Ԧ = ‫ݍ‬൫‫ܧ‬ሬԦ + ‫ݒ‬Ԧ × ‫ܤ‬
2.
ሬԦ first appear just a convenient replacements for force produced by
Although ‫ܧ‬ሬԦ and ‫ܤ‬
ߩ and ‫ܬ‬Ԧ, they have other important aspects
A.
Their introduction decouples conceptually the sources from the test bodies
experiencing EM forces.
B.
3.
EM fields can exist in regions of space where there are no sources: vacuum.
ሬԦ as ordinary, continuously varying fields is a classical notion.
The concept of ‫ܧ‬ሬԦ and ‫ܤ‬
A.
It can be thought of as the classical limit of quantum mechanical description in
terms of real or virtual photons.
B.
For example, an isotropic FM antenna with a power of 100 W at 10଼ Hz
(100 MHz) produces an electric field of 0.5 mV/݉ in rms at a distance of
100 ݇݉, but this still corresponds to a flux of ~10ଵଶ ܿ݉ଶ or ~10ଵଶ photons/ߣଷ .
- Concept of the potential
(The details of the vector analysis will be discussed in detail in the following lectures. Here, I
would just like to introduce the concept of the scalar and vector potentials.)
1.
Scalar potential
A.
The behavior of electrostatic field is described by the following two differential
equations
ߩ
and ߘ × ‫ܧ‬ሬԦ = 0
ߝ଴
The equation specifying curl ‫ܧ‬ሬԦ is equivalent to the statement that ‫ܧ‬ሬԦ is the
‫ܧ ڄ ׏‬ሬԦ =
B.
gradient of a scalar function, the scalar potential ߶:
‫ܧ‬ሬԦ = െ‫߶׏‬
C.
The two equations of ‫ܧ‬ሬԦ can be combined into one partial differential equation
for the single function ߶, the Poisson equation:
ߩ
‫׏‬ଶ ߶ = െ
ߝ଴
D.
The solution to the Poisson equation is well-known as
1
ߩ(‫ݔ‬Ԧ ᇱ ) ଷ ᇱ
Ԅ(‫ݔ‬Ԧ) =
න
݀ ‫ݔ‬Ԧ
4ߨߝ଴ ௔௟௟ |‫ݔ‬Ԧ െ ‫ݔ‬Ԧ ᇱ |
E.
It clearly shows the linear combination of the contributions of the charge
components.
2.
Vector potential
A.
ሬԦ = 0, any curl of a vector is divergence-free:
From ‫ܤ ڄ ׏‬
ሬԦ = ‫ܣ × ׏‬Ԧ
‫ܤ‬
B.
Employing the vector potential, Ampere’s law become
ሬԦ = ‫ × ׏‬൫‫ܣ × ׏‬Ԧ൯ = ‫׏‬൫‫ܣ ڄ ׏‬Ԧ൯ െ ‫׏‬ଶ ‫ܣ‬Ԧ = ߤ଴ ‫ܬ‬Ԧ
‫ܤ×׏‬
C.
When we chose a gauge, ‫ܣ ڄ ׏‬Ԧ = 0, called as the Coulomb gauge for electro/magneto-statics, we have
‫׏‬ଶ ‫ܣ‬Ԧ = െߤ଴ ‫ܬ‬Ԧ
D.
3.
This is the Poisson equation for the vector potential, whose solution is given as
ߤ଴
‫ݔ(ܬ‬Ԧ ᇱ ) ଷ ᇱ
֜ ‫ܣ‬Ԧ =
න
݀ ‫ݔ‬Ԧ
4ߨ |‫ݔ‬Ԧ െ ‫ݔ‬Ԧ ᇱ |
Freedom of gauge choice:
A.
ሬԦ = 0, we determine the vector potential as
From ‫ܤ ڄ ׏‬
ሬԦ = ‫ܣ × ׏‬Ԧ
‫ܤ‬
B.
Faraday’s law gives
‫ܧ × ׏‬ሬԦ = െ
֜ ‫ × ׏‬ቆ‫ܧ‬ሬԦ +
ሬԦ
߲‫ܤ‬
߲‫ܣ‬Ԧ
= െ‫× ׏‬
߲‫ݐ‬
߲‫ݐ‬
߲‫ܣ‬Ԧ
߲‫ܣ‬Ԧ
ቇ = 0 ֜ ‫ܧ‬ሬԦ = െ‫ ߶׏‬െ
߲‫ݐ‬
߲‫ݐ‬
C.
The curl of the magnetic field gives
ሬԦ = ‫ܣ × ׏ × ׏‬Ԧ = ‫׏‬ଶ ‫ܣ‬Ԧ െ ‫׏‬൫‫ܣ ڄ ׏‬Ԧ൯
‫ܤ×׏‬
D.
Combining the vector and scalar potentials with the Maxwell equations, we have
the equations equivalently converted as
߲
ߩ
‫ۓ‬
‫ܧ ڄ ׏‬ሬԦ = ‫׏‬ଶ ߶ + ൫‫ܣ ڄ ׏‬Ԧ൯ = െ
ۖ
߲‫ݐ‬
ߝ଴
ଶ Ԧ
ሬԦ
߲‫ܧ‬
߲
‫ܣ‬
1
1
1 ߲߶
‫ܬ‬Ԧ
‫۔‬
ሬԦ െ
= ‫׏‬ଶ ‫ܣ‬Ԧ െ ଶ ଶ െ ‫ ׏‬൬‫ܣ ڄ ׏‬Ԧ + ଶ ൰ = െ
ۖ‫ܤ × ׏‬
ଶ
ܿ ߲‫ݐ‬
ܿ ߲‫ݐ‬
ܿ ߲‫ݐ‬
ߝ଴ ܿ ଶ
‫ە‬
E.
Fortunately, we can now make use of our freedom to choose arbitrarily the
divergence of ‫ܣ‬Ԧ. What we are going to do is to use our choice to fix things so
that the equations for ‫ܣ‬Ԧ and for ߶ are separated but have the same form.
F.
We can do this by taking
G.
1 ߲߶
=0
ܿ ଶ ߲‫ݐ‬
Using this so-called Lorenz gauge, named by a Danish physicist, Ludvig Lorenz
‫ܣ ڄ ׏‬Ԧ +
(1829-1891. Not to be confused with Hendrik Lorentz or Edward Norton Lorenz),
H.
the Maxwell equations become
1 ߲ ଶ ‫ܣ‬Ԧ
ଔԦ
1 ߲ଶ߶
ߩ
ଶ
‫׏‬ଶ ‫ܣ‬Ԧ െ ଶ ଶ = െ
and
‫׏‬
߶
െ
=െ
ଶ
ଶ
ଶ
ܿ ߲‫ݐ‬
ߝ଴ ܿ
ܿ ߲‫ݐ‬
ߝ଴
What a beautiful, symmetric set of wave equations! First, they are nicely
separated-with the charge density, goes ߶; with the current, goes ‫ܣ‬Ԧ . Second,
ଵ డమ
using the D’Alembertian, ߲ఓ ߲ఓ = ߲ఓ ߲ఓ = మ
௖ డ௧ మ
െ ‫׏‬ଶ ‫ ؠ‬ǹ , a single equation can
describe everything in the classical electrodynamics as
ǹ‫ܣ‬ఓ = ߲ఓ ߲ఓ ‫ܣ‬ఓ = ‫ܬ‬ఓ
- 4-vectors: Covariance of electrodynamics
(At this moment, you don’t need to understand the following contents, of which the detail will
be discussed in the class of PH232 electromagnetism II in the next semester. We will first discuss
invariant physical quantities, such as scalars and vectors, in the following lecture.)
1.
The inner product of two 4-vector quantities is an invariant scalar physical quantity.
2.
4-vector current density
‫ܬ‬ఈ ‫ ؠ‬൫ܿߩ, ‫ܬ‬Ԧ൯
3.
4.
4-vector differential operator
߲
߲
߲ఈ ‫ ؠ‬൬
, െ‫׏‬൰ and ߲ఈ ‫ ؠ‬൬
, +‫׏‬൰
߲ܿ‫ݐ‬
߲ܿ‫ݐ‬
The inner product of the 4-vector differential operator and current density provides
the continuity equation as
߲ఈ ‫ܬ‬ఈ =
5.
6.
4-vector potential
߲ߩ
+ ‫ܬ ڄ ׏‬Ԧ = 0
߲‫ݐ‬
߶
‫ܣ‬ఈ ‫ ؠ‬൬ , ‫ܣ‬Ԧ൰
ܿ
(SI)
‫ܣ‬ఈ ‫ ؠ‬൫߶, ‫ܣ‬Ԧ൯
(CGS)
The inner product of the 4-vector differential operator and potential provides the
condition for the Lorentz gauge as
߲ఈ ‫ܣ‬ఈ = 0
֜
1 ߲߶
+ ‫ܣ ڄ ׏‬Ԧ = 0
ܿ ଶ ߲‫ݐ‬
(SI)
7.
The inner product between the covariant and contravariant 4-vector differential
8.
operators defines the d'Alembertian, the Laplace operator in the Minkowski space, as
1 ߲ଶ
߲ఈ ߲ ఈ = ଶ ଶ െ ‫׏‬ଶ ‫ ؠ‬ǹ
ܿ ߲‫ݐ‬
The Maxwell equations are equivalently converted into the wave-equations for the
components of the 4-vector potential under the Lorentz condition as
1 ߲ଶ߶
ߩ
ଶ
‫ۓ‬
െ
‫׏‬
߶
=
ଶ
ଶ
ۖ ܿ ߲‫ݐ‬
1 ఈ
ߝ଴
(SI)
߲ఈ ߲ ఈ ‫ܣ‬ఈ =
‫֜ ܬ‬
ଶ
ଶ
Ԧ
ߝ଴ ܿ
‫ܬ‬Ԧ
‫۔‬1 ߲ ‫ܣ‬
ଶ Ԧ
െ‫=ܣ ׏‬
ۖ
ߝ଴ ܿ ଶ
‫ ܿە‬ଶ ߲‫ ݐ‬ଶ
߲ఈ ߲ ఈ ‫ܣ‬ఈ =
9.
4ߨ ఈ
‫ܬ‬
ܿ
1 ߲ଶ߶
‫ ۓ‬ଶ ଶ െ ‫׏‬ଶ ߶ = 4ߨߩ
ܿ ߲‫ݐ‬
֜
4ߨ
‫ ۔‬1 ߲ ଶ ‫ܣ‬Ԧ
ଶ Ԧ
Ԧ
‫ ܿ ە‬ଶ ߲‫ ݐ‬ଶ െ ‫ܬ ܿ = ܣ ׏‬
(CGS)
Finding the Green function of the wave-equation in the Minkowski space and
employing the proper boundary condition, one can solve any problem of CEM
analytically or numerically, which is one of the essential purposes of this class.
- Quantum nature of the electromagnetic fields
1.
How to decide a priori when a classical description is adequate: when the number of
photons can be taken as large but the momentum ‫݌‬Ԧ of photon is small compared to
the momentum of the material system. For example, the elastic scattering of EM
waves by a free charged particle at low frequency (radio-frequency), the Thomson
scattering, is readily to be described by the classical theory. The inelastic scattering at
high frequencies, the Compton scattering (X-ray), can be semi-classically modelled.
2.
The photoelectric effect is non-classical for the matter system, since the quasi-free
electrons in the metal change their individual energies by amounts equal to those of
the absorbed photons, but the photoelectric current, which is a macroscopic quantity,
can be calculated quantum mechanically for the electrons using a classical description
of the electromagnetic fields.
3.
How about the interaction between single photons? In the Hong-Ou-Mandel
interference as illustrated in the figure below, if the two photons are identical in their
physical properties (polarization, spatial and temporal mode structures, and
wavelength), we cannot distinguish between the output states of possibilities 2 and 3,
and their relative minus sign ensures that these two terms cancel [Phys. Rev. Lett. 59,
2044 (1987)]. This can be interpreted as destructive interference or, in other words,
the output states of 2 and 3 do not occur.
4.
In the same reason, the spontaneous emission of radiation by atoms cannot be
described by the classical EM theory either. The absolute value of the density of
optical states (DOS) cannot be calculated. The classical EM theory just allows for the
calculation of the relative ratio of the DOS (Jsys) in the environmental system
(resonators, mirrors, and so on) to that (J0) in the vacuum: the Purcell factor (Jsys/J0).
5.
In this class, we aim to discuss the phenomena where the photons can be considered
as classical EM fields. The quantum nature of matters can be involved semi-classically.
- Lack of symmetry
1.
There is a lack of symmetry in the appearance of the source terms in the Maxwell
equations.
2.
The first two equations have sources; the second two do not.
3.
Conventionally, we say that no magnetic charges or currents exist. But, who knows…
1.3 Inverse Square Law or the Mass of the Photon
- Even though CEM is a limit of QED, experimental validation of Maxwell's equations and their
results is still a very important issue in Physics.
- Test of Coulomb’s law
1.
Two typical ways to test Coulomb’s law
A.
Assume that the force varies as 1/‫ ݎ‬ଶାఌ and measure ߝ precisely.
B.
Assume that the Yukawa potential
ଵ
௥
݁ ିఓ௥ and measure ߤ precisely. Here, the
mass of photon ݉ఊ can be virtually determined as ߤ = ݉ఊ ܿ/԰.
2.
Results of various tests of Coulomb's law and tests for a nonzero photon rest mass,
e.g. G. Feinberg, Science 166, 879 (1969) and E. R. Vhlliams, et al., PRL 26 721 (1971).
3.
A high-frequency test of Coulomb's law gives ߝ ~ (2.7 ± 3.1) × 10ିଵ଺ or ߤ ~ (3.22 ±
3.5) × 10ି଼ ݉ିଵ [PRL 26 721 (1971)].
1.4 Linear Superposition
(Related to assumptions in General Physics, e.g. Coulomb’s law)
ሬԦ . There are, of course, circumstances
- The Maxwell equations in vacuum are linear in ‫ܧ‬ሬԦ and ‫ܤ‬
where nonlinear (NL) effects occur: Magnetization, NL optics, mutual inductors, and so on. But,
here we are concerned with fields in vacuum or the microscopic fields inside atoms and nuclei.
At the microscopic and even at the atomic level, linear superposition is remarkable valid, if the
self-energies of point particles are infinite: the regime of CEM.
- Example: Coulomb’s law, the solution of the Poisson equation
1.
We already have a solution for the scalar potential as
Ԅ(‫ݔ‬Ԧ) =
2.
1
ߩ(‫ݔ‬Ԧ ᇱ ) ଷ ᇱ
න
݀ ‫ݔ‬Ԧ
4ߨߝ଴ ௔௟௟ |‫ݔ‬Ԧ െ ‫ݔ‬Ԧ ᇱ |
Here, the integration is over all charges in the universe, and ߶ is arbitrary only to the
extent that a constant can be added to the RHS.
3.
The singular nature of the Laplacian of 1/‫ ݎ‬can be exhibited formally in terms of a
Dirac delta function as
1
‫׏‬ଶ ൬
൰ = െ4ߨߜ(‫ݔ‬Ԧ െ ‫ݔ‬Ԧ ᇱ )
|‫ݔ‬Ԧ െ ‫ݔ‬Ԧ ᇱ |
- Quantum mechanical nonlinearity
1.
The inverse square law has a singularity as ‫ ݎ‬՜ 0. It is natural to suppose that there
is some upper limit.
2.
Born’s proposal: the vacuum is given electric and magnetic permeabilities as
ିଵ/ଶ
ߝ
ߤ଴
1
=
= ൤1 + ଶ (ܿ ଶ ‫ܤ‬ଶ െ ‫ ܧ‬ଶ )൨
ߤ
ߝ଴
ܾ
But, such theories suffer from arbitrariness in the manner of how the nonlinearity
occurs and, furthermore, there is no evidence of this kind of classical nonlinearity.
3.
Feynman’s path integral formulation: Vacuum polarization effects
A.
There is a QM NL of EM fields that arises because the uncertainty principle
permits the momentary creation of an electron-positron pair by two photons
and vice versa.
B.
This NL feature of QED can be expressed, at least for slowly varying fields, in
terms of ߝ and ߤ tensors of vacuum as
‫ܦ ۓ‬௜ = ߝ଴ ෍ ߝ௜௞ ‫ܧ‬௞
ۖ
௞
C.
4.
‫ܤ۔‬௜ = ߤ଴ ෍ ߤ௜௞ ‫ܪ‬௞
ۖ
‫ە‬
௞
Here, the electric and magnetic permeabilities in Gaussian unit, where the
Coulomb force is represented as ‫ݍ = ܨ‬ଵ ‫ݍ‬ଶ /‫ ݎ‬ଶ , are determined as
݁ ସ԰
‫ߝۓ‬௜௞ = ߜ௜௞ +
[2(‫ ܧ‬ଶ െ ܿ ଶ ‫ܤ‬ଶ )ߜ௜௞ + 7ܿ ଶ ‫ܤ‬௜ ‫ܤ‬௞ ] + ‫ڮ‬
45ߨ݉ସ ܿ ଻
݁ ସ԰
‫۔‬
[2(ܿ ଶ ‫ܤ‬ଶ െ ‫ ܧ‬ଶ )ߜ௜௞ + 7‫ܧ‬௜ ‫ܧ‬௞ ] + ‫ڮ‬
ߤ
=
ߜ
+
௜௞
‫ ە‬௜௞
45ߨ݉ସ ܿ ଻
In the classical limit ԰ ՜ 0 , the nonlinearity goes zero. For small nonlinearities,
comparison of Born’s expression and Feynman integration model employing a series
expansion provides the quantum mechanical field strength
ଵ/ଶ
45ߨ݉ସ ܿ ଻
ܾ=ቆ
ቇ
4݁ ସ ԰
~ 0.92 × 10ଶ଴ ܸ/݉
1.5 Maxwell Equations in Macroscopic Media
ሬԦ can be thought of as equations giving the fields
- The Maxwell equations for ‫ܧ‬ሬԦ and ‫ܤ‬
everywhere in space, if all sources of ߩ and ‫ܬ‬Ԧ are specified.
- Averaged quantities in macroscopic Maxwell equations (we will discuss the macroscopic view in
detail in the lecture related to Chap. 5 of the main textbook.)
1.
ሬԦ can be the averaged ‫ܧ‬ሬԦ and ‫ܤ‬
ሬԦ of the microscopic Maxwell equations.
‫ܧ‬ሬԦ and ‫ܤ‬
2.
ሬԦ and ‫ܪ‬
ሬԦ are no longer simply multiples of ‫ܧ‬ሬԦ and ‫ܤ‬
ሬԦ;
‫ܦ‬
߲ܳఈఉ
‫ۓ‬
ۖ‫ܦ‬ఈ = ߝ଴ ‫ܧ‬ఈ + ቌܲఈ െ ෍ ߲‫ ݔ‬+ ‫ ڮ‬ቍ
‫۔‬
ۖ
‫ە‬
3.
ఉ
‫ܪ‬ఈ =
ఉ
1
‫ ܤ‬+ (‫ܯ‬ఈ + ‫) ڮ‬
ߤ଴ ఈ
The components in blanket represent the macroscopically averaged electric dipole,
electric quadrupole, magnetic dipole, and higher-order moments of the material
medium in the presence of applied fields, describing the EM properties of material.
4.
One can easily see that, the ferromagnetism independent on the applied fields,
originating from the spin, a pure quantum mechanical quantity, cannot be simply
explained by the classical EM. But, the classical EM still provides an efficient way to
describe the ferromagnetic medium phenomenalistically.
- Linearity
1.
ሬԦ induces an
In case for weak enough fields, the presence of an applied ‫ܧ‬ሬԦ or ‫ܤ‬
ሬሬԦ) linearly proportional to the magnitude of
electric or magnetic polarization (ܲሬԦ or ‫ܯ‬
the applied field:
2.
‫ܦ ۓ‬ఈ = ෍ ߝఈఉ ‫ܧ‬ఉ
ۖ
ఉ
‫ = ܪ۔‬෍ 1 ‫ܤ‬
ۖ ఈ
ߤఈఉ ఉ
‫ە‬
ఉ
Using the Fourier transforms to include the nonlocality and generalize the equation,
we have (to be discussed in detail in the lecture related to Chap. 19 of the main text
book in the next semester)
ሬԦ , ߱൯ = ෍ ߝఈఉ ൫݇
ሬԦ , ߱൯‫ܧ‬ఉ ൫݇
ሬԦ, ߱൯ ฻ ‫ܦ‬ఈ (‫ݔ‬Ԧ, ‫ = )ݐ‬෍ න ݀ ଷ ‫ ݔ‬ᇱ න ݀‫ ݐ‬ᇱ ߝఈఉ (‫ݔ‬Ԧ ᇱ , ‫ ݐ‬ᇱ )‫ܧ‬ఉ (‫ݔ‬Ԧ െ ‫ݔ‬Ԧ ᇱ , ‫ ݐ‬െ ‫ ݐ‬ᇱ )
‫ܦ‬ఈ ൫݇
ఉ
ఉ
3.
At high enough field strengths, nonlinear behavior appears as
(ଵ)
(ଶ)
‫ܦ‬ఈ = ෍ ߝఈఉ ‫ܧ‬ఉ + ෍ ߝఈఉఊ ‫ܧ‬ఉ ‫ܧ‬ఊ
ఉ
ఉ,ఊ
- Momentum and energy of EM field in media
1.
In addition, linear momentum, orbital and spin angular momentum, and energy of
EM fields (photons) within media have not yet been generally defined.
2.
In fact, it is not clear how to distinguish the “electromagnetic” part from the “matter”
part. How much of
3.
ଵ
ଶ
ߝ‫ ܧ‬ଶ is the part of the energy stored in the EM fields?
The Abraham–Minkowski controversy concerning the momentum of EM fields inside
dielectric media is a long-history debate in Physics (see also a short news article,
“Momentum in an uncertain light,” Nature 444 823 (2006)).
1.6 Boundary Conditions at Interfaces between Different Media
(Related to Chap. 4 of the main text book)
- Boundary conditions in the Macroscopic view
1.
Applying the divergence theorem to the Maxwell equations,
ሬԦଶ െ ‫ܦ‬
ሬԦଵ ൯ = ߪ
݊ො ‫ ڄ‬൫‫ܦ‬
ቊ
ሬԦଶ െ ‫ܤ‬
ሬԦଵ ൯ = 0
݊ො ‫ ڄ‬൫‫ܤ‬
2.
Applying Stoke’s theorem to the Maxwell equations,
݊ො × ൫‫ܧ‬ሬԦଶ െ ‫ܧ‬ሬԦଵ ൯ = 0
ቊ
ሬԦଶ െ ‫ܪ‬
ሬԦଵ ൯ = ‫ܭ‬
ሬԦ
݊ො × ൫‫ܪ‬
3.
ሬԦ are the surface charge and current density, respectively.
Here, ߪ and ‫ܭ‬
- Microscopic, quantum mechanical view
1.
An idealization in macroscopic EM is the idea of surface charge and current densities.
The physical reality is that the charge or current is confined to the immediate
neighborhood of the surface. If this region has a thickness small compared to the
length scale of interest, we may use the approximation of the infinitesimal thickness.
2.
Also, quantum-mechanical effects should be considered. At the microscopic level, the
charge is not exactly at the surface and the field does not change discontinuously,
e.g. distribution of excess charge at the surface of a conductor and of the electric
field as shown below.
3.
To include such microscopic, quantum effects to the macroscopic picture, the efforts
on employing the density functional theory (DFT), semi-classical hydrodynamic model,
and so on, have been reported in the field of computational electromagnetism.
1.7 Short Conclusion
- As we have seen, the theory of classical electromagnetism has many limitations and areas it
does not cover. However, it has provided the starting point and inspiration for a variety of
important problems in modern physics.
- in the lectures of this semester, we will delve into the core principles and basic applications of
the theory of CEM, taking into account the following conditions:
1.
Linear phenomena (Nonlinear phenomena will only be addressed to a limited extent.)
2.
The regime of low-energy physics
3.
The framework of continuously varying fields and potentials, excluding quantum
mechanical attributes.
4.
The use of well-determined boundary conditions
(HW #1)
Due: 20240313
ሬԦ, and ‫ܥ‬Ԧ are vector fields in three dimensions. Prove the following relations.
1. Triple products. ‫ܣ‬Ԧ, ‫ܤ‬
ሬԦ × ‫ܥ‬Ԧ ൯ = ൫‫ܣ‬Ԧ × ‫ܤ‬
ሬԦ൯ ή ‫ܥ‬Ԧ
‫ܣ‬Ԧ ή ൫‫ܤ‬
ሬԦ × ‫ܥ‬Ԧ ൯ = ‫ܤ‬
ሬԦ൫‫ܣ‬Ԧ ή ‫ܥ‬Ԧ൯ െ ‫ܥ‬Ԧ൫‫ܣ‬Ԧ ή ‫ܤ‬
ሬԦ൯
‫ܣ‬Ԧ × ൫‫ܤ‬
These scalar and vector triple products in three dimensions are frequently encountered in Physics.
ሬԦ given by
2. An electric charge ‫ݍ‬ଵ moving with velocity ‫ݒ‬Ԧଵ produces a magnetic induction ‫ܤ‬
ߤ଴ ‫ݍ‬ଵ ‫ݒ‬Ԧଵ × ‫ݎ‬Ƹ
4ߨ
‫ݎ‬ଶ
ሬԦ is measured. This is Biot and
Here, ‫ݎ‬Ƹ is a unit vector that points from ‫ݍ‬ଵ to the point at which ‫ܤ‬
ሬԦ =
‫ܤ‬
Savart law.
(a) Show that the magnetic force exerted by ‫ݍ‬ଵ on a second charge ‫ݍ‬ଶ with a velocity of ‫ݒ‬Ԧଶ is
given by the vector triple product
ߤ଴ ‫ݍ‬ଵ ‫ݍ‬ଶ
‫ݒ‬Ԧ × (‫ݒ‬Ԧଵ × ‫ݎ‬Ƹ )
4ߨ ‫ ݎ‬ଶ ଶ
(b) Write out the corresponding magnetic force ‫ܨ‬Ԧଵ that ‫ݍ‬ଶ exerts on ‫ݍ‬ଵ . Do ‫ܨ‬Ԧଵ and ‫ܨ‬Ԧଶ satisfy
‫ܨ‬Ԧଶ =
Newton’s third law?
(c) Calculate and compare ‫ܨ‬Ԧଵ and ‫ܨ‬Ԧଶ for the caser of ‫ݍ‬ଵ and ‫ݍ‬ଶ moving along parallel trajectories
side by side.
(Note) Rigorous consideration on the interaction between two moving point charges requires
including the effect by the electric field and the special relativity. If you are interested in it, you may
first refer to the Liénard-Wiecher potential, that we will discuss in the next semester, from the
following sources:
- Chapter 21 of Reitz, Chapter 10 of Griffiths, Chapter 21 of Feynman Vol II
- https://en.wikipedia.org/wiki/Li%C3%A9nard%E2%80%93Wiechert_potential
- (Advanced level) Chapter 14 of Jackson
3. Rotational coordinate transformation in 3D.
(a) Show that the rotational coordinate transformation about ࢋଷ through angle ߙ is represented
by the matrix as
cos ߙ
ࡿଷ = ൭െ sin ߙ
0
sin ߙ
cos ߙ
0
0
0൱
1
(b) Show that the rotational transformation about ࢋଶ through angle ߚ is represented by
cos ߚ
ࡿଶ = ൭ 0
sin ߚ
0
1
0
െ sin ߚ
0 ൱
cos ߚ
(c) The following figure shows Euler angle rotations: (a) about ࢋଷ through angle ߙ, (b) about ࢋᇱଶ
through angle ߚ, and (c) about ࢋᇱᇱ
ଷ through angle ߛ.
Show that the rotational coordinate transformation in 3D is represented by the matrix in terms of
Euler angles as
ࣅ = ࡿଷ (ߛ)ࡿଶ (ߚ)ࡿଷ (ߙ)
cos ߛ cos ߚ cos ߙ െ sin ߛ sin ߙ
= ൭െ sin ߛ cos ߚ cos ߙ െ cos ߛ sin ߙ
sin ߚ cos ߙ
cos ߛ cos ߚ sin ߙ + sin ߛ cos ߙ
െ sin ߛ cos ߚ sin ߙ + cos ߛ cos ߙ
sin ߚ sin ߙ
െ cos ߛ sin ߚ
sin ߛ sin ߚ ൱
cos ߚ
PH231 Electromagnetism I (Spring 2024)
Lecture note 02: Vector Analysis for Electromagnetism
(Chapter 1 of the textbook by Reitz, Chapters 2 and 3 of the book by Feynman)
2.1 Understanding physics
- The physicist needs a facility in looking at problems from several points of view. The exact analysis
of real physical problems is usually quite complicated, and any particular physical situation may
be too complicated to analyze directly by solving the differential equation. Ordinally, you may
learn the physical ideas gradually by starting with simple situations and going on to more and
more complicated situations. This makes you forget that the ideas/results are true in certain
situations. Thus, mathematicians fail because the actual physical situations in the real world are so
complicated that it is necessary to have a much broader understanding of the equations.
- One can still get a very good idea of the behavior of a system if one has some feel for the
character of the solution in different circumstances. Ideas such as the capacitance, resistance, and
inductance are, for some purposes, very useful. However, such quantities are not universal or
general for understanding the behavior of objects in nature. Historically, physicists have preferred
to establish complete laws and apply them to simple situations. Physics is an empirical discipline
and is based on measurements. In this manner, we need well-determined physical quantities.
2.2 Physical quantities: Scalar and vector (in fact, tensor) fields
- In physics, a field is a physical quantity that has a value for each point in spacetime. A field can
be classified as a scalar field, a vector field, a tensor field and so on. For example, the Newtonian
gravitational field is a vector field: specifying its value at a point in spacetime requires three
numbers, the components of the gravitational field vector at that point.
- Physical quantity:
1.
In fact, it is really an important and difficult issue to define a physical quantity. Formally,
it is defined as “Property of a phenomenon, body, or substance, where the property
has a magnitude that can be expressed as a number and a reference.”
2.
Physicists want to understand matter and its behaviors through space and time. In
practice, note that different observers may get different values of a quantity depending
on the frame of reference; in turn the coordinate system and metric. Physical properties
such as length, mass or time, by themselves, are not physically invariant.
3. However, the laws of physics which include these properties are invariant.
4.
In order to construct the invariant laws, we need to employ the quantities, e.g. 4vectors, which can be transformed in the correct way, such as the Lorentzian
transformation according to the theory of special relativity.
2.3 Concept of a Scalar
- Quantities that are invariant under coordinate transformation. For example,
M(‫ݔ‬, ‫ ݔ(ܯ = )ݕ‬ᇱ , ‫ݕ‬Ԣ)
- Scalars in non-relativistic physics
1.
Examples: mass, charge, volume, time, speed, temperature …
2.
The distance between two points in three-dimensional space is a scalar, but the
direction from one of those points to the other is not: the speed of an object is a scalar
but its velocity is not.
- Scalars in relativity theory
1.
In the theory of relativity, one considers changes of coordinate systems that trade space
for time. As a consequence, several physical quantities that are scalars in "classical"
(non-relativistic) physics c to be combined with other quantities and treated as fourdimensional vectors or tensors.
2.
Examples: counts/integers (1, 2, 3, …), electric charge, space-time interval, and invariant
mass (rest mass).
2.4 Concept of a Vector
- Quantities that are transformed correctly under coordinate transformation
1.
In the non-relativistic regime, a vector quantity, ‫ܣ‬Ԧ, does transform in the way when the
coordinate system is rotated along the third axis as follows:
‫ܣ‬ᇱ
cos ߠ sin ߠ ‫ܣ‬ଵ
൬ ᇱଵ ൰ = ቀ
ቁ൬ ൰
‫ܣ‬ଶ
െ sin ߠ cos ߠ ‫ܣ‬ଶ
2.
Here, one can easily know that the norm of the vector does not change with respect
to the rotation; the norm of a vector is a scalar quantity.
‫ܣ‬Ԧᇱ ‫ܣ ڄ‬Ԧᇱ = (‫ܣ‬ଵ cos ߠ + ‫ܣ‬ଶ sin ߠ)(െ‫ܣ‬ଵ sin ߠ + ‫ܣ‬ଶ cos ߠ) = ‫ܣ‬ଵଶ + ‫ܣ‬ଶଶ = ‫ܣ‬Ԧ ‫ܣ ڄ‬Ԧ
(invariant)
- Transformation matrix for three-dimensional rotation
1.
In general, for three dimensions, we have
ଷ
‫ݔ‬௜ᇱ = ෍ ߣ௜௝ ‫ݔ‬௝
௝ୀଵ
2.
The transformation matrix for three-dimensional rotation is given as:
cos(‫ݔ‬ଵᇱ , ‫ݔ‬ଵ ) cos(‫ݔ‬ଵᇱ , ‫ݔ‬ଶ ) cos(‫ݔ‬ଵᇱ , ‫ݔ‬ଷ )
ߣଵଵ ߣଵଶ ߣଵଷ
ࣅ = ൭ߣଶଵ ߣଶଶ ߣଶଷ ൱ = ቌcos(‫ݔ‬ଶᇱ , ‫ݔ‬ଵ ) cos(‫ݔ‬ଶᇱ , ‫ݔ‬ଶ ) cos(‫ݔ‬ଶᇱ , ‫ݔ‬ଷ )ቍ
ߣଷଵ ߣଷଶ ߣଷଷ
cos(‫ݔ‬ଷᇱ , ‫ݔ‬ଵ ) cos(‫ݔ‬ଷᇱ , ‫ݔ‬ଶ ) cos(‫ݔ‬ଷᇱ , ‫ݔ‬ଷ )
3.
The inverse transformation is
ଷ
‫ݔ‬௜ = ෍ ߣ௝௜ ‫ݔ‬௝ᇱ
௝ୀଵ
4.
For orthogonal coordinate systems, the following condition must be satisfied:
෍ ߣ௜௝ ߣ௞௝ = ߜ௜௞ = ൜
௝
5.
0, ݂݅ ݅ ് ݇
1, ݂݅ ݅ = ݇
Here, ߜ௜௞ is the Kronecker delta symbol.
2.5 Definitions of a Scalar and a Vector in Terms of Transformation Properties
- Scalar
1.
Consider a coordinate transformation of the type
‫ݔ‬௜ᇱ = ȭ௝ ߣ௜௝ ‫ݔ‬௝
2.
Here, ȭ௝ ߣ௜௝ ߣ௞௝ = ߜ௜௞ so that the ‫ݔ‬௜ᇱ -axes are orthogonal.
3.
If, under such a transformation, a quantity ߶ is unaffected, then ߶ is called a scalar.
- Vector
1.
If a set of quantities (‫ܣ‬ଵ , ‫ܣ‬ଶ , ‫ܣ‬ଷ ) is transformed from the ‫ݔ‬௜ system to the ‫ݔ‬௜ᇱ system
by a transformation matrix ૃ as
‫ܣ‬ᇱ௜ = ȭ௝ ߣ௜௝ ‫ܣ‬௝
2.
Then the quantity ‫ܣ‬Ԧ = (‫ܣ‬ଵ , ‫ܣ‬ଶ , ‫ܣ‬ଷ ) is termed a vector.
- Four Vector in the Lorentz transformation
1.
If a set of quantities (‫ܣ‬௧ , ‫ܣ‬ଵ , ‫ܣ‬ଶ , ‫ܣ‬ଷ ) is transformed from the ‫ݔ‬௜ system to the ‫ݔ‬௜ᇱ
system by a transformation by a Lorentz transformation matrix ࡸ with the result
‫ܣ‬ᇱ௜ = ȭ௝ ‫ܮ‬௜௝ ‫ܣ‬௝
2.
Then the quantity ࡭ = (‫ܣ‬௧ , ‫ܣ‬ଵ , ‫ܣ‬ଶ , ‫ܣ‬ଷ ) is termed a four vector in the Minkowski space.
3.
The Lorentz transformations preserve this magnitude and include spatial rotations,
boosts (a change by a constant velocity to another inertial reference frame), and
temporal and spatial inversions.
4.
This is the topic in the next semester.
2.6 Scalar Product of Two Vectors
ሬԦ to form the scalar product, or called the dot product,
- The multiplication of two vectors ‫ܣ‬Ԧ and ‫ܤ‬
is defined to be
ሬԦ = ෍ ‫ܣ‬௜ ‫ܤ‬௜
‫ܣ‬Ԧ ‫ܤ ڄ‬
௜
ሬԦ is indeed a scalar.
- The dot product ‫ܣ‬Ԧ ‫ܤ ڄ‬
1.
ሬԦ are transformed as vectors:
‫ܣ‬Ԧ and ‫ܤ‬
‫ܣ‬ᇱ௜ = ȭ௝ ߣ௜௝ ‫ܣ‬௝ ܽ݊݀ ‫ܤ‬௜ᇱ = ȭ௝ ߣ௜௝ ‫ܤ‬௝
2.
ሬԦ ᇱ becomes
The product ‫ܣ‬Ԧᇱ ‫ܤ ڄ‬
ሬԦ ᇱ = ෍ ‫ܣ‬ᇱ௜ ‫ܤ‬௜ᇱ = ෍ ቌ෍ ߣ௜௝ ‫ܣ‬௝ ቍ ൭෍ ߣ௜௞ ‫ܤ‬௞ ൱
‫ܣ‬Ԧᇱ ‫ܤ ڄ‬
௜
௜
௝
௞
ሬԦ
= ෍ ෍ ൭෍ ߣ௜௝ ߣ௜௞ ൱ ‫ܣ‬௝ ‫ܤ‬௞ = ෍ ෍ ߜ௝௞ ‫ܣ‬௝ ‫ܤ‬௞ = ෍ ‫ܣ‬௝ ‫ܤ‬௝ = ‫ܣ‬Ԧ ‫ܤ ڄ‬
௝
3.
௞
௜
௝
௞
௝
This quantity is invariant under coordinate transformation and thus a scalar.
ሬԦ is commutative and distributive:
- The scalar product ‫ܣ‬Ԧ ‫ܤ ڄ‬
1.
2.
ሬԦ = σ௜ ‫ܣ‬௜ ‫ܤ‬௜ = σ௜ ‫ܤ‬௜ ‫ܣ‬௜ = ‫ܤ‬
ሬԦ ‫ܣ ڄ‬Ԧ
‫ܣ‬Ԧ ‫ܤ ڄ‬
ሬԦ + ‫ܥ‬Ԧ൯ = σ௜ ‫ܣ‬௜ (‫ܤ‬௜ + ‫ܥ‬௜ ) = σ௜(‫ܣ‬௜ ‫ܤ‬௜ + ‫ܣ‬௜ ‫ܥ‬௜ ) = ‫ܣ‬Ԧ ‫ܤ ڄ‬
ሬԦ + ‫ܣ‬Ԧ ‫ܥ ڄ‬Ԧ
‫ܣ‬Ԧ ‫ ڄ‬൫‫ܤ‬
2.7 Unit Vectors
- Vectors having a length equal to the unit of length
- One can choose unit vectors along the particular coordinate axes, such as (࢏, ࢐, ࢑), (ࢋ࢘ , ࢋࣂ , ࢋࢠ ), and
൫ࢋ࢘ , ࢋࣂ , ࢋࣘ ൯, to describe a vector in terms of the components along the axes
‫ܣ‬Ԧ = ȭ୧ ‫ܣ‬௜ ࢋ௜ ‫ܣ ݎ݋‬௜ = ‫ܣ‬Ԧ ‫ࢋ ڄ‬௜
- If any two unit vectors are orthogonal, we have ࢋ࢏ ‫ߜ = ࢐ࢋ ڄ‬௜௝
2.8 Vector Product of Two Vectors
ሬԦ to form the vector product, or called the cross product,
- The multiplication of two vectors ‫ܣ‬Ԧ and ‫ܤ‬
is defined to be
ሬԦ ൯ = ෍ ෍ ߝ௜௝௞ ‫ܣ‬௝ ‫ܤ‬௞
൫‫ܣ‬Ԧ × ‫ܤ‬
௜
௝
௞
- The symbol ߝ௜௝௞ is the permutation symbol (or Levi-Civita symbol) using the right-hand rule :
1.
in three dimensions:
0,
ߝ௜௝௞ = ቐ+1,
െ1,
2.
if any index is equal to any other index
if ݅, ݆, ݇ form an ݁‫ ݊݁ݒ‬permutation of 1,2,3
if ݅, ݆, ݇ form an ‫ ݀݀݋‬permutation of 1,2,3
in four dimensions:
0,
ߝ௜௝௞௟ = ቐ+1,
െ1,
if any index is equal to any other index
if ݅, ݆, ݇, ݈ form an ݁‫ ݊݁ݒ‬permutation of 1,2,3,4
if ݅, ݆, ݇, ݈ form an ‫ ݀݀݋‬permutation of 1,2,3,4
Some examples:
ߝଵସଷଶ = െߝଵଶଷସ = െ1
ߝଶଵଷସ = െߝଵଶଷସ = െ1
ߝସଷଶଵ = െߝଵଷଶସ = െ(െߝଵଶଷସ ) = 1
ߝଷଶସଷ = െߝଷଶସଷ = 0
- The orthogonality of the unit vectors
1.
ሬԦ has to be orthogonal to ‫ܣ‬Ԧ and ‫ܤ‬
ሬԦ:
The direction of the vector product ‫ܣ‬Ԧ × ‫ܤ‬
ሬԦ ൯ = ‫ܤ‬
ሬԦ ‫ ڄ‬൫‫ܣ‬Ԧ × ‫ܤ‬
ሬԦ ൯ = ૙
‫ܣ‬Ԧ ‫ ڄ‬൫‫ܣ‬Ԧ × ‫ܤ‬
2.
The vector product of the unit vectors
ࢋ௜ × ࢋ௝ = ࢋ௞ = ȭ௞ ߝ௜௝௞ ࢋ௞
3.
ሬԦ can be expressed as
Now, the vector product ‫ܣ‬Ԧ × ‫ܤ‬
ࢋଵ
ሬԦ = ȭ௜,௝,௞ ߝ௜௝௞ ࢋ௜ ‫ܣ‬௝ ‫ܤ‬௞ = อ‫ܣ‬ଵ
‫ܥ‬Ԧ = ‫ܣ‬Ԧ × ‫ܤ‬
‫ܤ‬ଵ
ࢋଶ
‫ܣ‬ଶ
‫ܤ‬ଶ
࢏
ࢋଷ
‫ܣ‬ଷ อ = ቮ‫ܣ‬௫
‫ܤ‬ଷ
‫ܤ‬௫
࢐
‫ܣ‬௬
‫ܤ‬௬
࢑
‫ܣ‬௭ ቮ
‫ܤ‬௭
- Magnitude of the vector product
1.
ሬԦ ൯൧
Consider the expansion of the quantity ൣ‫ ܤܣ‬sin൫‫ܣ‬Ԧ, ‫ܤ‬
ଶ
in the three-dimensional
orthogonal coordinate system:
ሬԦ ൯ = ‫ܣ‬ଶ ‫ ܤ‬ଶ െ ‫ܣ‬ଶ ‫ ܤ‬ଶ cosଶ ߠ
‫ܣ‬ଶ ‫ ܤ‬ଶ sinଶ ൫‫ܣ‬Ԧ, ‫ܤ‬
= (ȭ௜ ‫ܣ‬ଶ௜ )(ȭ௜ ‫ܤ‬௜ଶ ) െ (ȭ௜ ‫ܣ‬௜ ‫ܤ‬௜ )ଶ
ଶ
= ȭ௜ ൫ȭ௝ ȭ௞ ߝ௜௝௞ ‫ܣ‬௝ ‫ܤ‬௞ ൯ = ȭ௜ ‫ܥ‬௜ଶ = ห‫ܥ‬Ԧห
2.
ଶ
The magnitude of the vector product in three dimensions is equal to the area of the
ሬԦ ൯.
parallelogram, ‫ ܤܣ‬sin൫‫ܣ‬Ԧ, ‫ܤ‬
ሬԦ is indeed a vector.
- There still remains the problem of verifying that the cross product ‫ܥ‬Ԧ = ‫ܣ‬Ԧ × ‫ܤ‬
1.
Starting in a rotated (primed) system
‫ܥ‬௜ᇱ = ‫ܣ‬௝ᇱ ‫ܤ‬௞ᇱ െ ‫ܣ‬ᇱ௞ ‫ܤ‬௝ᇱ ݅, ݆, ݇ in cyclic order
= ȭ௟ ߣ௝௟ ‫ܣ‬௟ ȭ௠ ߣ௞௠ ‫ܤ‬௠ െ ȭ௟ ߣ௞௟ ‫ܣ‬௟ ȭ௠ ߣ௝௠ ‫ܤ‬௠
= ȭ௟,௠ ൫ߣ௝௟ ߣ௞௠ െ ߣ௞௟ ߣ௝௠ ൯‫ܣ‬௟ ‫ܤ‬௠
2.
We have the following direction cosine combinations and their negatives:
ߣଵଵ ߣଶଶ െ ߣଵଶ ߣଶଵ = ߣଷଷ
ߣଵଷ ߣଶଵ െ ߣଶଷ ߣଵଵ = ߣଷଶ
ߣଵଶ ߣଶଷ െ ߣଶଶ ߣଵଷ = ߣଷଵ
3.
This identity holds for rotations because they preserve volumes. For a more general
orthogonal transformation, the RHS of the identity is multiplied by the determinant of
the transformation matrix. (You may study this topic in Mathematical Physics courses.)
4.
Substituting the combinations into the first equation,
‫ܥ‬ଷᇱ = ߣଷଷ ‫ܣ‬ଵ ‫ܤ‬ଶ + ߣଷଶ ‫ܣ‬ଷ ‫ܤ‬ଵ + ߣଷଵ ‫ܣ‬ଶ ‫ܤ‬ଷ െ ߣଷଷ ‫ܣ‬ଶ ‫ܤ‬ଵ െ ߣଷଶ ‫ܣ‬ଵ ‫ܤ‬ଷ െ ߣଷଵ ‫ܣ‬ଷ ‫ܤ‬ଶ
= ɉଷଵ ‫ܥ‬ଵ + ߣଷଶ ‫ܥ‬ଶ + ߣଷଷ ‫ܥ‬ଷ = ȭ௝ ߣଷ௝ ‫ܥ‬௝
5.
Therefore
‫ܥ‬௜ᇱ = ȭ௝ ߣ௜௝ ‫ܥ‬௝
- Triple products: The following scalar and vector triple products in three dimensions are frequently
encountered in Physics.
ሬԦ × ‫ܥ‬Ԧ൯ = ൫‫ܣ‬Ԧ × ‫ܤ‬
ሬԦ ൯ ή ‫ܥ‬Ԧ
‫ܣ‬Ԧ ή ൫‫ܤ‬
ሬԦ × ‫ܥ‬Ԧ൯ = ‫ܤ‬
ሬԦ ൫‫ܣ‬Ԧ ή ‫ܥ‬Ԧ൯ െ ‫ܥ‬Ԧ൫‫ܣ‬Ԧ ή ‫ܤ‬
ሬԦ ൯
‫ܣ‬Ԧ × ൫‫ܤ‬
2.9 Derivatives of fields
- If a scalar function ߶(‫ )ݏ‬is differentiated with respect to the scalar variable ‫ݏ‬, then the derivative
itself is a scalar. It is because the part of the derivative cannot change under a coordinate
transformation either.
݀߶ ݀߶ ᇱ
݀߶ ᇱ
=
=
൬
൰
݀‫ݏ‬
݀‫ ݏ‬ᇱ
݀‫ݏ‬
- Similarly, the differentiation of a vector ‫ܣ‬Ԧ with respect to a scalar ‫ݏ‬.
1.
Starting from ‫ܣ‬ᇱ௜ = ȭ௝ ߣ௜௝ ‫ܣ‬௝
2.
Since the direction cosines, ߣ௜௝ , are independent on ‫ ݏ‬ᇱ ,
݀‫ܣ‬௝
݀‫ܣ‬ᇱ௜
݀
= ᇱ ቌ෍ ߣ௜௝ ‫ܣ‬௝ ቍ = ෍ ߣ௜௝ ᇱ
ᇱ
݀‫ݏ‬
݀‫ݏ‬
݀‫ݏ‬
௝
௝
3.
‫ ݏ‬and ‫ ݏ‬ᇱ are identical and thus
݀‫ܣ‬௝
݀‫ܣ‬ᇱ௜
݀‫ܣ‬௜ ᇱ
=
൬
൰ = ෍ ߣ௜௝ ቆ
ቇ
݀‫ ݏ‬ᇱ
݀‫ݏ‬
݀‫ݏ‬
௝
4.
Therefore, the differentiation of a vector ‫ܣ‬Ԧ, which can be written as ݀‫ܣ‬Ԧ/ds, is a vector,
of which the geometrical interpretation is given as follows
݀‫ܣ‬Ԧ
ȟ‫ܣ‬Ԧ
= lim
݀‫୼ ݏ‬௦՜଴ ȟ‫ݏ‬
2.10 Gradient Operator: Gradient, Divergence, and Curl
- Gradient: differentiation of a scalar as a function of multiple variables
1.
When applying the concept of scalars and vectors to derivatives and integration, i.e.,
vector calculus, the simplest is the relation of a particular vector field to the derivatives
of a scalar field as a function of three-dimensional position.
2.
The coordinate transform of the differentiation of a scalar ߮(‫ݔ‬ଵ , ‫ݔ‬ଶ , ‫ݔ‬ଷ ) = ߮ ᇱ (‫ݔ‬ଵᇱ , ‫ݔ‬ଶᇱ , ‫ݔ‬ଷᇱ ):
߲߮ ᇱ
߲߮ ߲‫ݔ‬௝
ᇱ =෍
߲‫ݔ‬௜
߲‫ݔ‬௝ ߲‫ݔ‬௜ᇱ
௝
3.
Differentiating the inverse transformation of orthogonal coordinate systems,
߲‫ݔ‬௝
߲
߲‫ݔ‬௞ᇱ
ᇱ
=
ቌ෍
ߣ
‫ݔ‬
ቍ
=
෍
ߣ
= ෍ ߣ௞௝ ߜ௜௞ = ߣ௜௝
௞௝
௞௝
௞
߲‫ݔ‬௜ᇱ ߲‫ݔ‬௜ᇱ
߲‫ݔ‬௜ᇱ
௞
4.
௞
௞
Then, we obtain
߲߮ ᇱ
߲߮
= ෍ ߣ௜௝
߲‫ݔ‬௜ᇱ
߲‫ݔ‬௝
௝
5.
డఝ
డఝ
డఝ
భ
మ
య
Consequently, three derivatives, డ௫ , డ௫ , and డ௫ , can form a vector, termed the
gradient of the function ߮, as:
߲߮ ߲߮ ߲߮
܏‫ = ߮׏ = ߮ ܌܉ܚ‬൬
,
,
൰
߲‫ݔ‬ଵ ߲‫ݔ‬ଶ ߲‫ݔ‬ଷ
6.
The complete vector gradient operator is expressed as:
‫ = ׏‬෍ ࢋ௜
௜
߲
߲
߲
߲
=൬
,
,
൰
߲‫ݔ‬௜
߲‫ݔ‬ଵ ߲‫ݔ‬ଶ ߲‫ݔ‬ଷ
- The gradient is normal to the level surface.
1.
From the chain rule,
݀߮ = ෍
߲߮
݀‫ݔ‬௜ = ෍ (‫)߮׏‬௜ ݀‫ݔ‬௜ = ‫ܛ݀ ڄ ߮׏‬
߲‫ݔ‬
௜
௜
௜
2.
If ݀‫ ܛ‬lies on the level surface of ߮, ݀߮ = 0 = ‫ܛ݀ ڄ ߮׏‬.
3.
Therefore, ‫ ߮׏‬has to be normal to the level surface, also called equipotential surface,
as shown in the figures below.
4.
Because any direction in space can be specified in terms of the unit vector ‫ ܛ‬in that
direction, the rate of change of ߮ in the direction of ࢙, that is the directional derivative
of ߮, can be found as
‫ؠ ܛ݀ ڄ ߮׏‬
߲߮
߲‫ݏ‬
- Representation of the gradient in three dimensions
1.
It is obvious that in the Cartesian coordinate, the gradient of a scalar is represented as
‫࢏ = ߮׏‬
߲߮
߲߮
߲߮
+࢐
+࢑
߲‫ݔ‬
߲‫ݕ‬
߲‫ݖ‬
൬or ‫ݔ‬ො
߲߮
߲߮
߲߮
+ ‫ݕ‬ො
+ ‫ݖ‬Ƹ ൰
߲‫ݔ‬
߲‫ݕ‬
߲‫ݖ‬
2.
In the cylindrical coordinate, (ߩ, ߠ, ‫)ݖ‬, the gradient is represented as
‫ࢋ = ߮׏‬ఘ
3.
൬or ߩො
߲߮
1 ߲߮
߲߮
+ ߠ෠
+ ‫ݖ‬Ƹ ൰
߲ߩ
ߩ ߲ߠ
߲‫ݖ‬
In the spherical coordinate, (‫ݎ‬, ߠ, ߶), we have
‫ࢋ = ߮׏‬௥
4.
߲߮
1 ߲߮
߲߮
+ ࢋఏ
+ ࢋ௭
߲ߩ
ߩ ߲ߠ
߲‫ݖ‬
߲߮
1 ߲߮
1 ߲߮
+ ࢋఏ
+ ࢋథ
߲‫ݎ‬
‫ߠ߲ ݎ‬
‫ ݎ‬sin ߠ ߲߶
൬or ‫ݎ‬Ƹ
߲߮
1 ߲߮
1 ߲߮
+ ߠ෠
+ ߶෠
൰
߲‫ݎ‬
‫ߠ߲ ݎ‬
‫ ݎ‬sin ߠ ߲߶
Derivation of the representation in the cylindrical and spherical coordinates is H.W.
- Divergence
1.
Another important derivative operation is the dot product of ‫ ׏‬with a vector field ‫ܨ‬Ԧ :
div ‫ܨ‬Ԧ = ‫ܨ ڄ ׏‬Ԧ = ෍
2.
߲‫ܨ‬௜
߲‫ݔ‬
௜
௜
In 3D, this product is invariant under a coordinate transformation and thus a scalar.
߲‫ݔ‬௝ ߲‫ܨ‬௜ᇱ
߲‫ݔ‬௝ ߲
߲‫ܨ‬௜ᇱ
=
෍
෍
=෍ ෍
෍ ߣ௜௞ ‫ܨ‬௞
ᇱ
ᇱ
ᇱ
௜ ߲‫ݔ‬௜
௜
௝ ߲‫ݔ‬௜ ߲‫ݔ‬௝
௜
௝ ߲‫ݔ‬௜ ߲‫ݔ‬௝
௞
߲‫ݔ‬௝
߲
߲
߲
=෍
෍ ෍
෍ ෍ ߣ௜௝ ߣ௜௞ ‫ܨ‬௞ = ෍
෍ ߜ௝௞ ‫ܨ‬௞
ᇱ ߣ௜௞ ‫ܨ‬௞ = ෍
௝ ߲‫ݔ‬௝
௜
௞ ߲‫ݔ‬௜
௝ ߲‫ݔ‬௝
௞
௜
௝ ߲‫ݔ‬௝
௞
߲‫ܨ‬௝
=෍
= ‫ܨ ڄ ׏‬Ԧ
௝ ߲‫ݔ‬௝
‫׏‬ᇱ ‫ܨ ڄ‬Ԧ ᇱ = ෍
డ௫
3.
Here, the chain rule and డ௫ೕᇲ = ߣ௜௝ were used.
4.
The divergence can be interpreted as the limit of the source strength per unit volume.
5.
The divergence in 3D is represented in various coordinate systems as
೔
߲‫ܨ‬௫ ߲‫ܨ‬௬ ߲‫ܨ‬௭
(Cartesian coordinate)
+
+
߲‫ݔ‬
߲‫ݕ‬
߲‫ݖ‬
1 ߲(‫ܨݎ‬௥ ) 1 ߲‫ܨ‬ఏ ߲‫ܨ‬௭
(Cylindrical coordinate)
=
+
+
‫ݎ߲ ݎ‬
‫ߠ߲ ݎ‬
߲‫ݖ‬
1 ߲(‫ ݎ‬ଶ ‫ܨ‬௥ )
1 ߲(sin ߠ ‫ܨ‬ఏ )
1 ߲‫ܨ‬థ
(Spherical coordinate)
= ଶ
+
+
‫ݎ‬
߲‫ݎ‬
‫ ݎ‬sin ߠ
߲ߠ
‫ ݎ‬sin ߠ ߲߶
‫ܨ ڄ ׏‬Ԧ =
6.
Laplacian, the successive operation of the gradient operator with a scalar or a vector.
(ߘ ή ‫ ߘ = ߶)׏‬ή (‫׏ = )߶׏‬ଶ ߶ = ෍
(ߘ ή ‫ܨ)׏‬Ԧ = ‫׏‬ଶ ‫ܨ‬Ԧ = ෍
߲ଶ߶
ଶ = a scalar field
௜ ߲‫ݔ‬௜
߲ ଶ ‫ܨ‬௜
ଶ ࢋ௜ = a vector field
௜ ߲‫ݔ‬௜
- Curl
1.
The cross SURGXFW RI ø ZLWK D YHFWRU ILHOG ‫ܨ‬Ԧ is also invariant under a coordinate
transformation (H.W.) and thus a vector.
curl ‫ܨ‬Ԧ = ‫ܨ × ׏‬Ԧ = ෍
௜,௝,௞
࢏
߲
= ተተ
߲‫ݔ‬
‫ܨ‬௫
2.
࢐
߲
߲‫ݕ‬
‫ܨ‬௬
ߝ௜௝௞
߲‫ܨ‬௞
ࢋ
߲‫ݔ‬௝ ௜
࢑
߲
߲‫ܨ‬௬ ߲‫ܨ‬௭
߲‫ܨ‬௭ ߲‫ܨ‬௬
߲‫ܨ‬௫ ߲‫ܨ‬௬
െ
ቇ࢏ + ቆ
െ
ቇ࢐ + ቆ
െ
ቇ࢑
ተተ = ቆ
߲‫ݖ‬
߲‫ݕ‬
߲‫ݖ‬
߲‫ݖ‬
߲‫ݕ‬
߲‫ݕ‬
߲‫ݔ‬
‫ܨ‬௭
(Cartesian coordinate)
The curl represents the rotation/twisting of a vector field and points in the direction of
the axis of the maximum rotation.
3.
Note that the curl is not generally orthogonal to the vector field itself.
4.
Curls have interesting properties.
A.
The curl of the gradient is always zero.
(In fact, ‫ =׏ × ׏‬0)
‫ = ߶׏ × ׏‬0
B.
The divergence of any curl is also zero.
‫ ڄ ׏‬൫‫ܨ × ׏‬Ԧ ൯ = 0
C.
The curl of a curl is given as
‫ × ׏‬൫‫ܨ × ׏‬Ԧ ൯ = ‫׏‬൫‫ܨ ڄ ׏‬Ԧ ൯ െ ‫׏‬ଶ ‫ܨ‬Ԧ
D.
For a vector satisfying ‫ܨ ڄ ׏‬Ԧ = 0, ‫ × ׏‬൫‫ܨ × ׏‬Ԧ ൯ becomes the Laplacian of the original
vector, which will be encountered in the discussions of (magneto-)statics.
- Vector calculus identities
1.
The vector differential operators may be applied them to various products of two
vectors and scalars. Here are some examples of vector calculus identities:
‫ × ׏‬൫‫ܨ × ׏‬Ԧ ൯ = ‫׏‬൫‫ܨ ڄ ׏‬Ԧ ൯ െ ‫׏‬ଶ ‫ܨ‬Ԧ
‫ ߰)߮׏( = )߰߮(׏‬+ ߮‫߰׏‬
‫׏‬൫‫ܨ‬Ԧ ‫ܩ ڄ‬Ԧ ൯ = ൫‫ܨ‬Ԧ ‫׏ ڄ‬൯‫ܩ‬Ԧ + ‫ܨ‬Ԧ × ൫‫ܩ × ׏‬Ԧ ൯ + ൫‫ܩ‬Ԧ ‫׏ ڄ‬൯‫ܨ‬Ԧ + ‫ܩ‬Ԧ × ൫‫ܨ × ׏‬Ԧ ൯
‫ ڄ ׏‬൫߮‫ܨ‬Ԧ ൯ = (‫ܨ ڄ )߮׏‬Ԧ + ߮‫ܨ ڄ ׏‬Ԧ
‫ ڄ ׏‬൫‫ܨ‬Ԧ × ‫ܩ‬Ԧ ൯ = ൫‫ܨ × ׏‬Ԧ ൯ ‫ܩ ڄ‬Ԧ െ ൫‫ܩ × ׏‬Ԧ ൯ ‫ܨ ڄ‬Ԧ
‫ × ׏‬൫߮‫ܨ‬Ԧ ൯ = (‫ܨ × )߮׏‬Ԧ + ߮‫ܨ × ׏‬Ԧ
‫ × ׏‬൫‫ܨ‬Ԧ × ‫ܩ‬Ԧ ൯ = ൫‫ܩ ڄ ׏‬Ԧ ൯‫ܨ‬Ԧ െ ൫‫ܨ ڄ ׏‬Ԧ ൯‫ܩ‬Ԧ + (‫ܨ)׏ ڄ ܩ‬Ԧ െ ൫‫ܨ‬Ԧ ‫׏ ڄ‬൯‫ܩ‬Ԧ
2.
There are some useful types of functions coming up often enough in EM theory:
‫ݎ ڄ ׏‬Ԧ = 3 ֜ ‫׏‬ଶ ‫ݎ‬Ԧ = 0
‫ݎ × ׏‬Ԧ = 0
൫‫ܨ‬Ԧ ‫׏ ڄ‬൯‫ݎ‬Ԧ = ‫ܨ‬Ԧ
1
1
‫׏‬ଶ ൬
൰ = െ4ߨߜ(‫ݎ‬Ԧ െ ‫ݎ‬Ԧ଴ ) ֜ ‫׏‬ଶ ൬ ൰ = െ4ߨߜ(‫ݎ‬Ԧ) (Green function for Laplace operator in freespace)
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ଴ |
‫ݎ‬
2.11 Vector Integrals, Gauss’ theorem, and Stokes’ theorem
- Definition of divergence at the limit point
1.
The divergence of a vector is the limit of its surface integral per unit volume as the
volume ܸ enclosed by the surface ܵ goes to zero: the strength of the source of the
vector field per unit volume.
1
ර ‫ܨ‬Ԧ ‫݊ ڄ‬ො݀ܽ
௏՜଴ ܸ ௌ
div ‫ܨ‬Ԧ = lim
2.
Let us now consider an infinitesimal box with a volume of ܸ = ȟ‫ݔ‬ȟ‫ݕ‬ȟ‫ ݖ‬in rectangular
coordinates.
3.
Here, ݊ො = െࢋ௫ and ࢋ௫ for the left and right surfaces, respectively.
4.
The surface integral over ܵ is then evaluated as
ර ‫ܨ‬Ԧ ‫݊ ڄ‬ො݀ܽ = ൫‫ܨ‬௫ (‫ ݔ‬+ ȟ‫ݔ‬, ‫ݕ‬, ‫ )ݖ‬െ ‫ܨ‬௫ (‫ݔ‬, ‫ݕ‬, ‫)ݖ‬൯ȟ‫ݕ‬ȟ‫ ݖ‬+ ቀ‫ܨ‬௬ (‫ݔ‬, ‫ ݕ‬+ ȟ‫ݕ‬, ‫ )ݖ‬െ ‫ܨ‬௬ (‫ݔ‬, ‫ݕ‬, ‫)ݖ‬ቁ ȟ‫ݔ‬ȟ‫ݖ‬
ௌ
+ ൫‫ܨ‬௭ (‫ݔ‬, ‫ݕ‬, ‫ ݖ‬+ ȟ‫ )ݖ‬െ ‫ܨ‬௭ (‫ݔ‬, ‫ݕ‬, ‫)ݖ‬൯ȟ‫ݔ‬ȟ‫ݕ‬
5.
When ȟ‫ݔ‬, ȟ‫ݕ‬, and ȟ‫ ݖ‬are infinitesimally small, we can ignore higher order terms of
ȟ‫ ݔ‬in approximations:
߲‫ܨ‬௫ (‫ݔ‬, ‫ݕ‬, ‫)ݖ‬
߲‫ܨ‬௫ (‫ݔ‬, ‫ݕ‬, ‫)ݖ‬
ȟ‫ ݔ‬+ ܱ(ȟ‫ ݔ‬ଶ ) ؆ ‫ܨ‬௫ (‫ݔ‬, ‫ݕ‬, ‫ )ݖ‬+
ȟ‫ݔ‬
߲‫ݔ‬
߲‫ݔ‬
߲‫ܨ‬௫ (‫ݔ‬, ‫ݕ‬, ‫)ݖ‬
֜ ‫ܨ‬௫ (‫ ݔ‬+ ȟ‫ݔ‬, ‫ݕ‬, ‫ )ݖ‬െ ‫ܨ‬௫ (‫ݔ‬, ‫ݕ‬, ‫= )ݖ‬
ȟ‫ݔ‬
߲‫ݔ‬
‫ܨ‬௫ (‫ ݔ‬+ ȟ‫ݔ‬, ‫ݕ‬, ‫ܨ = )ݖ‬௫ (‫ݔ‬, ‫ݕ‬, ‫ )ݖ‬+
6.
We thus have
߲‫ܨ‬௫ ߲‫ܨ‬௬ ߲‫ܨ‬௭
ර ‫ܨ‬Ԧ ‫݊ ڄ‬ො݀ܽ = ቆ
+
+
ቇ ȟ‫ݔ‬ȟ‫ݕ‬ȟ‫ݖ‬
߲‫ݔ‬
߲‫ݕ‬
߲‫ݖ‬
ௌ
1 ߲‫ܨ‬௫ ߲‫ܨ‬௬ ߲‫ܨ‬௭
߲‫ܨ‬௫ ߲‫ܨ‬௬ ߲‫ܨ‬௭
֜ div ‫ܨ‬Ԧ = lim ቆ
+
+
ቇ ȟ‫ݔ‬ȟ‫ݕ‬ȟ‫= ݖ‬
+
+
= ‫ܨ ڄ ׏‬Ԧ
௏՜଴ ܸ ߲‫ݔ‬
߲‫ݕ‬
߲‫ݖ‬
߲‫ݔ‬
߲‫ݕ‬
߲‫ݖ‬
7.
In fact, as we have already seen in Section 2.10, the divergence in the form of the dot
product of ‫ ׏‬with a vector field ‫ܨ‬Ԧ is applicable to any coordinate system and invariant
under coordinate transformations.
- Definition of curl at the limit point
1.
The curl of a vector is the limit of the surface integral of its cross product with the
outward drawn normal per unit volume as the volume enclosed by the surface goes to
zero, giving the magnitude of the rotation/twisting of a vector field and pointing the
direction of the axis of the maximum rotation.
1
ර ݊ො × ‫ܨ‬Ԧ ݀ܽ
௏՜଴ ܸ ௌ
curl ‫ܨ‬Ԧ = lim
8.
For the ‫ݔ‬-component in rectangular coordinates, the surfaces normal to the ‫ݔ‬-axis do
not contribute to the surface integral so that
ර ൫݊ො × ‫ܨ‬Ԧ ൯௫ ݀ܽ = ൣࢋ௭ × ‫ܨ‬Ԧ (‫ݔ‬, ‫ݕ‬, ‫ ݖ‬+ ȟ‫ )ݖ‬െ ࢋ௭ × ‫ܨ‬Ԧ (‫ݔ‬, ‫ݕ‬, ‫)ݖ‬൧ȟ‫ݔ‬ȟ‫ݕ‬
ௌ
+ ൣࢋ௬ × ‫ܨ‬Ԧ (‫ݔ‬, ‫ ݕ‬+ ȟ‫ݕ‬, ‫ ݖ‬+ ȟ‫ )ݖ‬െ ࢋ௬ × ‫ܨ‬Ԧ (‫ݔ‬, ‫ݕ‬, ‫)ݖ‬൧ȟ‫ݔ‬ȟ‫ݖ‬
= ൣെ‫ܨ‬௬ (‫ݔ‬, ‫ݕ‬, ‫ ݖ‬+ ȟ‫ )ݖ‬+ ‫ܨ‬௬ (‫ݔ‬, ‫ݕ‬, ‫)ݖ‬൧ȟ‫ݔ‬ȟ‫ ݕ‬+ [‫ܨ‬௭ (‫ݔ‬, ‫ ݕ‬+ ȟ‫ݕ‬, ‫ )ݖ‬െ ‫ܨ‬௭ (‫ݔ‬, ‫ݕ‬, ‫])ݖ‬ȟ‫ݔ‬ȟ‫ݖ‬
9.
In similar to the case of the divergence, we may ignore higher order terms of ȟ‫ ݔ‬in
approximations:
߲‫ܨ‬௭ ߲‫ܨ‬௬
߲‫ܨ‬௭ ߲‫ܨ‬௬
െ
ቇ ȟ‫ݔ‬ȟ‫ݕ‬ȟ‫ ֜ ݖ‬൫curl ‫ܨ‬Ԧ ൯࢞ = ቆ
െ
ቇ
߲‫ݕ‬
߲‫ݖ‬
߲‫ݕ‬
߲‫ݖ‬
ௌ
߲‫ܨ‬௬ ߲‫ܨ‬௭
߲‫ܨ‬௭ ߲‫ܨ‬௬
߲‫ܨ‬௫ ߲‫ܨ‬௬
֜ curl ‫ܨ‬Ԧ = ቆ
െ
ቇ࢏ + ቆ
െ
ቇ࢐ + ቆ
െ
ቇ ࢑ = ‫ܨ × ׏‬Ԧ
߲‫ݕ‬
߲‫ݖ‬
߲‫ݖ‬
߲‫ݕ‬
߲‫ݕ‬
߲‫ݔ‬
ර ൫݊ො × ‫ܨ‬Ԧ ൯௫ ݀ܽ ؆ ቆ
10. The definition of curl at the limit point is identical to the form of the cross product of
ø ZLWK D YHFWRU ILHOG ‫ܨ‬Ԧ , applicable to any coordinate system and invariant under
coordinate transformations.
11. Another equivalent definition is more useful, especially for the proof of Stokes’ theorem.
A.
The component of curl ‫ܨ‬Ԧ in the direction of the unit vector ܽො is the limit of a line
integral per unit area normal to ܽො, as the area ܵ enclosed by ‫ ܥ‬goes to zero:
1
ܽො ‫ ڄ‬curl ‫ܨ‬Ԧ = lim ර ‫ܨ‬Ԧ ‫݈݀ ڄ‬Ԧ
ௌ՜଴ ܵ ஼
B.
To show the equivalence, let us consider a vertical pillar with an infinitesimal
thickness of Ɍ along the axis of ܽො as shown below.
C.
It is obvious that the top and bottom surfaces with ݊ො = ܽො do not contribute to
the component of the curl along ܽො and only the side surface does.
1
1
ර ܽො ‫ ڄ‬൫݊ො × ‫ܨ‬Ԧ ൯݀ܽ = lim
න ܽො ‫ ڄ‬൫݊ො × ‫ܨ‬Ԧ ൯݀ܽ
௏՜଴ ܸ ௌ
ௌ,క՜଴ ߦܵ ୱ୧ୢୣ
ܽො ‫ ڄ‬curl ‫ܨ‬Ԧ = lim
D.
Here, ܵ is the area of the top and bottom surfaces. Considering that on the side
surface, ܽො ‫ ڄ‬൫݊ො × ‫ܨ‬Ԧ ൯݀ܽ = (ܽො × ݊ො) ‫ܨ ڄ‬Ԧ ݀ܽ = ‫ܨ‬Ԧ ‫݈݀ߦ ڄ‬Ԧ, we finally have
ܽො ‫ ڄ‬curl ‫ܨ‬Ԧ = lim
1
1
ර ‫ܨ‬Ԧ ‫݈݀ߦ ڄ‬Ԧ = lim ර ‫ܨ‬Ԧ ‫݈݀ ڄ‬Ԧ
ௌ՜଴ ܵ ஼
ௌ,క՜଴ ߦܵ ஼
- Gauss’ theorem
1.
Let us consider the integral of the divergence of a vector over a finite volume ܸ
enclosed by a surface ܵ.
2.
The volume can be thought of as a continuous, dense stack of identical, infinitesimally
small rectangular cells with a volume of ȟܸ௜ = ȟ‫ݔ‬ȟ‫ݕ‬ȟ‫ ݖ‬as shown below. (In fact, each
cell can have a different shape and size, as long as they fill the volume tightly and
without duplication.)
3.
Since outward to one cell is inward to the adjacent cell, the sum of the surface integral
of the vector over all cells is equal to the surface integral over ܵ:
෍ ර ‫ܨ‬Ԧ ‫݊ ڄ‬ො݀ܽ = ර ‫ܨ‬Ԧ ‫݊ ڄ‬ො݀ܽ
ௌ೔
௜
4.
ௌ
As the volume of each cell goes to zero,
lim
1
ර ‫ܨ‬Ԧ ‫݊ ڄ‬ො݀ܽ = div ‫ܨ‬Ԧ = ‫ܨ ڄ ׏‬Ԧ
୼௏೔ ՜଴ ȟܸ௜ ௌ
೔
1
ර ‫ܨ‬Ԧ ‫݊ ڄ‬ො݀ܽ ȟܸ௜ = lim ෍ൣ‫ܨ ڄ ׏‬Ԧ ൧ȟܸ௜
୼௏೔ ՜଴ ȟܸ௜ ௌ
୼௏೔ ՜଴
೔
֜ ර ‫ܨ‬Ԧ ‫݊ ڄ‬ො݀ܽ = ෍ ර ‫ܨ‬Ԧ ‫݊ ڄ‬ො݀ܽ = ෍ lim
ௌ
௜
ௌ೔
௜
௜
= න ‫ܨ ڄ ׏‬Ԧ ݀‫ݒ‬
௏
5.
This is Gauss’ theorem, also called the divergence theorem: The integral of the
divergence of a vector over a volume ܸ is equal to the surface integral of the normal
component of the vector over the surface bounding ܸ.
- Stokes’ theorem
1.
The proof of Stokes’ theorem is quite analogous to the proof of Gauss’ theorem.
2.
Consider a surface ܵ enclosed by a loop ‫ ܥ‬as shown below. The surface, not necessary
to be flat, can be divided into a large number of cells ȟܵ௜ bounding ‫ܥ‬௜ .
3.
Since each of the cells is traversed in the same sense (see the right-handed circular
arrows), the sum of line integrals of a vector field around the ‫ܥ‬௜ ’s is just the line integral
around the bounding curve; all of the other contributions cancel. Thus
ර ‫ܨ‬Ԧ ‫݈݀ ڄ‬Ԧ = ෍ ර ‫ܨ‬Ԧ ‫݈݀ ڄ‬Ԧ = lim ෍
஼
4.
୼ௌ೔ ՜଴
஼೔
௜
௜
1
ර ‫ܨ‬Ԧ ‫݈݀ ڄ‬Ԧ ȟܵ௜
ȟܵ௜ ஼೔
Using the definition of the curl at the limit point (the second one) and ܽො = ݊ො,
lim
1
ර ‫ܨ‬Ԧ ‫݈݀ ڄ‬Ԧ = ݊ො ‫ ڄ‬curl ‫ܨ‬Ԧ
୼ௌ೔ ՜଴ ȟܵ௜ ஼
೔
ර ‫ܨ‬Ԧ ‫݈݀ ڄ‬Ԧ = lim ෍ ݊ො ‫ ڄ‬curl ‫ܨ‬Ԧ ȟܵ௜ = න curl ‫ܨ‬Ԧ ‫݊ ڄ‬ො݀ܽ = න ൫‫ܨ × ׏‬Ԧ ൯ ‫݊ ڄ‬ො݀ܽ
஼
5.
୼ௌ೔ ՜଴
ௌ
௜
ௌ
Stokes’ theorem: The line integral of a vector around a closed curve is equal to the
integral of the normal component of its curl over any surface bounded by the curve.
2.12 Green’s theorem
- Green’s first identity
1.
Let ‫ܣ‬Ԧ = ߶‫߰׏‬, where ߶ and ߰ are arbitrary scalar fields. The vector field ‫ܣ‬Ԧ is defined
in the volume of ܸ bounded by the closed surface ܵ.
2.
The divergence of ‫ܣ‬Ԧ is
‫׏߶ = )߰׏߶( ڄ ׏‬ଶ ߰ + ‫߰׏ ڄ ߶׏‬
3.
డట
The normal component of ‫ܣ‬Ԧ at the surface ܵ is ߶‫݊ ڄ ߰׏‬ො = ߶ డ௡
4.
Here, ݊ො directs outward from inside the volume ܸ.
5.
Applying the equations above to the divergence theorem,
න ‫ܣ ڄ ׏‬Ԧ݀ଷ ‫ݔ‬Ԧ = ර ‫ܣ‬Ԧ ‫݊ ڄ‬ො݀ܽ
௏
ௌ
֜ න ߶‫׏‬ଶ ߰ + ‫ ݀߰׏ ڄ ߶׏‬ଷ ‫ݔ‬Ԧ = ර ߶
௏
ௌ
߲߰
݀ܽ
߲݊
ቆ= ර ߶‫݊ ڄ ߰׏‬ො݀ܽቇ
ௌ
- Green’s second identity, called Green’s theorem
1.
If we write Green’s first identity with ߶ and ߰ interchanged,
න ߰‫׏‬ଶ ߶ + ‫ ݀߰׏ ڄ ߶׏‬ଷ ‫ݔ‬Ԧ = ර ߰
௏
2.
ௌ
߲߶
݀ܽ ቆ= ර ߰‫݊ ڄ ߶׏‬ො݀ܽቇ
߲݊
ௌ
Subtracting the obtained equations, the ‫ ߰׏ ڄ ߶׏‬terms cancel and we have Green’s
second identity as
න (߶‫׏‬ଶ ߰ െ ߰‫׏‬ଶ ߶)݀ଷ ‫ݔ‬Ԧ = ර ൤߶
௏
3.
ௌ
߲߰
߲߶
െ ߰ ൨ ݀ܽ ቆ= ර [߶‫ ߰׏‬െ ߰‫݊ ڄ ]߶׏‬ො݀ܽቇ
߲݊
߲݊
ௌ
This theorem will be encountered when we discuss the uniqueness of the solution of
boundary value problems in electrostatics.
4.
But, this theorem is more important and universal. You may see Green’s theorem in the
courses of mathematical physics, graduate level electromagnetism, etc., in the future.
(HW #2)
Due: 20240320
1. (a) Show that curl of a vector field is a vector field.
ଵ
(b) From the definition at the limit, curl ‫ܨ‬Ԧ = lim ‫ׯ‬ௌ ݊ො × ‫ܨ‬Ԧ ݀ܽ , Show that
௏՜଴ ௏
൫curl ‫ܨ‬Ԧ ൯࢞ = ቆ
߲‫ܨ‬௭ ߲‫ܨ‬௬
െ
ቇ
߲‫ݕ‬
߲‫ݖ‬
(c) Derive the followings in the spherical coordinate, (‫ݎ‬, ߠ, ߶):
‫ࢋ = ߮׏‬௥
߲߮
1 ߲߮
1 ߲߮
+ ࢋఏ
+ ࢋథ
߲‫ݎ‬
‫ߠ߲ ݎ‬
‫ ݎ‬sin ߠ ߲߶
‫ܨ ڄ ׏‬Ԧ =
൬or ‫ݎ‬Ƹ
߲߮
1 ߲߮
1 ߲߮
+ ߠ෠
+ ߶෠
൰
߲‫ݎ‬
‫ߠ߲ ݎ‬
‫ ݎ‬sin ߠ ߲߶
1 ߲(sin ߠ ‫ܨ‬ఏ )
1 ߲‫ܨ‬థ
1 ߲(‫ ݎ‬ଶ ‫ܨ‬௥ )
+
+
߲ߠ
‫ ݎ‬sin ߠ
‫ ݎ‬sin ߠ ߲߶
‫ ݎ‬ଶ ߲‫ݎ‬
߲
߲ଶ߮
1 ߲ ଶ ߲߮
1
߲߮
1
൬‫ݎ‬
൰
+
൬sin
ߠ
൰
+
߲‫ݎ‬
‫ ݎ‬ଶ ߲‫ݎ‬
‫ ݎ‬ଶ sin ߠ ߲ߠ
߲‫ݎ‬
‫ ݎ‬ଶ sinଶ ߠ ߲߶ ଶ
ଶ
1
1 ߲
߲
߲ଶ߮
߲߮
1
(‫ )߮ݎ‬+ ଶ
=
൬sin ߠ ൰ + ଶ ଶ
ଶ
‫ ݎ‬sin ߠ ߲ߠ
‫ݎ߲ ݎ‬
߲‫ݎ‬
‫ ݎ‬sin ߠ ߲߶ ଶ
‫׏‬ଶ ߮ =
2. Achievement of a very uniform field
(a) Two rings with radius ‫ ݎ‬have charge ܳ and െܳ uniformly distributed around them. The rings
are parallel and concentric and located a distance ݄ apart. Let ‫ ݖ‬be the vertical coordinate, with
‫ = ݖ‬0 at the point between the centers of the lower and upper rings. Show that the electric field at
points on the axis of the rings as a function of ‫ ݖ‬is given as
‫ܧ‬ሬԦ = ‫ݖ‬Ƹ
ܳ
‫ ݖ‬+ ݄/2
‫ ݖ‬െ ݄/2
൤ ଶ
െ ଶ
൨ = ‫ܧ‬௓ ‫ݖ‬Ƹ
ଶ
ଷ/ଶ
(‫ ݎ‬+ (‫ ݖ‬െ ݄/2)ଶ )ଷ/ଶ
4ߨߝ଴ (‫ ݎ‬+ (‫ ݖ‬+ ݄/2) )
(b) The field is an even function of ‫ ݖ‬and thus the first derivative of the electric field at ‫ = ݖ‬0 is
zero, which implies that the field is fairly uniform there; there are no variations to first order in the
distance along the axis from the midpoint. To get further uniform field, one can consider the
condition that makes the second derivative be also zero. Show that the condition is given as
‫݄ = ݎ‬/ξ6
(c) Plot ‫ܧ‬௓ for three different cases: ‫݄ = ݎ‬/10, ‫݄ = ݎ‬/ξ6, and ‫݄ = ݎ‬. You can use any plotting
program, including Wolfram|Alpha, Origin, Mathematica, Matlab, and so on.
3. A hole of radius ܴ is cut out from a very large flat sheet with uniform charge density ߪ.
(a) Let ‫ ܮ‬be the line perpendicular to the sheet, passing through the center of the hole. Calculate
the electric field at a point on ‫ܮ‬, a distance ‫ ݖ‬from the center of the hole from Coulomb’s law.
(b) If a point charge of a weak enough magnitude of െ‫ ݍ‬with mass ݉ is released from rest on ‫ܮ‬,
very close to the center of the hole, show that it undergoes oscillatory motion, and the frequency
߱ is given as
߱=ඨ
‫ߪݍ‬
2ߝ଴ ܴ݉
PH231 Electromagnetism I (Spring 2024)
Lecture note 03: Electrostatics – Coulomb’s law and Gauss’ law
(Chapter 2 of the textbook by Reitz, Chapters 4-6 of the book by Feynman)
3.1 Introduction
- There are two typical ways to solve electrostatic problems. 1) Coulomb's law does indeed provide
a complete solution to an electrostatic problem, calculating the electric field and potential from
the given charge distribution in space. However, the charge distribution is often unknown or
partially determined, especially in experiments where we typically measure the electric field or
potential. 2) Alternatively, one can calculate the electric potential distribution in space by solving
the Laplace/Poisson equation combined with the boundary condition of the potential and its
derivative. The charge distribution can be calculated from the electric field or potential distribution.
- In this lecture, we will discuss the force law between charges (Coulomb’s law) and the concept of
electrostatic potential. We will also work through calculations of some electrostatic problems which
can be made with Gauss' law directly and discuss the equivalence between Gauss’ law and
Coulomb’s law in electrostatics.
- The boundary value problem in electrostatics and its solution will be discussed in the next lecture
(Chapter 3 of Reitz).
3.2 Statics in Electromagnetism
- All of electromagnetism is contained in the Maxwell equations:
Gauss’s law: ‫ ׏‬ή ‫ܧ‬ሬԦ =
஡
஫బ
ሬԦ = 0
Gauss’s law in magnetism: ‫ ׏‬ή ‫ܤ‬
ሬԦ
డ஻
Maxwell-Faraday’ law: ‫ܧ × ׏‬ሬԦ = െ
డ௧
ሬԦ
ሬԦ = డா + ఫԦ
Ampere’s law with Maxwell’s correction: ܿ ଶ ‫ܤ × ׏‬
డ௧
ఢబ
- The situations that are described by these equations can be very complicated. We will consider
first the easiest circumstance in which nothing depends on the time-called the static case. In this
circumstance, all of the terms in the Maxwell equations which are time derivatives of the field are
zero and the Maxwell equations become:
‫ ׏‬ή ‫ܧ‬ሬԦ =
ߩ
߳଴
Electrostatics: ቐ
‫ܧ × ׏‬ሬԦ = 0
&
Magnetostatics:
ሬԦ = 0
‫׏‬ή‫ܤ‬
ଔԦ
ቐ
ሬԦ =
‫ܤ×׏‬
߳଴ ܿ ଶ
- Note that the two fields are not interconnected. This means that electricity and magnetism are
ሬԦ
distinct phenomena so long as charges and currents are static. The interdependence of ‫ܧ‬ሬԦ and ‫ܤ‬
does not appear until there are changes in charges or currents. Only when there are sufficiently
rapid changes, so that the time derivatives in Maxwell's equations become significant, will ‫ܧ‬ሬԦ and
ሬԦ depend on each other. For example, electromagnetic waves; light.
‫ܤ‬
- Electrostatics is a neat example of a vector field with zero curl and a given divergence. On the
other hand, magnetostatics is an example of a vector field with zero divergence and a given curl.
1.
It follows that any vector field whose curl is zero is equal to the gradient of some scalar
function.
ߘ × ‫ܥ‬Ԧ = ‫ = )߰׏( × ׏‬0
2.
Similarly, it follows that any vector field whose divergence is zero is equal to the curl
of some vector function.
ߘ ή ‫ܥ‬Ԧ = ‫ ׏‬ή ൫‫ܣ × ׏‬Ԧ൯ = 0
3.
We can consider a scalar field for the electric field and a vector field for the magnetic
field respectively.
4.
The scalar field for the electric field is the well-known electric scalar potential
‫ܧ‬ሬԦୣ୪ୣୡ୲୰୧ୡ = െ‫׏‬Ԅ
5.
The vector field for the magnetic field is the vector potential which will be discussed
in detail in the lecture on magnetostatics (Chapter 8 of Reitz). If ‫ܣ‬Ԧ is continuous and
well-defined everywhere, then it is guaranteed that there is no magnetic monopole.
ሬԦ = ‫ܣ × ׏‬Ԧ
‫ܤ‬
6.
If we consider Maxwell-Faraday’s law and the superposition principle, we can roughly
obtain a further general equation for the electric field:
‫ܧ × ׏‬ሬԦ୫ୟ୥୬ୣ୲୧ୡ = െ
ሬԦ
߲‫ܤ‬
߲‫ܣ × ׏‬Ԧ
߲‫ܣ‬Ԧ
=െ
= െ‫× ׏‬
߲‫ݐ‬
߲‫ݐ‬
߲‫ݐ‬
߲‫ܣ‬Ԧ
߲‫ݐ‬
We will find the general and complete discussion of the scalar and vector potentials in
֜ ‫ܧ‬ሬԦ = ‫ܧ‬ሬԦୣ୪ୣୡ୲୰୧ୡ + ‫ܧ‬ሬԦ୫ୟ୥୬ୣ୲୧ୡ = െ‫׏‬Ԅ െ
7.
the lecture on Maxwell’s Equations and Poynting’s Theorem (Chapter 16 of Reitz)
3.3 Coulomb’s law and Electric field (Sections 2-2 and 2-3 of Reitz)
- The observations on the forces between electric charges can be summarized in three statements.
1.
There are two and only two kinds of electric charge, now known as positive and
negative.
2.
Two point charges exert on each other forces that act along the line joining them and
are inversely proportional to the square of the distances between them.
3.
These forces are also proportional to the product of the charges, are repulsive for like
charges, and attractive for unlike charges.
- Coulomb’s law: one of the inverse square laws
1.
The last two statements are known as Coulomb’s law, named by Charles Augustin de
Coulomb (1736-1806).
2.
For point charges, ‫ݍ‬ଵ and ‫ݍ‬ଶ , Coulomb’s law is formulated as
‫ܨ‬Ԧଵଶ =
3.
1 ‫ݍ‬ଵ ‫ݍ‬ଶ
1 ‫ݍ‬ଵ ‫ݍ‬ଶ
Ԧଵଶ =
ଷ ‫ݎ‬
ଶ ‫ݎ‬Ƹଵଶ
4ߨߝ଴ ‫ݎ‬ଵଶ
4ߨߝ଴ ‫ݎ‬ଵଶ
(SI)
Here, ‫ܨ‬Ԧଵଶ is the electrostatic force on charge ‫ݍ‬ଵ exerted by ‫ݍ‬ଶ and ‫ݎ‬Ԧଵଶ = ‫ݎ‬Ԧଵ െ ‫ݎ‬Ԧଶ =
|‫ݎ‬Ԧଵ െ ‫ݎ‬Ԧଶ |‫ݎ‬Ƹଵଶ = ‫ݎ‬ଵଶ ‫ݎ‬Ƹଵଶ .
4.
Coulomb’s law of course gives the electrostatic force on charge ‫ݍ‬ଶ exerted by ‫ݍ‬ଵ as
‫ܨ‬Ԧଶଵ =
1 ‫ݍ‬ଵ ‫ݍ‬ଶ
1 ‫ݍ‬ଵ ‫ݍ‬ଶ
Ԧ
ଶ ‫ݎ‬Ƹଶଵ = െ
ଶ ‫ݎ‬Ƹଵଶ = െ‫ܨ‬ଵଶ
4ߨߝ଴ ‫ݎ‬ଵଶ
4ߨߝ଴ ‫ݎ‬ଵଶ
5.
Inverse square law: The equation above is an experimental law. Nevertheless,
6.
There is both theoretical and experimental evidence to indicate that the inverse square
law is exact – that is, the exponent of ‫ݎ‬ଵଶ is exactly 2. By an experimental work [Phys.
Rev. Lett., 26 721 (1971)], is has been shown that the exponent can differ from 2 by no
more than one par in 1015! This topic will be discussed further in Section 3.6 of this
lecture note.
7.
The Coulomb constant, ݇ =
ଵ
ସగఌబ
, is experimentally determined. Since the 2019
redefinition of the SI base units, the Coulomb constant, as calculated from Committee
on Data of the International Science Council (CODATA) 2018 recommended values, is
8.9875517923 × 10ଽ ܰ݉ଶ ‫ି ܥ‬ଶ .
- Superposition and charge density
1.
If more than two point charges are present, the mutual forces are determined by the
superposition of Coulomb’s law. If a system of ܰ charges is considered, the force on
the ݅th charge is given by
ே
‫ݍ‬௝
‫ݍ‬௜
‫ܨ‬Ԧ௜ =
෍ ଶ ‫ݎ‬Ƹ௜௝
4ߨߝ଴
‫ݎ‬௜௝
where ‫ݎ‬Ԧ௜௝ = ‫ݎ‬Ԧ௜ െ ‫ݎ‬Ԧ௝
௝ஷ௜
2.
A simple extension of the idea of ܰ interacting point charges is the interaction of a
point charge with a continuous charge distribution.
3.
It is well known that electric charge is found in multiples of a basic charge, such as the
electron. The electronic charge is one of the seven fundamental physical constants and
is defined as 1.602176634 × 10ିଵଽ ‫ ܥ‬since the 2019 redefinition of the SI base units.
4.
For the purposes of macroscopic physics, the discreteness of charge is extremely small.
The smallness of the basic unit means that in a macroscopic charge distribution, any
small element of volume contains a large number of electronic charges.
5.
We may thus describe a charge distribution in terms of a charge density function
defined as the limit of the charge per unit volume/surface as the volume/surface
becomes infinitesimal.
ߩ = lim
ȟ‫ݍ‬
୼௏՜଴ ȟܸ
ߪ = lim
ȟ‫ݍ‬
୼ௌ՜଴ ȟܵ
6.
(volume charge density)
(surface charge density)
From what has been said about ‫ݍ‬, it is evident that ߩ and ߪ are net charge, or excess
charge, densities. It is worth noting that in typical solid materials even a very large
charge density ߩ will involve a change in the local electron density of only about one
part in 10ଽ .
7.
The Coulomb force on a point charge ‫ ݍ‬at ‫ݎ‬Ԧ exerted by a charge distribution through
a volume ܸ with a density ߩ and on the surface ܵ that bounds ܸ with a density ߪ
is obtained by replacing ‫ݍ‬௝ with ߩ௝ ݀‫ݒ‬௝ᇱ and ߪ௝ ݀ܽ௝ᇱ and proceeding to the limit
‫ܨ‬Ԧ௤ =
‫ݍ‬
‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ
‫ݍ‬
‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ
ᇱ )݀‫ ݒ‬ᇱ
න
ߩ(‫ݎ‬
Ԧ
+
න
ߪ(‫ݎ‬Ԧ ᇱ )݀ܽᇱ
4ߨߝ଴ ௏ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |ଷ
4ߨߝ଴ ௏ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |ଷ
- Electric field
1.
The electric field at a point is defined operationally as the limit of the force on a test
charge at the point as the magnitude of the test charge goes to zero:
‫ܨ‬Ԧ௤
‫ܧ‬ሬԦ = lim
௤՜଴ ‫ݍ‬
2.
Let us consider a charge distribution consisting of ܰ point charges, ‫ݍ‬ଵ , ‫ݍ‬ଶ , … , ‫ݍ‬ே ,
located at the points, ‫ݎ‬Ԧଵ , ‫ݎ‬Ԧଶ , … , ‫ݎ‬Ԧே , a volume distribution ߩ(‫ݎ‬Ԧ ᇱ ) in ܸ, and a surface
distribution ߪ(‫ݎ‬Ԧ ᇱ ) on ܵ. If a test charge ‫ ݍ‬is located at the point ‫ݎ‬Ԧ, it experiences a
force as
ே
‫ܨ‬Ԧ௤ =
‫ݎ‬Ԧ െ ‫ݎ‬Ԧ௜
‫ݍ‬
‫ݍ‬
‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ
‫ݍ‬
‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ
ᇱ )݀‫ ݒ‬ᇱ
෍ ‫ݍ‬௜
+
න
ߩ(‫ݎ‬
Ԧ
+
න
ߪ(‫ݎ‬Ԧ ᇱ )݀ܽᇱ
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ௜ |ଷ 4ߨߝ଴ ௏ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |ଷ
4ߨߝ଴
4ߨߝ଴ ௌ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |ଷ
௜ୀଵ
3.
The electric field is the limit of the ratio of this force to the test charge ‫ݍ‬, which is
independent of ‫ݍ‬:
ே
‫ܧ‬ሬԦ (‫ݎ‬Ԧ) =
1
‫ݎ‬Ԧ െ ‫ݎ‬Ԧ௜
1
‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ
1
‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ
ᇱ )݀‫ ݒ‬ᇱ
෍ ‫ݍ‬௜
+
න
ߩ(‫ݎ‬
Ԧ
+
න
ߪ(‫ݎ‬Ԧ ᇱ )݀ܽᇱ
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ௜ |ଷ 4ߨߝ଴ ௏ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |ଷ
4ߨߝ଴
4ߨߝ଴ ௌ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |ଷ
௜ୀଵ
4.
The quantity we have just defined, the electric field, may be calculated at each point
in space in the vicinity of a system of charges or of a charge distribution; ‫ܧ‬ሬԦ (‫ݎ‬Ԧ) is a
vector field as a function of position.
5.
As an aid of visualizing the electric field, Michael Faraday introduced the concept of
lines of force, an imaginary line drawn in such a way that its direction at any point is
the direction of the electric field at that point and it terminates on the sources of the
electric field. The figure below shows an example of the calculation of the electric field
as a vector field and the visualization of lines of force.
3.4 Electrostatic potential (Section 2-4 of Reitz)
- Conservative force and line integral
1.
The electrostatic force is curl free; ‫ܨ × ׏‬Ԧ = ‫ܧ × ׏ݍ‬ሬԦ = 0. Stokes’ theorem leads to the
corollary: The line integral of ‫ܨ‬Ԧ (or ‫ܧ‬ሬԦ ) around any closed path ‫ ܥ‬is zero:
ර ‫ܨ‬Ԧ ‫ݎ݀ ڄ‬Ԧ = ‫ ݍ‬ර ‫ܧ‬ሬԦ ‫ݎ݀ ڄ‬Ԧ = ‫ ݍ‬න ൫‫ܧ × ׏‬ሬԦ ൯ ‫݊ ڄ‬ො݀ܽ = 0
஼
2.
஼
ௌ
Indeed, the electrostatic force is a conservative force and the line integral between any
two points ܲଵ and ܲଶ is independent on the path.
3.
Because the line integral in the electrostatic field is path-independent, we can use it to
define a scalar quantity ߮ଶଵ , without specifying any particular path:
௉మ
߮ଶଵ = െ න ‫ܧ‬ሬԦ ‫ݎ݀ ڄ‬Ԧ
௉భ
4.
With the minus sign included here, ߮ଶଵ is the work per unit charge done by an external
agency in moving a positive charge from ܲଵ to ܲଶ : the potential energy associated
with the conservative force.
- Electric potential (Scalar potential)
1.
The curl-free property of the electrostatic force can also be directly obtained from
Coulomb’s law:
‫×׏‬ቆ
2.
‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ
ቇ=0
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |ଷ
֜
‫ܧ × ׏‬ሬԦ = 0
Using the vector identity, ‫ = )߮׏( × ׏‬0, and considering the work per unit charge, one
can easily see that
‫ܧ‬ሬԦ = െ‫߮׏‬
3.
For a charge component at the point ‫ݎ‬Ԧ ᇱ , the ingredient of integral/summation of
Coulomb’s law is a gradient of a scalar as
‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ
1
= െ‫ ׏‬൬
൰
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |ଷ
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
4.
Finally, we can represent Coulomb’s law in the form of the scalar potential as
ே
߮(‫ݎ‬Ԧ) =
1
‫ݍ‬௜
1
ߩ(‫ݎ‬Ԧ ᇱ )
1
ߪ(‫ݎ‬Ԧᇱ )
ᇱ
෍
+
න
݀‫ݒ‬
+
න
݀ܽᇱ
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ௜ | 4ߨߝ଴ ௏ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
4ߨߝ଴
4ߨߝ଴ ௌ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
௜ୀଵ
5.
Once the electrostatic potential is known to exist, we can write, where ߮(ref) = 0,
௥Ԧ
௥Ԧ
௥Ԧ
න ‫ܧ‬ሬԦ (‫ݎ‬Ԧᇱ ) ‫ݎ݀ ڄ‬Ԧ ᇱ = െ න ‫׏‬ᇱ ߮(‫ݎ‬Ԧ ᇱ ) ‫ݎ݀ ڄ‬Ԧ ᇱ = െ න ݀߮ = െ߮(‫ݎ‬Ԧ)
୰ୣ୤
6.
୰ୣ୤
୰ୣ୤
In electrostatics and Coulomb’s law, there is no question about the equivalence of the
two approaches using the electric field and the scalar potential.
3.5 Gauss’ law and Dirac delta function (Sections 2-5, 2-6, 2-7, and 2-10 of Reitz)
- Gauss’ law
1.
There is an important relationship between the integral of the normal component of
the electric field over a closed surface and the total charge enclosed by the surface.
2.
Take the simplest case imaginable: a single point charge ‫ ݍ‬at the origin. Coulomb’s
law gives the electric field at point ‫ݎ‬Ԧ = ‫ݎݎ‬Ƹ by the single point charge as
‫ܧ‬ሬԦ (‫ݎ‬Ԧ) =
3.
‫ݎ ݍ‬Ԧ
4ߨߝ଴ ‫ ݎ‬ଷ
The integral of the normal component of this electric field over a closed surface ܵ that
encloses the origin is then
ර ‫ܧ‬ሬԦ ‫݊ ڄ‬ො݀ܽ =
ௌ
4.
‫ݍ‬
‫ݎ‬Ԧ ‫݊ ڄ‬ො
ර
݀ܽ
4ߨߝ଴ ௌ ‫ ݎ‬ଷ
The quantity ‫ݎ‬Ԧ/‫݊ ڄ ݎ‬ො݀ܽ = ‫ݎ‬Ƹ ‫݊ ڄ‬ො݀ܽ is the projection of ݀ܽ on a plane perpendicular to ‫ݎ‬Ԧ.
Thus,
௥Ԧ‫ڄ‬௡ො
௥య
݀ܽ represents the solid angle subtended by ݀ܽ, which is written as ݀ȳ. In the
figure below, the solid angle subtended by ݀ܽ is the same to that by ݀‫ܣ‬.
5.
It is obvious that for a spherical area ܵ ᇱ with a radius of ‫ ݎ‬ᇱ , the integral of the solid
angle is 4ߨ:
ර
‫ݎ‬Ԧ ᇱ ‫݊ ڄ‬ොᇱ
‫ݎ‬ᇱଷ
ௌᇲ
6.
ௌᇲ
Finally, we have the following relationship
ර ‫ܧ‬ሬԦ ‫݊ ڄ‬ො݀ܽ =
ௌ
7.
݀ܽᇱ = ර ݀ȳᇱ = 4ߨ
‫ݍ‬
‫ݎ‬Ԧ ‫݊ ڄ‬ො
‫ݍ‬
‫ݍ‬
ර
݀ܽ =
‫ ڄ‬4ߨ =
ଷ
4ߨߝ଴ ௌ ‫ݎ‬
4ߨߝ଴
ߝ଴
If several point charges ‫ݍ‬ଵ , ‫ݍ‬ଶ , … , ‫ݍ‬ே are enclosed by the surface ܵ, the total electric
field is determined by the superposition and thus the surface integral of its normal
component becomes
ே
ර ‫ܧ‬ሬԦ ‫݊ ڄ‬ො݀ܽ =
ௌ
1
෍ ‫ݍ‬௜
ߝ଴
௜ୀଵ
8.
This result can be readily generalized to the case of a continuous distribution of charge
characterized by a charge density. If each element of charge ߩ݀‫ ݒ‬is considered as a
point charge, it contributes ߩ݀‫ݒ‬/ߝ଴ to the surface integral:
ර ‫ܧ‬ሬԦ ‫݊ ڄ‬ො݀ܽ =
ௌ
9.
1
න ߩ݀‫ݒ‬
ߝ଴ ௏
This is known as Gauss’ law.
10. The divergence theorem (Gauss’ theorem) states that
ර ‫ܧ‬ሬԦ ‫݊ ڄ‬ො݀ܽ = න ‫ܧ ڄ ׏‬ሬԦ ݀‫= ݒ‬
ௌ
௏
1
න ߩ݀‫ݒ‬
ߝ଴ ௏
֜
‫ܧ ڄ ׏‬ሬԦ =
ߩ
ߝ଴
11. This result is thought of as a differential form of Gauss’ law.
- The use of Dirac delta function
1.
It would be useful to be able to express point charges as a special case of a general
charge-density function.
2.
The Dirac delta function can serve this purpose. For a point charge ‫ ݍ‬at point ‫ݎ‬Ԧ଴ ,
ߩ(‫ݎ‬Ԧ) = ‫ݎ(ߜݍ‬Ԧ െ ‫ݎ‬Ԧ଴ )
3.
Here, the Dirac delta function satisfies
ߜ(‫ݎ‬Ԧ െ ‫ݎ‬Ԧ଴ ) = 0
for ‫ݎ‬Ԧ ് ‫ݎ‬Ԧ଴
and
‫ݎ(ܨ‬Ԧ ᇱ )ߜ(‫ݎ‬Ԧ ᇱ െ ‫ݎ‬Ԧ଴ ) ݀‫ ݒ‬ᇱ = ‫ݎ(ܨ‬Ԧ଴ )
න
௏ ୧୬ୡ୪୳ୢ୧୬୥ ௥Ԧబ
4.
The Dirac delta function is obviously normalized as
ߜ(‫ݎ‬Ԧᇱ െ ‫ݎ‬Ԧ଴ ) ݀‫ ݒ‬ᇱ = 1
න
௏ ୧୬ୡ୪୳ୢ୧୬୥ ௥Ԧబ
5.
Thus, for ߩ(‫ݎ‬Ԧ
߮(‫ݎ‬Ԧ) =
‫ܧ‬ሬԦ (‫ݎ‬Ԧ) =
6.
ᇱ)
= σ௜ ‫ݍ‬௜ ߜ(‫ݎ‬Ԧ ᇱ െ ‫ݎ‬Ԧ௜ ), we have
σ௜ ‫ݍ‬௜ ߜ(‫ݎ‬Ԧ ᇱ െ ‫ݎ‬Ԧ௜ ) ᇱ
1
ߩ(‫ݎ‬Ԧᇱ )
1
1
‫ݍ‬௜
ᇱ
න
݀‫ݒ‬
=
න
݀‫= ݒ‬
෍
ᇱ
ᇱ
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ |
Ԧ െ ‫ݎ‬Ԧ௜ |
4ߨߝ଴ ୟ୪୪ ୱ୮ୟୡୣ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧ |
4ߨߝ଴ ୟ୪୪ ୱ୮ୟୡୣ
4ߨߝ଴
௜ |‫ݎ‬
(‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ )
1
1
‫ݍ‬௜
(‫ݎ‬Ԧ െ ‫ݎ‬Ԧ௜ ) = െ‫ݎ(߮׏‬Ԧ)
න
෍ ‫ݍ‬௜ ߜ(‫ݎ‬Ԧ ᇱ െ ‫ݎ‬Ԧ௜ )
݀‫ ݒ‬ᇱ =
෍
ᇱ |ଷ
|‫ݎ‬
|‫ݎ‬
Ԧ
െ
‫ݎ‬Ԧ௜ |ଷ
Ԧ
െ
‫ݎ‬
Ԧ
4ߨߝ଴ ୟ୪୪ ୱ୮ୟୡୣ
4ߨߝ
௜
௜
଴
Some other properties of the delta function can be obtained as consequences of Gauss’
law in differential form. For a point charge ‫ݍ‬௜ at ‫ݎ‬Ԧ௜ ,
‫ڄ׏‬
7.
‫ݎ‬Ԧ െ ‫ݎ‬Ԧ௜
= 4ߨߜ(‫ݎ‬Ԧ െ ‫ݎ‬Ԧ௜ )
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ௜ |ଷ
‫ݍ‬௜ ‫ݎ‬Ԧ െ ‫ݎ‬Ԧ௜
1
ቇ = ‫ݍ‬௜ ߜ(‫ݎ‬Ԧ െ ‫ݎ‬Ԧ௜ )
ଷ
4ߨߝ଴ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧ௜ |
ߝ଴
Alternatively,
െ‫ڄ ׏‬
8.
֜ ‫ܧ ڄ ׏‬ሬԦ = ‫ ڄ ׏‬ቆ
In fact, െ
ଵ
‫ݎ‬Ԧ െ ‫ݎ‬Ԧ௜
1
= ‫׏ ڄ ׏‬൬
൰ = ‫׏‬ଶ ߮ = െ4ߨߜ(‫ݎ‬Ԧ െ ‫ݎ‬Ԧ௜ )
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ௜ |ଷ
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ௜ |
ଵ
ସగ |௥Ԧି௥Ԧ ᇲ|
is the Green function ‫ݎ(ܩ‬Ԧ, ‫ݎ‬Ԧᇱ ) of the Laplace operator, which gives
the solution of the Poisson equation as
‫׏‬ଶ ߮ = െ
9.
ߩ
ߝ଴
֜
߮(‫ݎ‬Ԧ) = െ න
ୟ୪୪ ୱ୮ୟୡୣ
‫ݎ(ܩ‬Ԧ, ‫ݎ‬Ԧᇱ )
ߩ(‫ݎ‬Ԧ ᇱ ) ᇱ
1
ߩ(‫ݎ‬Ԧᇱ )
݀‫= ݒ‬
න
݀‫ ݒ‬ᇱ
ߝ଴
4ߨߝ଴ ୟ୪୪ ୱ୮ୟୡୣ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
Further details of the Green function are expected to be discussed in the course of
Mathematical Physics or Advanced Electromagnetism.
- Examples of the application of Gauss’ law (skipped)
1.
There are a variety of examples to apply Gauss’ law to calculate the electric field or
electrostatic potential from a given charge distribution, including the followings:
2.
A.
Field of a spherical charge distribution (Section 1.11 of Purcell)
B.
Field of a long line charge (Section 2-7 of Reitz, Section 1.12 of Purcell)
C.
Field of an infinite flat sheet of charge (Section 1.13 of Purcell)
You might have encountered these examples in the course of General Physics. So, in
this lecture, we shall skip them.
3.6 Electric dipole and Multipole expansion (Sections 2-8 and 2-9 of Reitz)
- Localized distribution of charge and multipole moments
1.
A localized distribution of charge is described by the charge density ߩ(‫ݎ‬Ԧ ᇱ ), which is
nonvanishing only inside a finite volume ܸ around some origin.
2.
From Coulomb’s law, the potential at ‫ݎ‬Ԧ outside the volume can be given by
1
ߩ(‫ݎ‬Ԧ ᇱ )
߮(‫ݎ‬Ԧ) =
න
݀‫ ݒ‬ᇱ
4ߨߝ଴ ௏ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
3.
The Taylor expansion brings
ଵ
ଶ ି
ଶ
ଶ
ଶ
1
1
2‫ݎ‬Ԧ ‫ݎ ڄ‬Ԧ ᇱ ‫ ݎ‬ᇱ
1
1 2‫ݎ‬Ԧ ‫ݎ ڄ‬Ԧ ᇱ ‫ ݎ‬ᇱ
3 2‫ݎ‬Ԧ ‫ݎ ڄ‬Ԧ ᇱ ‫ ݎ‬ᇱ
=
ቆ1
െ
+
ቇ
=
ቊ1
െ
ቈെ
+
቉
+
ቈെ
+ ଶ቉+‫ڮ‬ቋ
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ | ‫ݎ‬
‫ݎ‬ଶ
‫ݎ‬ଶ
‫ݎ‬ଶ
‫ݎ‬ଶ
‫ݎ‬ଶ
‫ݎ‬
‫ݎ‬
2
8
ଶ
=
4.
1 2‫ݎ‬Ԧ ‫ݎ ڄ‬Ԧ ᇱ 1 3(‫ݎ‬Ԧ ‫ݎ ڄ‬Ԧ ᇱ )ଶ െ ‫ ݎ‬ଶ ‫ ݎ‬ᇱ
ଷ
+
+ ቈ
቉ + ܱ൫‫ ݎ‬ᇱ ൯
‫ݎ‬ଷ
‫ݎ‬ହ
2
‫ݎ‬
Omitting terms involving the cube and higher powers of ‫ ݎ‬ᇱ yields
߮(‫ݎ‬Ԧ) =
‫ݔ‬௜ ‫ݔ‬௝
1 ‫݌ ݍ‬Ԧ ‫ݎ ڄ‬Ԧ 1
቎ + ଷ + ෍ ܳ௜௝ ହ + ‫ ڮ‬቏
‫ݎ‬
4ߨߝ଴ ‫ݎ‬
‫ݎ‬
2
௜,௝
5.
Here, ‫ ݍ‬, ‫݌‬Ԧ , and ܳ௜௝ are the monopole, dipole, and quadrupole moments of the
localized charge distribution ߩ(‫ݎ‬Ԧᇱ ) defined as
‫ ؠ ݍ‬න ߩ(‫ݎ‬Ԧ ᇱ )݀‫ ݒ‬ᇱ
(In fact, the net charge)
௏
‫݌‬Ԧ ‫ ؠ‬න ‫ݎ‬Ԧᇱ ߩ(‫ݎ‬Ԧ ᇱ )݀‫ ݒ‬ᇱ
௏
ଶ
ܳ௜௝ ‫ ؠ‬න ൫3‫ݔ‬௜ᇱ ‫ݔ‬௝ᇱ െ ‫ ݎ‬ᇱ ߜ௜௝ ൯ߩ(‫ݎ‬Ԧᇱ )݀‫ ݒ‬ᇱ
௏
6.
Obviously, the contribution of the monopole, dipole, and quadrupole depends on ‫ି ݎ‬ଵ,
‫ି ݎ‬ଶ , and ‫ି ݎ‬ଷ, respectively.
7.
Since the matrix of ܳ௜௝ is symmetric and its trace is zero, the quadrupole moment has
five independent components. (The octapole moment has 7 independent components.)
8.
In fact, the rigorous discussion on the multipole moments should be given in terms of
spherical harmonics ܻ௟௠ (ߠ, ߮) as represented below: ‫ݍ‬௟௠ . But, this topic is beyond the
scope of the early undergraduate course. If you are interested, please refer to books
of mathematical physics or graduate-level electromagnetism (e.g., Chapter 4 of Classical
electrodynamics by Jackson).
߮(‫ݎ‬Ԧ) =
ܻ௟௠ (ߠ, ߶)
1
4ߨ
෍
‫ݍ‬௟௠
‫ ݎ‬௟ାଵ
4ߨߝ଴
2݈ + 1
௟
‫כ‬
where ‫ݍ‬௟௠ = න ߩ(‫ݎ‬Ԧ ᇱ )‫ ݎ‬ᇱ ܻ௟௠ (ߠ ᇱ , ߶ ᇱ )݀ ଷ ‫ݎ‬Ԧ ᇱ
௟,௠
- Electric dipole
1.
Although it does not provides the ideal electric dipole, the simplest way to setup an
electric dipole is combining two equal and opposite charges ±‫ ݍ‬located at positions
±݈/2 on the z axis. The dipole setup is rotationally symmetric around the line
containing the two charges (See Fig. A below).
2.
The exact expression for the potential at point ‫ݎ‬Ԧ = ‫ݎݎ‬Ƹ is
3.
‫ݍ‬
1 1
൬ െ ൰
4ߨߝ଴ ‫ݎ‬ଵ ‫ݎ‬ଶ
In the limit of ‫݈ ب ݎ‬, we can approximate the potential as
4.
‫ ݍ‬1
݈ cos ߠ
݈ cos ߠ
1 ‫ ݈ݍ‬cos ߠ
൤൬1 +
൰ െ ൬1 െ
൰൨ =
4ߨߝ଴ ‫ݎ‬
2‫ݎ‬
2‫ݎ‬
4ߨߝ଴ ‫ ݎ‬ଶ
Regarding the dipole moment as ‫݌‬Ԧ = ‫ݖ݈ݍ‬Ƹ , we have
߮(‫ݎ‬Ԧ) =
߮(‫ݎ‬, ߠ) ؆
߮(‫ݎ‬Ԧ) =
5.
1 ‫݌‬Ԧ ‫ݎ ڄ‬Ԧ
1
1
=െ
‫݌‬Ԧ ‫ ׏ ڄ‬൬ ൰
4ߨߝ଴ ‫ ݎ‬ଷ
4ߨߝ଴
‫ݎ‬
Here, it is worth noting that an idealized dipole or point dipole is one where ݈ ՜ 0
and ‫ ݍ‬՜ λ, with the product ‫ ݈ݍ = ݌‬taking on a particular finite value.
6.
The electric field is calculated as (H.W.) and its line is represented in Fig. B.
1 2‫ ݌‬cos ߠ
‫ܧۓ‬௥ =
1 3‫ݎ‬Ƹ (‫݌‬Ԧ ‫ݎ ڄ‬Ƹ ) െ ‫݌‬Ԧ
4ߨߝ଴ ‫ ݎ‬ଷ
‫ܧ‬ሬԦ (‫ݔ‬Ԧ) = െ‫ݎ(߮׏‬Ԧ) =
=
ଷ
4ߨߝ଴
‫ݎ‬
‫ = ܧ۔‬1 ‫ ݌‬sin ߠ and ‫ = ܧ‬0
థ
‫ ە‬ఏ 4ߨߝ଴ ‫ ݎ‬ଷ
7.
It is also interesting to find the potential energy of an electric dipole under an external
electrostatic field ‫ܧ‬ሬԦ = െ‫( ߮׏‬you might have seen this in the general physics course):
݈
݈
‫݈ݍ‬
‫݈ݍ‬
ሬԦ൯ + ‫ݖ‬Ƹ ‫߮׏ ڄ‬൫0
ሬԦ൯ െ ൜‫߮ݍ‬൫0
ሬԦ൯ െ ‫ݖ‬Ƹ ‫߮׏ ڄ‬൫0
ሬԦ൯ൠ = ‫݌‬Ԧ ‫߮׏ ڄ‬൫0
ሬԦ൯ = െ‫݌‬Ԧ ‫ܧ ڄ‬ሬԦ
ܷ = ‫ ߮ݍ‬൬ ‫ݖ‬Ƹ ൰ െ ‫ ߮ݍ‬൬െ ‫ݖ‬Ƹ ൰ ؆ ‫߮ݍ‬൫0
2
2
2
2
- Dependence on the choice of origin
1.
For a given local charge distribution, the lowest nonvanishing multipole moment is
independent of the choice of origin of coordinates, but all higher multipole moments
do in general depend on the choice of the origin.
2.
3.
A point charge ݁ at point ‫ݎ‬Ԧ଴ : ߩ(‫ݎ‬Ԧ ᇱ ) = ݁ߜ(‫ݎ‬Ԧ ᇱ െ ‫ݎ‬Ԧ଴ ).
A.
The monopole moment is independent as ‫݁ = ݍ‬.
B.
The dipole moment ‫݌‬Ԧ = ‫׬‬௏ ‫ݎ‬Ԧ ᇱ ߩ(‫ݎ‬Ԧ ᇱ )݀‫ ݒ‬ᇱ = ݁‫ݎ‬Ԧ଴ depends on the location of the origin.
Two point charges of +݁ and െ݁: ߩ(‫ݎ‬Ԧ ᇱ ) = ݁[ߜ(‫ݎ‬Ԧ ᇱ െ ‫ݎ‬Ԧ଴ ) െ ߜ(‫ݎ‬Ԧ ᇱ െ ‫ݎ‬Ԧଵ )].
A.
The monopole moment vanishes: ‫ = ݍ‬0.
B.
The dipole moment ‫݌‬Ԧ = ‫׬‬௏ ‫ݎ‬Ԧ ᇱ ߩ(‫ݎ‬Ԧ ᇱ )݀‫ ݒ‬ᇱ = ݁(‫ݎ‬Ԧ଴ െ ‫ݎ‬Ԧଵ ) is independent.
C.
But, all higher moments depend on the location of the origin.
3.7 Additional reading material (Section 5-8, 5-9, and 5-10, vol II of Feynman)
(This topic will be very briefly discussed in the lecture.)
- Examines the validity of Gauss’ law and Coulomb’s law:
1.
We have seen clearly why it is that Gauss' law is true only because the coulomb force
depends exactly on the square of the distance: the equivalence of two laws.
2.
Benjamin Franklin’s experiment: We need only determine whether or not the field inside
of a uniformly charged spherical shell is precisely zero. If we touch a small metal ball
to a charged object and then touch it to an electrometer the meter will become charged
and the pointer will move from zero (Fig. (a), below). If you do the same experiment
by touching the little ball to the inside of the charged sphere, you find that no charge
is carried to the electrometer (Fig. (b), below). If we write that the electrostatic force
depends on ‫ି ݎ‬ଶାఢ , we can place an upper bound on ߳. By this method Maxwell
determined that H was less than 1/10,000.
3.
The experiment was repeated and improved upon in 1936 by Plimpton and Laughton.
They found that Coulomb's exponent differs from two by less than one part in a billion.
4.
Now that brings up an interesting question: How accurate do we know this Coulomb
law to be in different scale? (Let’s change the size of the sphere!!!)
5.
As a result of very careful measurements in 1947 by Lamb and Retherford on the
relative positions of the energy levels of hydrogen, we know that the exponent is correct
again to one part in a billion on the atomic scale-that is, at distances of the order of
one angstrom (10-10 meter). (Refer to the Lamb shift)
6.
Is the same exponent correct at still shorter distances? From measurements in nuclear
physics it is found that there are electrostatic forces at typical nuclear distances-at
about 10-15 meter (1 fm)-and that they still vary approximately as the inverse square.
7.
Together with the Lamb-Rutherford experiment, the approach to the smaller scale in
examination of Coulomb’s law triggered the study on the Quantum Electro-Dynamics
(QED) by Richard Feynman, Julian Schwinger and Sin-Itiro Tomonoga and they were
jointly awarded the Nobel Prize in Physics in 1965.
8.
As stated earlier, a high-frequency experimental work [Phys. Rev. Lett., 26 721 (1971)],
showed that the exponent can differ from 2 by no more than one par in 1015.
9.
How about 10-16 meter? This range can be investigated by bombarding protons with
very energetic electrons and observing how they are scattered. Results to date seem
to indicate that the law fails at these distances. Now there are two possible explanations.
One is that the Coulomb law does not work at such small distances; the other is that
our objects, the electrons and protons, are not point charges. And, as you know, the
proton is not an element particle and it is consisting of quarks!!!
10. One more question: The inverse square law is valid at distances like one meter and
also at 10-10 m; but is the coefficient 1/4SH0 the same? The answer is yes; at least to an
accuracy of 15 parts in a million.
- Comment on such a high-precision to check Coulomb’s law/Gauss’ law
1.
Let us consider the hollow container-a conductor with a cavity (figure below).
2.
Imagine a loop Ȟ that crosses the cavity along a line of force from some positive
charge to some negative charge, and returns to its starting point via the conductor.
3.
At first, in electrostatics, every conductor/metal is an equipotential region; there is no
electric field inside. Thus, the integral through the conductor is zero.
4.
What about in the cavity? The integral along such a line of force from the positive to
the negative charges would not be zero. So, we would have
ර ‫ܧ‬ሬԦ ή ݀‫ݏ‬Ԧ ് 0
୻
5.
But, we have already known that the line integral of ‫ܧ‬ሬԦ around any closed loop in an
electrostatic field is always zero, since ‫ܧ × ׏‬ሬԦ = 0.
6.
Consequently, there can be no fields inside the empty cavity, nor any charges on the
inside surface.
7.
This shows why it was possible to check Coulomb's law to such a great precision: The
shape of the hollow shell used doesn't matter. It doesn't need to be spherical; it could
be even square! If Gauss' law is exact, the field inside a metal is always zero.
8.
Note that in the Benjamin Franklin’s experiment, the size of the small hole of the hollow
conductor cavity to bring the probe inside relative to the scale of the cavity may result
in the precision of measurements.
(HW #3)
Due: 20240401
1. Application of Gauss’ law: Field due to a distribution.
Each of the objects described below has uniform volume charge density ߩ. There are no other
charges present in addition to the given object.
(a) A rectangular slab has thickness ݈ in the ‫ ݔ‬direction and infinite extent in the ‫ ݕ‬and ‫ݖ‬
directions. Show that in sider slab,
‫ܧ‬ሬԦ =
ߩ‫ݔ‬
‫ݔ‬ො
ߝ଴
Here, ‫ ݔ‬is measured from the midplane of the slab.
(b) An infinitely long cylinder has radius ܴ. Show that inside the cylinder
ߩ‫ݎ‬
‫ܧ‬ሬԦ =
‫ݎ‬Ƹ
2ߝ଴
(c) A sphere has radius ܴ. Show that inside the sphere
ߩ‫ݎ‬
‫ܧ‬ሬԦ =
‫ݎ‬Ƹ
3ߝ଴
2. The screened Coulomb potential appropriate for a charge ‫ ݍ‬in a semiconducting material can
be given as
߮=
‫ି ݁ ݍ‬௥/ఒ
4ߨߝ଴ ‫ݎ‬
Here, ‫ ݎ‬is the distance of the observation point from the charge.
(a) Calculate the electric field.
(b) Shor that the corresponding charge density is given as
ߩ(‫ݎ‬Ԧ) = ‫ ݍ‬൤ߜ(‫ݎ‬Ԧ) െ
1
൨ ݁ ି௥/ఒ
4ߨߣଶ ‫ݎ‬
(optional) Briefly discuss how such an electric potential and charge distribution can be obtained.
(Don't worry, your answer doesn't have to be rigorous. Also, this problem will not be included in
the evaluation)
Note that the result in (b) gives that for an external (or excess) charge density ߩୣ୶୲ (‫ݎ‬Ԧ), the following
equation is satisfied effectively
൬‫׏‬ଶ െ
1
ߩୣ୶୲ (‫ݎ‬Ԧ)
൰߮ = െ
ߝ଴
ߣଶ
3. (a) Show that the electric field generated by an ideal electric dipole ‫݌‬Ԧ is given as
‫ܧ‬ሬԦ (‫ݎ‬Ԧ) = െ‫ݎ(߮׏‬Ԧ) =
1 3‫ݎ‬Ƹ (‫݌‬Ԧ ‫ݎ ڄ‬Ƹ ) െ ‫݌‬Ԧ
4ߨߝ଴
‫ݎ‬ଷ
(b) Show that the potential energy of an electric dipole under an external electrostatic field ‫ܧ‬ሬԦୣ୶୲
(you might have seen this in the general physics course).
ܷ = െ‫݌‬Ԧ ‫ܧ ڄ‬ሬԦୣ୶୲
(c) Show that the force acting on a dipole ‫݌‬Ԧ placed in an external electric field ‫ܧ‬ሬԦୣ୶୲ is
‫ܨ‬Ԧୣ୶୲ = (‫݌‬Ԧ ‫ܧ)׏ ڄ‬ሬԦ
(d) Show that the torque acting on a dipole ‫݌‬Ԧ placed in the field is
߬Ԧୣ୶୲ = ‫ݎ‬Ԧ × ൣ(‫݌‬Ԧ ‫ܧ)׏ ڄ‬ሬԦୣ୶୲ ൧ + ‫݌‬Ԧ × ‫ܧ‬ሬԦୣ୶୲
Here, ‫ݎ‬Ԧ is the vector distance to the dipole from the point about which the torque is to be measured.
The quantity ‫݌‬Ԧ × ‫ܧ‬ሬԦୣ୶୲ , which is independent of the point about the torque is calculated, is called
the turning couple acting on the dipole, which you have seen this in the general physics course.
4. Laplace equation in 3D rectangular coordinates. Consider a hollow, rectangular box with five sides
at ߶ = 0 while the sixth (‫ )ܿ = ݖ‬has ߮ = ܸ(‫ݔ‬, ‫ )ݕ‬as shown below.
Using the separation of variables, show that the solution to the Laplace equation with the boundary
condition is given as
ஶ
߮(‫ݔ‬, ‫ݕ‬, ‫ = )ݖ‬෍ ‫ܣ‬௡௠ sin(ߙ௡ ‫ )ݔ‬sin(ߚ௠ ‫ )ݕ‬sinh(ߛ௡௠ ‫)ݖ‬
௡,௠ୀଵ
where ‫ܣ‬௡௠ =
௔
௕
4
න ݀‫ ݔ‬න ݀‫ݔ(ܸݕ‬, ‫ )ݕ‬sin(ߙ௡ ‫ )ݔ‬sin(ߚ௠ ‫)ݕ‬
ܾܽ sinh(ߛ௡௠ ܿ) ଴
଴
and ߙ௡ = ݊ߨ/ܽ, ߚ௠ = ݉ߨ/ܾ, ߛ௡௠ = ߨ(݊ଶ /ܽଶ + ݉ଶ /ܾ ଶ )ଵ/ଶ
PH231 Electromagnetism I (Spring 2024)
Lecture note 04: Solutions of Electrostatic Problems
(Chapter 3 of the textbook by Reitz, Chapter 2 of the book by Purcell)
4.1 Introduction
- The solution to an electrostatic problem is straightforward for the case in which the charge
distribution is everywhere specified. Then, Coulomb’s law gives the potential and electric field
directly as integrals over the charge distribution as
߮(‫ݎ‬Ԧ) =
‫ܧ‬ሬԦ (‫ݎ‬Ԧ) =
1
݀‫ ݍ‬ᇱ
1
ߩ(‫ݎ‬Ԧ ᇱ )
න
=
න
݀‫ ݒ‬ᇱ
4ߨߝ଴ ௌ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ | 4ߨߝ଴ ௏ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
(‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ )݀‫ ݍ‬ᇱ
1
1
‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ
න
=
න
ߩ(‫ݎ‬Ԧ ᇱ )݀‫ ݒ‬ᇱ
4ߨߝ଴ ௌ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |ଷ
4ߨߝ଴ ௏ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |ଷ
- If the charge distribution is not specified in advance, it may be necessary to determine (or measure)
the electric field first and calculate the charge distribution second. For example, an electrostatic
problem may involve several conductors, with either the potential or the electric field on their
surfaces, but the surface charge distribution is not known.
- Such a problem is known as a boundary-value problem, in which we may try to calculate the
solution (the potential or electric field) in the volume of interest satisfying the given differential
equation, here the Laplace equation (and originally the Poisson equation in advance).
- In this lecture, we aim to solve the cases in the free-space (the vacuum) bounded by conductors
or equivalently specified potentials on a closed surface. Problems regarding the dielectric materials
or macroscopic matters will be discussed in the next lecture.
4.2 Poisson and Laplace equations (Sections 3-1 and 3-2 of Reitz)
- Poisson and Laplace equation
1.
Gauss’ law and the concept of the electric potential gives a second-order differential
equation
‫ܧ ڄ ׏‬ሬԦ =
ߩ
ߝ଴
and ‫ܧ‬ሬԦ = െ‫߮׏‬
֜
‫׏‬ଶ ߮ = െ
ߩ
ߝ଴
2.
This inhomogeneous differential equation is known as Poisson’s equation.
3.
Poisson’s equation may be solved once we know the functional dependence of ߩ(‫ݔ‬Ԧ ᇱ )
and the appropriate boundary conditions.
4.
Where the charge density vanishes, the Poisson equation reduces to the simpler form,
called the Laplace equation,
‫׏‬ଶ ߮ = 0
5.
The Laplace equation is homogeneous. If ߮ଵ , ߮ଶ , …, ߮௡ are all solutions of Laplace’s
equation, their linear superposition with arbitrary constants ‫ܥ‬௜ , is also a solution:
௡
߮ = ‫ܥ‬ଵ ߮ଵ + ‫ܥ‬ଶ ߮ଶ + ‫ ڮ‬+ ‫ܥ‬௡ ߮௡ = ෍ ‫ܥ‬௜ ߮௜
௡
௜ୀଵ
֜ ‫׏‬ଶ ߮ = ෍ ‫ܥ‬௜ ‫׏‬ଶ ߮௜ = 0
௜ୀଵ
- Classification of boundary conditions
1.
Typically, the boundary conditions for 2nd-order partial derivative equations (PDEs),
including the Poisson and Laplace equations, take one of three forms:
A.
Cauchy boundary conditions. The value of ߮ and normal derivative ߲߮/߲݊
specified on the boundary ܵ . In electrostatics, the potential and the normal
component of the electric field are applied.
2.
B.
Dirichlet boundary conditions. The value of ߮ specified on ܵ
C.
Neumann boundary conditions. ߲߮/߲݊ specified on ܵ
The arbitrary specification of both ߮ and ߲߮/߲݊ is an “over-specification” of the
electrostatic problem in a finite volume with a closed surface. Specification of either
the potential ( ߮ ) or electric field ( ߲߮/߲݊ ) everywhere on the boundary surface
determines a unique problem.
3.
Note that the boundary conditions, required to specify (in a consistent way) a unique
solution, depend on the equation class of 2nd-order PDEs as summarized as follows.
(See also the additional material about PDE and characteristics uploaded in klms.)
4.
Further details of this topic are beyond the level of an early sophomore class. You may
learn advanced topics and contents of 2nd-order PDEs and their appropriate boundary
conditions and Green functions in the classes of Mathematical Physics (PH211 and
PH212), Differential Equations and Applications (MAS201), graduate-level advanced
electromagnetism (PH507), and so on.
- Uniqueness of the solution to the Poisson/Laplace equation
1.
The uniqueness of the solution of the Poisson/Laplace equation can be easily verified.
A.
Suppose that ߮ଵ and ߮ are distinct two solution of the Poisson equation.
B.
Letting ܷ = ߮ଵ െ ߮ଶ , ‫׏‬ଶ ܷ = ‫׏‬ଶ ߮ଵ െ ‫׏‬ଶ ߮ଶ = 0 inside ܸ.
C.
Replacing both ߶ and ߰ by ܷ in the Green’s first identity,
න ܷ‫׏‬ଶ ܷ + ‫ܷ݀׏ ڄ ܷ׏‬ଷ ‫ݔ‬Ԧ = ර ܷ
௏
ௌ
߲ܷ
݀ܽ
߲݊
߲ܷ
֜ර ܷ
݀ܽ = න |‫|ܷ׏‬ଶ ݀ ଷ ‫ݔ‬Ԧ ൒ 0
߲݊
ௌ
௏
డ௎
D.
߮ଵ and ߮ଶ satisfy the same boundary conditions; ܷ = 0 or డ௡ = 0 on ܵ.
E.
For the Dirichlet boundary condition, ܷ vanishes everywhere on the closed
boundary surface ܵ and ‫׬‬௏ |‫|ܷ׏‬ଶ ݀ଷ ‫ݔ‬Ԧ = 0. Thus, ‫ ܷ׏‬has to be zero everywhere in
V and ߮ଵ = ߮ଶ by the continuity of ܷ.
F.
డ௎
Similarly, for the Neumann boundary condition, డ࢔ vanishes everywhere on the
closed boundary surface ߲ܸ and ‫׬‬௏ |‫|ܷ׏‬ଶ ݀߬ = 0. ‫ ܷ׏‬has to be zero everywhere
in ܸ, which can only be true if ߮ଵ െ ߮ଶ = ܿ; The solution for Neumann BC is
unique apart from an arbitrary additive constant.
2.
One can easily see that the Cauchy boundary condition is too restrictive since the
Dirichlet and Neumann boundary conditions are consistently satisfied for a given
solution of the Poisson equation.
4.3 Solutions to 1D / 2D Laplace equation (Section 3-3 of Reitz)
- Laplace equation in 1D
1.
In 1D, the Laplace equation is
݀ଶ ߮
=0
݀‫ ݔ‬ଶ
It is obvious that the general solution of the 2nd-order differential equation is
߮(‫ ݔܽ = )ݔ‬+ ܾ
‫׏‬ଶ ߮ =
2.
3.
The two unknown constants, ܽ and ܾ, are determined by the boundary conditions at
‫ݔ = ݔ‬ଵ and ‫ݔ = ݔ‬ଶ for the region of interest ‫ݔ( א ݔ‬ଵ , ‫ݔ‬ଶ ).
4.
As an example, consider two conducting plates at ‫ = ݔ‬0 and ‫ ݀ = ݔ‬that are charged
as ߮(‫ = ݔ‬0) = 0 and ߮(‫ܸ = )݀ = ݔ‬.
A.
The boundary condition determines the unknown constants as
B.
߮(‫ = ݔ‬0) = ܾ = 0 and ߮(‫ ݀ܽ = )݀ = ݔ‬+ ܾ = ܸ
ܸ
֜ܽ=
and ܾ = 0
݀
Finally, we have
ܸ
ܸ
߮(‫ܧ ֜ ݔ = )ݔ‬ሬԦ = െ‫ = ߮׏‬െ ‫ݔ‬ො
݀
݀
This is the result as you may study in the general physics.
C.
- Laplace equation in the isotropic case in spherical coordinates
1.
If there is no angular dependence, the Laplace equation in spherical coordinates also
becomes single-variable 2nd-order differential equation:
1 ߲
߲߮
1
߲
߲߮
1
߲ ଶ߮
1 ݀
݀߮
‫׏‬ଶ ߮ = ଶ ൬‫ ݎ‬ଶ ൰ + ଶ
൬sin ߠ ൰ + ଶ ଶ
= ଶ ൬‫ ݎ‬ଶ ൰ = 0
ଶ
‫ݎ߲ ݎ‬
߲‫ݎ‬
‫ ݎ‬sin ߠ ߲ߠ
߲ߠ
‫ ݎ‬sin ߠ ߲߶
‫ݎ݀ ݎ‬
݀‫ݎ‬
2. Its general solution is given as
ܽ
߮(‫ = )ݎ‬+ ܾ
‫ݎ‬
3. The two unknown constants, ܽ and ܾ, are determined by the boundary conditions.
4.
For example, consider a spherical conductor with a radius ܴ charged as ߮(ܴ) = ܸ.
A.
Here, we just have only one boundary condition at ‫ܴ = ݎ‬.
B.
However, we can consider another boundary condition at the limit ‫ ݎ‬՜ λ as
߮(λ) = 0. In fact, as we have discussed previously, the addition of a constant value
does not cause any change the electrostatic problem, since ‫ܧ‬ሬԦ = െ‫ ߮(׏‬+ ܿ) = െ‫߮׏‬.
C.
D.
Then, we have
ܽ
ܸܴ
߮(‫ = )ܴ = ݎ‬+ ܾ = ܸ and ߮(λ) = ܾ = 0 ֜ ߮(‫= )ݎ‬
ܴ
‫ݎ‬
This is also one of the well-known results in General Physics.
4.4 Solutions to 2D Laplace equation and Separation of variables (Sections 3-6 and 3-8 of Reitz)
- Laplace equation in 2D rectangular coordinates
1.
If the potential is a function of only two rectangular, ߮(‫ݔ‬, ‫)ݕ‬, the Laplace equation is
expressed as
‫׏‬ଶ ߮(‫ݔ‬, ‫= )ݕ‬
2.
߲ଶ
߲ଶ
߮(‫ݔ‬, ‫ )ݕ‬+ ଶ ߮(‫ݔ‬, ‫ = )ݕ‬0
ଶ
߲‫ݔ‬
߲‫ݕ‬
Now, we have to use an important mathematical technique, known as the separation
of variables.
A.
B.
C.
Letting ߮(‫ݔ‬, ‫)ݕ(ܻ)ݔ(ܺ = )ݕ‬, the Laplace equation becomes
݀ଶ ܺ
݀ଶ ܻ
ܻ ଶ +ܺ ଶ =0
݀‫ݔ‬
݀‫ݕ‬
Dividing by ߮ = ܻܺ and rearranging terms, we obtain
1 ݀ଶ ܺ
1 ݀ଶ ܻ
=
െ
ܺ ݀‫ ݔ‬ଶ
ܻ ݀‫ ݕ‬ଶ
The equality of two sides that depend on difference variables can only be attained
if each side must be equal to the same constant, a constant of separation.
D.
Thus, with a constant of separation ݇ ଶ , we have two separated equations as
݀ଶ ܺ
݀ଶ ܻ
ଶ
െ
݇
ܺ
=
0
and
+ ݇ଶܻ = 0
݀‫ ݔ‬ଶ
݀‫ ݕ‬ଶ
E.
Then, the general solution to the Laplace equation can be given by the
combination of the following functions:
ܺ (‫ܣ = )ݔ‬௞ cos ݇‫ ݔ‬+ ‫ܤ‬௞ sin ݇‫ݔ‬
൜ ௞
֜ ߮(‫ݔ‬, ‫ = )ݕ‬෍ ܽ௞ ܺ௞ (‫ܻ)ݔ‬௬ (‫)ݕ‬
ܻ௞ (‫ܥ = )ݕ‬௞ cosh ݇‫ ݕ‬+ ‫ܦ‬௞ sinh ݇‫ݕ‬
௞
F.
Note that there are no restrictions on ݇ as far. The constant of separation and the
unknown coefficients are determined by the boundary condition.
3.
The general solution to the Laplace equation can be represented by a variety of
different sets of functions, including the followings:
ܺ (‫ܣ = )ݔ‬ᇱ௞ ݁ ௜௞௫ + ‫ܤ‬௞ᇱ ݁ ି௜௞௫
ܺ (‫ܣ = )ݔ‬఑ cosh ߢ‫ ݔ‬+ ‫ܤ‬఑ sinh ߢ‫ݔ‬
(Here, ߢ ଶ = െ݇ ଶ ) , …
ቊ ௞
, ൜ ఑
ܻ఑ (‫ܥ = )ݕ‬఑ cos ߢ‫ ݕ‬+ ‫ܦ‬఑ sin ߢ‫ݕ‬
ܻ௞ (‫ܥ = )ݕ‬௞ᇱ ݁ ௞௬ + ‫ܦ‬௞ᇱ ݁ ି௞௬
4.
Determination of the constants
A.
Let us consider an example, a 2D rectangular conducting shell as illustrated below
B.
The boundary condition is given as
߮(‫ = ݔ‬0, ‫ܽ = ݔ(߮ = )ݕ‬, ‫ݔ(߮ = )ݕ‬, ‫ = ݕ‬0) = 0
C.
and
߮(‫ݔ‬, ‫ܸ = )ܾ = ݕ‬
Since ߮ = 0 for ‫ = ݔ‬0 and ‫ܽ = ݔ‬, it is convenient to choose the set of
൜
D.
ܺ௞ (‫ܣ = )ݔ‬௞ cos ݇‫ ݔ‬+ ‫ܤ‬௞ sin ݇‫ݔ‬
ܻ௞ (‫ܥ = )ݕ‬௞ cosh ݇‫ ݕ‬+ ‫ܦ‬௞ sinh ݇‫ݕ‬
Then, the boundary condition determines the unknown constants.
i.
The boundary condition at ‫ = ݔ‬0 gives
ܺ௞ (‫ = ݔ‬0) = ‫ܣ‬௞ = 0
ii.
In similar, the boundary condition at ‫ = ݕ‬0 gives
ܻ௞ (‫ = ݕ‬0) = ‫ܥ‬௞ = 0
iii.
The boundary condition at ‫ ܽ = ݔ‬determines ݇ in a discrete manner with
integers ݉ = 1,2,3, …
݉ߨ
ܽ
Then, the solution to the given problem becomes
ܺ௞ (‫ܤ = )ܽ = ݔ‬௞ sin ݇ܽ = 0 ֜ ݇ =
iv.
ஶ
݉ߨ
݉ߨ
‫ݔ‬ቁ sinh ቀ
‫ݕ‬ቁ
ܽ
ܽ
߮(‫ݔ‬, ‫ = )ݕ‬෍ ‫ܨ‬௞ sin ቀ
௠ୀଵ
v.
At last, the boundary condition at ‫ ܾ = ݕ‬give
ஶ
߮(‫ݔ‬, ‫ = )ܾ = ݕ‬෍ ‫ܨ‬௞ sin ቀ
௠ୀଵ
vi.
݉ߨ
݉ߨ
‫ݔ‬ቁ sinh ቀ
ܾቁ = ܸ
ܽ
ܽ
This is actually the Fourier series of ܸ in the region of ‫( א ݔ‬0, ܽ). Using
௔
௠గ௫
௡గ௫
௔
‫׬‬଴ sin ቀ ௔ ቁ sin ቀ ௔ ቁ ݀‫ = ݔ‬ଶ ߜ௠௡ , we finally have
௔ ஶ
ஶ
௔
݉ߨ
݉ߨ
݊ߨ‫ݔ‬
ܽ
݉ߨ
݊ߨ‫ݔ‬
න ෍ ‫ܨ‬௞ sinh ቀ
ܾቁ sin ቀ
‫ݔ‬ቁ sin ቀ
ቁ ݀‫ = ݔ‬෍ ‫ܨ‬௞ sinh ቀ
ܾቁ ߜ௠௡ = න ܸ sin ቀ
ቁ ݀‫ݔ‬
ܽ
ܽ
ܽ
2
ܽ
ܽ
଴
଴
௠ୀଵ
௠ୀଵ
0
for ݊ = even
݊ߨ
2ܸ ௔
݊ߨ‫ݔ‬
֜ ‫ܨ‬௞ sinh ቀ ܾቁ =
න sin ቀ
ቁ ݀‫ = ݔ‬൝4ܸ
for ݊ = odd
ܽ
ܽ ଴
ܽ
ߨ݊
୭ୢୢ
4ܸ
݊ߨ
݊ߨ
݊ߨ sin ቀ ܽ ‫ݔ‬ቁ sinh ቀ ܽ ‫ݕ‬ቁ
௡ୀଵ ߨ݊ sinh ቀ ܽ ܾቁ
When we employ the method of separation of variables, the strategy does not
‫ݔ(߮ ׵‬, ‫ = )ݕ‬෍
E.
change regardless of the dimensions and coordinate systems
5.
In fact, the general solution to the Laplace equation in a 2D coordinate system can be
given as ߮(‫ݎ‬Ԧ) = ‫ܨ‬ଵ (‫ ݔ‬+ ݅‫ )ݕ‬+ ‫ܨ‬ଶ (‫ ݔ‬െ ݅‫)ݕ‬, which is very useful for several 2D problems.
But, this is beyond the scope of this lecture, and we shall skip it. If you are interested
in this topic, you may read Section 3-9 of Reitz or relevant sections in Chapter 9 of
Mathematical methods for physicists a comprehensive guide (7th edition) by Arfken.
- Laplace equation in 2D polar coordinates
1.
In terms of the polar coordinates (ߩ, ߠ), the Laplace equation in 2D is
1 ߲
߲߮
1 ߲ ଶ߮
‫׏‬ଶ ߮ =
൬ߩ ൰ + ଶ ଶ = 0
ߩ ߲ߩ ߲ߩ
ߩ ߲ߠ
2.
Using the separation of variables, ߮(ߩ, ߠ) = ܴ(ߩ)ȣ(ߠ),
ߩ ݀
ܴ݀
1 ߲ ଶȣ
൬ߩ ൰ = െ
= ߥଶ
ܴ ݀ߩ ݀ߩ
ȣ ߲ߠ ଶ
3.
The solutions are then, except ߥ = 0,
ܴ(ߩ) = ܽߩఔ + ܾߩିఔ
൜
ȣ(ߠ) = ‫ ܣ‬cos(ߥߠ) + ‫ ܤ‬sin(ߥߠ)
4.
In case ߥ = 0,
൜
ܴ(ߩ) = ܽ଴ + ܾ଴ ln ߩ
ȣ(ߠ) = ‫ܣ‬଴ + ‫ܤ‬଴ ߠ
5.
The general solution is given by linear superposition of the obtained building blocks.
6.
Particularly, if there is no restriction on 0 ൑ ߠ ൑ 2ߨ, it has to be ensured that ߮(ߩ, ߠ) =
߮(ߩ, ߠ + 2ߨ); ߥ is an integer or zero as ɋ = ݊, and ‫ܤ‬଴ = 0. In this case, the general
solution is of the form with the phases ߙ௡ and ߚ௡
ஶ
߮(ߩ, ߮) = ܽ଴ + ܾ଴ ln ߩ + ෍{ܽ௡ ߩ௡ sin(݊߮ + ߙ௡ ) + ܾ௡ ߩି௡ sin(݊߮ + ߚ௡ )}
௡ୀଵ
4.5 Solutions to Laplace’s equation in 3D rectangular coordinates (Section 3-7 of Reitz)
- Laplace equation in 3D rectangular coordinates
1.
The Laplace equation in rectangular coordinates is
߲ ଶ߮ ߲ ଶ߮ ߲ ଶ߮
+
+
=0
߲‫ ݔ‬ଶ ߲‫ ݕ‬ଶ ߲‫ ݖ‬ଶ
2.
Assuming a nontrivial solution, ߮(‫ݔ‬, ‫ݕ‬, ‫)ݖ(ܼ)ݕ(ܻ)ݔ(ܺ = )ݖ‬,
1 ݀ଶ ܺ
1 ݀ଶ ܻ
1 ݀ଶ ܼ
+
+
=0
ଶ
ଶ
ܺ(‫ݔ݀ )ݔ‬
ܻ(‫ݕ݀ )ݕ‬
ܼ(‫ ݖ݀ )ݖ‬ଶ
3.
If the equation above is to hold for arbitrary values of the independent coordinates,
each of the three terms must be separated:
1 ݀ଶ ܺ
1 ݀ଶ ܻ
= ߙଶ,
= ߚଶ ,
ଶ
ܺ ݀‫ݔ‬
ܻ ݀‫ ݕ‬ଶ
4.
5.
1 ݀ଶ ܼ
= ߛଶ
ܼ ݀‫ ݖ‬ଶ
The equation connecting the constants of separation is
ߙ ଶ + ߚଶ + ߛ ଶ = 0
Note that ߙ , ߚ , and ߛ can be real, pure imaginary, or complex values, that are
determined by the boundary conditions. Typically, for the electrostatic potential of real
value, ߙ, ߚ, and ߛ are real or pure imaginary, so that ߙ ଶ , ߚ ଶ , and ߛ ଶ are real.
6.
To determine ߙ, ߚ, and ߛ, it is necessary to impose specific boundary conditions. Here
is an example: A hollow, rectangular box with five sides at ߶ = 0 while the sixth (‫)ܿ = ݖ‬
has ߮ = ܸ(‫ݔ‬, ‫( )ݕ‬See the figure below).
A.
B.
C.
Let the function ܺ be ܺ(‫ ݁ܣ = )ݔ‬ఈ௫ + ‫ି ݁ܤ‬ఈ௫ . Then, we have the following
equations from the boundary conditions at ‫ = ݔ‬0 and ‫ܽ = ݔ‬:
݊ߨ
ܺ(‫ = ݔ‬0) = ‫ ܣ‬+ ‫ = ܤ‬0
൜
֜ ‫ = ܤ‬െ‫ ܣ‬and ߙ =
݅
ఈ௔
ିఈ௔
ܺ(‫ ݁ܣ = )ܽ = ݔ‬+ ‫݁ܤ‬
=0
ܽ
݊ߨ
֜ ܺ(‫ܣ = )ݔ‬௡ sin
‫ݔ‬
ܽ
In similar,
݉ߨ
ܻ(‫ܥ = )ݕ‬௠ sin
‫ݕ‬
ܾ
Using the connecting equation,
݊ଶ ݉ ଶ
ߛ = ߨඨ ଶ + ଶ
ܽ
ܾ
D.
and ܼ(‫ ܦ = )ݖ‬sinh ߛ‫ ݖ‬+ ‫ ܧ‬cosh ߛ‫ݖ‬
The boundary condition at ‫ = ݖ‬0 provides ‫ = ܧ‬0. Thus, where ߙ௡ = ݊ߨ/ܽ, ߚ௠ =
݉ߨ/ܾ, and ߛ௡௠ = ߨ(݊ଶ /ܽଶ + ݉ଶ /ܾ ଶ )ଵ/ଶ ,
ஶ
߮(‫ݔ‬, ‫ݕ‬, ‫ = )ݖ‬෍ ‫ܣ‬௡௠ sin(ߙ௡ ‫ )ݔ‬sin(ߚ௠ ‫ )ݕ‬sinh(ߛ௡௠ ‫)ݖ‬
௡,௠ୀଵ
E.
‫ܣ‬௡௠ is determined by the boundary condition at ‫ ܿ = ݖ‬as
ஶ
߮(‫ݔ‬, ‫ݕ‬, ‫ݔ(ܸ = )ܿ = ݖ‬, ‫ = )ݕ‬෍ ‫ܣ‬௡௠ sin(ߙ௡ ‫ )ݔ‬sin(ߚ௠ ‫ )ݕ‬sinh(ߛ௡௠ ܿ)
௡,௠ୀଵ
௔
௕
4
֜ ‫ܣ‬௡௠ =
න ݀‫ ݔ‬න ݀‫ݔ(ܸݕ‬, ‫ )ݕ‬sin(ߙ௡ ‫ )ݔ‬sin(ߚ௠ ‫)ݕ‬
ܾܽ sinh(ߛ௡௠ ܿ) ଴
଴
4.6 Solutions to Laplace’s equation in 3D rectangular coordinates (Sections 3-4 and 3-5 of Reitz)
(We will not discuss the base in 3D cylindrical coordinates)
- Laplace equation in 3D spherical coordinates and separation of variables
1.
2.
3.
4.
5.
6.
In spherical coordinates (‫ݎ‬, ߠ, ߶), the Laplace equation is written as
1 ߲ଶ
1
߲
߲߮
1
߲ଶ߮
(‫)߮ݎ‬
‫׏‬ଶ ߮ =
+
൬sin
ߠ
൰
+
=0
‫ ݎ߲ ݎ‬ଶ
‫ ݎ‬ଶ sin ߠ ߲ߠ
߲ߠ
‫ ݎ‬ଶ sinଶ ߠ ߲߶ ଶ
We may employ a product form for the potential given by
ܷ(‫)ݎ‬
߮(‫ݎ‬, ߠ, ߶) = ܼ(‫= )߶(ܳ)ߠ(ܲ)ݎ‬
ܲ(ߠ)ܳ(߶)
‫ݎ‬
The Laplace equation can be converted to three ordinary differential equations as
1 ݀ଶ ܳ
‫ۓ‬
= െ݉ଶ
ܳ ݀߶ ଶ
ۖ
ۖ
݀ଶ ܷ ݈(݈ + 1)
െ
ܷ=0
݀‫ ݎ‬ଶ
‫ݎ‬ଶ
‫۔‬
ଶ
ۖ
ۖ 1 ݀ ൬sin ߠ ݀ܲ൰ + ቈ݈(݈ + 1) െ ݉ ቉ ܲ = 0
݀ߠ
sinଶ ߠ
‫ە‬sin ߠ ݀ߠ
The solutions of the first two differential equations are
ܳ(߶) = ‫ ݁ܣ‬௜௠థ + ‫ି ݁ܤ‬௜௠థ
ቐ
ܷ(‫)ݎ‬
ܼ(‫= )ݎ‬
= ‫ ݎܣ‬௟ + ‫(ି ݎܤ‬௟ାଵ)
‫ݎ‬
The third equation is the associated Legendre equation with ‫ = ݔ‬cos ߠ:
݀
݀ܲ
݉ଶ
൤(1 െ ‫ ݔ‬ଶ ) ൨ + ቈ݈(݈ + 1) െ
቉ܲ = 0
݀‫ݔ‬
݀‫ݔ‬
1 െ ‫ݔ‬ଶ
There are no restrictions on ݈ and ݉ as far. But, if ݉ and ݈ are zero or positive
integers and satisfy ݈ ൒ ݉, we obtain acceptable (converging) solutions known as
associated Legendre functions. In a further particular case of the azimuthal symmetry
(݉ = 0), the Legendre polynomials will give acceptable solutions.
- The case with the azimuthal symmetry: Brief summary of Legendre polynomials/functions
(For more details, you may need to refer to textbooks of mathematical physics.)
1.
Let us now consider the case with the azimuthal symmetry: ݉ = 0.
2.
Then we need to solve the ordinary Legendre equation as given below:
(1 െ ‫ ݔ‬ଶ )ܲᇱᇱ (‫ )ݔ‬െ 2‫ܲݔ‬ᇱ (‫ )ݔ‬+ ݈(݈ + 1)ܲ(‫ = )ݔ‬0
3.
The Legendre equation has regular singular points at ‫ = ݔ‬cos ߠ = ±1. A series solution
about ‫ = ݔ‬0 converges in the region of |‫ < |ݔ‬1 for all ݈. On the other hand, the
solution will diverge at ‫ = ݔ‬±1 in general, unless the power series is truncated.
4.
Frobenius’ method for general solution to Legendre equation
A.
௠
Let us substitute ‫ = ݕ‬σஶ
into the Legendre equation as
௠ୀ଴ ܽ௠ ‫ݔ‬
ஶ
(1 െ ‫ݔ‬
ଶ)
෍ ݉(݉ െ 1)ܽ௠ ‫ݔ‬
௠ୀଶ
B.
ஶ
௠ିଶ
െ 2‫ ݔ‬෍ ݉ܽ௠ ‫ݔ‬
ஶ
௠ିଵ
+ ݈(݈ + 1) ෍ ܽ௠ ‫ ݔ‬௠ = 0
௠ୀଵ
Then, we can get a recurrence relation
(݈ െ ‫ ݈()ݏ‬+ ‫ ݏ‬+ 1)
ܽ௦ାଶ = െ
ܽ
(‫ ݏ‬+ 2)(‫ ݏ‬+ 1) ௦
௠ୀ଴
(‫ = ݏ‬0,1,2, … )
C.
D.
E.
Using the recurrence relation, we find successively
(݈ െ 1)(݈ + 2)
݈(݈ + 1)
ܽଷ = െ
ܽଵ
ܽଶ = െ
ܽ଴
3!
2!
(݈ െ 2)(݈ + 3)
(݈ െ 3)(݈ + 4)
|
ܽସ = െ
ܽଶ
ܽହ = െ
ܽଷ
4‫ڄ‬3
5‫ڄ‬4
(݈ െ 2)݈(݈ + 1)(݈ + 3)
(݈ െ 3)(݈ െ 1)(݈ + 2)(݈ + 4)
=
ܽ଴
=
ܽଵ
4!
5!
By inserting these values for the coefficients for the series, we obtain
݈(݈ + 1) ଶ (݈ െ 2)݈(݈ + 1)(݈ + 2) ସ
‫ݕ‬ଵ = 1 െ
‫ ݔ‬+
‫ ݔ‬+‫ڮ‬
2!
4!
(݈ െ 1)(݈ + 2) ଷ (݈ െ 3)(݈ െ 1)(݈ + 2)(݈ + 4) ହ
‫ݕ‬ଶ = ‫ ݔ‬െ
‫ ݔ‬+
‫ ݔ‬+‫ڮ‬
3!
5!
These series converge for |‫ < |ݔ‬1. The ratio ‫ݕ‬ଶ /‫ݕ‬ଵ is not a constant, so that ‫ݕ‬ଵ
and ‫ݕ‬ଶ are linearly independent and ‫ܿ = )ݔ(ݕ‬ଵ ‫ݕ‬ଵ (‫ )ݔ‬+ ܿଶ ‫ݕ‬ଶ (‫ )ݔ‬is a general
solution to the Legendre equation with a given ݈.
5.
If ݈ is an integer ݈, the series is truncated after ‫ ݔ‬௟ , leaving a polynomial of degree ݈,
the Legendre polynomial:
A.
B.
Setting ܽଵ = 0 or ܽ଴ = 0, the first few of these polynomials, ܲ௟ (‫)ݔ‬, are
ܲ଴ (‫ = )ݔ‬1
ܲଵ (‫ݔ = )ݔ‬
1
1
ܲଶ (‫( = )ݔ‬3‫ ݔ‬ଶ െ 1)
ܲଷ (‫( = )ݔ‬5‫ ݔ‬ଷ െ 3‫)ݔ‬
|
2
2
1
1
ସ
ଶ
ܲସ (‫( = )ݔ‬35‫ ݔ‬െ 30‫ ݔ‬+ 3)
ܲହ (‫( = )ݔ‬63‫ ݔ‬ହ െ 70‫ ݔ‬ଷ + 15‫)ݔ‬
8
8
The Legendre polynomials converges for not only |‫ < |ݔ‬1 but also ‫ = ݔ‬±1 ;
|ܲ௟ (±1)| = 1. The first six Legendre polynomials are plotted as follows:
6.
C.
They are not normalized but orthogonal each other for the interval |‫ |ݔ‬൑ 1.
D.
If ݈ is even, ܲ௟ is even and if ݈ is odd ܲ௟ is odd.
As shown above, for a given integer ݈, one of ‫ݕ‬ଵ and ‫ݕ‬ଶ is terminated and provides
the Legendre polynomial. The other non-polynomial infinite series provide the second
solutions to the Legendre equation: the Legendre functions of the second kind
(݈ െ 1)(݈ + 2) ଷ (݈ െ 3)(݈ െ 1)(݈ + 2)(݈ + 4) ହ
‫ ݔ‬+
‫ ݔ‬+‫ڮ‬቉
for ݈ = 2‫݌‬
3!
5!
ܳ௟ (‫= )ݔ‬
݈(݈ + 1) ଶ (݈ െ 2)݈(݈ + 1)(݈ + 2) ସ
‫۔‬
‫ ݔ‬+‫ڮ‬቉
for ݈ = 2‫ ݌‬+ 1
ۖ ܾ௟ ቈ1 െ 2! ‫ ݔ‬+
4!
‫ە‬
‫ܾۓ‬௟ ቈ‫ ݔ‬െ
ۖ
A.
C.
Here, ܾ௟ is the scaling factor for convenience:
‫( ۓ‬െ1)௣ (2‫)݌‬ԥ
for ݈ = 2‫݌‬
ۖ
(2‫ ݌‬െ 1)ԥ
ܾ௟ =
‫( ۔‬െ1)௣ାଵ (2‫)݌‬ԥ
for ݈ = 2‫ ݌‬+ 1
ۖ
(2‫ ݌‬+ 1)ԥ
‫ە‬
Then, the first few of ܳ௟ (‫ )ݔ‬are
‫ݔ‬ସ
‫ݔ‬ଷ ‫ݔ‬ହ
ܳଵ (‫ = )ݔ‬െ1 + ‫ ݔ‬ଶ + + ‫ڮ‬
ܳ଴ (‫ ݔ = )ݔ‬+ + + ‫ڮ‬
3
3
5
|
2
4 ଷ 2 ହ
ଶ
ܳଷ (‫ = )ݔ‬െ 4‫ ݔ‬+ 2‫ ݔ‬ସ + ‫ڮ‬
ܳଶ (‫ = )ݔ‬െ2‫ ݔ‬+ ‫ ݔ‬+ ‫ ݔ‬+ ‫ڮ‬
3
3
5
The plots of the first five Legendre functions of the second kind are as follows:
D.
It is noted that the Legendre functions of the second kind diverge at ‫ = ݔ‬±1.
E.
In contrast to ܲ௟ , if ݈ is even, ܳ௟ is odd and if ݈ is odd ܳ௟ is even.
F.
Typically, if the interval of interest includes ‫ = ݔ‬±1, the Legendre function of the
B.
second kind are not used for the physically meaningful solution. But, if the interval
does not include the regular singular points at ‫ = ݔ‬±1, the combination of both
ܲ௟ and ܳ௟ has to be considered for the general solution.
- Orthogonality condition of Legendre polynomials
1.
Since the Legendre equation is self-adjoint and the coefficient (1 െ ‫ ݔ‬ଶ ) of ܲᇱᇱ (‫)ݔ‬
vanishes at ‫ = ݔ‬±1 , its solution of different eigenvalue ݈ will automatically be
orthogonal with unit weight on the interval (െ1,1):
ାଵ
గ
න ܲ௟ (‫ܲ)ݔ‬௠ (‫ = ݔ݀)ݔ‬න ܲ௟ (cos ߠ)ܲ௠ (cos ߠ) sin ߠ ݀ߠ = 0
ିଵ
2.
The orthonormality condition is given as
ାଵ
න ܲ௟ (‫ܲ)ݔ‬௠ (‫= ݔ݀)ݔ‬
ିଵ
3.
for ݈ ് ݉
଴
2ߜ௟௠
2݈ + 1
Using the orthonormality condition, we can obtain a basis for expansions of an arbitrary
function in the interval (െ1,1), Legendre series:
ஶ
݂(‫ = )ݔ‬෍ ܽ௟ ܲ௟ (‫)ݔ‬
௟ୀ଴
where ܽ௟ =
2݈ + 1 ାଵ
න ݂(‫ܲ)ݔ‬௟ (‫ݔ݀)ݔ‬
2
ିଵ
- Solution to Laplace’s equation with the azimuthal symmetry: no dependence on ߶
1.
We now have the building blocks to obtain the general solution to Laplace’s equation
for the case with the azimuthal symmetry (݉ = 0; no dependence on ߶) as follows
ஶ
߮(‫ݎ‬, ߠ) = ෍൫‫ܣ‬௟ ‫ ݎ‬௟ + ‫ܤ‬௟ ‫(ି ݎ‬௟ାଵ) ൯ܲ௟ (cos ߠ)
௟ୀ଴
2.
The coefficient ‫ܣ‬௟ and ‫ܤ‬௟ are determined by the boundary conditions on ‫ ݎ‬or ߠ.
3.
Note) The uniqueness of the solution provides an efficient way of obtaining the solution
from knowledge on the symmetric axis ‫ = ߠ( ݖ = ݎ‬0). For positive ‫ݖ‬, we can use
ܲ௟ (1) = 1 and thus have (For negative ‫ݖ‬, each term must be multiplied by (െ1)௟ ))
ஶ
߶(‫ݔ‬Ԧ = ‫ݖݖ‬Ƹ ) = ෍൫‫ܣ‬௟ ‫ ݖ‬௟ + ‫ܤ‬௟ ‫(ି ݖ‬௟ାଵ) ൯
௟ୀ଴
4.
Example 1: a conducting sphere with a radius ܽ grounded and placed in a uniform
external electric field ‫ܧ‬ሬԦ = ‫ܧ‬଴ ‫ݖ‬Ƹ
A.
The boundary condition can be represented as
߮(‫ܽ = ݎ‬, ߠ) = 0
߮(‫ ݎ‬՜ λ, ߠ) = െ‫ܧ‬଴ ‫ = ݖ‬െ‫ܧ‬଴ ‫ ݎ‬cos ߠ
B.
‫ܧ ׶‬ሬԦ (‫ ݎ‬՜ λ) = െ‫ܧ = ߮׏‬଴ ‫ݖ‬Ƹ
Since ܲଵ (cos ߠ) = cos ߠ, it is obvious that considering the orthogonal condition of
the Legendre polynomials, the contributions of the terms of ݈ ൒ 2 should vanish;
‫ܣ‬௟ = ‫ܤ‬௟ = 0 for ݈ ൒ 2. Also, the odd-parity symmetry of ߮ with respect to ‫ݖ‬
causes ‫ܣ‬௟ = ‫ܤ‬௟ = 0 for ݈ = even.
C.
Thus, we have
߮(‫ݎ‬, ߠ) = ‫ܣ‬ଵ ‫ ݎ‬cos ߠ + ‫ܤ‬ଵ ‫ି ݎ‬ଶ cos ߠ
D.
The boundary condition, ߮(‫ ݎ‬՜ λ, ߠ) = െ‫ܧ‬଴ ‫ ݎ‬cos ߠ, gives
‫ܣ‬ଵ = െ‫ܧ‬଴
E.
The boundary condition, ߮(ܽ, ߠ) = 0, gives
‫ܤ‬ଵ
െ‫ܧ‬଴ ܽ + ଶ = 0 for ߠ = 0
ܽ
Finally, we have
‫ܧ‬଴ ܽଷ
߮(‫ݎ‬, ߠ) = െ‫ܧ‬଴ ‫ ݎ‬cos ߠ + ଶ cos ߠ
‫ݎ‬
The electric field for ‫ ݎ‬൒ 0 is obtained as (see also the lines in the figure above)
ܽଷ
ܽଷ
‫ܧ‬ሬԦ (‫ = )ݎ‬െ‫ܧ = ߮׏‬଴ ቆ1 + 2 ଷ ቇ cos ߠ ‫ݎ‬Ƹ െ ‫ܧ‬଴ ቆ1 െ ଷ ቇ sin ߠ ߠ෠
‫ݎ‬
‫ݎ‬
F.
G.
H.
Also, the charge density induced on the surface is given as
ߪ(ߠ) = ߝ଴ ‫ܧ‬௥ |௥ୀ௔ = 3ߝ଴ ‫ܧ‬଴ cos ߠ
I.
The more detail of the induced surface charge density on the boundary of a
conductor will be discussed later.
5.
Example 2: Potential “inside” the sphere consisting of two hemispheres at +ܸ and െܸ
A.
The boundary conditions are
߮(‫ܽ = ݎ‬, ߠ) = ܸ(ߠ) = ൜
B.
+ܸ 0 ൑ ߠ < ߨ/2
െܸ ߨ/2 ൑ ߠ < ߨ
No charge at ‫ = ݎ‬0 thus the coefficients ‫ܤ‬௟ should be zero for all ݈:
ஶ
߮(‫ݎ‬, ߠ) = ෍ ‫ܣ‬௟ ‫ ݎ‬௟ ܲ௟ (cos ߠ)
௟ୀ଴
C.
Using the orthogonal conditions of the Legendre polynomials, we can determine
the coefficients ‫ܣ‬௟ as
2݈ + 1 ାଵ
න ܸ(ߠ)ܲ௟ (cos ߠ)݀(cos ߠ)
2ܽ௟ ିଵ
3‫ݎ‬
7 ‫ ݎ‬ଷ
֜ ߮(‫ݎ‬, ߠ) = ܸ ൤ ܲଵ (cos ߠ) െ ቀ ቁ ܲଷ (cos ߠ) + ‫ ڮ‬൨
2ܽ
8 ܽ
It is obvious that the Legendre polynomials of odd orders contribute to the
‫ܣ‬௟ =
D.
potential.
- Solution to Laplace’s equation in the case without the azimuthal symmetry
(This topic is beyond the scope of the early sophomore course. The following content will not
appear on the exam. Just enjoy the topic as a bridge to junior or senior undergraduate-level courses.
You may refer to further details of the associated Legendre polynomials/functions.)
1.
We need to extend our analysis to the associated Legendre equation to remove the
restriction to azimuthal symmetry so that ݉ଶ ൒ 0;
(1 െ ‫ ݔ‬ଶ )ܲᇱᇱ (‫ )ݔ‬െ 2‫ܲݔ‬ᇱ (‫ )ݔ‬+ ቈ݈(݈ െ 1) െ
2.
݉ଶ
቉ ܲ(‫ = )ݔ‬0
1 െ ‫ݔ‬ଶ
Associated Legendre functions:
A.
Just as for the original Legendre equation ( ݉ = 0 ), there are acceptable
(converging) solutions, ܲ௟௠ , for the range െ1 ൑ ‫ = ݔ‬cos ߠ ൑ 1, when ݈ and ݉ are
integers and satisfy the followings:
B.
For each ݈ ൒ 0, there are acceptable solutions with ݉ ranging from 0 to ݈. Or,
equivalently, for each ݉, there are an infinite number of acceptable solutions to
the associated Legendre ordinary differential equation with ݈ ൒ ݉.
C.
ܲ௟ି௠ and ܲ௟௠ are proportional and linearly dependent to each other. Consider that
in fact, the value ݉ enters only as ݉ଶ in the associated Legendre equation.
D.
The functions ܲ௟௠ are called as the associated Legendre functions and the first
few associated Legendre functions are as follows:
ܲ଴଴ = 1
for ݈ = 0
ቊ
ܲଵ଴ = ‫ = ݔ‬cos ߠ
ଵ
ܲଵଵ = െ(1 െ ‫ ݔ‬ଶ )ଶ = െ sin ߠ
for ݈ = 1
1
3
1
‫ܲۓ‬ଶ଴ = (3‫ ݔ‬ଶ െ 1) = cosଶ ߠ െ
ۖ
2
2
2
for ݈ = 2
ଵ
‫ܲ ۔‬ଶଵ = െ3‫(ݔ‬1 െ ‫ ݔ‬ଶ )ଶ = െ3 cos ߠ sin ߠ
ۖ
ܲଶଶ = 3(1 െ ‫ ݔ‬ଶ ) = 3 sinଶ ߠ
‫ە‬
…
E.
The associated Legendre functions satisfy the following orthogonality:
ାଵ
2 (݈ + ݉)!
න ܲ௟௠ (‫ܲ)ݔ‬௟௠
ߜ ᇲ
ᇲ (‫= ݔ݀)ݔ‬
2݈ + 1 (݈ െ ݉)! ௟௟
ିଵ
0
ܲ௟௠ (‫ܲ)ݔ‬௟௡ (‫)ݔ‬
(݈ + ݉)!
න
݀‫ = ݔ‬൞
ଶ)
(1
െ
‫ݔ‬
݉(݈
െ ݉)!
ିଵ
λ
ାଵ
F.
for ݉ ് ݊
for ݉ = ݊ ് 0
for ݉ = ݊ = 0
For more details, refer to textbooks for mathematical physics (e.g. Chap. 15, 7th
edition, Arfken).
3.
Spherical harmonics
A.
Separated-variable methods for solving the Laplace equation (also the Helmholtz
and Schrodinger equations including the Laplace operation, ‫׏‬ଶ ) in spherical polar
coordinates show that the possible angular solutions ܲ(ߠ)ܳ(߶) are always the
same in spherical coordinate problems as satisfying
1 ߲
߲
1 ߲ଶ
ቈ
൬sin ߠ ൰ + ଶ
+ ݈(݈ + 1)቉ ܲ(ߠ)ܳ(߶) = 0
sin ߠ ߲ߠ
߲ߠ
sin ߠ ߲߶ ଶ
B.
ଵ ௗమ ொ
From ொ ௗథమ = െ݉ଶ , it is obvious that
ܳ௠ (߶) =
1
ξ2ߨ
݁ ௜௠థ ,
݉ = 0, ±1, +2, …
ଶగ
‫߶݀)߶( ܳ)߶( כ‬
න ܳ௠
= ߜ௠భ ௠మ
௠మ
భ
଴
C.
The product
ܲ௟௠ ܳ௠
is called a spherical harmonic:
Y୪௠ (ߠ, ߮) = ඨ
D.
ଶ௟ାଵ (௟ି௠)!
The factor, ට ସగ (௟ା௠)!, is for the orthonormal condition as
௠
‫כ‬
௠
ଶగ
න ݀ȳ ቀY୪మ మ (ߠ, ߶)ቁ Y୪భ భ (ߠ, ߶) = න
ସగ
଴
E.
2݈ + 1 (݈ െ ݉)! ௠
ܲ (cos ߠ)݁ ௜௠థ
4ߨ (݈ + ݉)! ௟
ଵ
௠
‫כ‬
௠
݀߶ න ݀(cos ߠ) ቀY୪మ మ (ߠ, ߶)ቁ Y୪భ భ (ߠ, ߶) = ߜ௟భ ௟మ ߜ௠భ ௠మ
ିଵ
See the following table for the first few spherical harmonics (or see also
https://en.wikipedia.org/wiki/Table_of_spherical_harmonics).
F.
Visual representations of the first few real spherical harmonics are shown in the
figure below. The blue and yellow parts represent regions with opposite phases.
G.
Note that the spherical harmonics represent the components of the multipole
expansion such as the monopole (s-orbital), dipole (p-orbital), quadrupole (dorbital), and octupole (f-orbital) moments. You might encounter the spherical
harmonics when you solve the Schrodinger equation for the Hydrogen atom.
4.
Laplace expansion and the general solution in spherical coordinates
A.
One can get express any arbitrary function ݂(ߠ, ߶) in the form of expansion,
known as a Laplace series, as follows
௟
ஶ
݂(ߠ, ߶) = ෍ ෍ ܿ௟௠ Y௟௠ (ߠ, ߶)
௟ୀ଴ ௠ୀି௟
B.
Using the orthonormal property of the spherical harmonics, the coefficient is given
as
ଶగ
‫כ‬
ܿ௟௠ = න ݀ȳ൫Y௟௠ (ߠ, ߶)൯ ݂(ߠ, ߶) = න
ସగ
C.
଴
ଵ
‫כ‬
݀߶ න ݀(cos ߠ)൫Y௟௠ (ߠ, ߶)൯ ݂(ߠ, ߶)
ିଵ
This is called the Laplace expansion and used for finding the general solution for
the Laplace equation satisfying the boundary conditions on a spherical surface as
௟
ஶ
߮(‫ݎ‬, ߠ, ߶) = ෍ ෍ ൣ‫ܣ‬௟௠ ‫ ݎ‬௟ + ‫ܤ‬௟௠ ‫(ି ݎ‬௟ାଵ) ൧Y௟௠ (ߠ, ߶)
௟ୀ଴ ௠ୀି௟
D.
As briefly mentioned above, the Laplace expansion also gives the general solution
for other PDEs. For example, for the Helmholtz equation (‫׏‬ଶ + ݇ ଶ )߰ = 0, the
general solution in spherical coordinates is as below. Here, ݆௟ (݇‫ )ݎ‬and ‫ݕ‬௟ (݇‫ )ݎ‬are
the spherical Bessel functions replacing ‫ ݎ‬௟ and ‫(ି ݎ‬௟ାଵ) .
ஶ
௟
߰(‫ݎ‬, ߠ, ߶) = ෍ ෍ [ܽ௟௠ ݆௟ (݇‫ )ݎ‬+ ܾ௟௠ ‫ݕ‬௟ (݇‫])ݎ‬Y௟௠ (ߠ, ߶)
௟ୀ଴ ௠ୀି௟
4.7 Image charge method (Section 3-9 of Reitz, Sections 3.2 and 3.4 of Purcell)
- Electrostatic images
1.
As we have discussed, for a given set of boundary condition, the solution to Laplace’s
equation is unique; If one obtains a solution ߮(‫ݎ‬Ԧ) by any methods whatever, and if
this ߮(‫ݎ‬Ԧ) satisfies all boundary conditions given, it is a complete solution to the
boundary-value problem.
2.
&
The method of images is a procedure for accomplishing this result without specifically
solving a differential equation. It is not universally applicable to all types of electrostatic
problems, but is very effective for a variety of specific problems.
A.
Suppose the potential may be given in the following way:
߮(‫ݎ‬Ԧ) = ߮ଵ (‫ݎ‬Ԧ) +
1
ߪ(‫ݎ‬Ԧ ᇱ )
න
݀ܽᇱ = ߮ଵ (‫ݎ‬Ԧ) + ߮ଶ (‫ݎ‬Ԧ)
4ߨߝ଴ ௌ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
B.
Here, ߮ଵ is either a specified function or easily calculable.
C.
The integral part represents the contribution to the potential from surface charge
on all conductors or dielectrics appearing in the problem.
D.
The surface charge density ߪ(‫ݎ‬Ԧ ᇱ ) is not given, which is the essence of the imagecharge method, that the integral term can be replaced by a potential ߮ଶ due to
a specified charge distribution, such as image point or line charges. Then, ߪ can
be determined by the boundary condition of the electric field, which will be
డఝ
డఝ
discussed in detail in the lecture 05, ߝଶ ‫ܧ‬ଶ௡ െ ߝଵ ‫ܧ‬ଵ௡ = െߝଶ డ௡ ቚ + ߝଵ డ௡ ቚ = ߪ.
ଶ
E.
ଵ
The specified charges, image charges, producing ߮ଶ do not really exist. But, their
apparent location in inside the conductors/dielectrics, and the potential ߮(‫ݎ‬Ԧ) =
߮ଵ (‫ݎ‬Ԧ) + ߮ଶ (‫ݎ‬Ԧ) is a valid solution only in the exterior region.
3.
We shall see two examples of the method of image charge below.
- Example #1: Point charge near a conducting plane (Example 3-1 of Reitz)
1.
Consider the problem of a point charge ‫ ݍ‬distant by ‫ ݀ = ݔ‬from a grounded
conducting plane of infinite extent as shown in Fig. (a) below.
2.
The potential ߮ଵ by the point charge is apparent as
1 ‫ݍ‬
‫ݍ‬
߮ଵ (‫ݎ‬Ԧ) =
=
4ߨߝ଴ ‫ݎ‬ଵ 4ߨߝ଴ ඥ(‫ ݔ‬െ ݀)ଶ + ‫ ݕ‬ଶ + ‫ ݖ‬ଶ
3.
Considering the symmetry, let us suppose an image point charge inside the conductor
along the axis passing the original point charge normal to the plane.
4.
The boundary condition given in the problem is ߮ = 0 over the entire surface of the
conducting plate. To satisfy this boundary condition, the image charge should be
located at ‫ = ݔ‬െ݀ has a magnitude of െ‫;ݍ‬
߮ଶ (‫ݎ‬Ԧ) =
1 െ‫ݍ‬
‫ݍ‬
=െ
4ߨߝ଴ ‫ݎ‬ଶ
4ߨߝ଴ ඥ(‫ ݔ‬+ ݀)ଶ + ‫ ݕ‬ଶ + ‫ ݖ‬ଶ
֜ ߮(‫ = ݔ‬0, ‫ݕ‬, ‫= )ݖ‬
5.
‫ݍ‬
4ߨߝ଴ ඥ݀ ଶ + ‫ ݕ‬ଶ + ‫ ݖ‬ଶ
െ
‫ݍ‬
4ߨߝ଴ ඥ݀ ଶ + ‫ ݕ‬ଶ + ‫ ݖ‬ଶ
=0
The potential ߮ଶ (‫ݎ‬Ԧ) by the image charge is indeed equal to the potential by the
surface charge density distribution ߪ(‫ݕ‬, ‫ )ݖ‬induced by the original point charge ‫ݍ‬:
ߪ(‫ݕ‬, ‫ߝ = )ݖ‬଴ ‫ܧ‬௡ െ ߝୡ୭୬ ‫ܧ‬ୡ୭୬ = ߝ଴ ‫ܧ‬௡ = െߝ଴
6.
߲߮
‫݀ݍ‬
ฬ
=െ
߲‫ ݔ‬௫ୀ଴
2ߨ(݀ ଶ + ‫ ݕ‬ଶ + ‫ ݖ‬ଶ )ଷ/ଶ
Since the solutions of the image charge and the induced surface charge density
distribution satisfy the same boundary condition, they are equivalent in the exterior
region as shown in the figure below.
- Example #2: Point charge near a conducting sphere (Example 3-3 of Reitz)
1.
Let us consider the problem of a conducting, grounded sphere of radius ܽ and a point
charge ‫ ݍ‬as illustrated below
2.
There is the azimuthal symmetry about the axis passing the point charge ‫ ݍ‬and the
center of the sphere. So, one can consider the image charge inside the sphere on the
symmetric axis.
3.
When the distance of the original point charge and the image charge ‫ ݍ‬ᇱ is ݀ and ܾ,
respectively, the potential at the position ܲ(‫ݎ‬, ߠ, ߶) due to the charges ‫ ݍ‬and ‫ ݍ‬ᇱ is
1 ‫ݍ ݍ‬ᇱ
1
‫ݍ‬
‫ݍ‬ᇱ
߮(‫ݎ‬, ߠ, ߶) =
൤ + ൨=
൤
+
൨
4ߨߝ଴ ‫ݎ‬ଵ ‫ݎ‬ଶ
4ߨߝ଴ ξ‫ ݎ‬ଶ + ݀ ଶ െ 2‫ ݀ݎ‬cos ߠ ξ‫ ݎ‬ଶ + ܾ ଶ െ 2‫ ܾݎ‬cos ߠ
4.
The boundary condition, ߮ = 0 on the surface of the sphere (‫)ܽ = ݎ‬, should be satisfied
for all ߠ. For convenience, let us consider two cases, ߠ = 0 and ߠ = ߨ:
‫ݍ‬
‫ݍ‬ᇱ
‫ݍ‬
‫ݍ‬ᇱ
+
=
+
= 0 for ߠ = 0
ξܽଶ + ݀ଶ െ 2ܽ݀ ξܽଶ + ܾ ଶ െ 2ܾܽ ݀ െ ܽ ܽ െ ܾ
‫ݍ‬
‫ݍ‬ᇱ
‫ݍ‬
‫ݍ‬ᇱ
+
=
+
= 0 for ߠ = ߨ
ξܽଶ + ݀ଶ + 2ܽ݀ ξܽଶ + ܾ ଶ + 2ܾܽ ܽ + ݀ ܽ + ܾ
5.
6.
From the results, we can have
ܽଶ
ܾ=
݀
We finally have
߮(‫ݎ‬, ߠ, ߶) =
and
ܽ
‫ݍ‬ᇱ = െ ‫ݍ‬
݀
1
‫ݍ‬
‫ܽݍ‬
൤
െ
൨
4ߨߝ଴ ξ‫ ݎ‬ଶ + ݀ ଶ െ 2‫ ݀ݎ‬cos ߠ ξ‫ ݎ‬ଶ ݀ଶ + ܽସ െ 2‫ܽ݀ݎ‬ଶ cos ߠ
7.
One can easily see that for ‫ܽ = ݎ‬
‫ݍ‬
‫ܽݍ‬
‫ݍ‬
‫ݍ‬
െ
=
െ
=0
ଶ
ଶ
ଶ
ଶ
ସ
ଷ
ଶ
ଶ
ଶ
ଶ
ξܽ + ݀ െ 2ܽ݀ cos ߠ ξܽ ݀ + ܽ െ 2݀ܽ cos ߠ ξܽ + ݀ െ 2ܽ݀ cos ߠ ξܽ + ݀ െ 2ܽ݀ cos ߠ
8.
The actual charge density induced on the surface can be calculated from the normal
derivative of the potential as
ܽଶ
݀ଶ
ଷ/ଶ
ܽଶ
ܽ
൬1 + ଶ െ 2 cos ߠ൰
݀
݀
It is obvious that the total induced charge of the sphere is equal to the magnitude of
1െ
߲߮
‫ݍ‬
ܽ
ߪ(ߠ) = െߝ଴ ฬ
=െ
ቀ ቁ
߲‫ ݎ‬௥ୀ௔
4ߨܽଶ ݀
9.
the image charge, as it must be, according to Gauss’s law.
4.8 Short discussion on the Solution of the Poisson equation
(This topic is beyond the scope of the early sophomore course. The following content will not
appear on the exam. Just enjoy the topic and think of it as a bridge to senior undergraduate or
graduate-level courses.)
- Integration form of the Poisson equation
1.
The Poisson equation in the differential form can be converted into an integral equation.
2.
If we replace ߶ and ߰ in Green’s second identity (see Section 2.12 in the lecture note
ଵ
ଵ
02) by the electric potential ߮(‫ݔ‬Ԧ ᇱ ) and |௫Ԧି௫Ԧᇲ | ‫ ؠ‬ோ, respectively, we have
1
1
߲ 1
1 ߲߮(‫ݔ‬Ԧ ᇱ )
න ൤߮(‫ݔ‬Ԧ ᇱ )‫׏‬ଶ ൬ ൰ െ ‫׏‬ଶ ߮(‫ݔ‬Ԧ ᇱ )൨ ݀ ଷ ‫ݔ‬Ԧ ᇱ = ර ቈ߮(‫ݔ‬Ԧ ᇱ ) ᇱ ൬ ൰ െ
቉ ݀ܽᇱ
ᇱ
ܴ
ܴ
߲݊
ܴ
ܴ
߲݊
௏
ௌ
3.
ଵ
Using ‫׏‬ଶ ቀோቁ = െ4ߨߜ(‫ݔ‬Ԧ െ ‫ݔ‬Ԧ ᇱ ) and ‫׏‬ଶ ߮(‫ݔ‬Ԧ ᇱ ) = െߩ(‫ݔ‬Ԧ ᇱ )/ߝ଴ ,
න ൤െ4ߨ߮(‫ݔ‬Ԧ ᇱ )ߜ(‫ݔ‬Ԧ െ ‫ݔ‬Ԧ ᇱ ) +
௏
4.
1
߲ 1
1 ߲߮(‫ݔ‬Ԧ ᇱ )
ߩ(‫ݔ‬Ԧ ᇱ )൨ ݀ ଷ ‫ݔ‬Ԧ ᇱ = ර ቈ߮(‫ݔ‬Ԧ ᇱ ) ᇱ ൬ ൰ െ
቉ ݀ܽᇱ
ߝ଴ ܴ
߲݊ ܴ
ܴ ߲݊ᇱ
ௌ
Considering the properties of the Dirac delta function, if ‫ݔ‬Ԧ lies outside the volume, the
LHS of the equation is zero.
5.
If the point ‫ݔ‬Ԧ lies within the volume ܸ, we finally obtain
1
ߩ(‫ݔ‬Ԧ ᇱ ) ଷ ᇱ
1
1 ߲߮(‫ݔ‬Ԧ ᇱ )
߲ 1
߮(‫ݔ‬Ԧ) =
න
݀ ‫ݔ‬Ԧ +
ර ቈ
െ ߮(‫ݔ‬Ԧ ᇱ ) ᇱ ൬ ൰቉ ݀ܽᇱ
4ߨߝ଴ ௏
ܴ
4ߨ ௌ ܴ ߲݊ᇱ
߲݊ ܴ
6.
This is the solution to the Poisson equation with a given charge density distribution
ߩ(‫ݔ‬Ԧ ᇱ ) in the volume ܸ and the Cauchy boundary condition with ߮(‫ݔ‬Ԧ ᇱ ) and
డఝ൫௫Ԧ ᇲ ൯
డ௡ᇲ
given on the boundary surface ܵ enclosing the volume ܸ.
7.
If surface ܵ goes to infinity (open boundary) and the electric field on ܵ falls off faster
than 1/ܴ, then the integral terms vanish and we can see the well-known solution:
1
ߩ(‫ݔ‬Ԧ ᇱ ) ଷ ᇱ
߮(‫ݔ‬Ԧ) =
න
݀ ‫ݔ‬Ԧ
4ߨߝ଴ ୅୪୪ ୱ୮ୟୡୣ ܴ
8.
For a charge-free volume, the potential inside the volume is expressed in terms of the
potential and its normal derivative only on the boundary surface: Laplace equation’s
solution:
߮(‫ݔ‬Ԧ) =
1
1 ߲߮(‫ݔ‬Ԧ ᇱ )
߲ 1
ර ቈ
െ ߮(‫ݔ‬Ԧ ᇱ ) ᇱ ൬ ൰቉ ݀ܽᇱ
4ߨ ௌ ܴ ߲݊ᇱ
߲݊ ܴ
- Generalization of Green’s function for the Laplace operator
1.
ଵ
|௫Ԧି௫Ԧ ᇲ |
is indeed a Green function for the Laplace operator and, with a proper Cauchy
boundary condition, provides the solution of the Poisson equation.
2.
With an arbitrary function ‫ݔ(ܨ‬Ԧ, ‫ݔ‬Ԧ ᇱ ) satisfying the Laplace equation inside the volume
ܸ, Green’s function for the Poisson equation can be generalized as
1
‫ݔ(ܩ‬Ԧ, ‫ݔ‬Ԧ ᇱ ) =
+ ‫ݔ(ܨ‬Ԧ, ‫ݔ‬Ԧ ᇱ )
|‫ݔ‬Ԧ െ ‫ݔ‬Ԧ ᇱ |
ଶ
֜ ‫׏‬ᇱ ‫ݔ(ܩ‬Ԧ, ‫ݔ‬Ԧ ᇱ ) = ‫׏‬ᇱ
3.
ଶ
1
1
ଶ
ଶ
+ ‫׏‬ᇱ ‫׏ = ܨ‬ᇱ
= െ4ߨɁ(‫ݔ‬Ԧ െ ‫ݔ‬Ԧ ᇱ )
|‫ݔ‬Ԧ െ ‫ݔ‬Ԧ ᇱ |
|‫ݔ‬Ԧ െ ‫ݔ‬Ԧ ᇱ |
With Green’s theorem employing ߰ = ‫ݔ(ܩ‬Ԧ, ‫ݔ‬Ԧ ᇱ ) and ‫׏‬ᇱ ‫ݔ(ܩ‬Ԧ, ‫ݔ‬Ԧ ᇱ ) = െ4ߨɁ(‫ݔ‬Ԧ െ ‫ݔ‬Ԧ ᇱ ) , the
ଶ
integral equation converted from the Poisson equation is generalized as
1
1
߲߮(‫ݔ‬Ԧ ᇱ )
߲‫ݔ(ܩ‬Ԧ, ‫ݔ‬Ԧ ᇱ )
ᇱ)
߮(‫ݔ‬Ԧ) =
න ߩ(‫ݔ‬Ԧ ᇱ )‫ݔ(ܩ‬Ԧ, ‫ݔ‬Ԧ ᇱ )݀ ଷ ‫ݔ‬Ԧ ᇱ +
ර ቈ‫ݔ(ܩ‬Ԧ, ‫ݔ‬Ԧ ᇱ )
െ
߮(‫ݔ‬
Ԧ
቉ ݀ܽᇱ
4ߨߝ଴ ௏
4ߨ ௌ
߲݊ᇱ
߲݊ᇱ
4.
The freedom via the function ‫ݔ(ܨ‬Ԧ, ‫ݔ‬Ԧ ᇱ ) provides the possibility that we can use Green’s
theorem with ߰ = ‫ݔ(ܩ‬Ԧ, ‫ݔ‬Ԧ ᇱ ) and choose ‫ݔ(ܨ‬Ԧ, ‫ݔ‬Ԧ ᇱ ) to eliminate one or the other of the
two surface integrals, obtaining a result that involves only Dirichlet or Neumann
boundary conditions.
5.
For example, if we choose the function ‫ݔ(ܨ‬Ԧ, ‫ݔ‬Ԧ ᇱ ) resulting in ‫ݔ(ܩ‬Ԧ, ‫ݔ‬Ԧ ᇱ ) = 0 for ‫ݔ‬Ԧ or ‫ݔ‬Ԧ ᇱ
on ܵ, the solution in the integral form becomes
߮(‫ݔ‬Ԧ) =
6.
1
1
߲‫ݔ(ܩ‬Ԧ, ‫ݔ‬Ԧ ᇱ )
න ߩ(‫ݔ‬Ԧ ᇱ )‫ݔ(ܩ‬Ԧ, ‫ݔ‬Ԧ ᇱ )݀ ଷ ‫ݔ‬Ԧ ᇱ െ
ර ቈ߮(‫ݔ‬Ԧ ᇱ )
቉ ݀ܽᇱ
4ߨߝ଴ ௏
4ߨ ௌ
߲݊ᇱ
Thus, we need the boundary condition of only ߮(‫ݔ‬Ԧ ᇱ ) on ܵ : Dirichlet boundary
condition. We can consider a similar approach for the Neumann boundary condition.
But, we shall leave the rigorous discussions on the Green function for the Dirichlet and
Neumann boundary conditions for advanced undergraduate or graduate-level courses.
NOTE) The topics of complicated systems and numerical solution (Sections 3-11 and 3-12 of Reitz)
will be covered in the EM-II class next semester.
(HW #4)
Due: 20240408
1. Consider the sphere consisting of two hemispheres at +ܸ and െܸ
In the class, we have seen that the potential inside the sphere (‫ )ܽ < ݎ‬is given as
3‫ݎ‬
7 ‫ ݎ‬ଷ
߮(‫ݎ‬, ߠ) = ܸ ൤ ܲଵ (cos ߠ) െ ቀ ቁ ܲଷ (cos ߠ) + ‫ ڮ‬൨
2ܽ
8 ܽ
ஶ
= ܸ ෍(െ1)௡
௡ୀ଴
(4݊ + 3)(2݊ െ 1)ԥ ‫ ݎ‬ଶ௡ାଵ
ቀ ቁ
ܲଶ௡ାଵ (cos ߠ)
(2݊ + 2)ԥ
ܽ
(a) Show that the potential outside the sphere (‫ )ܽ > ݎ‬is given as
ஶ
߮(‫ݎ‬, ߠ) = ܸ ෍(െ1)௡
௡ୀ଴
(4݊ + 3)(2݊ െ 1)ԥ ܽ ଶ௡ାଶ
ቀ ቁ
ܲଶ௡ାଵ (cos ߠ)
(2݊ + 2)ԥ
‫ݎ‬
(b) Find the electric charge density ߪ on the outside surface.
Note) The result in (b) actually diverges at cos ߠ = ±1. This is due to the infinite capacitance of the
system, which requires zero thickness for the insulating barrier between the +ܸ-charged and െܸcharged conductors.
2. Consider a surface charge density ߪ(ߠ) = ߪ଴ cos ߠ glued to the surface of a spherical shell of
radius ܽ. There is a vacuum both inside and outside of the shell.
(a) Explain why the electrostatic potential inside and outside the shell can be expressed as
ஶ
߮(‫ݎ‬, ߠ) =
‫ ۓ‬෍ ‫(ି ݎ ܤ‬௟ାଵ) ܲ (cos ߠ)
௟
௟
ۖ
for ‫ܽ > ݎ‬
‫۔‬
ۖ෍ ‫ܣ‬௟ ‫ ݎ‬௟ ܲ௟ (cos ߠ)
‫ ە‬௟ୀ଴
for ‫ܽ < ݎ‬
௟ୀ଴
ஶ
(b) In the free space, the boundary condition for the normal component of the electric field on a
surface is given as ‫ܧ‬ଶ௡ െ ‫ܧ‬ଵ௡ = ߪ, where ݊ො represents the normal vector to the surface from the
region 1 to the region 2. (In general, ‫ܦ‬ଶ௡ െ ‫ܦ‬ଵ௡ = ߪ. The boundary conditions of the fields will be
discussed in detail in the lecture note 05.)
Using the boundary condition given for the normal component of the electric field, show that
ߪ ܽଷ
‫ ۓ‬଴ ଶ cos ߠ for ‫ܽ > ݎ‬
߮(‫ݎ‬, ߠ) = 3ߝ଴ ‫ݎ‬
‫ߪ۔‬଴ ‫ ݎ‬cos ߠ
for ‫ܽ < ݎ‬
‫ ە‬3ߝ଴
(c) Find the electric field both inside and outside of the spherical shell, and show that the electric
field is uniform inside and corresponds to that of the electric dipole outside of which the dipole
moment is
ସగ௔య
ଷ
× ߪ଴ .
3. Point charge near a conducting sphere. Consider the problem of a conducting, grounded sphere
of radius a and a point charge ‫ ݍ‬as illustrated below
(a) Find the electrostatic potential outside the sphere, employing the method of image charge.
(b) Find the surface charge distribution on the conducting sphere and show that the total induced
charge is equal to the magnitude of the image charge.
4. (optional). Consider the electrostatic potential produced by a thin conducting ring of radius ܽ
placed symmetrically in the equatorial plane of a spherical polar coordinates. The conducting ring
carries a total electric charge ‫ݍ‬.
(a) From Coulomb’s law, show that the electrostatic potential along the symmetric axis ‫ݖ‬Ƹ is
ஶ
(2݊ െ 1)ԥ ܽ ଶ௡ାଵ
1
‫ݍ‬
‫ݍ‬
=
෍(െ1)௡
ቀ ቁ
߮(‫ݎ‬, ߠ = 0) =
(2݊)ԥ
‫ݎ‬
4ߨߝ଴ ξ‫ ݎ‬ଶ + ܽଶ 4ߨߝ଴ ܽ
௡ୀ଴
(b) In similar to the result in the problem 2(a), for ‫ܽ > ݎ‬,
ஶ
߮(‫ݎ‬, ߠ) = ෍ ‫ܤ‬௟ ‫(ି ݎ‬௟ାଵ) ܲ௟ (cos ߠ)
௟ୀ଴
Comparing with the result in 4(a) along ‫ݖ‬Ƹ axis, show that the coefficients ‫ܤ‬௟ are equated such that
ஶ
(2݊ െ 1)ԥ ܽ ଶ௡ାଵ
‫ݍ‬
෍(െ1)௡
ቀ ቁ
ܲଶ௡ (cos ߠ)
߮(‫ݎ‬, ߠ) =
(2݊)ԥ
‫ݎ‬
4ߨߝ଴ ܽ
௡ୀ଴
(c) Show that the electrostatic potential for ‫ ܽ > ݎ‬is given as
ஶ
(2݊ െ 1)ԥ ‫ ݎ‬ଶ௡
‫ݍ‬
෍(െ1)௡
ቀ ቁ ܲଶ௡ (cos ߠ)
߮(‫ݎ‬, ߠ) =
(2݊)ԥ
ܽ
4ߨߝ଴ ܽ
௡ୀ଴
PH231 Electromagnetism I (Spring 2024)
Lecture note 05: Electrostatics in Dielectric Media
(Chapters 4 and 5 of the textbook by Reitz, Chapter 10 of the book by Purcell)
5.1 Introduction and Dielectric material
- Thus far, we have discussed the problems of charges and conductors (or equivalently specified
potentials) in the free space and ignored problems involving matter. In this section, we will discuss
the behavior of electric fields in matter, particularly dielectrics. We will study how electric fields
affect, and are affected by, matter. The topics in this section are somewhat simplified and
phenomenological, but they may give you some insights to study advanced topics in the future.
- An ideal dielectric material is one that has no free charges. Nevertheless, all material media are
composed of molecules, which are composed of charged entities (atomic nuclei and electrons).
The electric field affects the molecules of the dielectric and causes the distortion of the charge
distribution, such as displacement of the positive and negative parts of each molecule. These
displacements, however, are typically limited to very small fractions of a molecular diameter by
strong restoring forces. The term, bound charges, in contrast to the free charge in free space or
in a conductor, is used to emphasize the bound features of such molecular charges.
- Since a matter consists of innumerable molecules, one cannot take into account the position and
effect of each individual charges from the microscopic point of view. We thus need to develop the
theory in terms of overall effects and averaged fields from the macroscopic view. Even though a
dielectric is electrically neutral on the average, there can be a displacement of all the positive
bound charges relative to their counterparts and the dielectric is polarized. In this lecture, we will
study the macroscopic fields, the electric displacement and polarization, and their relations with
the electric fields. We will also see how to bridge the macroscopic and microscopic theories.
- The macroscopic equations for the electric field and displacement also provide boundary-value
problems. We will discuss the boundary condition at the interface between two piecewise
homogeneous dielectric media. Also, the method of images for problems involving dielectrics will
be discussed as in the lecture of the electrostatics with free carriers.
5.2 Polarization (Section 4-1 of Reitz)
- Much of electrostatics concerns itself with charges and fields in condensed media whose respective
electric (electrostatic) responses must be taken into account.
1.
The macroscopic electric field ‫ܧ‬ሬԦ୫ୟୡ୰୭ in a condensed medium is the average of the
microscopic electric field ‫ܧ‬ሬԦ୫୧ୡ୰୭ over a volume ȟ‫ ݒ‬sufficiently large compared to the
molecular scale and sufficiently infinitesimal from the microscopic point of view:
‫ܧ‬ሬԦ୫ୟୡ୰୭ (‫ݎ‬Ԧ) = ‫ܧۃ‬ሬԦ୫୧ୡ୰୭ ‫= ۄ‬
1
(‫ݎ‬Ԧᇱ )݀‫ ݒ‬ᇱ
න ‫ܧ‬ሬԦ
ȟ‫୼ ݒ‬௩ ୫୧ୡ୰୭
2.
Here, ‫ݎ‬Ԧ is a representative position inside ȟ‫ݒ‬.
3.
When an averaging is made of the homogeneous equation ‫ܧ × ׏‬ሬԦ୫୧ୡ୰୭ = 0, the same
equation ‫ܧ × ׏‬ሬԦ୫ୟୡ୰୭ = 0 holds for the averaged macroscopic electric field:
‫ܧ × ׏‬ሬԦ୫ୟୡ୰୭ = ‫ܧ × ׏ۃ‬ሬԦ୫୧ୡ୰୭ ‫= ۄ‬
1
න ‫׏‬ᇱ × ‫ܧ‬ሬԦ୫୧ୡ୰୭ (‫ݎ‬Ԧ ᇱ )݀‫ ݒ‬ᇱ = 0
ȟ‫୼ ݒ‬௩
4.
Thus, the electric field is still derivable from a potential ߮୫ୟୡ୰୭ .
5.
If an electric field applied to a medium made up of a large number of atoms or
molecules, the charges bound in each molecule will respond to the applied field and
will execute perturbed motions: the distorted charge density and multipole moments
of each molecule will be different from what they were in the absence of the field.
- Electric polarization and macroscopic charge density
1.
In simple substances other than ferroelectric materials and so on, if ‫ܧ‬ሬԦୟ୮୮୪୧ୣୢ = 0, the
multipole moments are all zero, at least when averaged over many molecules.
2.
If the medium is polarized by a field, a separation of positive and negative charge is
effected, and the volume element ȟ‫ ݒ‬is featured by an electric dipole moment:
ȟ‫݌‬Ԧ = න ‫ݎ‬Ԧ݀‫ݍ‬
୼௩
3.
It is sometimes desirable to speak about the electric dipole moment of a single
molecule, ‫݌‬Ԧ௠ = ‫׬‬୫୭୪ୣୡ୳୪ୣ ‫ݎ‬Ԧ݀‫ ݍ‬. Then, the dipole moment associated with ȟ‫ ݒ‬is given by
ȟ‫݌‬Ԧ = ෍
ୟ୪୪ ୫୭୪ୣୡ୳୪ୣୱ ୧୬ୱ୧ୢୣ ୼௩
‫݌‬Ԧ௠
4.
Since ȟ‫݌‬Ԧ depends on the size of the volume element, it is more convenient to work
5.
with the electric dipole moment per unit volume, called the electric polarization:
ȟ‫݌‬Ԧ
ܲሬԦ =
ȟ‫ݒ‬
For sufficiently infinitesimal ȟ‫ ݒ‬from the microscopic point of view, the polarization
can be considered as a vectorial field as a function of the position: ܲሬԦ(‫ݎ‬Ԧ). Where ‫݌‬Ԧ௜ and
ܰ௜ are the dipole moment and the macroscopic averaged number per unit volume of
the ݅th type of molecule in the medium, the polarization becomes
ܲሬԦ(‫ݎ‬Ԧ) = ෍ ܰ௜ ‫݌ۃ‬Ԧ௜ ‫ۄ‬
௜
6.
Meanwhile, the charge density at the macroscopic level is
ߩ୫ୟୡ୰୭ (‫ݎ‬Ԧ) = ෍ ܰ௜ ‫ݍۃ‬௜ ‫ ۄ‬+ ߩୣ୶ୡୣୱୱ
௜
7.
Usually, the averaged molecular charge ‫ݍۃ‬௜ ‫ ۄ‬is zero and the charge density is the excess
(or embedded) charge.
5.3 Field by a distribution of the polarization: a dielectric medium (Sections 4-2 and 4-3 of Reitz)
- Electric potential and field outside a finite dielectric medium
1.
As we have discussed in the lecture 03 (Coulomb’s law and Gauss’ law), the potential
at ‫ݎ‬Ԧ caused by the monopole and dipole moments, ‫ݍ‬୫ୟୡ୰୭ = ߩ୫ୟୡ୰୭ ȟ‫ ݒ‬ᇱ and ȟ‫݌‬Ԧ = ܲሬԦȟ‫ ݒ‬ᇱ ,
of a microscopic charge distribution in a volume element ȟ‫ ݒ‬ᇱ at ‫ݎ‬Ԧ ᇱ as
ȟ߮(‫ݎ‬Ԧ) =
=
2.
1
‫ݍ‬୫ୟୡ୰୭ ȟ‫݌‬Ԧ ‫ݎ( ڄ‬Ԧ െ ‫ݎ‬Ԧ ᇱ )
ቈ
+
቉
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |ଷ
4ߨߝ଴ |‫ݎ‬Ԧ െ ‫ ݎ‬ᇱ |
1 ߩ୫ୟୡ୰୭ (‫ݎ‬Ԧᇱ ) ܲሬԦ(‫ݎ‬Ԧᇱ ) ‫ݎ( ڄ‬Ԧ െ ‫ݎ‬Ԧ ᇱ )
ቈ
+
቉ ȟ‫ ݒ‬ᇱ
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |ଷ
4ߨߝ଴ |‫ݎ‬Ԧ െ ‫ ݎ‬ᇱ |
Now let us consider a dielectric medium with a volume of ܸ enclosed by a surface ܵ
and treat ȟ‫ ݒ‬macroscopically infinitesimal. Then, we have
߮(‫ݎ‬Ԧ) =
3.
1
ߩ୫ୟୡ୰୭ (‫ݎ‬Ԧᇱ ) ܲሬԦ(‫ݎ‬Ԧᇱ ) ‫ݎ( ڄ‬Ԧ െ ‫ݎ‬Ԧ ᇱ )
න ቈ
+
቉ ݀‫ ݒ‬ᇱ
|‫ݎ‬Ԧ െ ‫ ݎ‬ᇱ |
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |ଷ
4ߨߝ଴ ௏
The second term of the contribution of the polarization can be transformed by means
of vector identity, ‫׏‬ᇱ ‫ ڄ‬൫݂‫ܨ‬Ԧ ൯ = ݂‫׏‬ᇱ ‫ܨ ڄ‬Ԧ + ‫ܨ‬Ԧ ‫׏ ڄ‬ᇱ ݂, as
ܲሬԦ(‫ݎ‬Ԧᇱ ) ‫ݎ( ڄ‬Ԧ െ ‫ݎ‬Ԧ ᇱ )
1
ܲሬԦ
‫׏‬ᇱ ‫ܲ ڄ‬ሬԦ
ᇱ
ሬԦ(‫ݎ‬Ԧᇱ ) ‫׏ ڄ‬ᇱ ൬
=
ܲ
൰
=
‫׏‬
‫ڄ‬
ቆ
ቇ
െ
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |ଷ
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
4.
Considering the Gauss theorem, the following volume integral can be converted to the
closed surface integral, where ݊ො is the outward normal to the surface element ݀ܽᇱ :
ܲሬԦ
ܲሬԦ ‫݊ ڄ‬ො
න ‫׏‬ᇱ ‫ ڄ‬ቆ
ቇ ݀‫ ݒ‬ᇱ = ර
݀ܽᇱ
ᇱ
ᇱ|
|‫ݎ‬
|
|‫ݎ‬
Ԧ
െ
‫ݎ‬
Ԧ
Ԧ
െ
‫ݎ‬
Ԧ
௏
ௌ
5.
We thus have the macroscopic electrostatic potential as
߮(‫ݎ‬Ԧ) =
1
ߩ୫ୟୡ୰୭ (‫ݎ‬Ԧ ᇱ ) െ ‫׏‬ᇱ ‫ܲ ڄ‬ሬԦ ᇱ
1
ܲሬԦ ‫݊ ڄ‬ො
න
݀‫ݒ‬
+
ර
݀ܽᇱ
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
4ߨߝ଴ ௏
4ߨߝ଴ ௌ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
6.
Since the electrostatic potential is still defined by ‫ܧ × ׏‬ሬԦ୫ୟୡ୰୭ = 0, the macroscopic
electric field is obtained by the gradient of ߮
ߪ௣ (‫ݎ‬Ԧ ᇱ )
1
ߩ୫ୟୡ୰୭ (‫ݎ‬Ԧᇱ ) + ߩ௉ (‫ݎ‬Ԧ ᇱ ) ᇱ
1
‫ܧ‬ሬԦ୫ୟୡ୰୭ (‫ݎ‬Ԧ) = െ‫ = ߮׏‬െ
‫׏‬න
݀‫ݒ‬
െ
‫׏‬
ර
݀ܽᇱ
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
4ߨߝ଴ ௏
4ߨߝ଴ ௌ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
7.
=െ
1
1
1
1
(‫ݎ‬Ԧ ᇱ ) + ߩ௉ (‫ݎ‬Ԧᇱ )] ‫׏‬
න [ߩ
݀‫ ݒ‬ᇱ െ
ර ߪ (‫ݎ‬Ԧ ᇱ ) ‫׏‬
݀ܽᇱ
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
4ߨߝ଴ ௌ ௣
4ߨߝ଴ ௏ ୫ୟୡ୰୭
=
‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ
‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ
1
1
ᇱ
ᇱ)
(‫ݎ‬
Ԧ
න [ߩ୫ୟୡ୰୭ (‫ݎ‬Ԧᇱ ) + ߩ௉ (‫ݎ‬Ԧ ᇱ )]
݀‫ݒ‬
+
ර
ߪ
݀ܽᇱ
௣
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |ଷ
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |ଷ
4ߨߝ଴ ௌ
4ߨߝ଴ ௏
Without any excess charge,
‫ܧ‬ሬԦ୫ୟୡ୰୭ (‫ݎ‬Ԧ) =
1
‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ
‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ
1
ᇱ
න ߩ௉
݀‫ݒ‬
+
ර
ߪ
݀ܽᇱ
௣
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |ଷ
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |ଷ
4ߨߝ଴ ௏
4ߨߝ଴ ௌ
- Polarization charge densities
1.
As we have seen above, the polarization of a finite dielectric medium contributes to
the electrostatic potential with two scalar quantities
ߩ௉ = െ‫ܲ ڄ ׏‬ሬԦ
2.
and
ߪ௉ = ܲሬԦ ‫݊ ڄ‬ො
ߩ௉ and ߪ௉ are called as the volume and surface density of polarization charge,
respectively.
3.
One can easily see that a divergence-less polarization, including a uniform polarization,
does not support a volume polarization charge, and the absence of a surface, a sudden
termination of the dielectric, excludes the surface polarization charge.
4.
As illustrated in the figure below, the spatial change of the polarization vector field
causes the polarization charge densities. As an example, in a condensed matter, a defect
in its crystalline structure can cause the non-zero divergence.
- Electric potential and field inside a dielectric medium
1.
Now let us consider the case at the position inside a dielectric.
2.
Conceptually, the electric field in a dielectric is equal to the electric field inside a needleshaped cavity in the dielectric, with the cavity axis oriented parallel to the direction of
the electric field (see the following figure).
A.
First, the curl-free property of the electric field in statics gives
‫ܧ × ׏‬ሬԦ = 0
֜
ර
‫ܧ‬ሬԦ ‫݈݀ ڄ‬Ԧ = ‫ܧ‬ሬԦ஺஻ ‫ ڄ‬ሬሬሬሬሬԦ
‫ ܤܣ‬+ ‫ܧ‬ሬԦ஼஽ ‫ ڄ‬ሬሬሬሬሬԦ
‫ = ܦܥ‬൫‫ܧ‬ሬԦ஺஻ െ ‫ܧ‬ሬԦ஼஽ ൯ ‫ ڄ‬ሬሬሬሬሬԦ
‫ = ܤܣ‬0
஺஻஼஽
֜ ‫ܧ‬ሬԦ஺஻ = ‫ܧ‬ሬԦ஼஽
B.
In fact, this is one of the boundary conditions that the electric field should satisfy,
which will be discussed in detail soon.
C.
The electric field in the needle cavity, which can be considered as an exterior of
the dielectric, can be calculated as we have discussed earlier.
‫ܧ‬ሬԦ୫ୟୡ୰୭ (‫ݎ‬Ԧ) =
D.
1
‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ
‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ
1
ᇱ
න
ߩ௉
݀‫ݒ‬
+
ර
ߪ
݀ܽᇱ
4ߨߝ଴ ௏ି௏ౙ౗౬ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |ଷ
4ߨߝ଴ ௌିௌౙ౗౬ ௣ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |ଷ
Here, ܸୡୟ୴ and ܵୡୟ୴ are the volume and the surface of the needle cavity, and we
considered the polarized dielectric without excess charges.
E.
At the limit where the needle cavity becomes infinitesimally small, the integral
inside and outside the dielectric differs only by a negligible amount. It is because
the following reasons:
i.
‫ݎ‬Ԧ ᇱ does not exist on ܵୡୟ୴ , so 1Τ|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ | does not diverge in the evaluation of
the surface integral as the needle cavity becomes infinitesimally thin.
ii.
Although ܸୡୟ୴ contains the point ‫ݎ‬Ԧ = ‫ݎ‬Ԧ ᇱ , ݀‫ ݒ‬ᇱ Τ|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ | does not diverge. ݀‫ ݒ‬ᇱ
goes to zero faster than 1Τ|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ | diverges.
3.
In conclusion, the dielectric’s contribution to the electric potential and field at ‫ݎ‬Ԧ is
independent of whether ‫ݎ‬Ԧ is inside or outside the medium:
‫ܧ‬ሬԦ୫ୟୡ୰୭ (‫ݎ‬Ԧ) =
4.
1
‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ
‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ
1
ᇱ
න ߩ௉
݀‫ݒ‬
+
ර
ߪ
݀ܽᇱ
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |ଷ
4ߨߝ଴ ௏
4ߨߝ଴ ௌ ௣ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |ଷ
In fact, the discussion given in this lecture note and in the textbook of Reitz is strictly
true only for isotropic dielectrics. The further rigorous and generalized discussion is a
senior undergraduate or graduate level courses (If you are interested in it, you can refer
to section 4.1, section 6.6, and chapter 7 of the textbook by Jackson.)
5.4 Electric displacement and Gauss’ law in a dielectric (Sections 4-4 and 4-5 of Reitz)
- Electric displacement and macroscopic Gauss’ law
1.
Let us now consider the homogeneous dielectric so that there is no sudden boundary
surface where the surface polarization charge density can accumulate.
2.
Then, the electric potential is given as
߮୫ୟୡ୰୭ (‫ݎ‬Ԧ) =
3.
1
ൣߩ୫ୟୡ୰୭ (‫ݔ‬Ԧ ᇱ ) െ ‫׏‬ᇱ ‫ܲ ڄ‬ሬԦ(‫ݔ‬Ԧ ᇱ )൧ ᇱ
න
݀‫ݒ‬
|‫ݔ‬Ԧ െ ‫ݔ‬Ԧ ᇱ |
4ߨߝ଴ ୟ୪୪ ୱ୮ୟୡୣ
This is just the customary expression for the Coulomb potential caused by a charge
distribution ൫ߩ୫ୟୡ୰୭ െ ‫ܲ ڄ ׏‬ሬԦ൯. Thus, with ‫ܧ‬ሬԦ୫ୟୡ୰୭ = െ‫߮׏‬୫ୟୡ୰୭ (from ‫ܧ × ׏‬ሬԦ୫ୟୡ୰୭ = 0), the
Gauss’ law therefore reads
1
൫ߩ
െ ‫ܲ ڄ ׏‬ሬԦ൯
ߝ଴ ୫ୟୡ୰୭
֜ ‫ ڄ ׏‬൫ߝ଴ ‫ܧ‬ሬԦ + ܲሬԦ൯ = ߩ୫ୟୡ୰୭
‫ܧ ڄ ׏‬ሬԦ୫ୟୡ୰୭ = െ‫׏‬ଶ ߮୫ୟୡ୰୭ =
4.
We can now define a new vector, the electric displacement, as
ሬԦ = ߝ଴ ‫ܧ‬ሬԦ + ܲሬԦ and ‫ܦ ڄ ׏‬
ሬԦ = ߩ୫ୟୡ୰୭ (or = ߩୣ୶ୡୣୱୱ )
‫ܦ‬
5.
Further rigorous discussions of the development and validity of the macroscopic
ሬԦ, ‫ܧ‬ሬԦ୫ୟୡ୰୭ , ܲሬԦ, ߩ୫ୟୡ୰୭ , and so on, and the macroscopic equations is beyond
quantities, ‫ܦ‬
the scope of this course. But, you can refer to section 6.6 of the textbook by Jackson.
- Electric susceptibility and dielectric constant
1.
From the macroscopic point of view, the behavior of the material is completely specified
by an experimentally determined relationship, called a constitutive equation. For most
materials, ܲሬԦ vanishes when ‫ܧ‬ሬԦ vanishes. If we limit our discussion to materials of this
type and if the materials are isotropic, we have
ܲሬԦ = ߯(‫ܧ)ܧ‬ሬԦ
2.
Then, the displacement is proportional to the electric field as
ሬԦ = ߝ଴ (1 + ߯௘ )‫ܧ‬ሬԦ = ߝ‫ܧ‬ሬԦ
‫ܦ‬
3.
If the dielectric is not only isotropic but also homogeneous, ߝ is independent of
position and thus
ߩୣ୶ୡୣୱୱ
ߝ
The reduction of the electric field by a factor of ߝ଴ /ߝ can be understood in terms of a
‫ܧ ڄ ׏‬ሬԦ =
4.
polarization of the atoms (or molecules) producing fields in opposite to that of the
given charge: the screening effect.
5.
If the material is linear but anisotropic, one can consider the electric susceptibility in
the form of a tensor as
ܲሬԦ = ߝ଴ ߯ി௘ ‫ܧ‬ሬԦ
5.5 Boundary-value problems involving dielectrics (Sections 4-7 and 4-8 of Reitz)
- Solution to the Laplace equation
1.
As we discussed earlier, the electrostatic potential can still be determined as ‫ܧ‬ሬԦ = െ‫߮׏‬
from ‫ܧ × ׏‬ሬԦ୫ୟୡ୰୭ = 0.
2.
Let us consider a system consisting of piecewise homogeneous dielectrics; the system
is divided into distinct dielectrics, each of which has uniform (homogeneous) ߝ within
itself, but those characteristics can vary from one dielectric to another.
3.
It is obvious that the electrostatic potential inside one dielectric element in the system
4.
satisfies the Poisson equation as
ߩୣ୶ୡୣୱୱ
ߩୣ୶ୡୣୱୱ
‫ܧ ڄ ׏‬ሬԦ =
֜
‫׏‬ଶ ߮ = െ
ߝ
ߝ
If there is no excess charge, the potential satisfies the Laplace equation: ‫׏‬ଶ ߮ = 0.
5.
To solve the Poisson/Laplace equation involving dielectrics, we need to determine the
boundary conditions, particularly, between two dielectrics. In classical physics, the free
space (vacuum) is also a dielectric with ߝ୴ୟୡ = ߝ଴ .
- Boundary conditions
1.
The macroscopic equations for the electric field and displacement provide the require
boundary condition at the interface between ponderable media with a unit normal
vector ݊ොଶଵ:
ሬԦ = ߩ୫ୟୡ୰୭
‫ܦڄ׏‬
ቊ
‫ܧ × ׏‬ሬԦ = 0
2.
֜
ሬԦଶ െ ‫ܦ‬
ሬԦଵ ൯ ‫݊ ڄ‬ොଶଵ = ߪ୫ୟୡ୰୭
൫‫ܦ‬
ቊ
൫‫ܧ‬ሬԦଶ െ ‫ܧ‬ሬԦଵ ൯ × ݊ොଶଵ = 0
The boundary conditions can be easily understood from the following figure.
A.
Consider two dielectric media, 1 and 2.
B.
Left: The small disk enclosed by ܵ intersects the interface between two media. As
the height and volume of the disk becomes negligibly thin, Gauss’ law gives
ሬԦଶ ‫݊ ڄ‬ොଶ ȟܵ + ‫ܦ‬
ሬԦଵ ‫݊ ڄ‬ොଵ ȟܵ = ߪȟܵ ֜
ሬԦଶ െ ‫ܦ‬
ሬԦଵ ൯ ‫݊ ڄ‬ොଶଵ = ߪ୫ୟୡ୰୭
‫ܦ‬
൫‫ܦ‬
C.
The discontinuity of the normal component of the displacement is given by the
surface density of the macroscopic excess charge on the interface. ߪ୫ୟୡ୰୭ does
not include the polarization charge given by ܲሬԦ ‫݊ ڄ‬ො.
D.
Right: The closed line intersecting the interface. Since the electric field is curl free
(‫ܧ × ׏‬ሬԦ = 0), the line integral of the electric field around any closed path vanishes:
‫ܧ‬ሬԦଶ ‫ ڄ‬ȟ݈Ԧ + ‫ܧ‬ሬԦଵ ‫ ڄ‬൫െȟ݈Ԧ൯ = 0 ֜ ‫ܧ‬ଶ௧ = ‫ܧ‬ଵ௧ ֜ ൫‫ܧ‬ሬԦଶ െ ‫ܧ‬ሬԦଵ ൯ × ݊ොଶଵ = 0
E.
The tangential component of the electric field is continuous across an interface.
- Example of boundary-value problems involving dielectrics
1.
As a very conventional example, consider a linear, isotropic dielectric sphere of radius
ܽ with a dielectric constant ߝ/ߝ଴ in an initially uniform electric field.
2.
Using the axial symmetry and considering there is no excess charge inside the sphere,
we can expand the electrostatic potential, the solution to the Laplace equation, as
ஶ
‫ۓ‬
ۖ
߮(‫ݎ‬Ԧ) =
3.
෍ ‫ܣ‬௟ ‫ ݎ‬௟ ܲ௟ (cos ߠ)
ஶ
inside
௟ୀ଴
‫۔‬
ۖ෍ൣ‫ܤ‬௟ ‫ ݎ‬௟ + ‫ܥ‬௟ ‫(ି ݎ‬௟ାଵ) ൧ܲ௟ (cos ߠ)
‫ ە‬௟ୀ଴
outside
First, employing the boundary condition at infinity, ߮ ՜ െ‫ܧ‬଴ ‫ = ݖ‬െ‫ܧ‬଴ ‫ ݎ‬cos ߠ, (refer to the
lecture note 04)
for ݈ = 1
െ‫ܧ‬଴
0 otherwise
Second, the boundary conditions at ‫ ܽ = ݎ‬determine the other coefficients as
1 ߲߮୧୬
1 ߲߮୭୳୲
‫ۓ‬െ
ฬ
=െ
ฬ
ܽ ߲ߠ ௥ୀ௔
ܽ ߲ߠ ௥ୀ௔
߲߮୭୳୲
‫ ۔‬െߝ ߲߮୧୬ ฬ
= െߝ଴
ฬ
‫ە‬
߲‫ ݎ‬௥ୀ௔
߲‫ ݎ‬௥ୀ௔
‫ܤ‬௟ = ቄ
4.
ஶ
֜
ஶ
߲ܲ௟ (cos ߠ)
߲ܲ௟ (cos ߠ)
෍ ‫ܣ‬௟ ܽ
= ෍ൣ‫ܤ‬௟ ܽ௟ െ ‫ܥ‬௟ ܽ ି(௟ାଵ) ൧
߲ߠ
߲ߠ
‫ۓ‬
ۖ
௟
ஶ
௟ୀ଴
௟ୀ଴
ஶ
‫۔‬
ۖ෍ ߝ݈‫ܣ‬௟ ܽ௟ିଵ ܲ௟ (cos ߠ) = ‫ܥ‬଴ ln ܽ + ෍ ߝ଴ ൣ݈‫ܤ‬௟ ܽ௟ିଵ െ (݈ + 1)‫ܥ‬௟ ܽ ି(௟ାଶ) ൧ܲ௟ (cos ߠ)
‫ ە‬௟ୀଵ
௟ୀଵ
‫ܣۓ‬ଵ = െ ൬
֜
5.
൰ ‫ ܧ‬and ‫ܣ‬௟ஷଵ = 0
ߝ/ߝ଴ + 2 ଴
‫ = ܥ ۔‬൬ߝ/ߝ଴ െ 1൰ ܽଷ ‫ ܧ‬and ‫ = ܥ‬0
଴
௟ஷଵ
‫ ە‬ଵ
ߝ/ߝ଴ + 2
Finally, we have
߮(‫ݎ‬Ԧ) =
6.
3
3
3
‫ۓ‬െ ൬
ۖ ߝ/ߝ଴ + 2൰ ‫ܧ‬଴ ‫ ݎ‬cos ߠ = െ ൬ߝ/ߝ଴ + 2൰ ‫ܧ‬଴ ‫ݖ‬
inside
ܽଷ
ߝ/ߝ଴ െ 1
‫۔‬
ۖെ‫ܧ‬଴ ‫ ݎ‬cos ߠ + ൬ߝ/ߝ + 2൰ ‫ܧ‬଴ ‫ ݎ‬ଶ cos ߠ
଴
‫ە‬
outside
Defining ‫݌‬Ԧ = 4ߨߝ଴ ቀ
ఌ/ఌబ ିଵ
ఌ/ఌబ ାଶ
ቁ ܽଷ ‫ܧ‬଴ , the potential outside becomes
߮୭୳୲ (‫ݎ‬Ԧ) = െ‫ܧ‬଴ ‫ ݎ‬cos ߠ +
1 ‫݌‬Ԧ ‫ݎ ڄ‬Ԧ
4ߨߝ଴ ‫ ݎ‬ଷ
7.
The polarization and induced surface charge density are given as
3
ߝ/ߝ଴ െ 1
ሬԦ െ ߝ଴ ‫ܧ‬ሬԦ = (ߝ െ ߝ଴ )‫ܧ‬ሬԦ = (ߝ െ ߝ଴ ) ൬
‫ܲۓ‬ሬԦ = ‫ܦ‬
൰ ‫ܧ‬ሬԦ଴ = 3ߝ଴ ൬
൰ ‫ܧ‬ሬԦ
ߝ/ߝ଴ + 2
ߝ/ߝ଴ + 2 ଴
ߝ/ߝ଴ െ 1
‫۔‬
ߪ୮୭୪ = ܲሬԦ ‫ݎ ڄ‬Ƹ = 3ߝ଴ ൬
൰ ‫ ܧ‬cos ߠ
‫ە‬
ߝ/ߝ଴ + 2 ଴
8.
The arrow in the left and the right panels of the figure below represents the polarization
and the electric field by the induced surface charge െ ቀ
9.
ଷ
ఌ/ఌబ ାଶ
ቁ ‫ܧ‬ሬԦ଴ , respectively.
The figure below compares the lines of the displacement (left) and electric field (right).
Without excess charge, the lines of the displacement are connected everywhere, but
those of the electric field are not connected at the surface of the dielectric, which
ሬԦଶ െ ‫ܦ‬
ሬԦଵ ൯ ‫݊ ڄ‬ොଶଵ = ൫ߝଶ ‫ܧ‬ሬԦଶ െ ߝଵ ‫ܧ‬ሬԦଵ ൯ ‫݊ ڄ‬ොଶଵ = 0.
originates from the boundary condition: ൫‫ܦ‬
5.6 The method of images for problems involving dielectrics (Section 4-10 of Reitz)
- Generalization of the method of images
1.
In the previous lecture, we have focused on calculating the potential in a region of
space envisioned as being produced by point charges and charged conducting surfaces
surrounding the region and equivalently produced by fictitious image charges outside
the region.
2.
When we have two or more dielectrics present, the fictitious image charges may also
be located inside one of the other dielectrics not the on where the potential is being
calculated.
3.
But, we typically need to calculate the potentials across all the dielectrics, and different
fictitious image charges must be considered for calculating the potential in each
dielectric.
4.
Also, the boundary conditions at each dielectric-dielectric interface must be satisfied.
- Example
1.
Consider a system consisting of two dielectric media with a point charge in medium 1.
There is no excess charges in the system
2.
Boundary conditions. Since there is no excess charges, the normal component of the
displacement and the tangential component of the electric field are continuous:
߲߮ଵ
߲߮ଶ
‫ܦ‬ଵ௡ = ‫ܦ‬ଶ௡ ฻ ߝଵ
= ߝଶ
߲݊
߲݊
and
‫ܧ‬ଵ௧ = ‫ܧ‬ଶ௧
3.
฻
߲߮ଵ ߲߮ଶ
=
߲݊௧
߲݊௧
To calculate the potential in the medium 1, ߮ଵ , let us surmise the image charge ‫ ݍ‬ᇱ in
the region of the medium 2 to be located at the same distance ݀ from the interface
as the original point charge. From the point of view of the medium 1, the image charge
is also located in the medium with the same dielectric constant, so that
1 ‫ݍ ݍ‬ᇱ
1
‫ݍ‬
‫ݍ‬ᇱ
߮ଵ =
൤ + ᇱ൨ =
ቈ
+
቉
4ߨߝଵ ‫ݎ ݎ‬
4ߨߝଵ ඥ(‫ ݔ‬+ ݀)ଶ + ‫ ݕ‬ଶ + ‫ ݖ‬ଶ ඥ(‫ ݔ‬െ ݀)ଶ + ‫ ݕ‬ଶ + ‫ ݖ‬ଶ
4.
From the point of view of the medium 2, the image charge ‫ ݍ‬ᇱᇱ must be located in the
medium 1 but with the dielectric constant ߝଶ . If we assume that the image charge is
located at the position of the original charge, the potential in the medium 2 is
‫ ݍ‬ᇱᇱ
1 ‫ ݍ‬ᇱᇱ
1
߮ଶ =
=
4ߨߝଶ ‫ݎ‬
4ߨߝଶ ඥ(‫ ݔ‬+ ݀)ଶ + ‫ ݕ‬ଶ + ‫ ݖ‬ଶ
5.
Now let us apply the boundary condition to determine ‫ ݍ‬ᇱ and ‫ ݍ‬ᇱᇱ:
(‫ ݍ‬െ ‫ ݍ‬ᇱ )݀
߲߮ଵ
߲߮ଶ
‫ ݍ‬ᇱᇱ ݀
ߝଵ
ฬ
= ߝଶ
ฬ
֜
=
ଷ
ଷ
߲‫ ݔ‬௫ୀ଴
߲‫ ݔ‬௫ୀ଴
[݀ ଶ + ‫ ݕ‬ଶ + ‫ ݖ‬ଶ ]ଶ [݀ ଶ + ‫ ݕ‬ଶ + ‫ ݖ‬ଶ ]ଶ
and
߲߮ଶ
߲߮ଵ
ฬ
=
ฬ
߲‫ ݕ‬௫ୀ଴
߲‫ ݕ‬௫ୀ଴
֜
(‫ ݍ‬+ ‫ ݍ‬ᇱ )‫ݕ‬
‫ ݍ‬ᇱᇱ ‫ݕ‬
1
1
ଷ =
ଷ
ߝଵ [݀ ଶ
ߝଶ [݀ ଶ
+ ‫ ݕ‬ଶ + ‫ ݖ‬ଶ ]ଶ
+ ‫ ݕ‬ଶ + ‫ ݖ‬ଶ ]ଶ
6.
Solving the two equations, we can obtain
ߝଵ െ ߝଶ
2ߝଶ
‫ݍ‬ᇱ =
‫ ݍ‬and ‫ ݍ‬ᇱᇱ =
‫ݍ‬
ߝଵ + ߝଶ
ߝଵ + ߝଶ
7.
Recall that the solution to the Laplace equation satisfying the same boundary condition
is unique.
8.
The figure below shows the lines of the electric displacement for a point charge for
two difference cases, ߝଵ > ߝଶ and ߝଵ < ߝଶ.
5.7 Microscopic theory of dielectrics: Molecular Polarizability and Electric Susceptibility
(Sections 5-1 and 5-3 of Reitz)
- In this part, we aim to bridge the microscopic quantity and the macroscopic quantity: the
polarizability of molecule and the dielectric constant/electric susceptibility of medium. Although a
proper treatment (e.g. the shape of molecules inside condensed matters) necessarily would involve
quantum mechanical considerations, (semi-)classical model of the molecular properties is still
effective to describe the simple properties of dielectrics.
- Molecular field in a dielectric
1.
In rarefied media (e.g. gas), where molecular separations are large, there is little
difference between the macroscopic electric field ‫ܧ‬ሬԦ and the electric field acting on the
molecule or group of molecules.
2.
But, in dense media with closely packed molecules, the total field acting on the
molecule, the molecular field ‫ܧ‬ሬԦ௠ , is given by the sum of the average macroscopic
electric field ‫ܧ‬ሬԦ and the internal field ‫ܧ‬ሬԦ௜ at any molecule:
‫ܧ‬ሬԦ௠ = ‫ܧ‬ሬԦ + ‫ܧ‬ሬԦ௜
A.
The macroscopic electric field ‫ܧ‬ሬԦ ൫= ‫ܧ‬ሬԦ୫ୟୡ୰୭ ൯ is exactly what we discussed earlier.
B.
The internal field consists of the contribution of the near molecules to the target
molecule ‫ܧ‬ሬԦ௡௘௔௥ (‫ܧ‬ሬԦ ᇱ in Reitz) and that of the polarization charge on the surface
enclosing the molecule ‫ܧ‬ሬԦ௦ :
‫ܧ‬ሬԦ௜ = ‫ܧ‬ሬԦ௡௘௔௥ + ‫ܧ‬ሬԦ௦
3.
Contribution of near molecules
A.
‫ܧ‬ሬԦ௡௘௔௥ is difficult to be determined exactly but depends on the symmetry of the
problem. As an example, in a simple cubic lattice of the identical dipoles, ‫ܧ‬ሬԦ௡௘௔௥
vanishes as
ଶ
‫ܧ‬ሬԦ௡௘௔௥ =
3‫ݔ‬Ԧ௜௝௞ ൫‫݌‬Ԧ ‫ݔ ڄ‬Ԧ௜௝௞ ൯ െ ห‫ݔ‬Ԧ௜௝௞ ห ‫݌‬Ԧ
1
෍
=0
ହ
4ߨߝ଴
ห‫ݔ‬Ԧ െ ‫ݔ‬Ԧ ห
௜,௝,௞
B.
௜௝௞
‫ܧ‬ሬԦ௡௘௔௥ = 0 for a highly symmetric situation and for completely random situations.
For lattices other than simple cubic or highly-symmetric lattices, it is a good
working assumption that ‫ܧ‬ሬԦ௡௘௔௥ ؆ 0 for most materials.
4.
Contribution of the polarization charge
A.
Let us consider a cavity with a volume, macroscopically small but microscopically
large, that encloses the molecule (group) entirely. For example, a spherical cavity
is presented in the figure below.
B.
The contribution of the external system (e.g. the external charges ߪୣ୶୲ in the
planar electrodes in the figure above) and the screening by the induced
polarization charge density ߪ௉ is involved in ‫ܧ‬ሬԦ୫ୟୡ୰୭ = ‫ܧ‬ሬԦ .
C.
The contribution of the dielectric outside the cavity can be replaced by the
polarization charge density ߪ௦ .
D.
Although the exact evaluation of the contribution of ߪ௦ requires advanced
discussions beyond the scope of this class, the consideration of a spherical volume
would be effective for an isotropic, homogeneous dielectric as shown below.
E.
Using the spherical coordinates and taking the polar direction along the direction
of ܲሬԦ, the polarization charge density becomes ߪ௦ = െܲ cos ߠ. Note that the surface
charge density originates from the polarization outside the cavity and thus the
minus sign appears.
F.
The electric field by the polarization charge density is given by
‫ܧ‬ሬԦ௦ =
5.
ଶగ ାଵ
1
െܲ cos ߠ
1
െܲ cos ߠ ଶ
1
ර
‫ݎ‬
Ԧ݀ܽ
=
න
න
‫ݎ‬Ԧ ‫(݀ ݎ‬cos ߠ)݀߶ =
ܲሬԦ
ଷ
ଷ
4ߨߝ଴ ௌ
‫ݎ‬
4ߨߝ଴ ଴ ିଵ
‫ݎ‬
3ߝ଴
So far, we derived the elements of the molecular field in a dielectric with classical and
fairly reasonable approximations. In fact, ‫ܧ‬ሬԦ௦ depends on the shape of molecule or
group in the media rather than te sphere, which can be handled by quantum mechanics.
- Relation between the polarizability and susceptibility: Clausius-Mossotti equation
1.
Taking account of the internal field, we can determine the polarizability of molecule
ߙ௠ as
‫݌ۃ‬Ԧ௠ ‫ߝ = ۄ‬଴ ߙ௠ ‫ܧ‬ሬԦ௠ = ߝ଴ ߙ௠ ൫‫ܧ‬ሬԦ + ‫ܧ‬ሬԦ௜ ൯
2.
In principle, ߛ௠௢௟ is a function of the electric field but for a wide range of field strengths
is a constant characterizing the response of the molecule to an applied field.
3.
Solving for ܲሬԦ in terms of ‫ܧ‬ሬԦ , we find that in linear materials,
1
ܲሬԦ = ܰ‫݌ۃ‬Ԧ௠ ‫ߝܰ = ۄ‬଴ ߙ௠ ൫‫ܧ‬ሬԦ + ‫ܧ‬ሬԦ௜ ൯ = ܰߙ௠ ൬ߝ଴ ‫ܧ‬ሬԦ + ܲሬԦ൰
3
ܰߙ௠
ܰߙ௠
֜ ܲሬԦ = ߝ଴
‫ܧ‬ሬԦ ֜ ߯௘ =
1
1
1 െ ܰߙ௠
1 െ ܰߙ௠
3
3
4.
Since the dielectric constant is ߝ/ߝ଴ = 1 + ߯௘ ,
ܰߙ௠ = 3 ൬
5.
ߝ/ߝ଴ െ 1
൰
ߝ/ߝ଴ + 2
This is the Clausius-Mossotti equation, which relates (or bridges) the microscopic ߙ௠
and the macroscopic ߝ.
- Some classical models for the molecular polarizability
1.
2.
The polarization of a collection of atoms or molecules can arise in two ways:
A.
Applied field Æ distortion of charge distribution Æ induced dipole moment
B.
Applied field Æ line up the initially randomly oriented permanent dipole moments
Semiclassical model for the induced dipole moment
A.
The behavior of bound electrons can be effectively described by the spring model:
݀ ଶ ‫ݔ‬Ԧ
݀‫ݔ‬Ԧ
݁‫ܧ‬ሬԦ
ଶ
+
ߛ
+
߱
‫ݔ‬
Ԧ
=
଴
݉௘
݀‫ ݐ‬ଶ
݀‫ݐ‬
B.
The frequency-dependent induced electric dipole moment is given as
݁ ଶ /݉௘
݁ ଶ /݉௘
1
‫݌‬Ԧ = െ݁‫ݔ‬Ԧ = ଶ
‫ܧ‬ሬԦ ֜ ߙ௠ =
ଶ
ଶ
ߝ଴ ߱଴ െ ߱ ଶ + ݅߱ߛ
߱଴ െ ߱ + ݅߱ߛ
C.
If there are a set of charges ݁௝ with masses ݉௝ , the linear polarizability becomes
݁௝ଶ /݉௝
1
ߙ௠ = ෍ ଶ
ଶ
ߝ଴
௝ ߱଴௝ െ ߱ + ݅߱ߛ௝
D.
For electrostatics, ߱ = 0,
ߙ௠ =
݁௝ଶ /݉௝
1
෍
ଶ
ߝ଴
௝ ߱଴௝
3.
Statistical physics model for thermal dynamic effects
A.
In classical statistical mechanics, the probability distributions of particles is given
by the Hamiltonian as
ି
ு
݂(‫ ݁ = )ܪ‬௞ಳ்
B.
Harmonically bound charge (nonpolar medium) under an applied field ‫ܧ‬ሬԦ = ‫ݖܧ‬Ƹ
1
݉
|‫݌‬Ԧ|ଶ + ߱௢ଶ |‫ݔ‬Ԧ|ଶ െ ݁‫ܧ‬z
‫=ܪ‬
2݉
2
֜ ‫݌ۃ‬௠ ‫= ۄ‬
C.
݁ଶ
‫ ݀ ׬‬ଷ ‫݌‬Ԧ ‫ ݀ ׬‬ଷ ‫ݔ‬Ԧ(݁‫)ܪ(݂)ݖ‬
=
‫ܧ‬
݉߱଴ଶ
‫ ݀ ׬‬ଷ ‫݌‬Ԧ ‫ ݀ ׬‬ଷ ‫ݔ‬Ԧ݂(‫)ܪ‬
֜
ߙ௠ =
݁ଶ
= constant
݉ߝ଴ ߱଴ଶ
Contribution of partial orientation of random permanent dipoles ‫݌‬Ԧ଴ (polar
medium)
‫=ܪ‬
‫݌ۃ‬௠ ‫= ۄ‬
1
݉
|‫݌‬Ԧ|ଶ + ߱௢ଶ |‫ݔ‬Ԧ|ଶ െ ‫݌‬Ԧ଴ ‫ܧ ڄ‬ሬԦ = ‫ܪ‬଴ െ ‫݌‬Ԧ଴ ‫ܧ ڄ‬ሬԦ
2݉
2
‫݀ ׬‬ȳ‫݌‬௢ cos ߠ exp(‫݌‬௢ ‫ ܧ‬cos ߠ /݇஻ ܶ) 1 ‫݌‬଴ଶ
؆
‫ܧ‬
3 ݇஻ ܶ
‫݀ ׬‬ȳ exp(‫݌‬௢ ‫ ܧ‬cos ߠ /݇஻ ܶ)
֜
ߙ௠ ‫ן‬
1
ܶ
NOTE) The topics of point charge in a dielectric fluid (Sections 4-6 and 4-9 of Reitz) and
ferroelectricity (Section 5-4 of Reitz) will be skipped. The topics of induced dipoles as a semi-classical
model (Section 5-2 of Reitz) will be covered again in the EM-II class next semester together with
the topics of dispersion in media (Chapter 19 of Reitz).
(HW #5)
Due: 20240422
1. Prove the following relation ship between the polarization and the polarization charge densities
for a dielectric specimen of volume ܸ and surface ܵ:
න ܲሬԦ ݀‫ = ݒ‬න ߩ௉ ‫ݎ‬Ԧ݀‫ ݒ‬+ ර ߪ௉ ‫ݎ‬Ԧ݀ܽ
௏
௏
ௌ
Here, ‫ݎ‬Ԧ = ‫ݔ‬ଓƸ + ‫ݕ‬ଔƸ + ‫݇ݖ‬෠ is the position vector from any fixed origin. (Hint: you expand ‫ ڄ ׏‬൫‫ܲݔ‬ሬԦ൯
according to ‫ ڄ ׏‬൫݂‫ܨ‬Ԧ ൯ = ݂‫ܨ ڄ ׏‬Ԧ + ‫ܨ‬Ԧ ‫݂׏ ڄ‬.)
2. Consider an interface between two dielectrics of ߝଵ and ߝଶ without any excess charge. Let ߠଵ
(ߠଶ ) be the angle between the electric field in the dielectric #1 (#2) and the normal vector at the
interface. Show that
ߝଵିଵ tan ߠଵ = ߝଶିଵ tan ߠଶ
3. A plane slab of material with dielectric constant ߝଵ is bounded on both sides by material with
dielectric constant ߝଶ . The electric field in medium 2, ‫ܧ‬ሬԦଶ , is given to be uniform and perpendicular
to the boundaries. Find (a) the electric field in medium 1, ‫ܧ‬ሬԦଵ , (b) the polarization ܲሬԦ, and (c) the
polarization surface charge density ߪ௉ at the boundaries in terms of ߝଵ , ߝଶ , and ‫ܧ‬ሬԦଶ .
4. Consider a system consisting of two dielectric media with a point charge in medium 1. There is
no excess charges in the system. (You may refer to the lecture note.)
(a) Using the method of image, show that the electrostatic potential in the region of the dielectric
#1 is given as
߮ଵ =
1
‫ݍ‬
ߝଵ െ ߝଶ
‫ݍ‬
ቈ
+
቉
4ߨߝଵ ඥ(‫ ݔ‬+ ݀)ଶ + ‫ ݕ‬ଶ + ‫ ݖ‬ଶ ߝଵ + ߝଶ ඥ(‫ ݔ‬െ ݀)ଶ + ‫ ݕ‬ଶ + ‫ ݖ‬ଶ
(b) Show that the potential in the dielectric #2 is given as
1
‫ݍ‬
߮ଶ =
2ߨ(ߝଵ + ߝଶ ) ඥ(‫ ݔ‬+ ݀)ଶ + ‫ ݕ‬ଶ + ‫ ݖ‬ଶ
(Optional) We can also apply the discussions and results given in the lecture to permanently
polarized materials. Consider a sphere of radius ܽ that has a permanent polarization ܲሬԦ inside; the
direction and magnitude of ܲሬԦ are uniform. The permanently polarized sphere indeed gives rise to
an electric field.
(a) Show that the electric field inside the sphere is
‫ܧ‬ሬԦ = െ
1
ܲሬԦ
3ߝ଴
Inside the sphere, the electric field, which is in the opposite direction to the polarization, is called a
depolarizing field.
(b) Show that the electric field inside the sphere is
‫ܧ‬ሬԦ =
ܴ ଷ 3൫‫ݎ‬Ƹ ‫ܲ ڄ‬ሬԦ൯‫ݎ‬Ƹ െ ܲሬԦ
3ߝ଴
‫ݎ‬ଷ
Here, ‫ݎ‬Ԧ = ‫ݎݎ‬Ƹ is the position vector from the center of the sphere.
(Hint: Since ‫ܲ ڄ ׏‬ሬԦ vanishes at all points, the electrostatic potential satisfies Laplace’s equation both
inside and outside the sphere. But, do not assume that the material is characterized by a dielectric
constant or use the boundary condition associated with such a dielectric constant. Just think about
the source of the electric field.)
PH231 Electromagnetism I (Spring 2024)
Lecture note 06: Electrostatic Energy and Capacitor
(Chapter 6 of the textbook by Reitz, Sections 1.5, 1.6, 1.15, and 3.5~3.7 of the book by Purcell)
6.1 Introduction
- Many problems in physics are greatly simplified by means of energy consideration. In general, the
energy of a system is divided into its potential and kinetic contributions. Under static conditions,
however, the entire energy of a system of charges exists as potential energy that arises from
electrical interaction of the charges, the so-called electrostatic energy.
- In contrast to macroscopic mechanical systems, a desk, a bus, a ball, etc., we will see that
electrostatic energy can be represented as a function of position. This property raises an interesting
question: Can the electrostatic energy can be stored locally? The concept of the energy density
and the principle of the local conservation of energy are very essential in physics.
6.2 Electrostatic (potential) energy of a group of charges and a charge distribution
(Sections 6-1 and 6-2 of Reitz)
- Work-energy principle in electrostatics
1.
Let us consider to move a point charge ‫ ݍ‬from position ‫ ܣ‬to position ‫ ܤ‬under a given
electrostatic field ‫ܧ‬ሬԦ = െ‫߮׏‬. The work done by the electric field is then
஻
஻
஺
஺
ܹ = න ‫ܨ‬Ԧ ‫݈݀ ڄ‬Ԧ = െ‫ ݍ‬න ‫݈݀ ڄ ߮׏‬Ԧ = െ‫ݍ‬൫߮(‫ )ܤ‬െ ߮(‫)ܣ‬൯
2.
For statics, it is necessary that the total work is zero and the kinetic energy does not
change: The charge does not accelerate and the net force should be zero. This
condition requires an equal and opposite force balanced at each point by the
electrostatic force, which exerts work as
ܹ = ‫ݍ‬൫߮(‫ )ܤ‬െ ߮(‫)ܣ‬൯
3.
This result is equal to the increase in electrostatic energy of the charge over the path
integral from position ‫ ܣ‬to position ‫ܤ‬.
- Potential energy of a group of discrete charges (microscopic view point)
1.
Let us now consider an electrostatic system of ݉ point charges. The system can be
constructed by sequentially placing each point charge in its position.
2.
It is obvious that the work to bring the first charge ‫ݍ‬ଵ at its position ‫ݎ‬Ԧଵ without any
work: ܹଵ = 0.
3.
Placing the second charge ‫ݍ‬ଶ at the position ‫ݎ‬Ԧଶ requires work as
‫ݍ‬ଵ ‫ݍ‬ଶ
1
ܹଶ =
4ߨߝ଴ |‫ݎ‬Ԧଵ െ ‫ݎ‬Ԧଶ |
4.
If the ݅th point charge ‫ݍ‬௜ is brought from infinity to a point ‫ݎ‬Ԧ௜ , the work done on the
charge is given as
௜ିଵ
‫ݍ‬௝
‫ݍ‬௜
ܹ௜ = ‫ݍ‬௜ ߮(‫ݎ‬Ԧ௜ ) =
෍
4ߨߝ଴
ห‫ݎ‬Ԧ௜ െ ‫ݎ‬Ԧ௝ ห
௝ୀଵ
5.
The total potential energy of all the charges, the number of which is ݊, due to all the
forces acting between them is
௠
௠ ௜ିଵ
௜ୀଵ
௜ୀଵ ௝ୀଵ
‫ݍ‬௜ ‫ݍ‬௝
1
ܷ = ܹ = ෍ ܹ௜ =
෍෍
4ߨߝ଴
ห‫ݎ‬Ԧ௜ െ ‫ݎ‬Ԧ௝ ห
6.
A more symmetric form can be written dividing by a factor of 2 as
௠
ܷ=
௠
‫ݍ‬௜ ‫ݍ‬௝
1
෍෍
8ߨߝ଴
ห‫ݎ‬Ԧ௜ െ ‫ݎ‬Ԧ௝ ห
… (1)
௜ୀଵ ௝ୀଵ
௝ஷ௜
7.
Note that the terms with ݅ = ݆, which are actually diverging, are omitted.
8.
The work can be written in a different way by using the final value of the potential at
the ݆th point charge due to the other charges:
௠
‫ݍ‬௜ ‫ݍ‬௝
1
߮௝ =
෍
4ߨߝ଴
ห‫ݎ‬Ԧ௜ െ ‫ݎ‬Ԧ௝ ห
୧ୀଵ
௜ஷ௝
9.
௠
1
֜ ܷ = ෍ ‫ݍ‬௝ ߮௝ … (2)
2
௝ୀଵ
If the point charges had been assembled in a linear dielectric medium of infinite extent,
the permittivity ߝ would replace ߝ଴ in Eq. (1), but Eq. (2) would remain unchanged. In
fat, Eq. (2) has rather general validity applicable to a group of point charges in multiple
dielectric media and even conductors of finite size, which will be shown in the following
discussions. The only limitation on the validity is that all dielectrics be linear.
- Potential energy of a charge distribution (macroscopic view point / linear dielectrics + conductors)
1.
Now, we shall discuss the electrostatic energy of an arbitrary charge distribution in a
liner dielectric system involving some conductors.
2.
Considering that the dielectrics of the system are linear, we may evaluate the work,
equivalent to the stored electrostatic energy, to charge the system in its final state.
A.
It is obvious that the work to bring the charge increment ߜ‫ ݍ‬from a reference
potential ߮஺ = 0 to the potential at a particular point in the system, ߮ ᇱ (‫ݎ‬Ԧ) is
ߜܹ = ߮ ᇱ (‫ݎ‬Ԧ)ߜ‫ݍ‬
B.
Here, the charges can be bound inside the volume or on the surface in terms of
the volume and surface densities: ߜ‫ߩߜ = ݍ‬ȟ‫ ݒ‬or ߜ‫ߪߜ = ݍ‬ȟܽ.
C.
The continuous charging process can be modelled by a fraction factor 0 ൑ ߙ ൑ 1.
For the final state of ߩ(‫ݎ‬Ԧ) and ߪ(‫ݎ‬Ԧ), the charge increment can be represented as
ߜߩ = ߜ൫ߙߩ(‫ݎ‬Ԧ)൯ = ߩ(‫ݎ‬Ԧ)ߜߙ and ߜߪ = ߜ൫ߙߪ(‫ݎ‬Ԧ)൯ = ߪ(‫ݎ‬Ԧ)ߜߙ
D.
Letting ߮ ᇱ (ߙ; ‫ݎ‬Ԧ) be the potential at ‫ݎ‬Ԧ at a particular stage of the charging process,
the total electrostatic energy is
ଵ
ଵ
ܷ = න ݀ߙ න ߩ(‫ݎ‬Ԧ)߮ ᇱ (ߙ; ‫ݎ‬Ԧ)݀‫ ݒ‬+ න ݀ߙ න ߪ(‫ݎ‬Ԧ)߮ ᇱ (ߙ; ‫ݎ‬Ԧ)݀ܽ
଴
E.
௏
଴
ௌ
For a linear system, the potential builds up linearly in proportion to the stage of
the charging process: ߮ ᇱ (ߙ; ‫ݎ‬Ԧ) = ߙ߮(‫ݎ‬Ԧ).
F.
Finally, we have
ଵ
ଵ
ܷ = න ߙ݀ߙ න ߩ(‫ݎ‬Ԧ)߮(‫ݎ‬Ԧ)݀‫ ݒ‬+ න ߙ݀ߙ න ߪ(‫ݎ‬Ԧ)߮(‫ݎ‬Ԧ)݀ܽ
଴
௏
଴
ௌ
1
1
= න ߩ(‫ݎ‬Ԧ)߮(‫ݎ‬Ԧ)݀‫ ݒ‬+ න ߪ(‫ݎ‬Ԧ)߮(‫ݎ‬Ԧ)݀ܽ
2 ௏
2 ௌ
G.
This equation gives the desired result for the energy of a charge diction, in
analogue to the result in the microscopic view point.
3.
Although the result above covers the cases conductors are present in the system, it is
convenient to separate out the contribution of the conductors out. A conductor is an
equipotential region and neutral (no net charge) inside. The integration over the surface
(ܵ௝ ) of the ݆th conductor with a potential and total charge of ߮௝ and ܳ௝ is
1
1
1
න ߪ(‫ݎ‬Ԧ)߮(‫ݎ‬Ԧ)݀ܽ = න ߪ(‫ݎ‬Ԧ)݀ܽ ߮௝ = ܳ௝ ߮௝
2 ௌೕ
2 ௌೕ
2
4.
Where ܵ ᇱ is the nonconducting surfaces of the system, the electrostatic energy of a
charge distribution in a linear system including conductors becomes
1
1
1
ܷ = න ߩ(‫ݎ‬Ԧ)߮(‫ݎ‬Ԧ)݀‫ ݒ‬+ න ߪ(‫ݎ‬Ԧ)߮(‫ݎ‬Ԧ)݀ܽ + ෍ ܳ௝ ߮௝
2 ௏
2 ௌᇲ
2
௝
- A comparison between the cases of a group of point charges and a charge distribution
1.
If all space is filled with a single dielectric medium of ߝ, the potential is given by
1
ߩ(‫ݎ‬Ԧ ᇱ )
1
ߪ(‫ݎ‬Ԧ ᇱ )
ᇱ
߮(‫ݎ‬Ԧ) =
න
݀‫ݒ‬
+
න
݀ܽᇱ
4ߨߝ ௏ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
4ߨߝ ௌ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
2.
For a group of point charges, the charge densities should be
௠
ߩ(‫ݎ‬Ԧ) = ෍ ‫ݍ‬௜ ߜ(‫ݎ‬Ԧ െ ‫ݎ‬Ԧ௜ )
௜ୀଵ
௠
and ߩ(‫ݎ‬Ԧ ᇱ ) = ෍ ‫ݍ‬௝ ߜ൫‫ݎ‬Ԧ ᇱ െ ‫ݎ‬Ԧ௝ ൯
௝ୀଵ
௝ஷ௜
3.
If the term ݆ = ݅ is included, the integral diverges as
‫ݍ‬௜ଶ
ߜ(‫ݎ‬Ԧ െ ‫ݎ‬Ԧ௜ )ߜ(‫ݎ‬Ԧ ᇱ െ ‫ݎ‬Ԧ௜ ) ᇱ
‫ݍ‬௜ଶ
ߜ(‫ݎ‬Ԧ െ ‫ݎ‬Ԧ௜ )
න න
݀‫ݒ‬
݀‫ݒ‬
=
න
݀‫ = ݒ‬λ
ᇱ
|
|‫ݎ‬
8ߨߝ ௏ ௏
8ߨߝ ௏ ห‫ݎ‬Ԧ െ ‫ݎ‬Ԧ௜ ௜ ห
Ԧ െ ‫ݎ‬Ԧ
4.
In contrast, when ߩ is a continuous distribution, the vanishing of the denominator
does not make the integral diverge, and it is unnecessary to exclude the point ‫ݎ‬Ԧ ᇱ = ‫ݎ‬Ԧ.
6.3 Energy density of an electrostatic fields (Section 6-3 of Reitz)
- Electrostatic energy density in the free space and hint for the local energy conservation principle
1.
For a continuous charge distribution in the free space, the potential energy takes the
form:
ܷ=
1
ߩ(‫ݎ‬Ԧ)ߩ(‫ݎ‬Ԧ ᇱ ) ᇱ
1
ඵ
݀‫ = ݒ݀ ݒ‬න
ߩ(‫ݎ‬Ԧ)߶(‫ݎ‬Ԧ)݀‫ݒ‬
ୟ୪୪
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
8ߨߝ଴
2 ୟ୪୪
ୱ୮ୟୡୣ
2.
ୱ୮ୟୡୣ
Now, we may consider an interesting question: Where is the electrostatic energy located?
Or what carries the electrostatic energy? These questions may not make sense because
we really know only that the total energy of a closed system is conserved. The idea
that the energy is located somewhere is not necessary.
3.
4.
However, the integral enables us to define the electrostatic energy density given as
1
‫ݎ(ߩ = ݑ‬Ԧ)߮(‫ݎ‬Ԧ)
2
Using the Poisson equation in the free space, integration by parts, and divergence
theorem, we can also obtain an alternative form in terms of the electric field as
ܷ=െ
ߝ଴
න
߮(‫ݎ‬Ԧ)‫׏‬ଶ ߮(‫ݎ‬Ԧ)݀‫ݒ‬
2 ୟ୪୪
ୱ୮ୟୡୣ
ߝ଴
ߝ଴
‫ ݒ݀)߮׏߮( ڄ ׏‬+ න
‫ݒ݀߮׏ ڄ ߮׏‬
=െ න
2 ୟ୪୪
2 ୟ୪୪
ୱ୮ୟୡୣ
5.
ୱ୮ୟୡୣ
ߝ଴
1
ߝ଴
ଶ
|‫|߮׏‬ଶ ݀‫ = ݒ‬න
ߝ ห‫ܧ‬ሬԦ ห ݀‫ݒ‬
= െ ර ߮‫ ܽ݀߮׏‬+ න
2 ோ՜ஶ
2 ୟ୪୪
2 ୟ୪୪ ଴
ୱ୮ୟୡୣ
ୱ୮ୟୡୣ
1
ଶ
‫ߝ = ݑ ׵‬଴ ห‫ܧ‬ሬԦ ห
2
Note that, when we are very far away from all charges, ߶ and ‫ ߶׏‬vary as 1/ܴ and
1/ܴଶ , respectively. Since the surface area of the large sphere increases as ܴଶ , we see
that the surface integral falls off as 1/ܴ and finally goes to zero as ܴ ՜ λ.
6.
We might then extend our principle of the conservation of energy with the idea that if
the energy in a given volume changes, we should be able to account for the change
by the flow of energy into or out of that volume. We might call it a principle of the
local conservation of energy.
7.
For electrostatics, where Coulomb’s law is valid, no flow of energy exists and ‫= ݑ‬
ଵ
ଶ
ଵ
ଶ
ߩ(‫ݔ‬Ԧ)߶(‫ݔ‬Ԧ) is true. However, it is false in general. On the other hand, ‫ߝ = ݑ‬଴ ห‫ܧ‬ሬԦ ห (in
ଵ
ሬԦ) is always true in the free space.
fact, ‫ܧ = ݑ‬ሬԦ ‫ܦ ڄ‬
ଶ
ଶ
8.
The further details of the local electromagnetic energy density and its flow will be
discussed in the Poynting theorem (Chapter 16 of Reitz).
- Self-energy contribution
1.
ଶ
Considering the integrand ߝ଴ ห‫ܧ‬ሬԦ ห , the volume integral for the potential energy is
necessarily non-negative.
2.
This seems to contradict the result that the potential energy of two charges of opposite
sign is negative.
3.
It is due to the “self-energy”. The electric field by two point charges is
1 ‫ݍ‬ଵ (‫ݎ‬Ԧ െ ‫ݎ‬Ԧଵ )
1 ‫ݍ‬ଶ (‫ݎ‬Ԧ െ ‫ݎ‬Ԧଶ )
‫ܧ‬ሬԦ =
+
ଷ
4ߨߝ଴ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧଵ |
4ߨߝ଴ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧଶ |ଷ
4.
The energy density is then
‫=ݑ‬
5.
1
‫ݍ‬ଵଶ
‫ݍ‬ଶଶ
‫ݍ‬ଵ ‫ݍ‬ଶ (‫ݎ‬Ԧ െ ‫ݎ‬Ԧଵ ) ‫ݎ( ڄ‬Ԧ െ ‫ݎ‬Ԧଶ )
ቆ
+
+2
ቇ
ଶ
ସ
ସ
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧଶ |
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧଵ |ଷ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧଶ |ଷ
32ߨ ߝ଴ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧଵ |
Clearly, the first two terms are self-energy contributions, which are actually omitted in
the double sum. And the second term is the term of the interaction potential energy.
6.
The total interaction potential energy of two point charges, ܷ୧୬୲ , is indeed given as
(‫ݎ‬Ԧ െ ‫ݎ‬Ԧଵ ) ‫ݎ( ڄ‬Ԧ െ ‫ݎ‬Ԧଶ )
‫ݍ‬ଵ ‫ݍ‬ଶ
‫ݍ‬ଵ ‫ݍ‬ଶ
1
ܷ୧୬୲ =
න
݀‫= ݒ‬
ଶ
ଷ
ଷ
16ߨ ߝ଴ ୟ୪୪ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧଵ | |‫ݎ‬Ԧ െ ‫ݎ‬Ԧଶ |
4ߨߝ଴ |‫ݎ‬ሬሬሬԦଵ െ ‫ݎ‬Ԧଶ |
ୱ୮ୟୡୣ
- Consideration on the energy of a point charge
1.
Our new relation of the equation above says that even a single point charge ‫ ݍ‬will
have some electrostatic energy via electric fields. The energy density at the distance ‫ݎ‬
from the charge is
‫ݍ‬
߳଴ ‫ ܧ‬ଶ
‫ݍ‬ଶ
֜
=
ଶ
2
32ߨ ଶ ߳଴ ‫ ݎ‬ସ
4ߨ߳଴ ‫ݎ‬
Taking for an element of volume a spherical shell of thickness ݀‫ ݎ‬and area 4ߨ‫ ݎ‬ଶ , the
‫=ܧ‬
2.
total energy is
௥ୀஶ
‫ݍ‬ଶ
‫ݍ‬ଶ 1
ଶ
ܷ=න
4ߨ‫ݎ‬
݀‫ݎ‬
=
െ
ቤ
ଶ
ସ
8ߨ߳଴ ‫ ݎ‬௥ୀ଴
଴ 32ߨ ߳଴ ‫ݎ‬
ஶ
3.
Now the limit at ‫ = ݎ‬λ gives no difficulty. But for a point charge we are supposed to
integrate down to ‫ = ݎ‬0, which gives an infinite integral, as we have already discussed.
4.
Here, we must conclude that the idea of locating the energy in the field is inconsistent
with the assumption of the existence of point charges. One way out of the difficulty
would be to say that elementary charges, such as an electron, are not points but are
really small distributions of charge.
5.
Alternatively, we could say that there is something wrong in our theory of electricity at
very small distances, or with the idea of the local conservation of energy.
6.
These difficulties have never been overcome; they exist to this day.
- Electrostatic energy density in linear dielectrics (macroscopic view point)
1.
We shall consider the case of charging a system of linear dielectrics, which means that
excess charges are brought in from a reference potential ߮஺ = 0.
2.
The overall process is almost identical to the free space case. As we have discussed
earlier, the work to build up an excess charge density ߩ and ߪ at the final state ߮ is
ܹ=
3.
1
1
න ߩ(‫ݎ‬Ԧ)߮(‫ݎ‬Ԧ)݀‫ ݒ‬+ න ߪ(‫ݎ‬Ԧ)߮(‫ݎ‬Ԧ)݀ܽ = ܷ
2 ௏
2 ௌ
The volume ܸ includes the dielectrics of the system, and the surface ܵ consists of the
surfaces of all conductors in the system and the surface bounds the system from the
outside.
4.
ሬԦ = ߩ, and its boundary condition give
The macroscopic Gauss law, ‫ܦ ڄ ׏‬
1
1
ሬԦ݀‫ ݒ‬+ න ߮‫ܦ‬
ሬԦ ‫݊ ڄ‬ො݀ܽ
ܷ = න ߮‫ܦ ڄ ׏‬
2 ௏
2 ௌ
5.
Here, ݊ො is the normal into the dielectrics, so into ܸ . There is no electric field,
ሬԦ ‫݊ ڄ‬ො = ߪ.
polarization, or displacement inside conductors and out of the system: ‫ܦ‬
6.
ሬԦ = ‫ ڄ ׏‬൫߮‫ܦ‬
ሬԦ൯ െ ‫ܦ‬
ሬԦ ‫ ߮׏ ڄ‬, and the mathematical Gauss
Using the vector identity, ߮‫ܦ ڄ ׏‬
theorem, we have
7.
1
1
1
ሬԦ൯݀‫ ݒ‬െ න ‫ܦ‬
ሬԦ ‫ ݒ݀߮׏ ڄ‬+ න ߮‫ܦ‬
ሬԦ ‫݊ ڄ‬ො݀ܽ
ܷ = න ‫ ڄ ׏‬൫߮‫ܦ‬
2 ௏
2 ௏
2 ௌ
1
1
1
1
ሬԦ ‫݊ ڄ‬ො݀ܽ െ න ‫ܦ‬
ሬԦ ‫ ݒ݀߮׏ ڄ‬+ න ߮‫ܦ‬
ሬԦ ‫݊ ڄ‬ො݀ܽ = െ න ‫ܦ‬
ሬԦ ‫ݒ݀߮׏ ڄ‬
= െ න ߮‫ܦ‬
2 ௌ
2 ௏
2 ௌ
2 ௏
Finally, using ‫ܧ‬ሬԦ = െ‫߮׏‬, we have
1
ሬԦ ‫ܧ ڄ‬ሬԦ ݀‫ݒ‬
ܷ=െ න ‫ܦ‬
2 ௏
8.
9.
We are led to the concept of energy density:
1
ሬԦ ‫ܧ ڄ‬ሬԦ
‫ܦ =ݑ‬
2
For isotropic dielectrics,
1
1 ‫ܦ‬ଶ
‫ ܧߝ = ݑ‬ଶ =
2
2 ߝ
- The electrostatic energy of an ionic crystal (Refer to Section II-8-3, Lectures on Physics by Feynman)
1.
At the discussion from the macroscopic point of view, we only consider the work
necessary to charge the given system of (dielectric) media.
2.
Here is a question. When the macroscopic electric field in a linear dielectric is zero, the
electrostatic energy density becomes zero. However, the dielectric medium consists of
charged particles that are densely packed. Where is the electrostatic energy of the
interactions of the charged particles?
3.
In typical condensed media, such as ionic crystals, the microscopic charges are very
strongly bounded. The electrostatic energy required to bind them together does not
change, unless their molecular/atomic arrangement does not change.
4.
The electrostatic energy to construct the medium is just a reference value that does
not need to be added to the macroscopic electrostatic energy. Remind that the effects
of the distortion of the microscopic charge distribution induced by ‫ܧ‬ሬԦ , such as the
ሬԦ.
polarization, are included throughout the displacement ‫ܦ‬
5.
Nevertheless, we are often interested in the energy differences between one atomic
arrangement and another, as, for example, the energy of a chemical change.
6.
Let's consider, for example, the electrostatic energy of an ionic lattice. An ionic crystal
like NaCI consists of positive and negative ions which can be thought of as rigid spheres.
They attract electrically until they begin to touch; then there is a repulsive force which
goes up very rapidly if we try to push them closer together.
7.
The following picture shows a cross-sectional view of a NaCl crystal. The spacing of the
ions is 2.81 Å (= 2.81 × 10ିଵ଴ ݉).
8.
Then, how much energy will it take to pull all these ions apart that is, to separate the
crystal completely into ions? This energy should be equal to the energy to evaporate
NaCI plus the energy required to dissociate the molecules into ions. This total energy
to separate NaCI to ions is determined experimentally to be 7.92 ܸ݁ per molecule.
9.
According to our theory, the work to pull apart the crystal is the sum of the electrostatic
potential energies of all the pairs of ions.
10. Let us consider the electrostatic energy step by step.
A.
B.
C.
D.
The energy of an ion with one of its nearest neighbors is
1 ݁ଶ
= 5.12 ܸ݁
4ߨߝ଴ ܽ
If summing all the terms from the ions along a straight line, we have
1 ݁ଶ
2 2 2 2
1 ݁ଶ
1 1 1
൬െ + െ + + ‫ ڮ‬൰ = െ
൬1 െ + െ + ‫ ڮ‬൰
4ߨߝ଴ ܽ
1 2 3 4
2ߨߝ଴ ܽ
2 3 4
1 ݁ଶ
=െ
ln 2 = െ7.09 ܸ݁
2ߨߝ଴ ܽ
Now consider the next adjacent line of ions above. There are four such lines.
1 ݁ଶ
1
2
2
2
4×
൬െ +
െ
+
+‫ڮ‬൰
4ߨߝ଴ ܽ
1 ξ2 ξ5 ξ10
If you work patiently through for all the lines, and then take the sum, you find
that the grand total is
1 ݁ଶ
= 8.94 ܸ݁
4ߨߝ଴ ܽ
This answer is about 10% above the experimentally observed energy.
ܷ = 1.746 ×
E.
11. It shows that our idea that the whole lattice is held together by electrical Coulomb
forces is fundamentally correct; the major contribution to the energy of a crystal like
NaCl is electrostatic.
12. The subject that tries to understand the behavior of bulk matter in terms of the laws
of atomic behavior is called solid-state physics.
6.4 Capacitor (Sections 3-11, 6-4, 6-5, and 6-6 of Reitz / Sections 3.5 and 3.6 of E. M. Purcell)
- Capacitance of a conductor
1.
An isolated conductor carrying a charge ܳ has a certain potential ߮଴ with zero
potential at infinity.
2.
If ܳ = 0 (so that ߪ = 0), the normal component of the electric field on the surface of
the conductor should be zero as ‫ܧ‬௡ = െ
డఝ
ቚ =
డ௡ ௌ
ఙ
ఌబ
= 0, which gives the trivial solution,
߮ = 0 everywhere.
3.
Then, considering that the Laplace equation is linear, the solution should be
proportional to ܳ and vice versa.
ܳ = ‫߮ܥ‬଴
4.
We call the factor ‫ ܥ‬the capacitance of that conductor.
5.
It is obvious that the constant of proportionality is determined by the boundary
condition, and considering the linearity, it depends only on the geometry of the
boundary, the size and shape of the conductor.
- Potentials and charges of a system of charged conductors
1.
As a more general problem, suppose there are ܰ conductors in fixed geometry
surrounded by a zero-potential shell.
2.
At first, let us consider the case of three conductors as below.
A.
In tackling this problem, about all we can use is the uniqueness theorem and the
superposition principle of the solution to the Laplace equation.
B.
The uniqueness theorem guarantees that, with ߮ଵ , ߮ଶ , and ߮ଷ are given as the
boundary condition, the electric fields and the charges ܳଵ , ܳଶ , and ܳଷ on the
individual conductors are uniquely determined.
C.
Superposition principle
i.
Let us consider a possible state of the system, in which ߮ଶ , and ߮ଷ are zero
(State I in the figure). The conductors 2 and 3 are connected to the zeropotential shell by very thin wires that any charge residing on them is negligible.
ii.
In such a state, if ߮ଵ is doubled, the electric potential, field, and charges are
doubled.
iii.
Thus, with ߮ଶ = ߮ଷ = 0, the charges must be proportional to ߮ଵ :
State I: ܳଵ = ‫ܥ‬ଵଵ ߮ଵ ;
iv.
v.
ܳଶ = ‫ܥ‬ଶଵ ߮ଵ ;
ܳଷ = ‫ܥ‬ଷଵ ߮ଵ
In similar, for the states of ߮ଵ = ߮ଷ = 0 (State II) and ߮ଵ = ߮ଶ = 0 (State III),
State II: ܳଵ = ‫ܥ‬ଵଶ ߮ଶ ;
ܳଶ = ‫ܥ‬ଶଶ ߮ଶ ;
ܳଷ = ‫ܥ‬ଷଶ ߮ଶ
State III: ܳଵ = ‫ܥ‬ଵଷ ߮ଷ ;
ܳଶ = ‫ܥ‬ଶଷ ߮ଷ ;
ܳଷ = ‫ܥ‬ଷଷ ߮ଷ
It is straightforward that the superposition of the three states is the state that
we now address. Thanks to the linearity of the Laplace equation, we
‫ܥ‬ଵଵ ‫ܥ‬ଵଶ ‫ܥ‬ଵଷ ߮ଵ
ܳଵ
൭ܳଶ ൱ = ൭‫ܥ‬ଶଵ ‫ܥ‬ଶଶ ‫ܥ‬ଶଷ ൱ ൭߮ଶ ൱
ܳଷ
‫ܥ‬ଷଵ ‫ܥ‬ଷଶ ‫ܥ‬ଷଷ ߮ଷ
D.
The C’s are called the coefficient of capacitance.
E.
There is an equivalent set of linear relations of the following form. The P’s are
called the potential coefficient.
߮ଵ
‫ܥ‬ଵଵ ‫ܥ‬ଵଶ ‫ܥ‬ଵଷ ିଵ ܳଵ
ܲଵଵ
߮
൭ ଶ ൱ = ൭‫ܥ‬ଶଵ ‫ܥ‬ଶଶ ‫ܥ‬ଶଷ ൱ ൭ܳଶ ൱ = ൭ܲଵଶ
߮ଷ
ܳଷ
ܲଵଷ
‫ܥ‬ଷଵ ‫ܥ‬ଷଶ ‫ܥ‬ଷଷ
3.
ܲଶଵ
ܲଶଶ
ܲଶଷ
ܲଷଵ ܳଵ
ܲଷଶ ൱ ൭ܳଶ ൱
ܲଷଷ ܳଷ
For a general system of ܰ conductors,
ே
ே
ܳ௜ = ෍ ‫ܥ‬௜௝ ߮௝
฻
߮௜ = ෍ ܲ௜௝ ܳ௝
௝ୀଵ
௝ୀଵ
- Symmetry of capacitance and potential coefficients
1.
As we have discussed in Section 6.2, the electrostatic energy of a ݅th conductor with a
potential and total charge of ߮௝ and ܳ௝ is
1
1
1
ܷ௜ = න ߪ(‫ݎ‬Ԧ)߮(‫ݎ‬Ԧ)݀ܽ = න ߪ(‫ݎ‬Ԧ)݀ܽ ߮௜ = ܳ௜ ߮௜
2 ௌ೔
2 ௌ೔
2
2.
ே
1
֜ ܷ = ෍ ܳ௜ ߮௜
2
௜ୀଵ
Then, combining the equation of the potential coefficients, we have
ே
ே
ே
ே
1
1
ܷ = ෍ ෍ ܲ௜௝ ܳ௜ ܳ௝ = ෍ ෍ ܿ௜௝ ߮௜ ߮௝
2
2
௜ୀଵ ௝ୀଵ
௜ୀଵ ௝ୀଵ
3.
Thus, the energy is a quadratic function of the charges on the various conductors.
4.
There are three important general statements for the coefficient ܲ௜௝ :
5.
A.
ܲ௜௝ = ܲ௜௝
B.
All of the ܲ௜௝ are positive.
C.
ܲ௜௜ ൒ ܲ௜௝
Proof of the first statement
A.
B.
Expressing ܷ as ܷ(ܳଵ … ܳ௡ ), the chain rule gives
߲ܷ
߲ܷ
ܷ݀ = ൬
൰ ݀ܳଵ + ‫ ڮ‬+ ൬
൰ ݀ܳே
߲ܳଵ
߲ܳே
If ܳ௜ only is changed, from the quadratic equation, we have
ே
߲ܷ
1
ܷ݀ = ൬
൰ ݀ܳ௜ = ෍൫ܲ௜௝ + ܲ௝௜ ൯ܳ௝ ݀ܳ௜
߲ܳ௜
2
௝ୀଵ
C.
Now, let us directly consider the work to bring ݀ܳ௜ at ߮௜ :
ே
ܷ݀ = ܹ݀ = ߮௜ ݀ܳ௜ = ෍ ܲ௜௝ ܳ௝ ݀ܳ௜
௝ୀଵ
D.
E.
6.
Finally, we prove that
1
൫ܲ + ܲ௝௜ ൯ = ܲ௜௝ ֜
ܲ௜௝ = ܲ௝௜
2 ௜௝
డ
ଵ
(ܳ௜ଶ ) = ܲ௜௜ ܳ௜ ݀ܳ௜ and ܴ‫ܲ = ܵܪ‬௜௜ ܳ௜ ݀ܳ௜ : it is trivial.
Note) for ݅ = ݆, ‫ܲ = ܵܪܮ‬௜௜
ଶ
డொ೔
Proof of the second and third statements (H.W. See the page 149-150 of Reitz.)
- Capacitors
1.
A practical form of the system of conductors is a capacitor: Two conductors that can
store equal and opposite charges (±ܳ), with a potential difference between them
independent of whether other conductors in the system are charged.
2.
The independence of other charges implies that one of the pair of conductors is
shielded by the others: the ideal capacitor.
3.
If two conductors, 1 and 2, form a capacitor and are charged by ܳ and െܳ, we can
write
߮ଵ = ܲଵଵ ܳ െ ܲଵଶ ܳ + ߮௫
߮ଶ = ܲଵଶ ܳ െ ܲଶଶ ܳ + ߮௫
4.
Here, ߮௫ is the common potential contributed by other charges/systems.
5.
The potential difference is then
ȟ߮ = ߮ଵ െ ߮ଶ = (ܲଵଵ + ܲଶଶ െ 2ܲଵଶ )ܳ
6.
Thus, the difference in potential between the conductors of a capacitor is proportional
to the charge stored and the coefficient ‫ܲ( = ܥ‬ଵଵ + ܲଶଶ െ 2ܲଵଶ )ିଵ is called the
capacitance of the capacitor.
7.
Examples of capacitors are in the general physics level (skipped).
6.5 Electrostatic forces and torques (Section 6-7 of Reitz)
- For the isolated (Q is fixed) system
1.
Thus far, we have developed a number of alternative procedures for calculating the
electrostatic energy of a charge system.
2.
Let us suppose we are dealing with an isolated system composed of a number of parts,
and we will allow a small displacement ݀‫ݎ‬Ԧ influenced by the electrical force ‫ܨ‬Ԧ :
ܹ݀ = ‫ܨ‬Ԧ ‫ݎ݀ ڄ‬Ԧ
3.
In case, ‫ܨ‬Ԧ is a conservative force and the system is isolated,
ܹ݀ = െܷ݀
4.
and
‫ܨ‬Ԧ = െ(‫)ܷ׏‬ொ
In similar, for small displacement ݀ߠ௜ along an axis ଓƸ, the electrical torque ߬Ԧ is
߲ܷ
߬௜ = െ ൬ ൰
߲ߠ௜ ொ
- For the system maintained at fixed potentials
1.
The earlier results limited to isolated systems do not cover all case of interest.
2.
If the potentials of conductors are maintained by external systems (external batteries),
the work becomes
ܹ݀ = ܹ݀௕ െ ܷ݀
3.
Here, ܹ݀௕ is the work supplied by the batteries and necessary to be eliminated from
the equation.
4.
Fixing the electrostatic potentials, the electrostatic energy of a system of charged
conductors is given as
ே
1
ܷ = ෍ ܳ௜ ߮௜
2
௜ୀଵ
5.
ே
ே
ே
௜ୀଵ
௜ୀଵ
௜ୀଵ
1
1
1
֜ ܷ݀ = ෍ ܳ௜ ݀߮௜ + ෍ ߮௜ ݀ܳ௜ = ෍ ߮௜ ݀ܳ௜
2
2
2
Furthermore, the work supplied by the batteries is the work required to mode each of
charge increment ݀ܳ௜ from the zero-potential:
ே
ܹ݀௕ = ෍ ߮௜ ݀ܳ௜ = 2ܷ݀
௜ୀଵ
6.
Finally, we have
ܷ݀ = ‫ܨ‬Ԧ ‫ݎ݀ ڄ‬Ԧ
7.
֜
In similar fashion,
߬௜ = ൬
8.
‫ܨ‬Ԧ = (‫)ܷ׏‬ఝ
߲ܷ
൰
߲ߠ௜ ఝ
Here, the subscript ߮ denotes the fact all potentials are maintained constant during
the virtual displacement ݀‫ݎ‬Ԧ or ݀ߠ௜ .
- Note for the Maxwell stress tensor (to be discussed later)
1.
In fact, the force and torque in the electromagnetism should be considered in the point
of view of the conservation/change of linear and angular momentums.
2.
The rigorous consideration should be given in a more rigorous fashion, which is the
Poynting theorem and Maxwell stress tensor.
3.
We shall encounter and discuss the topics in the late this semester.
(HW #6)
Due: 20240508
1. Show that the total interaction potential energy of two point charges, ܷ୧୬୲ , is indeed given as
(‫ݎ‬Ԧ െ ‫ݎ‬Ԧଵ ) ‫ݎ( ڄ‬Ԧ െ ‫ݎ‬Ԧଶ )
‫ݍ‬ଵ ‫ݍ‬ଶ
‫ݍ‬ଵ ‫ݍ‬ଶ
1
ܷ୧୬୲ =
න
݀‫= ݒ‬
16ߨ ଶ ߝ଴ ୟ୪୪ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧଵ |ଷ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧଶ |ଷ
4ߨߝ଴ |‫ݎ‬ሬሬሬԦଵ െ ‫ݎ‬Ԧଶ |
ୱ୮ୟୡୣ
‫ݍ‬ଵ and ‫ݍ‬ଶ are located at the positions ‫ݎ‬Ԧଵ and ‫ݎ‬Ԧଶ , respectively.
2. Show/discuss that the following statements for the potential coefficients of a system of charged
conductors:
(a) All of the ܲ௜௝ are positive.
(b) ܲ௜௜ ൒ ܲ௜௝
3. two concentric, spherical, conducting shell of radii ‫ݎ‬ଵ and ‫ݎ‬ଶ are maintained at potentials ߮ଵ
and ߮ଶ , respectively. the region between the shells is filled with a dielectric medium. Show by direct
calculation that the energy stored in the dielectric is equal to ‫߮(ܥ‬ଵ െ ߮ଶ )ଶ /2, there by determine ‫ܥ‬,
the capacitance of the system.
4. A parallel-plate capacitor of plate separation ݀ has the region between its plates filled by a block
of solid dielectric of permittivity ߝ . Dimensions of each plate are length ݈ and width ‫ ݓ‬. The
dielectric block is withdrawn along the ݈ dimension until only the length ‫ ݔ‬remains between the
plates.
(a) If the plates are maintained at the constant potential difference ȟ߮, what is the force exerted
on the plate?
(b) If the charges on the plates are fixed (isolated) at +ܳ and െܳ, respectively, what is the force
exerted on the plate?
(Optional) Maximum energy storage between spheres. We want to design a spherical vacuum
capacitor, with a given radius ܽ for the outer spherical shell, that will be able to store the greatest
amount of electrical energy subject to the constraint that the electric field strength at the surface
of the inner sphere may not exceed ‫ܧ‬଴ .
(a) Show that when the radius ܾ of the inner spherical conductor is ܾ = 3ܽ/4, the stored energy
can be maximized.
(b) Find the capacitance of the system in (a).
(c) Show that the maximum energy is given as
ܷ୫ୟ୶ =
27
ߨߝ ܽଷ ‫ܧ‬଴ଶ
128 ଴
PH231 Electromagnetism I (Spring 2024)
Lecture note 07: Electric Current
(Chapter 7 of the textbook by Reitz, Chapter 4 of the book by Purcell)
7.1 Introduction
- So far, we have dealt with charges at rest. Moving charge constitutes a current, and the process
whereby charge is transported is called conduction. To be precise, the current ‫ ܫ‬is defined as the
rate at which net charge is transported through a given surface in a conducting system (e.g.,
through a given cross section of a wire). The SI unit of current is then the coulomb/second, which
is called ampere (A): 1 A = 1 CΤ1 sec. In this chapter and the following chapters on magnetostatics,
we shall focus mainly on charges in uniform motion: steady current.
7.2 Nature of the current (Section 7-1 of Reitz and Section 4.4 of Purcell)
- In nature, there are various media to support an electric current including not only the conventional
conductors, such as metals and alloys, but also semiconductors, electrolytes, ionized gases,
imperfect dielectrics, and even vacuums in the vicinity of a thermionic emitting cathode. In many
conductors, the charge carriers are electrons, and in other cases, the charge may be carried by
positive or negative ions.
- Metal
1.
In a metal, current is carried entirely by electrons, while the heavy positive ions are
fixed at regular positions in the crystal structure. In fact, the conduction electrons
originate from the electronic band structure of the crystal structure of metals, which is
a topic of solid-state physics.
2.
The valence electrons do not participate in the conduction process.
3.
Under steady-state conditions, electrons may be fed into the metal at on point/site
and removed at another, producing a current, but the metal as a whole is
electrostatically neutral.
- Electrolyte
1.
In an electrolyte, the current is carried by both positive and negative ions, cations and
anions. However, since some ions move faster than others, conduction by one type of
ion usually predominates.
2.
The net charge transported through a given surface, depending on both the sign of
the charge carrier and the direction in which it is moving, determine the current.
- Semiconductor
1.
In a semiconductor, such as the silicon crystal (see the figure below), there are two
mobile charges.
2.
One is the free electron that occupies the quantum state called the conduction band
and move like a conduction electron in a metal.
3.
The positive charge left behind is also mobile. We can think of it as an electron missing
in the bond between atoms A and B in the figure above and call it the hole. The hole
among the valence electrons could be transferred to the bond between B and C, thence
to the bond between C and D, and so on, and occupies the quantum state called the
valence band with a positive charge.
4.
At ܶ = 0, as shown in the figure (a) below, all of the valence states are occupied and
none of the conduction band state is: zero conduction and no current.
5.
At high temperature, the probability that electron states will be occupied is given by
the exponential factor ݁ ି୼ா/௞ಳ ் , where ݇஻ and ȟ‫ ܧ‬are the Boltzmann constant and
the energy band gap, respectively.
6.
In addition, if a semiconductor is doped, electrons or holes from the dopants (impurity
atoms) cause a high conductivity. For example, the silicon crystal doped by aluminum
(phosphorus) becomes a p-type (n-type) semiconductor, whose majority carriers are
holes (electrons).
7.
More details are the subjects of semiconductor physics (a suggested reference:
Semiconductor Physics, An Introduction by Karlheinz Seeger).
- Gas discharge
1.
In a gas discharge, the current is carried by both electrons and positive ions. However,
since the electrons are so much more mobile than the heavy ions, practically all of the
current is carried by electrons.
2.
Gas discharge conduction is somewhat complicated, depending on the experimental
conditions, such as the electron and ion populations (pressure), the speed of the charge
carriers (temperature), collisions, and so on, which is dealt in plasma physics.
- Thermionic emitting cathode
1.
Even in the vacuum, current can exist. An instructive example of a stationary current
distribution occurs in the plane diode, a two=electrode vacuum tube, as shown in the
figure below.
2.
One electrode, the cathode, is coated with a material that emits electrons when heated,
and the other is a simple metal. Electrons emerge from the hot, negatively charged
cathode with very low velocities and then are accelerated toward the positively charged
anode by the electric field between cathode and anode.
- Conduction current vs convection current
1.
Conduction currents represent the drift motion of charge carriers in a neutral medium.
2.
Although in this chapter we will focus only on the conduction currents, it is worth
noting that there is another type of current, convention currents. Such currents arise
from mass transport of a charged medium.
3.
The motion of electrons in a vacuum tube of thermionic emitting cathode is a
convention current. A characteristic feature of the convection current is that it is not
electrostatically neutral but it is often considered under the electrostatic condition.
4.
Convection currents are important to the subject of atmospheric electricity (the
dynamics of stellar and planetary atmospheres) and astronomical charged objects (the
interactions within ionized gases in space).
7.3 Current density and the Equation of continuity
(Section 7-2 of Reitz and Sections 4.1 and 4.2 of Purcell)
- Current density
1.
Let us first consider a conduction medium with only one type of charge carrier of ‫ݍ‬.
The number of these carriers per unit volume is ܰ.
2.
We here ignore the random thermal motion of the charge carriers and assign the same
drift velocity ‫ݒ‬Ԧ to each carrier, as shown in the figure below.
3.
During the time ߜ‫ݐ‬, each carrier moves a distance ‫ݒ‬Ԧߜ‫ ݐ‬and thus the current through
an area of ݊ො݀ܽ is
ߜܳ ‫ݒܰݍ‬Ԧ ‫݊ ڄ‬ොߜ‫ܽ݀ݐ‬
=
= ܰ‫ݒݍ‬Ԧ ‫݊ ڄ‬ො݀ܽ
ߜ‫ݐ‬
ߜ‫ݐ‬
In general, if there is more than one kind of charge carrier present, the current thought
݀‫= ܫ‬
4.
the area is obviously
݀‫ = ܫ‬൥෍ ܰ௜ ‫ݍ‬௜ ‫ݒ‬Ԧ௜ ൩ ‫݊ ڄ‬ො݀ܽ
௜
5.
The quantity in brackets is a vector that has dimensions of current per unit area, which
is called the current density ‫ܬ‬Ԧ:
‫ܬ‬Ԧ = ෍ ܰ௜ ‫ݍ‬௜ ‫ݒ‬Ԧ௜
֜
݀‫ܬ = ܫ‬Ԧ ‫݊ ڄ‬ො݀ܽ
௜
6.
For an arbitrary shaped surface area ܵ of macroscopic size, the current is given by
‫ = ܫ‬න ‫ܬ‬Ԧ ‫݊ ڄ‬ො݀ܽ
ௌ
- Equation of continuity: local conservation of charge
1.
The current density ‫ܬ‬Ԧ and the charge density ߩ are not independent quantities.
2.
The electric current entering an arbitrary volume ܸ enclosed by a closed surface ܵ is
given as (using the divergence theorem)
‫ = ܫ‬െ ර ‫ܬ‬Ԧ ‫݊ ڄ‬ො݀ܽ = െ න ‫ܬ ڄ ׏‬Ԧ݀‫ݒ‬
ௌ
3.
௏
The minus sign comes about because ݊ො is the outward normal, and we wish to call ‫ܫ‬
positive when the net flow of charge is from the outside of ܸ to within. So, the rate at
which charge is transported into ܸ is
‫=ܫ‬
݀ܳ
݀
߲ߩ
= න ߩ݀‫ = ݒ‬න
݀‫ݒ‬
݀‫ ݐ݀ ݐ‬௏
௏ ߲‫ݐ‬
߲ߩ
֜ න ൬ + ‫ܬ ڄ ׏‬Ԧ൰ ݀‫ = ݒ‬0
߲‫ݐ‬
௏
4.
ܸ is completely arbitrary and the integral holds for an arbitrary volume segment of
5.
the medium. Therefore, the integrand must vanish at each point:
߲ߩ
+ ‫ܬ ڄ ׏‬Ԧ = 0
߲‫ݐ‬
This is called the equation of continuity and presents the local conservation of charge.
- Derivation of the charge conservation from Maxwell’s equations and a relevant note
1.
Although we will deal with the time-dependent Maxwell’s equations later, it is worth
seeing that they already involve the charge conservation.
A.
First, the time derivative of Gauss’ law:
߲ߩ
߲
߲‫ܧ‬ሬԦ
= ൫ߝ଴ ‫ܧ ڄ ׏‬ሬԦ ൯ = ߝ଴ ‫ڄ ׏‬
߲‫ݐ‬
߲‫ݐ߲ ݐ‬
B.
2.
ଵ డாሬԦ
ሬԦ െ మ
Second, Maxwell-Ampere’s law, ‫ܤ × ׏‬
௖ డ௧
= ߤ଴ ‫ܬ‬Ԧ, gives
߲ߩ
ሬԦ െ ߤ଴ ‫ܬ‬Ԧ൯
= ߝ଴ ܿ ଶ ‫ ڄ ׏‬൫‫ܤ × ׏‬
߲‫ݐ‬
C. Using the vector identity ‫ ڄ ׏‬൫‫ܣ × ׏‬Ԧ൯ = 0 and ܿ ଶ = (ߝ଴ ߤ଴ )ିଵ , we finally have
߲ߩ
+ ‫ܬ ڄ ׏‬Ԧ = 0
߲‫ݐ‬
The value of zero in RHS is an invariant scalar. So, we may consider that the inner
product of two 4-vector quantities is an invariant scalar physical quantity as follows.
A.
4-vector current density
‫ܬ‬ఈ ‫ ؠ‬൫ܿߩ, ‫ܬ‬Ԧ൯
B.
4-vector differential operator
C.
߲
߲
, െ‫׏‬൰ and ߲ఈ ‫ ؠ‬൬
, +‫׏‬൰
߲ܿ‫ݐ‬
߲ܿ‫ݐ‬
The inner product of the 4-vector differential operator and current density
߲ఈ ‫ ؠ‬൬
provides the continuity equation as
߲ߩ
߲ఈ ‫ܬ‬ఈ =
+ ‫ܬ ڄ ׏‬Ԧ = 0
߲‫ݐ‬
7.4 Electrical conductivity and Ohm’s law (Section 7-3 of Reitz and Section 4.3 of Purcell)
- Phenomenological approach
1.
It is found experimentally that in a metal at constant temperature the current density
‫ܬ‬Ԧ is approximately linear to the electric field as
‫ܬ‬Ԧ = ߪ‫ܧ‬ሬԦ
2.
Here, the constant of proportionality ߪ is called the conductivity. (In Reitz, ߪ is
symbolled by ݃ in order not to confuse with the surface charge density.)
3.
The constitutive equation is called Ohm’s law, which is a very good approximation for
a large number of the common conducting materials.
4.
In the general isotropic case,
‫ܬ‬Ԧ = ߪ(‫ܧ)ܧ‬ሬԦ
5.
The most general linear relation would be expressed by a tensor as
ߪ௫௫ ߪ௫௬ ߪ௫௭
‫ܧ‬௫
‫ܬ‬௫
ߪ
ߪ
ߪ
‫ܬ‬
‫ܧ‬
௬௬
௬௭ ൱ ቌ ௬ ቍ
ቌ ௬ ቍ = ൭ ௬௫
ߪ௭௫ ߪ௭௬ ߪ௭௭
‫ܬ‬௭
‫ܧ‬௭
- Resistance
1.
The reciprocal of the conductivity is called the resistivity ߩ. (In Reitz, ߪ is symbolled by
2.
ߟ in order not to confuse with the volume charge density.)
1
ߩ=
ߪ
Consider a conduction specimen obeying Ohm’s law in the shape of a wire as shown
in the figure below.
3.
The wire is assumed to be homogenous and characterized by the constant conductivity.
4.
An electric field that exists in the wire under these conditions has a relation as
ܸ ‫ ؠ‬ȟ߮ = න ‫ܧ‬ሬԦ ‫݈݀ ڄ‬Ԧ
5.
Since the metal is statically neutral, it is evident that there can be the steady-state
electric field should be longitudinal to the wire. Otherwise, the side surface of the metal
wire will be charging.
6.
But, the longitudinal electric field implies a current density as ‫ܬ‬Ԧ = ߪ‫ܧ‬ሬԦ , so that the
current through any cross section of the wire is
‫ = ܫ‬න ‫ܬ‬Ԧ ‫݊ ڄ‬ො݀ܽ = න ߪ‫ܧ‬ሬԦ ‫݊ ڄ‬ො݀ܽ
஺
஺
7.
If the cross-section of the wire is constant as ‫ܣ‬, we have
ܸ = ‫ ݈ܧ‬and ‫ܣܧߪ = ܫ‬
8.
Finally, we can define a quantity, called as the resistance, as
ܸ
‫݈ܧ‬
݈
ܴ= =
=
‫ܣߪ ܣܧߪ ܫ‬
The unit of the resistance is 1 ‫( ݄݉݋‬ȳ) = 1 VΤ1 A
9.
7.5 Steady currents and Kirchhoff’s law (Sections 7-4 and 7-5of Reitz and Section 4.2 of Purcell)
- Statistic condition
1.
If the local charge density ߩ does not change in time (steady state), ߲ߩ/߲‫ = ݐ‬0, we
have
‫ܬ ڄ ׏‬Ԧ = 0
2.
When we consider a homogeneous, ohmic (linear), conducting medium under the
steady-state condition, we obtain
‫ ڄ ׏‬൫ߪ‫ܧ‬ሬԦ ൯ = 0
֜
‫ܧ ڄ ׏‬ሬԦ = 0
3.
Since ‫ܧ × ׏‬ሬԦ = 0 for a static field, we still have
‫ܧ‬ሬԦ = െ‫׏ ֜ ߮׏‬ଶ ߮ = 0
4.
The boundary conditions for such steady-state current conditions across the interface
between two conducting media are
฻
ߪଵ ‫ܧ‬ଵ௡ = ߪଶ ‫ܧ‬ଶ௡
‫ܬ = ܬ‬ଶ௡
൜ ଵ௡
‫ܧ‬ଵ௧ = ‫ܧ‬ଶ௧
5.
The boundary conditions from the longitudinal continuity of the steady-state current
density and the tangential continuity of the electric field by ‫ܧ × ׏‬ሬԦ = 0.
6.
In fact, the curl-free electric field in the steady-state condition gives one of Kirchhoff’s
law: the algebraic sum of the voltage differences around any loop of the electric circuit
is zero.
‫ܧ × ׏‬ሬԦ = 0
֜
ර ‫ܧ‬ሬԦ ‫݈݀ ڄ‬Ԧ = 0 ֜
෍ ȟ߮௜ = ෍ ܸ௜ = 0
௜
஼
7.
௜
Also, the divergence-less charge density gives the other one of Kirchhoff’s law: the
algebraic sum of the currents flowing toward a branch point is zero.
‫ܬ ڄ ׏‬Ԧ = 0
֜
ර ‫ܬ‬Ԧ ‫݊ ڄ‬ො݀ܽ = 0 ֜
ௌ
෍ ‫ܫ‬௜ = 0
௜
- Relation to the magnetostatics
1.
A current wire or loop supporting a constant current is a good example of the
divergence-less current density, ‫ܬ ڄ ׏‬Ԧ = 0, for the steady-state condition.
2.
For steady-state magnetic phenomena (magnetostatics), the electric field
is
independent on time and Ampere’s law, the steady-state version of Maxwell-Ampere’s
law, reads
ሬԦ = ߤ଴ ‫ܬ‬Ԧ
‫ܤ×׏‬
3.
Noting that the divergence of the curl of any continuously twice-differentiable vector
4.
field is always zero, we again obtain
1
1
ሬԦ൰ = ‫ ڄ ׏‬൫‫ܤ × ׏‬
ሬԦ൯ = 0
‫ܬ ڄ ׏‬Ԧ = ‫ ڄ ׏‬൬ ‫ܤ × ׏‬
ߤ଴
ߤ଴
Thus, although we will see the topics in magnetostatics from the following chapter, the
classical magnetostatics is mainly based on for the localized, divergence-less current
distribution:
7.6 Microscopic theory of conduction (Section 7-7 of Reitz and Section 4.4 of Purcell)
- Semi-classical model
1.
As we have already seen in the lecture note 5, the behavior of charges ‫ ݍ‬in molecules
can be effectively described by the spring model:
݀ ଶ ‫ݔ‬Ԧ
݀‫ݔ‬Ԧ
‫ܧݍ‬ሬԦ
ଶ
+
ߛ
+
߱
‫ݔ‬
Ԧ
=
଴
݉௘
݀‫ ݐ‬ଶ
݀‫ݐ‬
2.
For conductors featured by conducting (free) carriers with an effective mass of ݉௘ , we
do not need to consider the term of the restoring force ߱଴ଶ ‫ݔ‬Ԧ, so that
݀ ଶ ‫ݔ‬Ԧ
݀‫ݔ‬Ԧ ‫ܧݍ‬ሬԦ
+
ߛ
=
݀‫ ݐ‬ଶ
݀‫݉ ݐ‬௘
ௗ మ ௫Ԧ
3.
For the steady-state current condition,
4.
Then, we finally have
5.
ܰ‫ ݍ‬ଶ
ܰ‫ ݍ‬ଶ
‫ܧ‬ሬԦ ֜ ߪ =
ߛ݉௘
ߛ݉௘
Here, ܰ is the density of the single kind of charge carrier.
6.
This is the Drude model of electrical conduction that was proposed in 1900 by the
ௗ௧ మ
= 0.
‫ܬ‬Ԧ = ܰ‫ݒݍ‬Ԧ =
German physicist, Paul Karl Ludwig Drude (1863-1906).
7.
However, the model should be corrected with the consideration of the temperature,
which is the topic of the statistical physics.
(Note: Sections 7-5 of the textbook by Reitz discuss the time scale of the approach to electrostatic
equilibrium. This topic is too minor and will be skipped in the lecture).
(HW #7)
Due: 20240515
1. Two infinite, plane, parallel plates of metal are separated by the distance ݀. The space between
the plates is filled with two conducting media, the interface between the media being a plane that
is parallel to the metal plates. The first medium (conductivity ߪଵ , permittivity ߝଵ ) is of thickness ܽ,
and the second (conductivity ߪଶ , permittivity ߝଶ ) is of thickness ݀ െ ܽ . The metal plates are
maintained at potentials ߮ଵ and ߮ଶ , respectively, in the steady state.
(a) show that the potential of the interface separating the two media is
߮୧୬୲ =
߮ଵ ߪଵ (݀ െ ܽ) + ߮ଶ ߪଶ ܽ
ߪଶ ܽ + ߪଵ (݀ െ ܽ)
(b) Show that the surface density of charge on the interface is
ߪ୧୬୲ =
(ߪଵ ߝଶ െ ߪଶ ߝଵ )(߮ଵ െ ߮ଶ )
ߪଶ ܽ + ߪଵ (݀ െ ܽ)
2. Two long cylindrical shells of metal (radii ‫ݎ‬ଵ and ‫ݎ‬ଶ , with ‫ݎ‬ଶ > ‫ݎ‬ଵ ) are arranged coaxially. The
plates are maintained at the potential difference ȟ߮. The region between the shells is filled with a
medium of conductivity ߪ and permittivity ߝ.
(a) Show that the electric current between unit lengths of the shells is
‫=ܫ‬
2ߨߪȟ߮
ln(‫ݎ‬ଶ /‫ݎ‬ଵ )
(b) The capacitance of the system may be computed from the definition ‫ܳ = ܥ‬/ȟ߮. Show for this
system that the product of resistance per unit length and capacitance per unit length is ߝ/ߪ.
3. A system of charges and currents is completely contained inside the fixed volume ܸ. As we have
studied earlier, the dipole moment of the charge-current distribution is defined by
‫݌‬Ԧ = න ‫ݎ‬Ԧߩ(‫ݎ‬Ԧ)݀‫ݒ‬
௏
Here, ‫ݎ‬Ԧ is the position vector from a certain fixed origin. Show that
݀‫݌‬Ԧ
= න ‫ܬ‬Ԧ(‫ݎ‬Ԧ)݀‫ݒ‬
݀‫ݐ‬
௏
This result will be importantly revisited in the topics of the radiation of electromagnetic waves
PH231 Electromagnetism I (Spring 2024)
Lecture note 08: Magnetostatics of Steady Currents
(Chapter 8 of the textbook by Reitz, Chapters 6 and 11 of the book by Purcell)
8.1 Introduction
- The second kind of field in electricity and magnetism is the magnetic field. The effects of the
magnetic fields have been known since ancient times. The effects of permanent magnets such as
magnetite (Fe3O4) were first observed. However, in fact, the magnetic fields of such materials
originate from the spin of the electron, which cannot be explained by classical electromagnetic
theory. It is worth noting that the orbital angular momentum ‫ܮ‬ሬԦ = ‫ݎ‬Ԧ × ‫݌‬Ԧ depends on the reference
frame but the spin angular momentum ܵԦ is the invariant, fundamental angular moment in nature.
- In 1820, Hans Christian Ørsted (1777-1851), a Danish physicist, discovered that an electric current
produces a magnetic field: a compass needle was affected by a nearby electric current. This work,
together with the later work of Gauss, Henry, Faraday, and others, brought the magnetic field into
ሬԦ field), oersted unit
prominence as a partner to the electric field. The unit of the magnetic field (‫ܪ‬
(Oe), was named by Ørsted. Then the theoretical work of Maxwell and others, discussed in Lectures
10 (Chapter 11 of Reitz) and 12 (Chapter 16 of Reitz), showed that the partnership was real and
that the electric and magnetic fields were inextricably intertwined.
- In fact, the magnetic field is a necessary condition when the special relativity, one of the pivotal
results of Maxwell’s equations, is combined with the theory of electricity and the conservation of
charge. In this light, in some textbook, the topic of moving charges is covered before the topic of
magnetism: Purcell’s book covers the field of moving charges and the magnetic field in chapters
5 and 6, respectively. In this lecture, we will follow the conventional order, and the relativistic
effects will be covered next semester.
ሬሬԦ field) and force
8.2 Magnetic induction (࡮
(Sections 8-1, 8-2, and 8-9 of Reitz / Sections 6.1 and 11.2 of Purcell)
- The definition of magnetic induction: the Lorentz force
1.
In experiments, it is observed that a charge moving parallel to a current of other
charges experiences a force perpendicular to its own velocity.
2.
Just as we defined the electric field ‫ܧ‬ሬԦ as the vector force on unit charge at rest, so we
ሬԦ by the velocity-dependent part of the force, called the
can define another field ‫ܤ‬
Lorentz force, that acts on a charge in motion:
ሬԦ
‫ܨ‬Ԧ = ‫ܧݍ‬ሬԦ + ‫ݒݍ‬Ԧ × ‫ܤ‬
3.
or
‫ܨ‬Ԧ
௤՜଴ ‫ݍ‬
ሬԦ = lim
‫ܧ‬ሬԦ + ‫ݒ‬Ԧ × ‫ܤ‬
In fact, the velocity-dependent force is consistent with – indeed, is required by –
Coulomb’s law combined with charge invariance and special relativity. When taking the
relativistic expression of the momentum, ‫݌‬Ԧ =
௠బ ௩ሬԦ
ඥଵି௩ మ /௖ మ
= ߛ݉଴ ‫ݒ‬Ԧ, the Lorentz force law is
precisely given as
݀‫݌‬Ԧ ݀(ߛ݉଴ ‫ݒ‬Ԧ)
݀
݉଴ ‫ݎ‬Ԧሶ
ሬԦ
=
= ቆ
ቇ = ‫ܧݍ‬ሬԦ + ‫ݎݍ‬Ԧሶ × ‫ܤ‬
݀‫ݐ‬
݀‫ݐ‬
݀‫ ݐ‬ඥ1 െ ‫ݎ‬ሶ ଶ /ܿ ଶ
4.
As mentioned earlier, the special relativity in classical electromagnetism will be
discussed in detail next semester.
- Basic element of the magnetic field
1.
The magnetic field can be created by moving charges – currents – and magnetic spins.
2.
The magnetic field is divergence-free not only for the classical but also for the quantum
theory of electromagnetism (Section 8-9 of Reitz. Section 11.2 of Purcell).
ሬԦ = 0
‫ܤڄ׏‬
3.
This magnetic Gauss law obviously shows that there is no magnetic monopole.
4.
Thus, the basic entity in magnetic studies is magnetic dipole ݉
ሬሬԦ.
A.
In classical electromagnetism, an infinitesimally small current loop of charge
currents acts as an ideal magnetic dipole (Amperian loop model), which will be
discussed later in this lecture note.
B.
The magnetic field due to the spins of elementary particles can be treated
phenomenologically by the classical theory: e.g. permanent magnets.
C.
In the presence of magnetic materials, the dipole ݉
ሬሬԦ tends to align itself in the
ሬԦ by the mechanical torque ܰ
ሬԦ = ݉
ሬԦ.
ሬሬԦ × ‫ܤ‬
direction of magnetic-flux density ‫ܤ‬
- Magnetic force between two moving charges (Nonrelativistic approximation)
1.
A moving charge is a good example of the classical sources of the magnetic field/force.
2.
In the previous lecture, the Coulomb force on a charge ‫ ݍ‬located at ‫ݎ‬Ԧ due to a charge
‫ݍ‬ଵ at the origin was given by
‫ܨ‬Ԧ௘ =
3.
1 ‫ݍݍ‬ଵ ‫ݎ‬Ԧ
4ߨߝ଴ ‫ ݎ‬ଶ ‫ݎ‬
If the charges are moving with constant velocities ‫ݒ‬Ԧ and ‫ݒ‬Ԧଵ , respectively, an additional
magnetic force ‫ܨ‬Ԧ௠ is exerted on ‫ ݍ‬by ‫ݍ‬ଵ as
ߤ଴ ‫ݍݍ‬ଵ
‫ݎ‬Ԧ
‫ܨ‬Ԧ௠ =
‫ݒ‬Ԧ × ቆ‫ݒ‬Ԧଵ × ቇ
ଶ
4ߨ ‫ݎ‬
‫ݎ‬
4.
The magnetic induction is then determined from its definition in the Lorentz force law:
ߤ ‫ݍ‬
‫ݎ‬Ԧ
ሬԦ = ଴ ଵ ቆ‫ݒ‬Ԧଵ × ቇ
‫ܤ‬
4ߨ ‫ ݎ‬ଶ
‫ݎ‬
5.
Of course, all three of the above equations are only first approximations to the correct
relativistic expressions, which will be discussed next semester. If you are interested in
the correct results/expressions, you can refer to chapter 21 of Reitz, chapter 5 of Purcell,
and section 6.5 of Jackson.
- Forces between two current-carrying conducting wires
1.
The use of point charges in real experiments is often wasteful and inefficient.
2.
The magnetic force and field exerted by one current-carrying conductor on another
and vice versa would be examined more conveniently.
3.
Now, let us consider one of the simplest cases: two parallel conducting wires carrying
currents ‫ܫ‬ଵ and ‫ܫ‬ଶ , respectively.
4.
As we will see in the discussion of Biot-Savart’s law and Ampere’s circuital law in
magnetostatics, the current in wire 1 causes a magnetic field of strength ‫ܤ‬ଵ =
ఓబ ூభ
ଶగ௥
at
the location of wire 2.
5.
Within wire 2, there are ݊ଶ moving charges per meter length of wire, each with charges
‫ݍ‬ଶ and speed ‫ݒ‬ଶ , resulting in the current
‫ܫ‬ଶ = ݊ଶ ‫ݍ‬ଶ ‫ݒ‬ଶ
6.
According to the Lorentz force equation, the force on each charge is ‫ݍ = ܨ‬ଶ ‫ݒ‬ଶ ‫ܤ‬ଵ , and
the total force on a length ݈ of wire 2 is then
݈
ߤ଴
‫ܫ = ܨ‬ଶ ‫ܤ‬ଵ ݈ =
‫ܫܫ‬
2ߨ ଵ ଶ ‫ݎ‬
- Determination of the magnetic constant
ఓబ
The factor
2.
The magnetic constant ߤ଴ also known as the magnetic permeability of free space, is
ସగ
plays the same role as
ଵ
1.
ସగఌబ
in electrostatics.
the proportionality between the magnetic induction and its force in a classical vacuum.
3.
From 1948 to 2019, leading to the primary definition of the ampere (A) in terms of the
coulomb (C), which reads the constant current which, if maintained in two straight
parallel conductors of infinite length, of negligible circular cross section, and placed
1 ݉ apart in vacuum, would produce between these conductors a force equal to
4.
2 × 10ି଻ N/݉ (Refer to the previous topic), ߤ଴ was exactly defined as
ߤ଴
N
H
‫ ؠ‬10ି଻ ଶ = 10ି଻
4ߨ
A
݉
In the 2019 redefinition of the SI base units, the ampere is defined exactly in terms of
the elementary charge (݁) and the second (‫)ݏ‬, and the value of ߤ଴ now has to be
5.
determined experimentally:
ߤ଴
ܰ
؆ 1.000 000 000 55(15) × 10ି଻ ଶ
4ߨ
‫ܣ‬
It is well known that the full set of Maxwell’s equations gives the following equation
between the electric and magnetic constants
1
݉
ܿ=
‫ ؠ‬299,792,458
‫ݏ‬
ඥߝ଴ ߤ଴
6.
By 2019, the speed of light was not exactly defined but experimentally measured.
- Forces and torques on current-carrying conductors
1.
Let us now calculate the Lorentz force and torque on current-carrying conductors under
a given magnetic field distribution.
2.
Force
A.
From the Lorentz force, an expression for the force on an element ݈݀Ԧ of a currentcarrying conductor can be found. If ݈݀Ԧ is parallel to the drift velocity ‫ݒ‬Ԧ of the
charge carriers, whose density per unit volume is ܰ, and has a cross-sectional area
‫ܣ‬, the force on the element ݈݀Ԧ is
ሬԦ = ܰ‫ݒ|ݍܣ‬Ԧ|݈݀Ԧ × ‫ܤ‬
ሬԦ
݀‫ܨ‬Ԧ = ܰ‫ܣ‬ห݈݀Ԧห‫ݒݍ‬Ԧ × ‫ܤ‬
B.
For a steady current,
‫ݒ|ݍܣܰ = ܫ‬Ԧ|
C.
֜
ሬԦ
݀‫ܨ‬Ԧ = ‫݈݀ܫ‬Ԧ × ‫ܤ‬
If the circuit in question is a closed loop by the contour ‫ܥ‬, the force on the
complete circuit is
ሬԦ = ‫ ܫ‬ර ݈݀Ԧ × ‫ܤ‬
ሬԦ
‫ܨ‬Ԧ = ර ‫݈݀ܫ‬Ԧ × ‫ܤ‬
஼
D.
஼
If the magnetic field is uniform,
ሬԦ × ‫ܤ‬
ሬԦ
ሬԦ = ‫ܫ‬0
ሬԦ = 0
‫ܨ‬Ԧ = ‫ ܫ‬ቈර ݈݀Ԧ቉ × ‫ܤ‬
஼
3.
Torque
A.
The torque on an element and the complete circuit is readily given by
ሬԦ൯
݀߬Ԧ = ‫ݎ‬Ԧ × ݀‫ܨ‬Ԧ = ‫ݎܫ‬Ԧ × ൫݈݀Ԧ × ‫ܤ‬
ሬԦ൯
and ߬Ԧ = ‫ ܫ‬ර ‫ݎ‬Ԧ × ൫݈݀Ԧ × ‫ܤ‬
஼
B.
If the magnetic field is uniform, we can simply the equation further as
C.
1
ሬԦ = ‫ܣܫ‬Ԧ × ‫ܤ‬
ሬԦ
߬Ԧ = ‫ ܫ‬ቈ ර ‫ݎ‬Ԧ × ݈݀Ԧ቉ × ‫ܤ‬
2 ஼
It is easy to show that the integral of ‫ݎ‬Ԧ × ݈݀Ԧ around a closed path gives twice the
area vector ‫ܣ‬Ԧ enclosed by the curve, whose the components, ‫ܣ‬௫ , ‫ܣ‬௬ , and ‫ܣ‬௭ ,
correspond to the areas enclosed by projections of the curve ‫ ܥ‬on the ‫ݖݕ‬-, ‫ݔݖ‬-,
and ‫ݕݔ‬-planes, respectively. (H.W. You can refer to Section 8-2 of Reitz.)
D.
We can now define the magnetic dipole moment of a closed loop carrying a
steady current as
1
݉
ሬሬԦ = ‫ܣܫ‬Ԧ = ‫ ܫ‬ර ‫ݎ‬Ԧ × ݈݀Ԧ
2 ஼
and
ሬԦ
߬Ԧ = ݉
ሬሬԦ × ‫ܤ‬
E.
Of course, if the magnetic field varies over the region of the current loop, we
should consider the contributions of higher-order terms, such as the magnetic
quadrupole, to the torque.
F.
We will return to this topic when we discuss the field of a distant circuit and the
magnetic dipole in Section 8.6 of this lecture note.
4.
The use of current density
A.
Instead of being confined to wires, the current exists in a medium, then the
following identification is appropriate:
‫݈݀ܫ‬Ԧ ՜ ‫ܬ‬Ԧ݀‫ݒ‬
B.
Then, we have the following representations, which are useful in discussing the
magnetic properties of matter.
ሬԦ(‫ݎ‬Ԧ)݀‫ݒ‬
‫ܨ‬Ԧ = න ‫ܬ‬Ԧ(‫ݎ‬Ԧ) × ‫ܤ‬
ሬԦ(‫ݎ‬Ԧ)ቁ ݀‫ݒ‬
߬Ԧ = න ‫ݎ‬Ԧ × ቀ‫ܬ‬Ԧ(‫ݎ‬Ԧ) × ‫ܤ‬
ሬሬԦ =
݉
1
න ‫ݎ‬Ԧ × ‫ܬ‬Ԧ(‫ݎ‬Ԧ)݀‫ݒ‬
2
8.3 Biot and Savart law (Sections 8-3 and 8-4 of Reitz / Section 6.4 of Purcell)
- Magnetic interaction of two current circuits
1.
In 1820, just a few weeks after Ørsted’s announcement of his discovery, André-Marie
Ampère, a French physicist, presented the results of a series of experiments.
2.
The magnetostatic interaction of two current circuits is in general expressed in modern
mathematical language as follows:
A.
Although it is empirical and phenomenological at this moment, the force on the
closed current loop #2 by another current loop #1 is given by
‫ܨ‬Ԧଶ =
B.
݈݀Ԧଶ × ቀ݈݀Ԧଵ × (‫ݎ‬Ԧଶ െ ‫ݎ‬Ԧଵ )ቁ
ߤ଴
‫ܫ‬ଵ ‫ܫ‬ଶ ර ර
|‫ݎ‬Ԧଶ െ ‫ݎ‬Ԧଵ |ଷ
4ߨ
ଵ
ଶ
The force depends on the mutual geometry of the loops, but due to the lack of
symmetry it seems to violate Newton's third law.
C.
However, Newton's third law is indeed valid for the magnetic forces.
i.
ሬԦ × ‫ܥ‬Ԧ ൯ = ൫‫ܣ‬Ԧ ‫ܥ ڄ‬Ԧ ൯‫ܤ‬
ሬԦ െ ൫‫ܣ‬Ԧ ‫ܤ ڄ‬
ሬԦ൯‫ܥ‬Ԧ, to
Using the vector triple product identity, ‫ܣ‬Ԧ × ൫‫ܤ‬
the integrand, we have
݈݀Ԧଶ × ቀ݈݀Ԧଵ × (‫ݎ‬Ԧଶ െ ‫ݎ‬Ԧଵ )ቁ
|‫ݎ‬Ԧଶ െ ‫ݎ‬Ԧଵ |ଷ
ii.
݈݀Ԧଶ ‫ݎ( ڄ‬Ԧଶ െ ‫ݎ‬Ԧଵ )
‫ݎ‬Ԧଶ െ ‫ݎ‬Ԧଵ
ቇ െ ൫݈݀Ԧଶ ‫݈݀ ڄ‬Ԧଵ ൯
ଷ
|‫ݎ‬Ԧଶ െ ‫ݎ‬Ԧଵ |ଷ
|‫ݎ‬Ԧଶ െ ‫ݎ‬Ԧଵ |
The integral of the first term in RHS along a closed loop should vanish
ර
ଶ
iii.
= ݈݀Ԧଵ ቆ
݈݀Ԧଶ ‫ݎ( ڄ‬Ԧଶ െ ‫ݎ‬Ԧଵ )
1
= െ ර ‫׏‬ଶ ൬
൰ ‫݈݀ ڄ‬Ԧଶ = 0
ଷ
|‫ݎ‬Ԧଶ െ ‫ݎ‬Ԧଵ |
|‫ݎ‬Ԧଶ െ ‫ݎ‬Ԧଵ |
ଶ
Now the expression is clearly symmetric and It is therefore obvious that
‫ܨ‬Ԧଶ = െ
ߤ଴
‫ݎ‬Ԧଶ െ ‫ݎ‬Ԧଵ
‫ܫ‬ଵ ‫ܫ‬ଶ ර ර ൫݈݀Ԧଶ ‫݈݀ ڄ‬Ԧଵ ൯
= െ‫ܨ‬Ԧଵ
|‫ݎ‬Ԧଶ െ ‫ݎ‬Ԧଵ |ଷ
4ߨ
ଵ
ଶ
- Biot-Savart law
1.
From the Lorentz force theory, the magnetic field exerted on the loop #2 by the loop
#1 is given as
ሬԦଶଵ
‫ܨ‬Ԧଶ = ර ‫ܫ‬ଶ ݈݀Ԧଶ × ‫ܤ‬
֜
ሬԦଶଵ (‫ݎ‬Ԧଶ ) =
‫ܤ‬
ଶ
2.
ߤ଴
݈݀Ԧଵ × (‫ݎ‬Ԧଶ െ ‫ݎ‬Ԧଵ )
‫ܫ‬ଵ ර
|‫ݎ‬Ԧଶ െ ‫ݎ‬Ԧଵ |ଷ
4ߨ
ଵ
This is known as the Biot-Savart law, named after Jean-Baptiste Biot and Félix Savart,
of which the differential form is
ߤ଴ ‫ܫ‬ଵ ݈݀Ԧଵ × (‫ݎ‬Ԧଶ െ ‫ݎ‬Ԧଵ )
|‫ݎ‬Ԧଶ െ ‫ݎ‬Ԧଵ |ଷ
4ߨ
There are several examples to calculate the magnetic field by current-carrying
ሬԦଶଵ (‫ݎ‬Ԧଶ ) =
݀‫ܤ‬
3.
conductors in the textbook (Examples 8-1, 8-2, 8-3, and 8-4). I recommend you to solve
the examples.
- A generalized expression of the Biot and Savart Law
1.
Exchanging the current element ‫݈݀ܫ‬Ԧ with the current density ‫ܬ‬Ԧ(‫ݔ‬Ԧ ᇱ ) , we have the
2.
generalized Biot and Savart Law given as
(‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ )
ߤ
ሬԦ(‫ݎ‬Ԧ) = ଴ න ‫ܬ‬Ԧ(‫ݎ‬Ԧ ᇱ ) ×
‫ܤ‬
݀‫ ݒ‬ᇱ
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |ଷ
4ߨ
This expression is the magnetic analog of the electric field by Coulomb’s law in terms
of the charge density
(‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ )
1
න ߩ(‫ݎ‬Ԧ ᇱ )
݀‫ ݒ‬ᇱ
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |ଷ
4ߨߝ଴
This generalized expression satisfies the divergence-free characteristic of the magnetic
‫ܧ‬ሬԦ (‫ݔ‬Ԧ) =
3.
field as
ሬԦ =
‫ܤڄ׏‬
4.
(‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ )
(‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ )
ߤ଴
ߤ଴
ᇱ
Ԧ(‫ݎ‬Ԧ ᇱ ) ‫ ڄ‬ቆ‫× ׏‬
න ‫ ڄ ׏‬ቆ‫ܬ‬Ԧ(‫ݎ‬Ԧ ᇱ ) ×
ቇ
݀‫ݒ‬
=
െ
න
‫ܬ‬
ቇ ݀‫ ݒ‬ᇱ = 0
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |ଷ
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |ଷ
4ߨ
4ߨ
The Biot-Savart law is valid in the magnetostatic approximation and consistent with
Ampère's circuital law, which will be discussed in the next section. When magnetostatics
does not apply, the Biot-Savart law should be replaced by Jefimenko's equations, which
are the relativistic, correct expression of the Coulomb and Biot-Savart laws (Sections
21-2 and 21-3 of Reitz, Section 6.5 of Jackson), to be discussed next semester.
8.4 Magnetostatics and Ampere’s circuital law (Section 8-5 of Reitz / Section 6.2 of Purcell)
- Condition of magnetostatics
1.
2.
Continuity equation for charge conservation
߲ߩ
+ ‫ܬ ڄ ׏‬Ԧ = 0
߲‫ݐ‬
Steady-state magnetic phenomena are characterized by no change in the net charge
density anywhere in space:
߲ߩ
=0
߲‫ݐ‬
The required condition is indeed supported by a circulating current in a closed loop or
‫ܬ ڄ ׏‬Ԧ = െ
3.
point magnetic dipoles.
4.
Additionally, we can readily consider a localized current distribution that can be entirely
enclosed by a closed surface. The closed surface is chosen to lie outside a bounded
region where ‫ܬ‬Ԧ is nonvanishing.
- Equivalence between the generalized Biot and Savart Law and Ampere’s law
1.
Let us consider a current distribution ‫ܬ‬Ԧ(‫ݎ‬Ԧ ᇱ ) completely localized in a volume ܸ.
2.
Using |௥Ԧି௥Ԧ ᇲ|య = െ‫ ׏‬ቀ|௥Ԧି௥Ԧ ᇲ|ቁ and ‫ܽ߰( × ׏‬Ԧ) = ‫ܽ × ߰׏‬Ԧ + ߰‫ܽ × ׏‬Ԧ , the generalized Biot and
௥Ԧି௥Ԧ ᇲ
ଵ
Savart Law becomes
ߤ଴
1
න ‫ܬ‬Ԧ(‫ݎ‬Ԧ ᇱ ) × ‫ ׏‬൬
൰ ݀‫ ݒ‬ᇱ
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
4ߨ ௏
ߤ଴
‫ܬ‬Ԧ(‫ݎ‬Ԧ ᇱ )
‫ܬ × ׏‬Ԧ(‫ݎ‬Ԧ ᇱ ) ᇱ
=
න ‫×׏‬ቆ
ቇ
െ
݀‫ݒ‬
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
4ߨ ௏
ሬԦ(‫ݎ‬Ԧ) = െ
‫ܤ‬
=
ߤ଴
‫ݔ(ܬ‬Ԧ ᇱ )
‫×׏‬න ቆ
ቇ ݀‫ ݒ‬ᇱ
ᇱ|
|‫ݔ‬
4ߨ
Ԧ
െ
‫ݔ‬
Ԧ
௏
3.
ሬԦ = 0, since ‫ܽ × ׏( ڄ ׏‬Ԧ) = 0.
Again, this result is clearly consistent with ‫ܤ ڄ ׏‬
4.
Now, let us consider the curl of the magnetic field.
A.
Using ‫ܣ × ׏ × ׏‬Ԧ = ‫׏‬൫‫ܣ ڄ ׏‬Ԧ൯ െ ‫׏‬ଶ ‫ܣ‬Ԧ,
ሬԦ(‫ݎ‬Ԧ) =
‫ܤ×׏‬
=
B.
ߤ଴
‫ܬ‬Ԧ(‫ݎ‬Ԧ ᇱ )
‫×׏×׏‬න ቆ
ቇ ݀‫ ݒ‬ᇱ
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
4ߨ
௏
1
ߤ଴
1
ߤ଴
‫ ׏‬න ‫ܬ‬Ԧ(‫ݎ‬Ԧ ᇱ ) ‫ ׏ ڄ‬൬
൰ ݀‫ ݒ‬ᇱ െ
න ‫ܬ‬Ԧ(‫ݎ‬Ԧ ᇱ )‫׏‬ଶ ൬
൰ ݀‫ ݒ‬ᇱ
ᇱ
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ |
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
4ߨ ௏
4ߨ ௏
ଵ
௥Ԧି௥Ԧ ᇲ
௥Ԧ ᇲି௥Ԧ
ଵ
ି௥
ି௥
Using ‫ ׏‬ቀ|௥Ԧି௥Ԧ ᇲ|ቁ = െ |௥Ԧି௥Ԧ ᇲ|య = |௥Ԧ ᇲ Ԧ|య = െ‫׏‬ᇱ ቀ|௥Ԧ ᇲ Ԧ|ቁ and integrating by parts, the first
term becomes
1
ߤ଴
ߤ଴
1
‫ ׏‬න ‫ܬ‬Ԧ(‫ݎ‬Ԧ ᇱ ) ‫ ׏ ڄ‬൬
൰ ݀‫ ݒ‬ᇱ = െ
‫ ׏‬න ‫ܬ‬Ԧ(‫ݎ‬Ԧ ᇱ ) ‫׏ ڄ‬ᇱ ൬
൰ ݀‫ ݒ‬ᇱ
ᇱ
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ |
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
4ߨ ௏
4ߨ ௏
=െ
C.
ߤ଴
‫ܬ‬Ԧ(‫ݎ‬Ԧ ᇱ )
‫׏‬ᇱ ‫ܬ ڄ‬Ԧ(‫ݎ‬Ԧ ᇱ ) ᇱ
‫ ׏‬න ‫׏‬ᇱ ‫ ڄ‬ቆ
ቇെ
݀‫ݒ‬
ᇱ
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ |
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
4ߨ ௏
For the localized current distribution, ‫ܬ‬Ԧ(‫ݎ‬Ԧ ᇱ ) vanishes on the surface ܵ enclosing
the volume ܸ, so that the divergence theorem leads to
න ‫׏‬ᇱ ‫ ڄ‬ቆ
௏
D.
ଵ
ሬԦ(‫ݎ‬Ԧ) becomes
Using ‫ ׏‬ቀ|௫Ԧି௫Ԧ ᇲ |ቁ = െ4ߨߜ(‫ݔ‬Ԧ െ ‫ݔ‬Ԧ ᇱ ), the second term for ‫ܤ × ׏‬
ଶ
െ
E.
‫ܬ‬Ԧ(‫ݎ‬Ԧᇱ )
‫ܬ‬Ԧ(‫ݎ‬Ԧ ᇱ )
ᇱ
ቇ
݀‫ݒ‬
=
ර
‫݊ ڄ‬ොᇱ ݀ܽᇱ = 0
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
Ԧ െ ‫ݎ‬Ԧ ᇱ |
ௌ |‫ݎ‬
ߤ଴
1
න ‫ܬ‬Ԧ(‫ݎ‬Ԧ ᇱ )‫׏‬ଶ ൬
൰ ݀‫ ݒ‬ᇱ = ߤ଴ ‫ܬ‬Ԧ(‫ݎ‬Ԧ)
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
4ߨ ௏
Finally, we have
ሬԦ(‫ݎ‬Ԧ) = ߤ଴ ‫ܬ‬Ԧ(‫ݎ‬Ԧ) +
‫ܤ×׏‬
ߤ଴
‫׏‬ᇱ ‫ܬ ڄ‬Ԧ(‫ݎ‬Ԧ ᇱ ) ᇱ
‫׏‬න
݀‫ݒ‬
4ߨ ௏ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
5.
For steady-state magnetic phenomena (magnetostatics), ‫ܬ ڄ ׏‬Ԧ = 0 so that
ሬԦ(‫ݔ‬Ԧ) = ߤ଴ ‫ܬ‬Ԧ(‫ݔ‬Ԧ)
‫ܤ×׏‬
6.
This is the differential form of Ampere’s circuital law.
8.5 Vector potential (Section 8-6 of Reitz / Section 6.3 of Purcell)
- Vector potential and the solution in the microscopic view
1.
ሬԦ = 0 and the generalized Biot and Savart Law, we have
From ‫ܤ ڄ ׏‬
ሬԦ = ‫ܣ × ׏‬Ԧ
‫ܤ‬
֜ ‫ܣ‬Ԧ =
‫ݎ(ܬ‬Ԧᇱ )
ߤ଴
න
݀‫ ݒ‬ᇱ + ‫ݎ(߰׏‬Ԧ)
4ߨ ୟ୪୪ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
ୱ୮ୟୡୣ
2.
Here, the added gradient of an arbitrary function ߰ enables the vector potential ‫ܣ‬Ԧ to
be freely transformed (the gauge transformation) as
‫ܣ‬Ԧ ՜ ‫ܣ‬Ԧ + ‫߰׏‬
3.
Coulomb gauge
A.
Employing the vector potential, Ampere’s law becomes
ሬԦ = ‫ × ׏‬൫‫ܣ × ׏‬Ԧ൯ = ‫׏‬൫‫ܣ ڄ ׏‬Ԧ൯ െ ‫׏‬ଶ ‫ܣ‬Ԧ = ߤ଴ ‫ܬ‬Ԧ
‫ܤ×׏‬
B.
Let us make the convenient choice of gauge, ‫ܣ ڄ ׏‬Ԧ = 0: Coulomb gauge. Then
‫׏‬ଶ ‫ܣ‬Ԧ = െߤ଴ ‫ܬ‬Ԧ
C.
This is a Poisson equation, so that the solution of ‫ܣ‬Ԧ in unbounded space (open
boundary condition) is given again as
‫ܣ‬Ԧ(‫ݎ‬Ԧ) =
ߤ଴
‫ݎ(ܬ‬Ԧ ᇱ )
න
݀‫ ݒ‬ᇱ
4ߨ ୟ୪୪ ୱ୮ୟୡୣ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
D.
This result is identical to the Coulomb’s law in the electrostatics:
1
ߩ(‫ݎ‬Ԧᇱ )
߮(‫ݎ‬Ԧ) =
න
݀‫ ݒ‬ᇱ
4ߨߝ଴ ୟ୪୪ ୱ୮ୟୡୣ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
E.
For a given current density distribution, you can use the same technique to the
electrostatic problems with a given charge density distribution (‫׏‬ଶ ߮ = െߩ/ߝ଴). Of
course, the Poisson equation is valid only for the electro- and magneto-static cases.
- Example: a circular current loop (Direct calculation of ‫ܣ‬Ԧ from the integral)
1.
Consider a circular current loop of radius ܽ carrying a current ‫ ܫ‬as illustrated below
2.
Since ‫ܬ‬Ԧ ‫߮ צ‬ො, we have
‫ܣ‬௥ = ‫ܣ‬ఏ = 0
ᇲ ି௔൯
ఋ൫௥
‫ܬ‬ఝ = ‫ ܫ‬sin ߠ ᇱ ߜ(cos ߠ ᇱ )
, the current density distribution is given as
௔
‫ܬ‬Ԧ(‫ݎ‬Ԧ ᇱ ) = െ‫ܬ‬ఝ sin ߮ ᇱ ‫ݔ‬ො + ‫ܬ‬ఝ cos ߮ ᇱ ‫ݕ‬ො
3.
Letting
4.
To calculate ‫ܣ‬ఝ , we may choose the observation point at ߮ = 0 so that ‫ܣ‬ఝ = ‫ܣ‬௬ .
5.
We finally have
‫ܣ‬ఝ = ‫ܣ‬௬ (at ߮ = 0) =
=
ߤ଴ ‫ܫ‬
න
4ߨܽ
sin ߠ ᇱ ߜ(cos ߠ ᇱ ) cos ߮ ᇱ ߜ(‫ ݎ‬ᇱ െ ܽ)
ଶ
ଵ/ଶ
ቀ‫ ݎ‬ଶ + ‫ ݎ‬ᇱ ଶ െ 2‫ ݎݎ‬ᇱ (cos ߠ cos ߠ ᇱ + sin ߠ sin ߠ ᇱ cos ߮ ᇱ )ቁ
=
‫ ݎ‬ᇱ ݀‫ ݎ‬ᇱ ݀ȳᇱ
cos ߮ ᇱ ߜ(‫ ݎ‬ᇱ െ ܽ)
ߤ଴ ‫ܫ‬
ଶ
න
‫ ݎ‬ᇱ ݀‫ ݎ‬ᇱ ݀߮ ᇱ
4ߨܽ ൫‫ ݎ‬ଶ + ‫ ݎ‬ᇱ ଶ െ 2‫ ݎݎ‬ᇱ sin ߠ cos ߮ ᇱ ൯ଵ/ଶ
=
6.
ߤ଴ ‫ݎ(ܬ‬Ԧᇱ )
݀‫ ݒ‬ᇱ
4ߨ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
ߤ଴ ‫ ܫ‬ଶ ଶగ
cos ߮ ᇱ
ܽ න
݀߮ ᇱ
ଶ
ଶ
ᇱ ଵ/ଶ
4ߨܽ
଴ (‫ ݎ‬+ ܽ െ 2‫ ܽݎ‬sin ߠ cos ߮ )
The integration is the combination of complete elliptic integrals, the solution of which
requires numerical calculations.
7.
So, let us consider the approximation for ‫ܽ ب ݎ‬:
‫ܣ‬ఝ =
؆
ߤ଴ ‫ܽ ܫ‬ଶ ଶగ
න
4ߨܽ ‫ ݎ‬଴
cos ߮ ᇱ
ଵ/ଶ
ܽ ଶ
ܽ
൬ቀ ቁ െ 2 sin ߠ cos ߮ ᇱ + 1൰
‫ݎ‬
‫ݎ‬
݀߮ ᇱ
ܽ
ߤ଴ ‫ܽ ܫ‬ଶ ଶగ ܽ
ߤ଴ ‫ܽ ܫ‬ଶ ଶగ
න ቀ1 + sin ߠ cos ߮ ᇱ ቁ cos ߮ ᇱ ݀߮ ᇱ =
න
sin ߠ cos ଶ ߮ ᇱ ݀߮ ᇱ
4ߨܽ ‫ ݎ‬଴
4ߨܽ ‫ ݎ‬଴ ‫ݎ‬
‫ݎ‬
ߤ଴ ‫ܽܫ‬ଶ sin ߠ
4‫ ݎ‬ଶ
sin ߠ ߤ଴ |݉
ሬሬԦ||‫ݎ‬Ԧ| sin ߠ
ߤ଴
(‫ܽߨܫ‬ଶ ) ଶ =
=
4ߨ
4ߨ
‫ݎ‬ଷ
‫ݎ‬
=
8.
Where ݊ො is the normal vector to the plane of the current loop, the magnetic dipole
moment by a closed current loop can be determined as
݉
ሬሬԦ = ‫ܽߨܫ‬ଶ ݊ො = ‫݊)ܽ݁ݎܣ( × ܫ‬ො
9.
And, the vector potential by a single magnetic dipole is generalized as
‫ܣ‬Ԧ =
ሬሬԦ × ‫ݎ‬Ԧ
ߤ଴ ݉
4ߨ |‫ݎ‬Ԧ|ଷ
10. The more rigorous determination of the magnetic dipole of a current distribution will
be discussed in the next section.
- Problems solving the Poisson equation will be discussed in the next chapter (lecture note – 09).
8.6 Field of a distant circuit and magnetic dipole
(Section 8-7 of Reitz / Sections 11.3 and 11.4 of Purcell)
- Multipole expansion
1.
As in electrostatics, we can consider the multipole expansion of the contribution of a
localized distribution of currents ‫ݎ(ܬ‬Ԧ ᇱ ) nonvanishing only inside a finite volume ܸ
around some origin.
2.
Consider a localized circuitry as shown in the figure below
3.
Using ‫ ݒ݀ܬ‬ᇱ ՜ ‫ݎ݀ܫ‬Ԧ ᇱ , we have
‫ܣ‬Ԧ =
4.
ߤ଴
‫ݎ(ܬ‬Ԧᇱ )
ߤ଴ ‫ܫ‬
݀‫ݎ‬Ԧ ᇱ
ᇱ
න
݀‫ݒ‬
=
ර
4ߨ ௏ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
4ߨ ஼ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
ଵ
ଶ௥Ԧ‫ڄ‬௥Ԧ ᇲ
௥
௥య
The Taylor expansion of |‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |ିଵ = +
‫ݎ(ܣ‬Ԧ) =
+ ‫ ڮ‬gives
ߤ଴ ‫ ܫ‬1
1
ቈ ර ݀‫ݎ‬Ԧ ᇱ + ଷ ර (‫ݎ‬Ԧ ‫ݎ ڄ‬Ԧ ᇱ )݀‫ݎ‬Ԧ ᇱ + ‫ ڮ‬቉
4ߨ ‫ ݎ‬஼
‫ ݎ‬஼
5.
Obviously the first term vanishes; no magnetic monopole.
6.
The second term can be simplified to bring the part of ‫ݎ‬Ԧ outside the integral as follows:
A.
Considering that ‫ݎ‬Ԧ is fixed,
(‫ݎ‬Ԧ ‫ݎ ڄ‬Ԧᇱ )݀‫ݎ‬Ԧ ᇱ = ݀൫(‫ݎ‬Ԧ ‫ݎ ڄ‬Ԧ ᇱ )‫ݎ‬Ԧ ᇱ ൯ െ (‫ݎ‬Ԧ ‫ݎ݀ ڄ‬Ԧ ᇱ )‫ݎ‬Ԧᇱ
B.
ሬԦ × ‫ܥ‬Ԧ ൯ = ൫‫ܣ‬Ԧ ‫ܥ ڄ‬Ԧ ൯‫ܤ‬
ሬԦ െ ൫‫ܣ‬Ԧ ‫ܤ ڄ‬
ሬԦ൯‫ܥ‬Ԧ, gives
The vector triple product identity, ‫ܣ‬Ԧ × ൫‫ܤ‬
(‫ݎ‬Ԧᇱ × ݀‫ݎ‬Ԧ ᇱ ) × ‫ݎ‬Ԧ = െ‫ݎ‬Ԧ × (‫ݎ‬Ԧ ᇱ × ݀‫ݎ‬Ԧ ᇱ ) = െ(‫ݎ‬Ԧ ‫ݎ݀ ڄ‬Ԧ ᇱ )‫ݎ‬Ԧ ᇱ + (‫ݎ‬Ԧ ᇱ ‫ݎ ڄ‬Ԧ ᇱ )݀‫ݎ‬Ԧ
֜ (‫ݎ‬Ԧ ᇱ ‫ݎ݀ ڄ‬Ԧ ᇱ )‫ݎ‬Ԧ = (‫ݎ‬Ԧ ᇱ ‫ݎ ڄ‬Ԧ)݀‫ݎ‬Ԧ ᇱ െ (‫ݎ‬Ԧᇱ × ݀‫ݎ‬Ԧ ᇱ ) × ‫ݎ‬Ԧ
C.
֜ (‫ݎ‬Ԧ ‫ݎ ڄ‬Ԧ ᇱ )݀‫ݎ‬Ԧ ᇱ = ݀൫(‫ݎ‬Ԧ ‫ݎ ڄ‬Ԧ ᇱ )‫ݎ‬Ԧ ᇱ ൯ െ (‫ݎ‬Ԧᇱ ‫ݎ ڄ‬Ԧ)݀‫ݎ‬Ԧ ᇱ + (‫ݎ‬Ԧᇱ × ݀‫ݎ‬Ԧ ᇱ ) × ‫ݎ‬Ԧ
1
1
֜ (‫ݎ‬Ԧ ‫ݎ ڄ‬Ԧ ᇱ )݀‫ݎ‬Ԧ ᇱ = ݀൫(‫ݎ‬Ԧ ‫ݎ ڄ‬Ԧ ᇱ )‫ݎ‬Ԧᇱ ൯ + (‫ݎ‬Ԧ ᇱ × ݀‫ݎ‬Ԧ ᇱ ) × ‫ݎ‬Ԧ
2
2
ᇱ )‫ ݎ‬ᇱ
Since ݀൫(‫ݎ‬Ԧ ‫ݎ ڄ‬Ԧ Ԧ ൯ is an exact differential, its line integral over a closed loop must
vanish. Finally, using we have
1
1 1
ර (‫ݎ‬Ԧ ‫ݎ ڄ‬Ԧ ᇱ )݀‫ݎ‬Ԧ ᇱ = ଷ ቈ ර ‫ݎ‬Ԧᇱ × ݀‫ݎ‬Ԧ ᇱ ቉ × ‫ݎ‬Ԧ
‫ݎ‬ଷ ஼
‫ ݎ‬2 ஼
7.
Therefore, the magnetic dipole moment of a localized circuitry can be given by
‫ݎ(ܣ‬Ԧ) =
ሬሬԦ × ‫ݎ‬Ԧ
ߤ଴ ‫ ܫ‬1
‫ݎ‬Ԧ
ߤ଴ ‫݉ ܫ‬
ቈ ර ‫ݎ‬Ԧᇱ × ݀‫ݎ‬Ԧ ᇱ ቉ × ଷ =
4ߨ 2 ஼
4ߨ ‫ ݎ‬ଷ
‫ݎ‬
1
ሬሬԦ = ර ‫ݎ‬Ԧ ᇱ × ݀‫ݎ‬Ԧ ᇱ
݉
2 ஼
8.
This is the same result to what we have already obtained in the discussion of the torque
on a localized current loop.
9.
It is also possible to derive the magnetic dipole moment of a localized current density
distribution ‫ݎ(ܬ‬Ԧᇱ ) , using the Taylor expansion. The result is given as follows (the
derivation is given in the additional reading material):
‫ݎ(ܣ‬Ԧ) =
ሬሬԦ × ‫ݎ‬Ԧ
ߤ଴ ‫ ܫ‬1
‫ݎ‬Ԧ
ߤ଴ ‫݉ ܫ‬
ቈ න ‫ݎ‬Ԧ ᇱ × ‫ܬ‬Ԧ(‫ݎ‬Ԧ ᇱ )݀‫ ݒ‬ᇱ ቉ × ଷ =
4ߨ 2 ௏
4ߨ ‫ ݎ‬ଷ
‫ݎ‬
and
݉
ሬሬԦ =
1
න ‫ݎ‬Ԧ ᇱ × ‫ܬ‬Ԧ(‫ݎ‬Ԧ ᇱ )݀‫ ݒ‬ᇱ
2 ௏
- Comparison of the electric and magnetic dipoles
1.
The magnetic field by the magnetic dipole moment is given as
ሬԦ = ‫ܣ × ׏‬Ԧ =
‫ܤ‬
2.
ሬሬԦ) െ ݉
ሬሬԦ
ߤ଴ 3‫ݎ‬Ƹ (‫ݎ‬Ƹ ‫݉ ڄ‬
ቈ
቉
ଷ
4ߨ
‫ݎ‬
Remind the electric field by the electric dipole moment:
‫ܧ‬ሬԦ (‫ݎ‬Ԧ) = െ‫ݎ(߮׏‬Ԧ) =
3.
1 3‫ݎ‬Ƹ (‫݌‬Ԧ ‫ݎ ڄ‬Ƹ ) െ ‫݌‬Ԧ
4ߨߝ଴
‫ݎ‬ଷ
The figure below shows the electric field of a pair of equal and opposite charges and
the magnetic field of a current ring. (These are not the ideal dipoles).
- The case of a moving charged particle (optional)
1.
The current density of a moving charged particle of ‫ݒ‬Ԧ at ‫ݎ‬Ԧ can be represented as
‫ܬ‬Ԧ(‫ݎ‬Ԧ ᇱ ) = ‫ݒݍ‬Ԧߜ(‫ݎ‬Ԧ ᇱ െ ‫ݎ‬Ԧ)
2.
The magnetic dipole moment becomes
1
1
‫ݍ‬
݉
ሬሬԦ = න ‫ݎ‬Ԧ ᇱ × ‫ܬ‬Ԧ(‫ݎ‬Ԧ ᇱ )݀‫ ݒ‬ᇱ = ‫ݎ(ݍ‬Ԧ × ‫ݒ‬Ԧ) =
‫ܮ‬ሬԦ
2 ௏
2
2‫ܯ‬
3.
If ‫ݍ‬/‫݁ = ܯ‬/݉௘ (electrons),
4.
݁
ߤ஻
‫ܮ‬ሬԦ =
‫ܮ‬ሬԦ
԰
2݉௘
Here, ߤ஻ = ݁԰/2݉௘ is known as the Bohr magneton.
5.
This is the well-known classical connection between angular momentum and magnetic
݉
ሬሬԦ =
moment, which holds for orbital motion even on the atomic scale.
6.
But, this classical connection fails for the intrinsic moment of electrons and other
elementary particles. The classical result should be corrected by a dimensionless
correction factor ݃, known as the g-factor
݉
ሬሬԦ = ݃
݁
െ2.0023
ܵԦ and ݃ = ൜
+5.5857
2݉
for electron
for proton
8.7 Magnetic scalar potential (In case, ࡶԦ = ૙) (Section 8-8 of Reitz)
- Solution in the space outside the region of the localized current distribution
1.
As a special case, we can consider the Ampere circuital law in the space without any
current, in which the magnetic field is curl free and can be represented by a gradient
of a scalar:
ሬԦ = ߤ଴ ‫ܬ‬Ԧ = 0
‫ܤ×׏‬
2.
֜
ሬԦ = െߤ଴ ‫כ ߮׏‬
‫ܤ‬
߮ ‫ כ‬is called the magnetic scalar potential, which satisfies the Laplace equation based
on the magnetic Gauss’ law:
ሬԦ = െߤ଴ ‫׏‬ଶ ߮ ‫ = כ‬0
‫ܤڄ׏‬
3.
Of course, the boundary conditions are required to obtain the solution to the Laplace
equation of the magnetic scalar potential.
4.
For the magnetic dipole, one can easily see that in analogy to the electric dipole,
݉
ሬሬԦ) െ ݉
ሬሬԦ
ሬሬԦ ‫ݎ ڄ‬Ԧ
ሬሬԦ ‫ݎ ڄ‬Ԧ
ߤ 3‫ݎ‬Ƹ (‫ݎ‬Ƹ ‫݉ ڄ‬
݉
ሬԦ = ଴ ቈ
ሬԦ = െߤ௢ ‫ ׏‬ቆ
‫ܤ‬
቉ ֜ ‫ܤ‬
ቇ ֜
߮ ‫ݎ( כ‬Ԧ) =
4ߨ
‫ݎ‬ଷ
4ߨ‫ ݎ‬ଷ
4ߨ‫ ݎ‬ଷ
5.
Particularly, the use of the magnetic scalar potential is efficient to solve the macroscopic
magnetostatics problems of the permanent magnets, which will be discussed further
in the next lecture note.
6.
In the macroscopic case, the magnetic scalar potential is determined in the viewpoint
ሬԦ field rather than ‫ܤ‬
ሬԦ field:
of ‫ܪ‬
ሬԦ = ‫ܬ‬Ԧ = 0
‫ܪ×׏‬
7.
ሬԦ = െ‫כ ߮׏‬
֜ ‫ܪ‬
In the case where we consider two different magnetic materials of permeabilities ߤଵ
and ߤଶ , the boundary conditions are determined as
ߤ
߲߮ଶ‫ߤ כ‬ଵ ߲߮ଶ‫כ‬
ሬԦଶ ‫݊ ڄ‬ො = ଵ ‫ܪ‬
ሬԦଵ ‫݊ ڄ‬ො
‫ܪ‬
=
ߤଶ
ቐ
֜ ቐ ߲݊
ߤଶ ߲݊
ሬԦଶ × ݊ො = ‫ܪ‬
ሬԦଵ × ݊ො
‫߮׏‬ଶ‫݊ × כ‬ො = ‫߮׏‬ଶ‫݊ × כ‬ො
‫ܪ‬
(HW #8)
Due: 20240522
ሬԦ is given
1. (a) Show that the torque on a closed circuit of ‫ ܥ‬under a magnetic field distribution ‫ܤ‬
by
ሬԦ ൯
߬Ԧ = ‫ ܫ‬ර ‫ݎ‬Ԧ × ൫݈݀Ԧ × ‫ܤ‬
஼
(b) Show that if the magnetic field is uniform, the torque becomes
1
ሬԦ = ݉
ሬԦ
߬Ԧ = ‫ ܫ‬ቈ ර ‫ݎ‬Ԧ × ݈݀Ԧ቉ × ‫ܤ‬
ሬሬԦ × ‫ܤ‬
2 ஼
Thus, the magnetic dipole moment is given as
1
݉
ሬሬԦ = ‫ ܫ‬ර ‫ݎ‬Ԧ × ݈݀Ԧ
2 ஼
Hint) You may refer to Section 8-2 of Reitz.
2. Consider a Helmholtz coil consisting of two identical circular coils carrying a current ‫ܫ‬. The radius
of the two coils is ܽ and the distance between them is 2ܾ. Set the axis of the coils as the ‫ݖ‬-axis
and the origin of the coordinates at the midpoint between the two coils.
(a) Using the Biot-Savart law, calculate the magnetic field at a point on the axis of the coil.
(b) Show that the condition to obtain a relatively uniform magnetic field, with the second-order
term vanishing at the midpoint, is ܽ = 2ܾ.
(c) Show that if the fourth derivative is evaluated under the condition in (b), the magnetic field at
the midpoint is
ሬԦ = ‫ݖܤ‬Ƹ =
‫ܤ‬
ߤ଴ ‫ ܫ‬8
144 ‫ ݖ‬ସ
൤1 െ
ቀ ቁ ൨ ‫ݖ‬Ƹ
ܽ ξ125
125 ܽ
Thus, for the region where |‫ |ݖ‬is less than ܽ/10, the magnetic field difference is only on the order
of one in ten thousand.
3. Consider an infinitely long solenoid of radius ܴ with ܰ turns per unit length carrying a constant
current ‫ܫ‬.
(a) Show that in the sylindrical coordinate (‫ݎ‬, ߠ, ‫ )ݖ‬with respect to the axis of the solenoid,
inside
ሬԦ (‫ = )ݎ‬ቄ ߤ଴ ܰ‫ݖܫ‬Ƹ
‫ܤ‬
0
outside
(b) Show that while the magnetic field vanishes, the vector potential outside the solenoid is
ߤ଴ ܰ‫ ܴܫ‬ଶ
‫ܣ‬Ԧ =
ߠ෠
outside
2‫ݎ‬
ሬԦ = 0.
(c) Show that the vector potential in (b) still gives ‫ܤ‬
PH231 Electromagnetism I (Spring 2024)
Lecture note 09: Magnetostatics in Matter
(Chapter 9 and Section 10-1 of the textbook by Reitz, Chapter 11 of the book by Purcell)
9.1 Introduction
ሬԦ field)
- In the previous lecture, we discussed techniques for finding the magnetic induction field (‫ܤ‬
and the vector potential ‫ܣ‬Ԧ by a (local) distribution of currents in the free space. It is now the time
to discuss the influence of matter on the magnetic field.
- We will discuss the theory of magnetostatics from the macroscopic point of view and how to solve
the problem in terms of the vector potential (general magnetostatics) or the magnetic scalar
potential (magnetostatics without any excess current density: ‫ܬ‬Ԧ = 0) from the point of view of
boundary value problems.
- A brief discussion of the molecular field inside magnetic matter will also be given and compared
with the case of electrostatics. However, the topics of the origin of diamagnetism, paramagnetism
and ferromagnetism, whose theory should include the concept of spin beyond the classical theory,
will not be discussed and will be left to the classes of solid-state physics.
9.2 Magnetization and magnetization current densities
(Sections 9-1, 9-2, and 9-3 of Reitz / Sections 11.7 and 11.8 of Purcell)
- Magnetization
1.
In the classical model, a magnetic material can be simplified as a combination of atomic
ሬሬԦ௜ .
loop currents of ݉
2.
As in the case of electrostatics, we can sum up vectorially all of the dipole moments in
an infinitesimally small volume element ȟ‫ݒ‬:
1
ሬሬԦ = lim
‫ܯ‬
෍݉
ሬሬԦ௜ = ܰ‫݉ۃ‬
ሬሬԦ‫ۄ‬
୼௩՜଴ ȟ‫ݒ‬
௜
3.
Here, ܰ is the volume density of the magnetic dipole moments, and if there are
different types of magnetic dipole moments,
ሬሬԦ = ෍ ܰ௝ ‫݉ۃ‬
‫ܯ‬
ሬሬԦ௝ ‫ۄ‬
௝
- Magnetization current density
1.
If the magnetization is uniform, the currents in the atomic loops cancel each other out,
and there is no net effective current in the interior of the material.
2.
But, if the magnetization is nonuniform, the cancellation will not be complete. As an
example, if there is an abrupt change in the magnetization distribution (macroscopically
abrupt change), there exist a net effective current (‫ܫ‬௠ in the figure below), called the
magnetization current, at the region of the abrupt change.
3.
Magnetization current density in volume
A.
Let us find the relation between the magnetization current density ‫ܬ‬Ԧெ and the
ሬሬԦ.
magnetization ‫ܯ‬
B.
The figure above shows the magnetization depending on the position. The volume
elements (ȟ‫ݔ‬ȟ‫ݕ‬ȟ‫ )ݖ‬are supposed to be infinitesimally small in the macroscopic
viewpoint, so that
ሬሬԦ
߲‫ܯ‬
ȟ‫ݕ‬
߲‫ݕ‬
The magnetic dipole moment by a closed loop of current ‫ ܫ‬and area ‫ ܣ‬is given
ሬሬԦ(‫ݔ‬, ‫ ݕ‬+ ȟ‫ݕ‬, ‫ܯ = )ݖ‬
ሬሬԦ(‫ݔ‬, ‫ݕ‬, ‫ )ݖ‬+
‫ܯ‬
C.
ሬሬԦ = ‫݊ܣܫ‬ො. With respect to the figure above, we have
as ݉
݉௫ of the first volume element = ‫ܯ‬௫ ȟ‫ݔ‬ȟ‫ݕ‬ȟ‫ܫ = ݖ‬௖ᇱ ȟ‫ݕ‬ȟ‫ݖ‬
߲‫ܯ‬௫
݉௫ of the second volume element = ൬‫ܯ‬௫ +
ȟ‫ݕ‬൰ ȟ‫ݔ‬ȟ‫ݕ‬ȟ‫ܫ = ݖ‬௖ᇱᇱ ȟ‫ݕ‬ȟ‫ݖ‬
߲‫ݕ‬
D.
At the interface of the elements, there is a net current in the ‫ ݖ‬axis as
߲‫ܯ‬௫
ȟ‫ܫ‬஼ = ‫ܫ‬஼ᇱ െ ‫ܫ‬஼ᇱᇱ = െ
ȟ‫ݔ‬ȟ‫ݕ‬
߲‫ݕ‬
E.
Another way of getting a ‫ݖ‬-directed current is to have a ‫ݕ‬-component of the
F.
magnetization that varies in the ‫ݔ‬-direction:
߲‫ܯ‬௬
ȟ‫ܫ‬஼ =
ȟ‫ݔ‬ȟ‫ݕ‬
߲‫ݔ‬
The effective area for each of the current is ȟ‫ݔ‬ȟ‫ݕ‬, and thus the effective current
G.
per unit area flowing in the ‫ ݖ‬direction is
߲‫ܯ‬௬ ߲‫ܯ‬௫
൫‫ܬ‬Ԧெ ൯௭ =
െ
߲‫ݔ‬
߲‫ݕ‬
Therefore, the magnetization current density is the curl of the magnetization:
ሬሬԦ
‫ܬ‬Ԧெ = ‫ܯ × ׏‬
- Vector potential by magnetization and magnetization current density
1.
Let us consider a finite magnetic medium in a volume ܸ଴ enclosed by a surface ܵ଴
2.
Each volume element ȟ‫ ݒ‬ᇱ of the magnetized matter is characterized by a magnetic
dipole moment
ሬሬԦ(‫ݔ‬, ‫ݕ‬, ‫)ݖ‬ȟ‫ ݒ‬ᇱ
ȟ݉
ሬሬԦ = ‫ܯ‬
3.
As we have discussed in the previous lecture, the vector potential at ‫ݎ‬Ԧ by the magnetic
dipole moment at ‫ݎ‬Ԧ ᇱ is given as
ȟ‫ܣ‬Ԧ(‫ݎ‬Ԧ) =
4.
ሬሬԦ(‫ݎ‬Ԧ ᇱ ) × (‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ )
ሬሬԦ × (‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ ) ߤ଴ ‫ܯ‬
ߤ଴ ݉
=
ȟ‫ ݒ‬ᇱ
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |ଷ
4ߨ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |ଷ
4ߨ
The total vector potential at ‫ݎ‬Ԧ is then
‫ܣ‬Ԧ(‫ݎ‬Ԧ) =
5.
ଵ
൫௥Ԧି௥Ԧ ᇲ ൯
Using ‫׏‬ᇱ ቀ|௥Ԧି௥Ԧᇲ |ቁ = |௥Ԧି௥Ԧᇲ|య and integrating by parts, we have
‫ܣ‬Ԧ(‫ݎ‬Ԧ) =
ሬሬԦ(‫ݎ‬Ԧ ᇱ ) × (‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ )
ߤ଴
‫ܯ‬
ߤ଴
1
ሬሬԦ(‫ݎ‬Ԧ ᇱ ) × ‫׏‬ᇱ ൬
න
݀‫ ݒ‬ᇱ =
න ‫ܯ‬
൰ ݀‫ ݒ‬ᇱ
ᇱ
ଷ
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ |
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
4ߨ ௏బ
4ߨ ௏బ
=
6.
ሬሬԦ(‫ݎ‬Ԧ ᇱ ) × (‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ )
ߤ଴
‫ܯ‬
න
݀‫ ݒ‬ᇱ
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |ଷ
4ߨ ௏బ
ሬሬԦ(‫ݎ‬Ԧ ᇱ )
ሬሬԦ(‫ݎ‬Ԧ ᇱ )
‫׏‬ᇱ × ‫ܯ‬
ߤ଴
‫ܯ‬
ߤ଴
ᇱ
ᇱ
න
݀‫ݒ‬
+
න
‫׏‬
×
ቆ
ቇ ݀‫ ݒ‬ᇱ
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
4ߨ ௏బ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
4ߨ ௏బ
Employing Stokes’ law to the second term,
ሬሬԦ(‫ݎ‬Ԧ ᇱ )
ሬሬԦ(‫ݎ‬Ԧ ᇱ ) × ݊ො
ߤ଴
‫׏‬ᇱ × ‫ܯ‬
‫ܯ‬
ᇱ
‫ܣ‬Ԧ(‫ݎ‬Ԧ) =
න
݀‫ݒ‬
+
ර
݀ܽᇱ
ᇱ|
|‫ݎ‬
Ԧ
െ
‫ݎ‬
Ԧ
4ߨ ௏బ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
ௌబ
7.
ሬሬԦ and defining an effective magnetic surface current density ‫ܭ‬
ሬԦெ =
Using ‫ܬ‬Ԧெ = ‫ܯ × ׏‬
ሬሬԦ × ݊ො (in the textbook ࢐ெ ), we finally have
‫ܯ‬
ሬԦெ
ߤ଴
‫ܬ‬Ԧெ
‫ܭ‬
‫ܣ‬Ԧ(‫ݎ‬Ԧ) =
න
݀‫ ݒ‬ᇱ + ර
݀ܽᇱ
ᇱ
4ߨ ௏బ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧ |
Ԧ െ ‫ݎ‬Ԧ ᇱ |
ௌబ |‫ݎ‬
8.
In addition, if the macroscopic excess current density ‫ܬ‬Ԧୣ୶ୡୣୱୱ from conventional currents
of the charge transport variety, the vector potential is given as
ሬሬԦ(‫ݎ‬Ԧ ᇱ )
ሬሬԦ(‫ݎ‬Ԧ ᇱ ) × ݊ො
ߤ଴
‫ܬ‬Ԧୣ୶ୡୣୱୱ
ߤ଴
‫׏‬ᇱ × ‫ܯ‬
ߤ଴
‫ܯ‬
ᇱ
ᇱ
‫ܣ‬Ԧ(‫ݎ‬Ԧ) =
න
݀‫ݒ‬
+
න
݀‫ݒ‬
+
ර
݀ܽᇱ
4ߨ ௏బ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
4ߨ ௏బ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
4ߨ ௌబ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
- Comparison between the electric and magnetic cases
1.
In electrostatics, the macroscopic electrostatic potential was
1
ߩୣ୶ୡୣୱୱ (‫ݎ‬Ԧ ᇱ ) ᇱ
1
െ‫׏‬ᇱ ‫ܲ ڄ‬ሬԦ ᇱ
1
ܲሬԦ ‫݊ ڄ‬ො
߮(‫ݎ‬Ԧ) =
න
݀‫ݒ‬
+
න
݀‫ݒ‬
+
ර
݀ܽᇱ
4ߨߝ଴ ௏ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
4ߨߝ଴ ௏ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
4ߨߝ଴ ௌ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
2.
One can thus easily find a similarity between the electric and magnetic cases.
3.
The following figure and description (a part of Fig. 11.19 of Purcell) briefly summarizes
the difference/similarity between the electric polarization and magnetic magnetization.
ሬሬሬԦ field) and macroscopic field equations
9.3 Magnetic field (ࡴ
(Sections 9-2, 9-4, and 9-5 of Reitz / Section 11.10 of Purcell)
ሬԦ field) and its master equation
- Determination of the magnetic field (‫ܪ‬
1.
ሬԦ୫୧ୡ୰୭ = 0 leads to the same equation
The averaging of the microscopic equation ‫ܤ ڄ ׏‬
in the macroscopic view as
ሬԦ୫୧ୡ୰୭ ‫ܤ ڄ ׏ = ۄ‬
ሬԦ୫ୟୡ୰୭ = 0
‫ܤ ڄ ׏ۃ‬
2.
ሬԦ:
Thus, we can still use the concept of a vector potential ‫ܣ‬Ԧ(‫ݔ‬Ԧ) where its curl gives ‫ܤ‬
ሬԦ୫ୟୡ୰୭ = ‫ܣ × ׏‬Ԧ.
‫ ڄ ׏‬൫‫ܣ × ׏‬Ԧ൯ = 0 ֜ ‫ܤ‬
3.
ሬԦ becomes
In the Coulomb gauge with ‫ܣ ڄ ׏‬Ԧ = 0, the curl of ‫ܤ‬
ሬԦ୫୧ୡ୰୭ ‫ܤ × ׏ = ۄ‬
ሬԦ୫ୟୡ୰୭ = ߤ଴ ‫ܬ‬Ԧ = ߤ଴ ൣ‫ܬ‬Ԧୣ୶ୡୣୱୱ + ‫ܬ‬Ԧெ ൧
‫ܤ × ׏ۃ‬
ሬԦ୫ୟୡ୰୭ = ߤ଴ ൣ‫ܬ‬Ԧୣ୶ୡୣୱୱ + ‫ܯ × ׏‬
ሬሬԦ൧
֜‫ܤ×׏‬
4.
Here, ‫ܬ‬Ԧୣ୶ୡୣୱୱ includes conventional currents of the charge transport variety, and we can
define a new macroscopic magnetic field
1
ሬԦ = ‫ܤ‬
ሬԦ െ ‫ܯ‬
ሬሬԦ
‫ܪ‬
ߤ଴
5.
The macroscopic differential equations thus are
ሬԦ = ‫ܬ‬Ԧୣ୶ୡୣୱୱ
‫ܪ×׏‬
6.
ሬԦ = 0
and ‫ܤ ڄ ׏‬
It has to be noted that the definition of the magnetic field in the macroscopic viewpoint
has to be given more rigorously. This topic as well as the electric case is beyond the
scope of this lecture and will not be discussed. However, you may encounter it in the
graduate course. If you are interested in it, you may refer to Section 6.6 and Problem
6.7 of Jackson’s book.
ሬԦ field
- Note on the ‫ܪ‬
1.
ሬԦ field as an effective, macroscopic field.
In this lecture we have regarded the ‫ܪ‬
2.
However, the origin of magnetization is not only classical currents but also spins, and
with respect to the special relativity, the effects of the classical currents can be
equivalent to the electrical effects depending on the frame of reference.
3.
Considering that the primary source in magnetism and spintronics is the spin and it
cannot be dealt with by the classical theory of electromagnetism, we may need to call
ሬԦ field as the magnetic field and ‫ܤ‬
ሬԦ field as the magnetic induction.
‫ܪ‬
4.
ሬԦ and ‫ܪ‬
ሬԦ field
In the non-magnetic materials, we don’t need to distinguish the ‫ܤ‬
ሬԦ = ߤ଴ ‫ܪ‬
ሬԦ = ‫ܣ × ׏‬Ԧ
‫ܤ‬
5.
In fact, we will see later that in the presence of magnetic material, the B field behaves
similarly to the D field, and the H field behaves similarly to the E field.
9.4 Magnetic susceptibility, permeability, and magnetic materials
(Section 9-6 of Reitz / Sections 11.1 and 11.7 of Purcell)
- Magnetic susceptibility and permeability
1.
In order to solve problems in magnetic theory, it is essential to have a relation ship
ሬԦ and ‫ܪ‬
ሬԦ or, equivalently, a relation ship between ‫ܯ‬
ሬሬԦ and one of the
between ‫ܤ‬
magnetic field vectors.
2.
In a large class of materials, there exists an approximately linear relationship between
ሬሬԦ and ‫ܪ‬
ሬԦ. For a linear, isotropic material,
‫ܯ‬
ሬሬԦ = ߯௠ ‫ܪ‬
ሬԦ
‫ܯ‬
֜
ሬԦ = ߤ଴ ൫‫ܪ‬
ሬԦ + ‫ܯ‬
ሬሬԦ൯ = ߤ଴ (1 + ߯௠ )‫ܪ‬
ሬԦ = ߤ‫ܪ‬
ሬԦ
‫ܤ‬
3.
Here, the dimensionless quantity ߯௠ is called the magnetic susceptibility and ߤ is
called the permeability.
4.
Meanwhile, there is another class of magnetic material called ferromagnets. They are
not linear, so that the equations with ߯ and ߤ do not apply. The best advice that can
be given is to consider each problem involving ferromagnetism separately, try to
ሬԦ-‫ܪ‬
ሬԦ diagram is appropriate for the particular problem,
determine which region of the ‫ܤ‬
and make an approximation for ߤ appropriate to that region.
ሬԦ and ‫ܪ‬
ሬԦ)
- Paramagnetic and diamagnetic materials (with a linear dependency between ‫ܤ‬
1.
Typical materials in nature interact weakly with an external magnetic field. Such weak
interactions can be approximated by a relative magnetic permeability ߤ/ߤ଴ slightly
different from 1.
2.
Paramagnetic materials have a low and positive magnetic susceptibility to magnetic
fields, which means that they are weakly attracted by a magnetic field. For example,
liquid oxygen can be suspended by magnets due to its paramagnetism. (You may also
see https://www.youtube.com/watch?v=KcGEev8qulA) Here, strong magnets are
required because ߯௠ is very small.
3.
On the other hands, diamagnetic materials are those that exhibit a weak and negative
susceptibility to magnetic fields. This means they are repelled by magnetic fields. For
example, bismuth can levitate a neodymium magnet due to its diamagnetism.
4.
The Radboud University Nijmegen, the Netherlands, has conducted experiments where
water and other substances were successfully levitated. Most spectacularly, a live frog
(see figure above) was levitated. However, the levitation of diamagnetic materials
should be distinguished from the levitation of a superconductor, the Meissner effect,
in which any magnetic field is expelled entirely inside the superconductor.
5.
The following table shows the magnetic susceptibility of some paramagnetic and
diamagnetic materials at room temperature. One can easily see that the magnitude of
the susceptibility is very small.
ሬԦ and ‫ܪ‬
ሬԦ)
- Ferromagnetic materials (Hysteresis between ‫ܤ‬
1.
A ferromagnetic material is not linear and is often characterized by a possible
ሬሬԦ.
permanent magnetization with a constant ‫ܯ‬
2.
Magnetic hysteresis
A.
Magnetization: Consider an unmagnetized sample of ferromagnetic material
ሬԦ field,
whose magnetic domains have randomly oriented magnetizations. As the ‫ܪ‬
initially zero, is increased, the microscopic magnetizations of the magnetic
ሬሬԦ-‫ܪ‬
ሬԦ and
domains become aligned parallel to the applied magnetic field and the ‫ܯ‬
ሬԦ-‫ܪ‬
ሬԦ relationships follow the dashed lines in the figure above.
‫ܤ‬
B.
Saturation Magnetization: Once the microscopic magnetizations of all magnetic
domains are aligned, the total magnetization of the sample must be maximized.
The maximized magnetization ‫ܯ‬ୱ is called the saturation magnetization. After
ሬԦ field increases proportionally to the ‫ܪ‬
ሬԦ field as
saturation, the ‫ܤ‬
ப୆
பு
= ߤ଴ , but in
typical ferromagnets, the contribution of ߤ଴ ‫ ܯ‬dominates in the magnetic
induction.
C.
ሬԦ field
Retentivity: Next, consider a magnetized ferromagnetic sample. As the ‫ܪ‬
ሬሬԦ - ‫ܪ‬
ሬԦ and ‫ܤ‬
ሬԦ - ‫ܪ‬
ሬԦ relationships do not follow back to the
decreases, the ‫ܯ‬
unmagnetized state. The magnetization, once established, does not disappear with
ሬԦ = 0൯ = ‫ܯ‬୰ and ‫ܤ‬൫‫ܪ‬
ሬԦ = 0൯ = ‫ܤ‬୰ , which is called retentivity
the removal of ‫ܯ ;ܪ‬൫‫ܪ‬
or residual magnetism.
D.
Coercivity: A certain reverse magnetic field, called the coercivity, is required to
ሬԦ = ‫ܪ‬
ሬԦୡ୧ ൯ = 0 and ‫ܤ‬൫‫ܪ‬
ሬԦ = ‫ܪ‬
ሬԦୡ ൯ = 0.
completely demagnetize the ferromagnet; ‫ܯ‬൫‫ܪ‬
E.
The
above
statements
are
indeed
phenomenological.
The
origin
of
ferromagnetism, as well as that of diamagnetism and paramagnetism, must be
discussed in terms of the concept of spins and quantum mechanics.
3.
The following table shows the properties of typical ferromagnetic materials at room
temperature. Here, ‫ܭ‬௠ is the ratio of ‫ܤ‬/ߤ଴ ‫ ܪ‬at the saturation state.
9.5 Boundary-value problems involving magnetic materials
(Sections 9-2, 9-7 and 9-8 of Reitz / No related section in Purcell)
- Boundary conditions
1.
ሬԦ = ‫ܬ‬Ԧ and ‫ܤ ڄ ׏‬
ሬԦ = 0, the fields ‫ܤ‬
ሬԦ and ‫ܪ‬
ሬԦ should
From the differential equations, ‫ܪ × ׏‬
satisfy the following boundary conditions, where ݊ො is a unit normal vector form the
ሬԦ is the surface free current density,
region #1 into #2 and ‫ܭ‬
ሬԦଶ െ ‫ܤ‬
ሬԦଵ ൯ ‫݊ ڄ‬ොଶ = 0
൫‫ܤ‬
ቊ
ሬԦଶ െ ‫ܪ‬
ሬԦଵ ൯ = ‫ܭ‬
ሬԦ
݊ොଶ × ൫‫ܪ‬
2.
ሬԦ = 0 and the media are isotropic and linear, the boundary conditions for ‫ܤ‬
ሬԦ and ‫ܪ‬
ሬԦ
If ‫ܭ‬
can be represented respectively as
ሬԦଶ ‫݊ ڄ‬ො = ‫ܤ‬
ሬԦଵ ‫݊ ڄ‬ො
‫ܤ‬
ߤଶ
ቐ ሬԦ
ሬԦ × ݊ො
‫ܤ‬ଶ × ݊ො = ‫ܤ‬
ߤଵ ଵ
3.
and
ߤ
ሬԦଶ ‫݊ ڄ‬ො = ଵ ‫ܪ‬
ሬԦ ‫݊ ڄ‬ො
‫ܪ‬
ߤଶ ଵ
ቐ
ሬԦଶ × ݊ො = ‫ܪ‬
ሬԦଵ × ݊ො
‫ܪ‬
ሬԦଶ becomes normal to the boundary surface
If ߤଵ ‫ߤ ب‬ଶ , the magnetic field ‫ܪ‬
ሬԦଵ . The surfaces of the high permeability material are
independent of the direction of ‫ܪ‬
approximately equipotential, which is a well-known analogy exploited in many magnetdesign problems.
- Generally applicable method of the vector potential
1.
ሬԦ = ‫ܬ‬Ԧ and ‫ܤ ڄ ׏‬
ሬԦ = ‫ ڄ ׏‬൫‫ܣ × ׏‬Ԧ൯ = 0,
From the macroscopic equations, ‫ܪ × ׏‬
ሬԦ ൣ‫ܤ‬
ሬԦ൧ = ‫ܪ × ׏‬
ሬԦ ൣ‫ܣ × ׏‬Ԧ൧ = ‫ܬ‬Ԧ
‫ܪ×׏‬
2.
This is a complicated differential equation but gives the general solution with the
ሬԦଶ െ ‫ܤ‬
ሬԦଵ ൯ ‫݊ ڄ‬ො = 0 and ݊ො × ൫‫ܪ‬
ሬԦଶ െ ‫ܪ‬
ሬԦଵ ൯ = ‫ܭ‬
ሬԦ.
boundary conditions as discussed previously: ൫‫ܤ‬
3.
Application to linear media
A.
B.
ሬԦ = ‫ܤ‬
ሬԦ/ߤ, the general equation becomes
For linear media, ‫ܪ‬
1
ሬԦ = ‫ × ׏‬൬ ‫ܣ × ׏‬Ԧ൰ = ‫ܬ‬Ԧ
‫ܪ×׏‬
ߤ
Supposing piecewise constant ߤ and using the Coulomb gauge (‫ܣ ڄ ׏‬Ԧ = 0) for
statistics, we finally have the Poisson equation as
‫׏‬ଶ ‫ܣ‬Ԧ = െߤ‫ܬ‬Ԧ
C.
The above equation is only a change from ߤ଴ to ߤ in the microscopic equation.
ሬԦ = 0:
And, the following boundary condition can be used for the case ‫ܭ‬
ሬԦଶ ‫݊ ڄ‬ො = ‫ܤ‬
ሬԦଵ ‫݊ ڄ‬ො
൫‫ܣ × ׏‬Ԧଶ ൯ ‫݊ ڄ‬ො = ൫‫ܣ × ׏‬Ԧଵ ൯ ‫݊ ڄ‬ො
‫ܤ‬
ሬԦ = 0൯
ߤଶ
ߤଶ
൫for ‫ܭ‬
֜
ቐ
ቐ ሬԦ
ሬԦଵ × ݊ො
‫ܤ‬ଶ × ݊ො = ‫ܤ‬
൫‫ܣ × ׏‬Ԧଶ ൯ × ݊ො = ൫‫ܣ × ׏‬Ԧଵ ൯ × ݊ො
ߤଵ
ߤଵ
- Useful method for ‫ܬ‬Ԧ = 0 of the magnetic scalar potential
1.
ሬԦ = ‫ܬ‬Ԧ = 0 so that we can introduce a magnetic scalar potential as
In this case, ‫ܪ × ׏‬
ሬԦ (‫ݎ‬Ԧ) = െ‫߮׏‬ெ (‫ݎ‬Ԧ)
‫ܪ‬
2.
ሬԦ[െ‫߮׏‬ெ ] = 0 , with the boundary
In general, we need to solve the equation, ‫ܤ ڄ ׏‬
ሬԦ (= െ‫߮׏‬ெ ).
conditions for ‫ܪ‬
3.
Application to linear media
A.
For the linear media, the equation becomes
‫߮׏ߤ( ڄ ׏‬ெ ) = 0
B.
If ߤ is piecewise constant, the magnetic scalar potential satisfies the Laplace
equation:
C.
‫׏‬ଶ ߮ெ = 0
ሬԦ can be used as (‫ܬ‬Ԧ = 0 ֜ ‫ܭ‬
ሬԦ = 0)
And, the boundary condition for ‫ܪ‬
ߤଵ
߲߮ெଶ ߤଵ ߲߮ெଵ
ሬԦଶ ‫݊ ڄ‬ො = ‫ܪ‬
ሬԦ ‫݊ ڄ‬ො
‫ܪ‬
=
ߤଶ ଵ
ቐ
֜ ቐ
߲݊
ߤଶ ߲݊
ሬԦଶ × ݊ො = ‫ܪ‬
ሬԦଵ × ݊ො
‫߮׏‬ெଶ × ݊ො = ‫߮׏‬ெଵ × ݊ො
‫ܪ‬
ሬሬԦ given and ‫ܬ‬Ԧ = 0): Permanent magnets
- Hard ferromagnets (‫ܯ‬
1.
ሬԦ is independent on the applied field. In this case, we can use either ߮ெ or ‫ܣ‬Ԧ.
‫ܪ‬
2.
ሬԦ = െ‫߮׏‬ெ )
Method #1: using the magnetic scalar potential (‫ܪ‬
A.
Let a hard ferromagnet has a finite volume ܸ and surface ܵ with ‫ܬ‬Ԧ = 0.
B.
Inside ܸ , the magnetic scalar potential satisfies, where ߩெ is the effective
magnetic-charge density,
ሬԦ = ߤ଴ ‫ ڄ ׏‬൫‫ܪ‬
ሬԦ + ‫ܯ‬
ሬሬԦ൯ = 0
‫ܤڄ׏‬
ሬԦ = ‫ܯ ڄ ׏‬
ሬሬԦ = െߩெ
֜ ‫׏‬ଶ ߶ெ = െ‫ܪ ڄ ׏‬
C.
On ܵ, we need to consider the contribution by the effective magnetic surfaceሬሬԦ.
charge density, ߪெ = ݊ො ‫ܯ ڄ‬
D.
Thus, the magnetic scalar potential is given as
ߩெ (‫ݎ‬Ԧ ᇱ )
1
ߪெ (‫ݎ‬Ԧ ᇱ )
1
ᇱ
߮ெ (‫ݎ‬Ԧ) =
න
݀‫ݒ‬
+
ර
݀ܽᇱ
4ߨ ௌ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
4ߨ ௏ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
=െ
ሬሬԦ(‫ݎ‬Ԧ ᇱ )
ሬሬԦ(‫ݎ‬Ԧ ᇱ )
1
‫׏‬ᇱ ‫ܯ ڄ‬
1
݊ොᇱ ‫ܯ ڄ‬
න
݀‫ ݒ‬ᇱ +
ර
݀ܽᇱ
ᇱ
4ߨ ௏ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧ |
4ߨ ௌ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
E.
This result can also be converted as
ሬሬԦ(‫ݎ‬Ԧ ᇱ )
ሬሬԦ(‫ݎ‬Ԧ ᇱ )
‫׏‬ᇱ ‫ܯ ڄ‬
1
‫ܯ‬
1
ᇱ
ᇱ
߮ெ (‫ݎ‬Ԧ) = െ න
݀‫ݒ‬
+
න
‫׏‬
‫ڄ‬
ቆ
ቇ ݀‫ ݒ‬ᇱ
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
4ߨ ௏
4ߨ ௏ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
using Gaussᇱ law
1
1
ሬሬԦ(‫ݎ‬Ԧ ᇱ ) ‫׏ ڄ‬ᇱ ൬
න ‫ܯ‬
൰ ݀‫ ݒ‬ᇱ integrating by parts
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
4ߨ ௏
ሬሬԦ(‫ݎ‬Ԧ ᇱ )
‫ܯ‬
1
1
1
݀‫ ݒ‬ᇱ
using ‫׏‬ᇱ ൬
൰ = െ‫ ׏‬൬
൰
=െ ‫ڄ׏‬න
ᇱ
ᇱ
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ |
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
Ԧ െ ‫ݎ‬Ԧ |
4ߨ
௏ |‫ݎ‬
=
3.
ሬԦ = ‫ܣ × ׏‬Ԧ)
Method #2: using the vector potential (‫ܤ‬
A.
Consider the identical case.
B.
Inside ܸ, the vector potential satisfies, where ‫ܬ‬Ԧெ is the effective magnetic-current
density,
ሬԦ
‫ܤ‬
ሬሬԦቇ = 0
െ‫ܯ‬
ߤ଴
ሬԦ = െߤ଴ ‫ܯ × ׏‬
ሬሬԦ = െߤ଴ ‫ܬ‬Ԧெ
֜ ‫׏‬ଶ ‫ܣ‬Ԧ = െ‫ܤ × ׏‬
ሬԦ = ‫ × ׏‬ቆ
‫ܪ×׏‬
C.
On ܵ, we need to consider the contribution by the effective magnetic surfaceሬԦெ = ‫ܯ‬
ሬሬԦ × ݊ො.
current density, ‫ܭ‬
D.
Thus, the vector potential is given as
ሬԦெ (‫ݎ‬Ԧ ᇱ )
ߤ଴
‫ܬ‬Ԧெ (‫ݎ‬Ԧ ᇱ )
ߤ଴
‫ܭ‬
‫ܣ‬Ԧ(‫ݎ‬Ԧ) =
න
݀‫ ݒ‬ᇱ +
ර
݀ܽᇱ
ᇱ
4ߨ ௏ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧ |
4ߨ ௌ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
=
ሬሬԦ(‫ݎ‬Ԧ ᇱ )
ሬሬԦ(‫ݎ‬Ԧ ᇱ ) × ݊ොᇱ
ߤ଴
‫׏‬ᇱ × ‫ܯ‬
ߤ଴
‫ܯ‬
ᇱ
න
݀‫ݒ‬
+
ර
݀ܽᇱ
4ߨ ௏ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
4ߨ ௌ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
4.
The two results above are equivalent.
5.
ሬሬԦ is uniform (‫ܯ ڄ ׏‬
ሬሬԦ = 0 and ‫ܯ × ׏‬
ሬሬԦ = 0), only the surface integral survives.
Note: if ‫ܯ‬
- Example #1: A magnetically permeable sphere in uniform magnetic field (H.W.)
(This is the example 9-1 of Reitz. Since no excess current exists, one can use the magnetic scalar
potential. The method and result to calculate the magnetic scalar potential are exactly same to
the case of a dielectric sphere in uniform electric field, which was discussed in Section 5.5 of the
lecture note 5 and Example 4-2 of Reitz.)
- Example #2: Uniformly magnetized sphere (permanent magnet)
1.
ሬሬԦ = ‫ݖܯ‬Ƹ ,
Consider a sphere of radius ܽ, with a uniform permanent magnetization ‫ܯ‬
embedded in a nonpermeable (nonmagnetic) medium as shown below.
2.
ሬԦ = െ‫߮׏‬ெ )
Method #1: using the magnetic scalar potential (‫ܪ‬
A.
ሬሬԦ = 0 and ‫ܬ‬Ԧ = 0, the magnetic scalar potential becomes, where ‫ݎ‬ழ (‫ݎ‬வ )
Since ‫ܯ ڄ ׏‬
is the smaller (larger) of ‫ݎ| = ݎ‬Ԧ| and ܽ,
ሬሬԦ(‫ݎ‬Ԧ ᇱ )
݊ොᇱ
݊ොᇱ ‫ܯ ڄ‬
ܽଶ
1
ሬሬԦ ‫ ڄ‬න ݀ȳᇱ
߮ெ (‫ݔ‬Ԧ) =
ර
݀ܽᇱ =
‫ܯ‬
ᇱ
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
4ߨ
4ߨ ௌ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧ |
ସగ
‫ݎ‬ழ
ܽଶ
4ߨ ‫ݎ‬ழ
1
ሬሬԦ ‫ ڄ‬ቆ
=
‫ܯ‬
݊ොቇ = ‫ܽܯ‬ଶ ଶ cos ߠ
4ߨ
3 ‫ݎ‬வଶ
3
‫ݎ‬வ
B.
௡ොᇲ
The following were used for ‫݀ ׬‬ȳᇱ |௥Ԧି௥Ԧᇲ| =
ସగ ௥ಬ
ଷ ௥ಭమ
݊ො. (This is slightly beyond the level
of the sophomore class. But, you can refer to the additional reading material):
i.
C.
ଵ
|௥Ԧି௥Ԧ ᇲ |
= 4ߨ σஶ
௟ୀ଴
௥ಬ೗
ଵ
ଶ௟ାଵ ௥ಭ೗శభ
௠‫כ‬
௠
ᇱ
ᇱ
σା௟
௠ୀି௟ ܻ௟ (ߠ , ߶ )ܻ௟ (ߠ, ߶)
ii.
݊ොᇱ = sin ߠ ᇱ cos ߶ ᇱ ‫ݔ‬ො + sin ߠ ᇱ sin ߶ ᇱ ‫ݕ‬ො + cos ߠ ᇱ ‫ݖ‬Ƹ
iii.
‫׬‬ସగ ݀ȳ ቀܻ௟మ మ (ߠ, ߶)ቁ ܻ௟భ భ (ߠ, ߶) = ߜ௟భ ௟మ ߜ௠భ௠మ
௠
‫כ‬
௠
Inside the sphere, ‫ݎ‬ழ = ‫ ݎ‬and ‫ݎ‬வ = ܽ so that
1
1
߮ெ (‫ݎ‬Ԧ) = ‫ܯ‬଴ ‫ ݎ‬cos ߠ = ‫ܯ‬଴ ‫ݖ‬
3
3
֜
1
ሬԦ௜௡ = െ‫߮׏‬ெ = െ ‫ܯ‬
ሬሬԦ
‫ܪ‬
3
൞
2ߤ
ሬԦ௜௡ = ߤ଴ ‫ܪ‬
ሬԦ௜௡ + ‫ܯ‬
ሬሬԦ = ଴ ‫ܯ‬
ሬሬԦ
‫ܤ‬
3
D.
Outside the sphere, ‫ݎ‬ழ = ܽ and ‫ݎ‬வ = ‫ ݎ‬so that
cos ߠ
ሬሬԦ ‫ݎ ڄ‬Ԧ
1 ݉
1
߮ெ (‫ݔ‬Ԧ) = ‫ܽܯ‬ଷ ଶ =
‫ݎ‬
4ߨ ‫ ݎ‬ଷ
3
E.
֜
ሬԦ௢௨௧ = ‫ܤ‬
ሬԦ௢௨௧ =
ߤ଴ ‫ܪ‬
ሬሬԦ ‫ݎ ڄ‬Ƹ ) െ ݉
ሬሬԦ
ߤ଴ 3‫ݎ‬Ƹ (݉
ଷ
4ߨ
‫ݎ‬
ሬሬԦ is the averaged magnetic dipole moment over the volume
that ‫ܯ‬
3.
ସగ௔య
This is the scalar potential by a single magnetic dipole moment ݉
ሬሬԦ =
ଷ
ସగ௔య
ଷ
ሬሬԦ. Note
‫ܯ‬
.
ሬԦ = ‫ܣ × ׏‬Ԧ)
Method #2: using the vector potential (‫ܤ‬
A.
The method using the vector potential also provides the same result.
B.
ሬሬԦ = 0 and ‫ܬ‬Ԧ = 0, the vector potential becomes
Since ‫ܯ ڄ ׏‬
ሬሬԦ(‫ݎ‬Ԧ ᇱ ) × ݊ොᇱ
ߤ଴
‫ܯ‬
ߤ଴ ‫ܯ‬
‫ݖ‬Ƹ × ݊ොᇱ
ᇱ
‫ܣ‬Ԧ(‫ݎ‬Ԧ) =
ර
݀ܽ
=
ර
݀ܽᇱ
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
4ߨ ௌ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
4ߨ ௌ
C.
Considering the azimuthal symmetry, the azimuthal component of the vector
potential survives as follows
ߤ଴ ‫ܽܯ‬ଶ
sin ߠ ᇱ cos ߶ ᇱ ᇱ
߶෠ න
݀ȳ
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
4ߨ
ସగ
ߤ଴
‫ ݎܯ‬sin ߠ inside
‫ݎ‬
ߤ଴
3
ழ
֜ ‫ܣ‬థ (‫ݎ‬Ԧ) = ‫ܽܯ‬ଶ ቆ ଶ ቇ sin ߠ = ൞
ߤ଴ ܽ ଷ
3
‫ݎ‬வ
‫ ܯ‬sin ߠ outside
3 ‫ݎ‬ଶ
The following were used here (see the additional reading material):
‫ܣ‬Ԧ(‫ݎ‬Ԧ) =
D.
i.
E.
ଵ
|௥Ԧି௥Ԧ ᇲ |
= 4ߨ σஶ
௟ୀ଴
௥ಬ೗
ଵ
ଶ௟ାଵ ௥ಭ೗శభ
଼గ
௠‫כ‬
௠
ᇱ
ᇱ
σା௟
௠ୀି௟ ܻ௟ (ߠ , ߶ )ܻ௟ (ߠ, ߶)
ii.
sin ߠ ᇱ cos ߶ ᇱ = െට
iii.
‫׬‬ସగ ݀ȳ ቀܻ௟మ మ (ߠ, ߶)ቁ ܻ௟భ భ (ߠ, ߶) = ߜ௟భ ௟మ ߜ௠భ௠మ
௠
ଷ
‫כ‬
Re[ܻଵଵ (ߠ ᇱ , ߶ ᇱ )]
௠
From the vector potential, we can get the magnetic induction and field. Inside the
sphere (‫ݎ‬ழ = ‫ ݎ‬and ‫ݎ‬வ = ܽ), they are given by
ߤ ‫ܯ‬
ߤ ‫ܯ‬
2ߤ
2ߤ
ሬԦ௜௡ = ‫ܣ × ׏‬Ԧ௜௡ = ଴ ‫ × ׏‬൫‫ ݎ‬sin ߠ ߶෠൯ = ଴ ൣ2‫ݎ‬Ƹ cos ߠ െ 2ߠ෠ sin ߠ൧ = ଴ ‫ݖܯ‬Ƹ = ଴ ‫ܯ‬
ሬሬԦ
‫ܤ‬
3
3
3
3
ሬԦ
‫ܤ‬
1
ሬԦ௜௡ = െ ‫ܯ‬
ሬሬԦ = െ ‫ܯ‬
ሬሬԦ
‫ܪ‬
ߤ଴
3
F.
Outside the sphere (‫ݎ‬ழ = ܽ and ‫ݎ‬வ = ‫)ݎ‬, we have
ሬሬԦ × ‫ݎ‬Ԧ
ߤ଴ 4ߨܽଷ 1
ߤ଴ ݉
‫ ܯ‬ଶ sin ߠ ߶෠ =
4ߨ 3
4ߨ ‫ ݎ‬ଷ
‫ݎ‬
ሬሬԦ × ‫ݎ‬Ԧ
ሬሬԦ ‫ݎ ڄ‬Ƹ ) െ ݉
ሬሬԦ
ߤ
݉
ߤ଴ 3‫ݎ‬Ƹ (݉
ሬԦ௢௨௧ = ‫ܤ‬
ሬԦ௢௨௧ = ‫ܣ × ׏‬Ԧ௢௨௧ = ଴ ‫ × ׏‬ቆ
ቇ=
ߤ଴ ‫ܪ‬
ଷ
ଷ
‫ݎ‬
4ߨ
4ߨ
‫ݎ‬
‫ܣ‬Ԧ(‫ݎ‬Ԧ) =
4.
For the uniformly magnetized sphere, the fields are not only dipole in character
asymptotically but also close to the sphere; there are no higher multipoles
ሬԦ and ‫ܪ‬
ሬԦ
- Comparison of ‫ܤ‬
1.
ሬԦ and ‫ܪ‬
ሬԦ further.
The result obtained above enables to discuss the difference of ‫ܤ‬
2.
ሬԦ are closed curves, the lines of ‫ܪ‬
ሬԦ originate on the surface of the
While the lines of ‫ܤ‬
sphere where the effective surface magnetic current/charge resides, as shown below.
3.
ሬԦ is exactly same to the result for ‫ܧ‬ሬԦ by a single electric dipole. So, if
The result for ‫ܪ‬
considering permanent magnetic dipoles, e.g. the spin of electron, the description
ሬԦ fields would be more realistic.
employing ‫ܪ‬
4.
ሬԦ are identical at the observation point in a nonmagnetic
ሬԦ and ߤ଴ ‫ܪ‬
But, note that ‫ܤ‬
medium and far from the source.
9.6 Molecular field inside matter (Section 10-1 of Reitz, optional)
- Molecular field inside matter
(This lecture note uses the vector potential ‫ܣ‬Ԧ, but the textbook of Reitz (Section 10-1) uses the
magnetic scalar potential ߮ெ . You can compare two methods, that actually give the same result.)
1.
Although understanding the microscopic features of magnetism requires the use of
quantum mechanics and the concept of electron spin as found in books on solid-state
physics, it is still useful to consider the concept of the molecular field inside magnetized
matter as in our earlier discussion of dielectric matter.
2.
Molecular field
A.
ሬԦ௠ may be approximated with the field inside a
Meanwhile, the molecular field ‫ܪ‬
ሬሬԦ (the figure
spherical empty (ߤ = ߤ଴ ) cavity of radius ܽ in a magnetic material of ‫ܯ‬
below) and differ from the magnetic field in the specimen by the contribution of
ሬԦ௦ and the contribution of the various
the surface density of magnetic currents ‫ܪ‬
ሬԦ ᇱ :
dipoles inside the cavity ‫ܪ‬
ሬԦ௠ = ‫ܪ‬
ሬԦ + ‫ܪ‬
ሬԦ௦ + ‫ܪ‬
ሬԦ ᇱ = ‫ܤ‬
ሬԦ௠ ൗߤ଴
‫ܪ‬
B.
ሬԦ௦ of this problem is identical to that of the spherical
The evaluation of ‫ܪ‬
permanent magnet discussed in the previous part, except that the inside and
outside of the magnetic medium change (݊ොᇱ changes to െ݊ොᇱ ).
C.
When ݊ොᇱ is the normal vector outward from the surface of the cavity, the
contribution of the surface current density induced by the uniformly magnetized
ሬሬԦ = ‫ݖܯ‬Ƹ and ‫ ܯ‬is constant) is given by
matter (‫ܯ‬
‫ܣ‬Ԧ௦ =
ሬሬԦ(‫ݎ‬Ԧ ᇱ ) × െ݊ොᇱ
ߤ଴
‫ܯ‬
ߤ଴ ‫ܽܯ‬ଶ
‫ݖ‬Ƹ × ݊ොᇱ
ߤ଴ ‫ܯ‬
න
ܽଶ ݀ȳᇱ = െ
න
݀ȳᇱ = െ
‫ ݎ‬sin ߠ ߶෠
ᇱ
3
4ߨ ସగ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧ |
4ߨ ସగ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
D.
From the vector potential, we can get the corresponding magnetic induction as
E.
2ߤ଴
2ߤ
ሬሬԦ ֜ ‫ܤ‬
ሬԦ௠ = ‫ܤ‬
ሬԦ + ‫ܤ‬
ሬԦ௦ = ‫ܤ‬
ሬԦ െ ଴ ‫ܯ‬
ሬሬԦ
‫ܯ‬
3
3
The molecular magnetic field in the empty cavity is then
ሬԦ௦ = ‫ܣ × ׏‬Ԧ௦ = െ
‫ܤ‬
ሬԦ௠ =
‫ܪ‬
3.
ሬԦ௠
ሬԦ 2
‫ܤ‬
‫ܤ‬
2
1
ሬሬԦ = ൣ‫ܪ‬
ሬԦ + ‫ܯ‬
ሬሬԦ൧ െ ‫ܯ‬
ሬሬԦ = ‫ܪ‬
ሬԦ + ‫ܯ‬
ሬሬԦ
= െ ‫ܯ‬
ߤ଴ ߤ଴ 3
3
3
The same result can also be obtained by using the magnetic scalar potential.
A.
ሬሬԦ ‫ڄ‬
If ݊ොᇱ is the normal vector outward from the surface of the cavity, ߪெ (‫ݎ‬Ԧ ᇱ ) = ‫ܯ‬
ሬሬԦ(‫ݎ‬Ԧ ᇱ ) = 0.
(െ݊ොᇱ ) = െ‫ ܯ‬cos ߠ. For a uniformly magnetized material, ߩெ (‫ݎ‬Ԧ ᇱ ) = െ‫׏‬ᇱ ‫ܯ ڄ‬
B.
Then, we have
(‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ ) ᇱ
ߪெ (‫ݎ‬Ԧ ᇱ )
1
1
ᇱ
ᇱ)
(‫ݎ‬
‫׏‬ර
݀ܽ
=
ර
ߪ
݀ܽ
Ԧ
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |ଷ
4ߨ ௌ ெ
4ߨ ௌ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
గ
(‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ ) ᇱ ‫ݖܯ‬Ƹ ଶగ
‫ܯ‬
1
ሬሬԦ
= െ ර cos ߠ
݀ܽ =
න ݀߶ ᇱ න cosଶ ߠ ᇱ sin ߠ ᇱ ݀ߠ ᇱ = ‫ܯ‬
ᇱ
ଷ
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ |
4ߨ ௌ
4ߨ ଴
3
଴
ሬԦ௦ (‫ݎ‬Ԧ) = െ
‫ܪ‬
4.
ሬԦ௠ = ‫ܪ‬
ሬԦ + ‫ܪ‬
ሬԦ௦ = ‫ܪ‬
ሬԦ + ଵ ‫ܯ‬
ሬሬԦ and ‫ܤ‬
ሬԦ௠ = ߤ଴ ‫ܪ‬
ሬԦ௠ give the molecular field in terms of
Equations ‫ܪ‬
ଷ
the macroscopic magnetic field and the magnetization in the sample. For most
diamagnetic and paramagnetic materials, the term
ଵ
ଷ
ሬሬԦ = ଵ ߯௠ ‫ܪ‬
ሬԦ is negligibly small, but
‫ܯ‬
ଷ
for ferromagnetic materials, the correction is quite important (Refer to the tables of the
properties of magnetic materials above).
(The topics of magnetic circuits (Sections 9-9~9-11 of Reitz) are too specific to be discussed in this
class. The topics of origins of diamagnetism, paramagnetism, and ferromagnetism in Sections from
10-2 to 10-5 of Reitz are needed to be more adequately covered in the course of solid-state physics
rather than classical electromagnetism.)
(HW #9)
Due: 20240529
1. Consider a sphere of linear magnetic material of radius ܽ and permeability ߤ placed in a region
ሬԦ଴ . There is no excess current.
of space containing an initially uniform magnetic field, ‫ܤ‬
ሬԦ and ‫ܤ‬
ሬԦ fields inside and outside the sphere.
(a) Using the magnetic scalar potential, find the ‫ܪ‬
ሬሬԦ induced inside the sphere.
(b) Find the magnetization ‫ܯ‬
(c) (optional) Solve the problem using the vector potential and show that the identical results are
obtained.
ሬԦ and ‫ܤ‬
ሬԦ fields inside and outside the spherical hard magnet with
2. In the class, we calculated the ‫ܪ‬
ሬሬԦ = ‫ݖܯ‬Ƹ , employing the magnetic scalar potential. Now, solve the problem
a constant magnetization ‫ܯ‬
using the vector potential.
(Hint: the lecture note and the additional reading material uploaded in the klms)
ሬሬԦ݀‫ݒ‬.
3. By definition, the magnetic moment of a macroscopic body of ܸ is given as ‫׬‬௏ ‫ܯ‬
(a) Show that where ܵ is the surface bounding ܸ,
ሬሬԦ݀‫ = ݒ‬න ‫ݎ‬Ԧߩெ ݀‫ ݒ‬+ ර ‫ݎ‬Ԧߪெ ݀ܽ
න ‫ܯ‬
௏
௏
ௌ
ሬሬԦ଴ .
(b) A permanent magnet in the shape of a sphere of radius ܴ has uniform magnetization ‫ܯ‬
Determine the magnetic moment of the magnet.
4. A permanent magnet has the shape of a right circular cylinder of length ‫ܮ‬. The magnetization
ሬሬԦ is uniform and has the direction of the cylinder axis.
‫ܯ‬
ሬԦெ .
(a) Find the magnetization current densities ‫ܬ‬Ԧெ and ‫ܭ‬
(b) Compare the current distribution with that of a solenoid.
PH231 Electromagnetism I (Spring 2024)
Lecture note 10: Electromagnetic Induction and Magnetic Energy
(Chapters 11 and 12 of the textbook by Reitz, Chapter 7 of the book by Purcell)
10.1 Introduction
- Once we've studied statics, it's a natural progression to study time dynamics. In this lecture, we
will study the effects of magnetic fields that change with time as the first time-dependent
phenomena in classical electromagnetism.
- We will begin by introducing the concept of electromotive force and discuss the case when a
current loop moves in a stationary magnetic field (constant in time). Then, we will see Faraday’s
brilliant experimental tactics and the universal law of induction, called Faraday’s law. The study of
electromagnetic induction leads to the concept of self and mutual inductance. We will also see
that the magnetic field, like the electric field, stores energy.
- Together with the Maxwell-Ampere’s law, the main topic of the next lecture note, Faraday’s law
reveals the relation between the electric and magnetic fields and the existence of electromagnetic
waves. The complete set of Maxwell's equations also initiates the study of special relativity.
10.2 Electromagnetic induction and Faraday’s law
(Section 11-1 of Reitz / Sections 7.1-7.5 of Purcell)
- Electromotive force
1.
Let us first define the electromotive force, or emf, around a circuit by
ࣟ = ර ‫ܧ‬ሬԦ ‫݈݀ ڄ‬Ԧ
஼
2.
With static electric and magnetic fields, the emf was always zero. Coulomb’s law based
on ‫ܧ × ׏‬ሬԦ = 0 cannot define the emf. Also, the laws of magnetostatics are completely
independent of the electric field and vice versa.
3.
The next logical way to produce an emf would be to move a loop dynamically in an
inhomogeneously distributed magnetic field in space or change the magnetic field
around the loop in time.
4.
ሬԦ൯, is always the
At this point, it is worth noting that the Lorentz force, ‫ܨ‬Ԧ = ‫ݍ‬൫‫ܧ‬ሬԦ + ‫ݒ‬Ԧ × ‫ܤ‬
electromagnetic force on a test charge ‫ ݍ‬and includes the force on the charge moving
in a magnetic field.
- Faraday’s efforts
1.
Faraday's law of electromagnetic induction is entirely a triumph
originating from numerous experimental observations.
2.
Michael Faraday tried to answer this question: If electricity,
charge currents, produces magnetism, why does magnetism
not produce electricity?
3.
He observed that magnetism can induce electricity by motion
(as mentioned above) and published the results in 1831.
4.
He continued to perform hundreds of experiments over next
several years and finally arrived at his law of electromagnetic
induction with the concept of magnetic field flux change.
5.
He collected his ideas and experimental results in his book (Experimental Researches
in Electricity, London, 1839), which has since been revised several times to include new
results. On the top right is the inside cover of the book and below are the contents.
6.
The first series (Series I) in the contents is about the first observation of electromagnetic
induction in 1831. You can find and enjoy the book (the 1839 edition) in the Internet
Archive (https://archive.org/details/in.ernet.dli.2015.500031/mode/2up).
7.
Here, I may emphasize that in his papers and book, he described his concepts and
experiments without any use of mathematics. Nevertheless, his brilliant mind was able
to come up with one of the most important achievements in the history of physics:
bridging the originally separate fields of electricity and magnetism.
- Faraday’s law of electromagnetic induction and Lenz’s law
1.
The results of a large number of experiments can be summarized with a change in the
2.
net magnetic flux Ȱ through the area enclosed by a closed circuit:
݀Ȱ
ࣟ=െ
݀‫ݐ‬
This result, known as Faraday’s law of electromagnetic induction, is independent of the
way the magnetic flux is changed: by moving the circuit or by changing the magnetic
field over time.
3.
From the point of view of the Lorentz force in a moving loop
A.
Consider a loop ‫ ܥ‬that occupies the position ‫ܥ‬ଵ at time ‫ ݐ‬and movies to the
position ‫ܥ‬ଶ at time ‫ ݐ‬+ ݀‫ ݐ‬with a velocity of ‫ݒ‬Ԧ in a non-uniform magnetic field
distribution in space, as shown in the figure below.
B.
The magnetic flux through the loop at the instant of time ‫ ݐ‬is
ሬԦ ‫݊ ڄ‬ො݀ܽ
Ȱ(‫ = )ݐ‬න ‫ܤ‬
ௌ
C.
At time ‫ ݐ‬+ ݀‫ݐ‬, a surface that spans the loop is the original surface ܵ augmented
by an infinitesimal change of ݀ܵ: ܵ + ݀ܵ:
ሬԦ ‫݊ ڄ‬ො݀ܽ = Ȱ(‫ )ݐ‬+ න ‫ܤ‬
ሬԦ ‫݊ ڄ‬ො݀ܽ
‫ܤ‬
Ȱ(‫ ݐ‬+ ݀‫ = )ݐ‬න
ௌାௗௌ
D.
On ݀ܵ, an element of surface area is ݊ො݀ܽ = ‫ݒ‬Ԧ݀‫݈݀ × ݐ‬Ԧ and thus
ሬԦ ‫ ڄ‬ൣ‫ݒ‬Ԧ݀‫݈݀ × ݐ‬Ԧ൧
‫ܤ‬
ሬԦ ‫݊ ڄ‬ො݀ܽ = ර
݀Ȱ = න ‫ܤ‬
ௗௌ
E.
ௗௌ
݀Ȱ
ሬԦ ‫ ڄ‬ൣ‫ݒ‬Ԧ × ݈݀Ԧ൧
=ර ‫ܤ‬
݀‫ݐ‬
஼
֜
஼ୀడௌ
Using ܽԦ ‫ ڄ‬൫ܾሬԦ × ܿԦ൯ = െ൫ܾሬԦ × ܽԦ൯ ‫ܿ ڄ‬Ԧ for the scalar triple product, we have
݀Ȱ
ሬԦ൯ ‫݈݀ ڄ‬Ԧ
= െ ර ൫‫ݒ‬Ԧ × ‫ܤ‬
݀‫ݐ‬
஼
F.
Now, the Lorentz force on a test charge ‫ ݍ‬that is carried along by the loop is just
ሬԦ, so the electromotive force, which is the line integral around the loop of
‫ݒݍ‬Ԧ × ‫ܤ‬
the force per unit charge, is given by
ሬԦ൯ ‫݈݀ ڄ‬Ԧ = െ
ࣟ = ර ൫‫ݒ‬Ԧ × ‫ܤ‬
஼
݀Ȱ
݀‫ݐ‬
G.
In fact, we did not even have to assume that the velocity ‫ݒ‬Ԧ is the same for all
parts of the loop.
4.
Lenz’s law: The direction of the electromotive force
A.
Considering the negative sign in the law, one can predict that an electromotive
force due to an increasing magnetic flux will tend to drive a positive charge around
the loop in a counter-clock-wise direction.
B.
The Biot-Savart law tells us such a current itself would create some flux through
the loop in a direction to counterpart the assumed flux change.
C.
It is summarized that the direction of the induced electromotive force is such that
the induced current creates a magnetic field that opposes the change in flux,
which is called Lenz’s law.
- Maxwell-Faraday equation: Generalization of Faraday's law
1.
Faraday found that a time-varying magnetic field also creates an emf without any
motion of the test loop.
2.
James Clerk Maxwell expanded on Faraday's concepts to build the quantitative
electromagnetic theory and revealed that a time-varying magnetic field is always paired
by a spatially-varying, non-conservative electric field, and vice versa.
A.
Faraday’s observation connects the magnetic flux and electromotive force as
ර ‫ܧ‬ሬԦ ‫݈݀ ڄ‬Ԧ = െ
஼
B.
݀
ሬԦ ‫݊ ڄ‬ො݀ܽ
න ‫ܤ‬
݀‫ ݐ‬ௌ
Using Stokes’ law, a mathematical representation of Faraday’s law, the MaxwellFaraday equation, is obtained.
ሬԦ
߲‫ܤ‬
න ቆ‫ܧ × ׏‬ሬԦ + ቇ ‫݊ ڄ‬ො݀ܽ = 0
߲‫ݐ‬
ௌ
C.
֜
‫ܧ × ׏‬ሬԦ +
ሬԦ
߲‫ܤ‬
=0
߲‫ݐ‬
In fact, the loop ‫ ܥ‬can be thought of as any closed geometric path in space, not
necessarily coincident with a real electrical circuit.
3.
Generalization of Faraday's law
A.
Although
we
derived
the
Maxwell-Faraday
equation
from
the
law
of
electromagnetic induction above, Maxwell’s equations (including the MaxwellFaraday equation) are actually the most fundamental, along with the Lorentz force
(the definition of the electric and magnetic field), that explains everything in
classical electromagnetism.
B.
That is, Faraday's law of electromagnetic induction must be derived and
generalized from the fundamental equations.
C.
Let us start with the time derivative of the magnetic flux at time ‫ݐ‬଴ . We have to
take into account that the integration domain as well as the integrand can change
in such a way that
݀Ȱ(‫)ݐ‬
݀
ሬԦ(‫݊ ڄ )ݐ‬ො݀ܽቤ
ቤ = න ‫ܤ‬
݀‫ ݐ‬௧
݀‫ ݐ‬ௌ(௧)
బ
௧బ
ሬԦ
߲‫ܤ‬
݀
ሬԦ(‫ݐ‬଴ ) ‫݊ ڄ‬ො݀ܽ
ቤ ‫݊ ڄ‬ො݀ܽ + න ‫ܤ‬
߲‫ݐ‬
݀‫ݐ‬
ௌ(௧బ )
ௌ(௧)
௧
=න
బ
D.
Employing the integral form of the Maxwell-Faraday equation to the first term,
݀Ȱ(‫)ݐ‬
݀
ሬԦ(‫ݐ‬଴ ) ‫݊ ڄ‬ො݀ܽ
ቤ = െර
‫ܧ‬ሬԦ ‫݈݀ ڄ‬Ԧ + න ‫ܤ‬
݀‫ ݐ‬௧
݀‫ݐ‬
డௌ(௧బ )
ௌ(௧)
బ
E.
The second term related to the motion of the loop has already been evaluated by
the Lorentz force in the previous part.
݀
ሬԦ(‫݊ ڄ ) ݐ‬ො݀ܽ = െ ර
ሬԦ(‫ݐ‬଴ )ቁ ‫݈݀ ڄ‬Ԧ
න ‫ܤ‬
ቀ‫ݒ‬Ԧ(‫ݐ‬଴ ) × ‫ܤ‬
݀‫ ݐ‬ௌ(௧) ଴
డௌ(௧బ )
F.
Putting the two terms and replacing ‫ݐ‬଴ by ‫ݐ‬, we obtain the generalization of the
law of induction as
݀Ȱ(‫)ݐ‬
ሬԦ(‫)ݐ‬൧ ‫݈݀ ڄ‬Ԧ
= െර
ൣ‫ܧ‬ሬԦ (‫ )ݐ‬+ ‫ݒ‬Ԧ(‫ܤ × )ݐ‬
݀‫ݐ‬
డௌ(௧)ୀ஼(௧)
G.
The first term is called the transformer emf due to a time-varying magnetic field
(governed by the Maxwell-Faraday equation), and the second term is called the
motional emf due to the Lorentz force on the charges by the motion/deformation
of the loop in the presence of the magnetic field.
10.2 Self and mutual inductance (Sections 11-2, 11-3, and 11-4 of Reitz / Sections 7.6-7.8 of Purcell)
- Self inductance
1.
The relationship between the magnetic flux and associated current in an isolated circuit
introduces a practical circuit parameter, called the self-inductance.
2.
3.
4.
For a rigid, stationary circuit, the only changes in flux result from changes in the current:
݀Ȱ ݀Ȱ ݀‫ܫ‬
=
݀‫ݐ‬
݀‫ݐ݀ ܫ‬
In any case, we can define a factor, called the self-inductance, as
݀Ȱ
‫=ܮ‬
݀‫ܫ‬
If the magnetic media in the problem are linear, Ȱ is directly proportional to the
current, and the self-inductance is a constant. The law of induction then gives
݀‫ܫ‬
ࣟ = െ‫ܮ‬
݀‫ݐ‬
- Example of self-inductance: toroidal coil of rectangular cross section.
1.
As an example of a circuit for which the self-inductance can be calculated, consider the
rectangular toroidal coil.
2.
A steady current ‫ ܫ‬flows in the coil of ܰ turns, which are completely winding.
3.
Using Ampere’s law, one can easily show that the magnetic fields exist only inside the
toroid. The produced magnetic field at a radial distance ‫ ݎ‬from the axis of the coil is
given as (This is actually a general physics problem)
ሬԦ = ߤ଴ ‫ܬ‬Ԧ
‫ܤ×׏‬
֜
2ߨ‫ߤ = ܤݎ‬଴ ܰ‫ܫ‬
֜
‫=ܤ‬
ߤ଴ ܰ‫ܫ‬
2ߨ‫ݎ‬
4.
The magnetic flux through one turn of the coil is then
௕
ߤ଴ ܰ‫ܫ‬
ߤ଴ ܰ‫݄ܫ‬
ܾ
Ȱ(one turn) = ݄ න
݀‫= ݎ‬
ln ൬ ൰
2ߨ‫ݎ‬
2ߨ
ܽ
௔
5.
The flux threading the entire circuit is ܰ times as great: Ȱ = ܰȰ(one turn)
6.
Hence, for a slowly varying current, the induced electromotive force and the selfinductance are given as
ࣟ=െ
݀Ȱ
ߤ଴ ܰ ଶ ݄
ܾ ݀‫ܫ‬
=െ
ln ൬ ൰
2ߨ
݀‫ݐ‬
ܽ ݀‫ݐ‬
֜
‫=ܮ‬
ߤ଴ ܰ ଶ ݄
ܾ
ln ൬ ൰
2ߨ
ܽ
- Mutual inductance
1.
In the previous case, we considered an isolated circuit. But, if there are ݊ circuits and
they have magnetic flux linking, the flux through ݅th circuit is written as
௡
Ȱ௜ = ෍ Ȱ௜௝
௝ୀଵ
2.
Here, Ȱ௜௝ is the flux through the ݅th circuit due to the ݆th circuit. The following figure
shows a schematic of the flux linking between two loop circuits, where the magnetic
field generated by a current flowing in the circuit 1 makes a flux through the circuit 2.
3.
The electromotive force in the ݅th circuit is then
௡
௡
௝ୀଵ
௝ୀଵ
݀Ȱ௜௝
݀Ȱ௜௝ ݀‫ܫ‬௝
݀Ȱ௜
ࣟ௜ = െ
= െ෍
= െ෍
݀‫ݐ‬
݀‫ݐ‬
݀‫ܫ‬௝ ݀‫ݐ‬
4.
In either case, linear or nonlinear, we can define the mutual inductance as
݀Ȱ௜௝
‫ܯ‬௜௝ =
for ݅ ് ݆
݀‫ܫ‬௝
5.
As the self-inductance, the mutual inductance is constant, if the magnetic media in the
problem are linear.
- Example of mutual inductance: concentric rings
1.
Consider two coplanar, concentric rings: a small ring ‫ܥ‬ଶ and a much larger ring ‫ܥ‬ଵ .
2.
Assuming ܴଵ ‫ܴ ب‬ଶ , we can neglect the variation of ‫ܤ‬ଵ over the interior of the small
ring. The flux trough the small ring is then
Ȱଶଵ = න ‫ܤ‬ଵ ݀ܽ ؆ (ߨܴଶଶ ) ‫ ڄ‬൬
஺మ
3.
ߤ଴ ߨܴଶଶ
ߤ଴ ‫ܫ‬ଵ
൰=
‫ܫ‬
2ܴଵ
2ܴଵ ଵ
The mutual inductance is therefore
݀Ȱଶଵ ߤ଴ ߨܴଶଶ
‫ܯ‬ଶଵ =
=
݀‫ܫ‬ଵ
2ܴଵ
- A reciprocal theorem: the Neumann formula
1.
In the previous example, we calculated ‫ܯ‬ଶଵ . Then, the next question is the mutual
inductance ‫ܯ‬ଵଶ about the emf induced in ‫ܥ‬ଵ by a changing current in ‫ܥ‬ଶ .
2.
Here is a remarkable reciprocal theorem: For any two circuits
‫ܯ‬௜௝ = ‫ܯ‬௝௜
3.
This is known as the Neumann formula.
4.
Proof (The Reitz textbook employs Biot-Savart’s law, but I here use the vector potential.
But, two methods are intrinsically the same. You may compare them):
A.
The emf in ‫ܥ‬ଵ can be written in terms of the
vector potential
ࣟଵ = െ
B.
݀
݀
ሬԦ ‫݊ ڄ‬ሬԦ݀ܽ = െ ර ‫ܣ‬Ԧ ‫݈݀ ڄ‬Ԧଵ
න ‫ܤ‬
݀‫ ݐ‬ௌభ
݀‫ ݐ‬஼భ
Let's assume that the vector potential at ‫ܥ‬ଵ comes
from currents in ‫ܥ‬ଶ . Then it can be written as a line
integral around ‫ܥ‬ଶ :
ߤ଴
‫ܬ‬Ԧଶ (‫ݎ‬Ԧଶ )
ߤ଴
‫ܫ‬ଶ ݈݀Ԧଶ
‫ܣ‬Ԧ =
න
݀‫ݒ‬ଶ =
ර
4ߨ ஼మ |‫ݎ‬Ԧଵ െ ‫ݎ‬Ԧଶ |
4ߨ ஼మ |‫ݎ‬Ԧଵ െ ‫ݎ‬Ԧଶ |
C.
Combining the two equations, we can express the
emf in ‫ܥ‬ଵ as a double line integral:
݀‫ܫ‬ଶ
ߤ଴ ݀
‫ܫ‬ଶ ݈݀Ԧଶ
ߤ଴
݈݀Ԧଶ ‫݈݀ ڄ‬Ԧଵ ݀‫ܫ‬ଶ
ࣟଵ = െ
ර ර
‫݈݀ ڄ‬Ԧଵ = െ ර ර
= െ‫ܯ‬ଵଶ
4ߨ ݀‫ ݐ‬஼భ ஼మ |‫ݎ‬Ԧଵ െ ‫ݎ‬Ԧଶ |
4ߨ ஼భ ஼మ |‫ݎ‬Ԧଵ െ ‫ݎ‬Ԧଶ | ݀‫ݐ‬
݀‫ݐ‬
֜ ‫ܯ‬ଵଶ =
ߤ଴
݈݀Ԧଶ ‫݈݀ ڄ‬Ԧଵ
ර ර
= ‫ܯ‬ଶଵ
4ߨ ஼భ ஼మ |‫ݎ‬Ԧଵ െ ‫ݎ‬Ԧଶ |
D.
Therefore, for a system with only two coils, the coefficients ‫ܯ‬ଵଶ and ‫ܯ‬ଶଵ are often
represented by the symbol ‫ ܯ‬without subscripts, simply the mutual inductance:
‫ܯ = ܯ‬ଵଶ = ‫ܯ‬ଶଵ
E.
We can see from this integral that ‫ܯ‬ଵଶ depends only on the circuit geometry. It
depends on a kind of average separation of the two circuits, with the average
weighted most for parallel segments of the two coils.
5.
We can encounter many of reciprocal theorems in the electromagnetism, including the
Lorentz reciprocity, the optical reciprocity theorem, and so on. If you are interested in
such reciprocities, visit https://en.wikipedia.org/wiki/Reciprocity_(electromagnetism).
10.3 Magnetic energy density (Sections 12-2 and 12-3 of Reitz / Section 7.10 of Purcell)
- Torque and energy for a localized current loop
1.
In the lecture note 08, we have derived that the lowest-order element of a localized
ଵ
ଵ
ሬሬԦ = ‫ׯ ܫ‬஼ ‫ݎ‬Ԧ × ݈݀Ԧ = ‫׬‬௏ ‫ݎ‬Ԧ × ‫ܬ‬Ԧ(‫ݎ‬Ԧ)݀‫ݒ‬.
current loop is the magnetic dipole moment ݉
ଶ
ଶ
2.
Also, we show that the Lorentz force on the current loop results in the torque given as
1
ሬԦ = ݉
ሬԦ
߬Ԧ = ‫ ܫ‬ቈ ර ‫ݎ‬Ԧ × ݈݀Ԧ቉ × ‫ܤ‬
ሬሬԦ × ‫ܤ‬
2 ஼
3.
Letting the angle ߠ between the directions of the applied magnetic field and the
4.
magnetic dipole moment, we get the torque and mechanical energy as
߲ܷ
ሬԦ
߬Ԧ = െ
ߠ෠ ֜
ܷ୫ୣୡ୦ = െ݉‫ ܤ‬cos ߠ = െ݉
ሬሬԦ ‫ܤ ڄ‬
߲ߠ
One can also show that for a localized, divergence-less current distribution (current
loops), the result of the mechanical energy is general not only for the rotational motion,
but also for any generalized motion (see the additional reading material).
- Mechanical and electrical energies
1.
ሬԦ, the previously obtained result, is not the total energy of a current
ܷ୫ୣୡ୦ = ܹ = െ݉
ሬሬԦ ‫ܤ ڄ‬
distribution (or loop).
2.
The reason is that the creation of a steady-state configuration of currents and
associated magnetic fields involves a transient period during which the currents and
fields are brought from zero to the final values. For such time-varying flux, there are
induced electromotive forces that cause the source of currents to do work, ܷୣ୪ୣୡ୲ .
3.
Faraday’s law provides an important fact between the mechanical and electrical
energies on a current loop:
ܷ୫ୣୡ୦ = െܷୣ୪ୣୡ୲
A.
As an example, consider a wire carrying an electrical current ‫ ܫ‬and moving with
ሬԦ = െ‫ݖܤ‬Ƹ as illustrated below.
‫ݒ‬Ԧ in a magnetic field ‫ܤ‬
B.
C.
From the Lorentz force, the chance of the mechanical energy is given as
ܷ݀୫ୣୡ୦
= ‫ܨ‬Ԧ ‫ݒ ڄ‬Ԧ = െ‫ݒܤܮܫ‬
݀‫ݐ‬
From Faraday’s law, the change of the electrical energy by an external source (e.g.
battery) is given as
ܷ݀ୣ୪ୣୡ୲
݀
ܷ݀
ሬԦ ‫݊ ڄ‬ො݀ܽ = െ‫( × ܫ‬െ‫ = )ܮݒܤ‬െ ୫ୣୡ୦
= ‫ ܫ‬ර ‫ܧ‬ሬԦ ‫݈݀ ڄ‬Ԧ = െ‫ ܫ‬න ‫ܤ‬
݀‫ݐ‬
݀‫ݐ‬
݀‫ ݐ‬ௌ
஼
֜ ݀(ܷୣ୪ୣୡ୲ + ܷ୫ୣୡ୦ ) = 0 ֜ ܷୣ୪ୣୡ୲ + ܷ୫ୣୡ୦ = constant
D.
Letting the constant be zero (this value is just a reference), we finally have
ܷ୫ୣୡ୦ = െܷୣ୪ୣୡ୲
E.
Here, we haven’t considered the work and energy on an external system carrying
the magnetic field, which are necessary to be included to address the total energy
of the entire system.
- Total energy
1.
Now, let us discuss the total energy of the entire system based on the principle of
relativity in the following way:
A.
Suppose we imagine a complete system consisting of a loop and a coil such as
that drawn below.
B.
When moving the loop toward the stationary coil, we know that the electrical
energy accumulated on the loop ܷୣ୪ୣୡ୲,ଵ is just equal and opposite to the
mechanical work done, so that
ܷ୫ୣୡ୦,ଵ + ܷୣ୪ୣୡ୲,ଵ = 0
C.
Note that the electrical energy accumulated on the coil ܷୣ୪ୣୡ୲,ଶ is not included.
D.
If the loop is at rest, and the coil is moved toward it, the coil is then moving into
the field produced by the loop. The same arguments would give that
ܷ୫ୣୡ୦,ଶ + ܷୣ୪ୣୡ୲,ଶ = 0
E.
The principle of relativity tells that the mechanical energy should be identical for
both the cases as
ܷ୫ୣୡ୦,ଵ = ܷ୫ୣୡ୦,ଶ = ܷ୫ୣୡ୦
F.
When moving the loop and coil at the same time to the final configuration, based
on the results above, the total energy consumption becomes
ܷ୲୭୲ୟ୪ = ܷ୫ୣୡ୦ + ܷୣ୪ୣୡ୲,ଵ + ܷୣ୪ୣୡ୲,ଶ = െܷ୫ୣୡ୦
2.
The work against the induced electromotive force is twice as large as, and of the
opposite sign to, the potential-energy change of the body: ܷୣ୪ୣୡ୲,ଵ + ܷୣ୪ୣୡ୲,ଶ = െ2ܷ୫ୣୡ୦ .
3.
If there is a mechanical change on a circuit but its current is maintained, the force on
the circuit is the gradient of the total (magnetic) energy:
‫ܨ‬Ԧ = െ‫ܷ׏‬୫ୣୡ୦ = ‫୲ܷ׏‬୭୲ୟ୪
4.
For the magnetic dipole moment, the total energy is then
ሬԦ
ܷ୲୭୲ୟ୪ = െܷ୫ୣୡ୦ = +݉
ሬሬԦ ‫ܤ ڄ‬
- Energy density in the magnetic field
1.
First, ‫ܬ ڄ ׏‬Ԧ = 0 holds to any desired degree of accuracy.
2.
For a closed circuit ‫ ܥ‬bounded by a surface ܵ and carrying ‫ܫ‬, the increment of work
ሬԦ in the magnetic
against the induced electromotive force ࣟ in terms of the change ߜ‫ܤ‬
field during a time interval ߜ‫ ݐ‬is given as
ߜܹ = െ‫ܫ = ݐߜࣟܫ‬
3.
݀Ȱ
ሬԦ ‫݊ ڄ‬ො݀ܽ
ߜ‫ ܫ = ݐ‬න ߜ‫ܤ‬
݀‫ݐ‬
ௌ
Expressing the magnetic field in terms of the vector potential,
ߜܹ = ‫ ܫ‬න ݊ො ‫ ڄ‬൫‫ܣߜ × ׏‬Ԧ൯݀ܽ = ‫ ܫ‬ර ߜ‫ܣ‬Ԧ ‫݈݀ ڄ‬Ԧ
ௌ
4.
஼
Case of magnetostatics
A.
For a general current distribution, ‫݈݀ܫ‬Ԧ = ‫ܬ‬Ԧ݀ ଷ ‫ݔ‬Ԧ and
ߜܹ = න ‫ܬ‬Ԧ ‫ܣߜ ڄ‬Ԧ݀‫ݒ‬
B.
For rigid circuits and linear media, let us call again the fraction 0 ൑ ߙ ൑ 1
representing an intermediate stage as ߜ‫ܣ‬Ԧ = ߙ‫ܣ‬Ԧ. Then we have
ଵ
1
1
ܹ = න ݀ߙ න ‫ܬ‬Ԧ ‫ܣߙ ڄ‬Ԧ݀‫ = ݒ‬න ‫ܬ‬Ԧ ‫ܣ ڄ‬Ԧ݀‫ݑ ֜ ݒ‬௠ = ‫ܬ‬Ԧ ‫ܣ ڄ‬Ԧ
2
2
଴
5.
Generalization
A.
Employing Ampere’s law, the expression above becomes
ሬԦ ൯ ‫ܣߜ ڄ‬Ԧ݀‫ݒ‬
ߜܹ = න ‫ܬ‬Ԧ ‫ܣߜ ڄ‬Ԧ݀‫ = ݒ‬න൫‫ܪ × ׏‬
ሬԦ ‫ ڄ‬൫‫ܣߜ × ׏‬Ԧ൯ + ‫ ڄ ׏‬൫‫ܪ‬
ሬԦ × ߜ‫ܣ‬Ԧ൯൧݀‫ݒ‬
= නൣ‫ܪ‬
B.
For a localized field distribution (or considering all space),
ሬԦ ‫ ڄ‬൫‫ܣߜ × ׏‬Ԧ൯݀‫ = ݒ‬න ‫ܪ‬
ሬԦ ‫ܤߜ ڄ‬
ሬԦ݀‫ݒ‬
ߜܹ = න ‫ܪ‬
C.
6.
For a linear para- or diamagnetic medium,
1
1
1
ሬԦ ‫ܤߜ ڄ‬
ሬԦ = ߜ൫‫ܪ‬
ሬԦ ‫ܤ ڄ‬
ሬԦ൯ ֜ ܷ = ܹ = න ‫ܪ‬
ሬԦ ‫ܤ ڄ‬
ሬԦ݀‫ݑ ֜ ݒ‬௠ = ‫ܪ‬
ሬԦ ‫ܤ ڄ‬
ሬԦ
‫ܪ‬
2
2
2
ଵ
ଵ
As commented earlier, ‫ݑ = ݑ‬௘ + ‫ݑ‬௠ = ߩ߮ + ‫ܬ‬Ԧ ‫ܣ ڄ‬Ԧ is only valid for statics, while ‫= ݑ‬
ଵ
ଶ
ଶ
ଶ
ሬԦ + ଵ ‫ܪ‬
ሬԦ ‫ܤ ڄ‬
ሬԦ is generally correct. We will see this point in detail in the discussion of
‫ܧ‬ሬԦ ‫ܦ ڄ‬
ଶ
the Poynting theorem in the next lecture
10.4 Magnetic energy of coupled circuits (Section 12-1 of Reitz / Section 7.10 of Purcell)
- The magnetic energy in coupled circuits
1.
A system of ܰ distinct current-carrying circuits, the ݅th one with total current ‫ܫ‬௜ , in
otherwise empty space.
2.
From the discussion in the previous section, the total energy is
1
1
ߤ଴
‫ܬ‬Ԧ(‫ݎ‬Ԧ ᇱ )
ܹ = න ‫ܬ‬Ԧ ‫ܣ ڄ‬Ԧ݀‫ = ݒ‬න ‫ܬ‬Ԧ(‫ݎ‬Ԧ) ‫ ڄ‬ቈ න
݀‫ ݒ‬ᇱ ቉ ݀‫ݒ‬
4ߨ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
2
2
‫ܬ‬Ԧ(‫ݎ‬Ԧ) ‫ܬ ڄ‬Ԧ(‫ݎ‬Ԧ ᇱ )
ߤ଴
න ݀‫ ݒ‬න ݀‫ ݒ‬ᇱ
=
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
8ߨ
3.
The integrals can be broken up into sums of separated integrals over each circuit ‫ܬ‬Ԧ(‫ݎ‬Ԧ) =
Ԧ
σே
௜ୀଵ ‫ܫ‬௜ ݈݀௜ so that
ே
ே
௜ୀଵ
௝ୀଵ
ே
ߤ଴
‫ܫ‬௜ ‫ܫ‬௝
ߤ଴
ܹ=
෍ ර න ݈݀Ԧ௜ ‫ ڄ‬෍ ර ݈݀Ԧ௝ᇱ
8ߨ
ห‫ݎ‬Ԧ௜ െ ‫ݎ‬Ԧ௝ᇱ ห
஼೔
஼ೕ
ே
ே
݈݀Ԧ௜ ‫݈݀ ڄ‬Ԧ௝ᇱ
ߤ଴
ଶ
෍ ‫ܫ‬௜ ර ර
+
෍
෍
‫ܫ‬
‫ܫ‬
ර
ර
=
௜
௝
8ߨ
Ԧ௜ െ ‫ݎ‬Ԧ௜ᇱ ห 4ߨ
Ԧ௜ െ ‫ݎ‬Ԧ௝ᇱ ห
஼೔ ஼ೕ ห‫ݎ‬
஼೔ ஼ೕ ห‫ݎ‬
݈݀Ԧ௜ ‫݈݀ ڄ‬Ԧ௜ᇱ
௜ୀଵ
௜ୀଵ ௝வଵ
ே
ே
ே
௜ୀଵ
௜ୀଵ ௝வଵ
1
‫ ؠ‬෍ ‫ܮ‬௜ ‫ܫ‬௜ଶ + ෍ ෍ ‫ܯ‬௜௝ ‫ܫ‬௜ ‫ܫ‬௝
2
4.
The self- and mutual inductances are determined as
‫ܮ‬௜ ‫ؠ‬
5.
ߤ଴
݈݀Ԧ௜ ‫݈݀ ڄ‬Ԧ௜ᇱ
ර ර
4ߨ ஼೔ ஼ೕ ห‫ݎ‬Ԧ௜ െ ‫ݎ‬Ԧ௜ᇱ ห
and ‫ܯ‬௜௝ ‫ؠ‬
݈݀Ԧ௜ ‫݈݀ ڄ‬Ԧ௝ᇱ
ߤ଴
ර ර
4ߨ ஼೔ ஼ೕ ห‫ݎ‬Ԧ௜ െ ‫ݎ‬Ԧ௝ᇱ ห
In fact, the integral for the self-inductance is not easy to be calculated due to the term
1/|‫ݎ‬Ԧ௜ െ ‫ݎ‬Ԧ௜ᇱ | diverging at ‫ݎ‬Ԧ௜ = ‫ݎ‬Ԧ௜ᇱ . Nevertheless, you can calculate the self-inductance. For
example, the solenoid with a length and cross-section of ݈ and ܵ has a self-inductance
given as
ଶ
ሬԦ ‫ܤ ڄ‬
ሬԦ
1
1
1 ‫ܤ‬
1 ߤܰ‫ܫ‬
ߤܰ ଶ ܵ
ଷ
ሬԦ ‫ܤ ڄ‬
ሬԦ݀ ଷ ‫ݔ‬Ԧ ֜ ‫ = ܮ‬න
ܹ = ‫ܫܮ‬ଶ = න ‫ܪ‬
݀
‫ݔ‬
Ԧ
=
൬
×
݈ܵ൰
=
݈
ߤ
2
2
‫ܫ‬ଶ
ߤ‫ܫ‬ଶ ݈
- The case of two coupled circuits
1.
We now wish to show that an interesting inequality exists between mutual Inductance
‫ ܯ‬and the self-inductances ‫ܮ‬ଵ and ‫ܮ‬ଶ of the two coils.
2.
Now our energy equation can equally well be written in the following form:
1
1
1
‫ ܯ‬ଶ 1
‫ܯ‬ଶ ଶ
ܷ = ‫ܮ‬ଵ ‫ܫ‬ଵଶ + ‫ܫܯ‬ଵ ‫ܫ‬ଶ + ‫ܮ‬ଶ ‫ܫ‬ଶଶ = ‫ܮ‬ଵ ൬‫ܫ‬ଵ + ‫ܫ‬ଶ ൰ + ቆ‫ܮ‬ଶ െ
ቇ‫ܫ‬
‫ܮ‬ଵ ଶ
2
2
2
‫ܮ‬ଵ
2
3.
If the energy is to be positive, the last term must be greater than zero. We have the
requirement that
‫ܮ‬ଵ ‫ܮ‬ଶ ൒ ‫ܯ‬ଶ
4.
The relation between ‫ ܯ‬and the self-inductances is usually written as
‫݇ = ܯ‬ඥ‫ܮ‬ଵ ‫ܮ‬ଶ
5.
The constant ݇ is called the coefficient of coupling, which can be positive or negative,
depending on the sign conventions for the currents ‫ܫ‬ଵ and ‫ܫ‬ଶ .
6.
The concept of the coupling coefficient is very crucial in lots of coupled systems, e.g.
coupled optical or microwave resonators.
7.
If most of the flux from one coil links the other coil, the coefficient of coupling is near
one; we say the coils are "tightly coupled." If the coils are far apart or otherwise
arranged so that there is very little mutual flux linkage, the coefficient of coupling is
near zero and the mutual inductance is very small.
(The topic of RLC circuits (Section 11-5 of Reitz) is for general physics, and the topic of hysteresis
loss (Section 12-4) is too specific. So, we will skip these two topics.)
(HW #10)
Due: 20240605
1. For magnetostatics, the magnetic energy is evaluated by
1
ܷ௠ = න
‫ܬ‬Ԧ ‫ܣ ڄ‬Ԧ݀‫ݒ‬
2 ୟ୪୪ ୱ୮ୟୡୣ
Show that for a linear para- or diamagnetic medium, the magnetostatic energy is equivalent to
1
ሬԦ ‫ܤ ڄ‬
ሬԦ݀‫ݒ‬
ܷ௠ = න ‫ܪ‬
2
2. Consider a system of ܰ coupled circuits. Show that the magnetic energy of the system is given
by
ே
ே
௜ୀଵ
௜ୀଵ ௝வଵ
ே
1
1
ܷ௠ = න ‫ܬ‬Ԧ ‫ܣ ڄ‬Ԧ݀‫ = ݒ‬෍ ‫ܮ‬௜ ‫ܫ‬௜ଶ + ෍ ෍ ‫ܯ‬௜௝ ‫ܫ‬௜ ‫ܫ‬௝
2
2
where the self- and mutual inductances are determined as
‫ܮ‬௜ ‫ؠ‬
ߤ଴
݈݀Ԧ௜ ‫݈݀ ڄ‬Ԧ௜ᇱ
ර ර
4ߨ ஼೔ ஼ೕ ห‫ݎ‬Ԧ௜ െ ‫ݎ‬Ԧ௜ᇱ ห
and ‫ܯ‬௜௝ ‫ؠ‬
݈݀Ԧ௜ ‫݈݀ ڄ‬Ԧ௝ᇱ
ߤ଴
ර ර
4ߨ ஼೔ ஼ೕ ห‫ݎ‬Ԧ௜ െ ‫ݎ‬Ԧ௝ᇱ ห
3. A square wire frame with side length ݈ has total resistance ܴ is being pulled with velocity ‫ݒ‬Ԧ out
ሬԦ = ‫ݖܤ‬Ƹ pointing out of the page (the shaded
of a region where there is a uniform magnetic field ‫ܤ‬
area in the figure below). Consider the moment when the left corner is a distance ‫ ݔ‬inside the
shaded area.
(a) Show that the force needed to apply to the square so that it moves with a constant speed ‫ ݒ‬is
given by
4‫ܤ‬ଶ ‫ ݔ‬ଶ ‫ݒ‬
‫ݔ‬ො
ܴ
(b) Verify that the work on does from ‫ݔ = ݔ‬଴ ൑ ݈/ξ2 down to ‫ = ݔ‬0 equals the energy dissipated
‫ܨ‬Ԧ = െ
in the resistor.
4. (optional) Consider a dielectric cylinder of permittivity ߝ, radius ܽ, and length ݈, which rotates
ሬԦ = ‫ݖܤ‬Ƹ exists parallel to the
ሬԦ = ߱‫ݖ‬Ƹ . A uniform magnetic field ‫ܤ‬
about its axis with angular velocity ߱
cylinder axis. The Lorentz force on charges in the dielectric causes the polarization. Since the amount
of surface charges bound by the induced polarization is insignificant, we can neglect the magnetic
field caused by their rotation.
(a) Find the Lorentz force per a unit charge depending on the radial position ‫ ݎ‬from the axis.
(b) The Lorentz force acts as an effective electric field ‫ܧ‬ሬԦୣ୤୤ . Find the contribution of the effective
electric field on the polarization.
(c) As we learned in electrostatics, the polarization produces a volume charge density ߩ௉ = െ‫ܲ ڄ ׏‬ሬԦ
and a surface charge density ߪ௉ = ܲሬԦ ‫݊ ڄ‬ො. Assuming that the polarization can be written as ܲሬԦ = ‫ݎݎܥ‬Ƹ ,
find the total polarization volume charge ܳ௏ inside the cylinder and the total polarization surface
charge ܳௌ on the side, and show that ܳ௏ + ܳௌ = 0. Note that the electric is initially neutral.
(d) According to Gauss’ law, the polarization charges also generate an electric field ‫ܧ‬ሬԦ ᇱ . Using the
result from (c), find the electric field ‫ܧ‬ሬԦ ᇱ with respect to ‫ܥ‬.
(e) The polarization is then given by ܲሬԦ = (ߝ െ ߝ଴ )‫ܧ‬ሬԦ = (ߝ െ ߝ଴ )൫‫ܧ‬ሬԦୣ୤୤ + ‫ܧ‬ሬԦ ᇱ ൯. Show that
ߝ െ ߝ଴
߱‫ݎݎܤ‬Ƹ
ܲሬԦ = ‫ݎݎܥ‬Ƹ = 2ߝ଴
ߝ + ߝ଴
(f) Find the induced polarization charge in the dielectric and the bound surface charge density.
PH231 Electromagnetism I (Spring 2024)
Lecture note 11: Maxwell’s Equations and Poynting Theorem
(Chapter 16 of the textbook by Reitz, Chapter 9 of the book by Purcell)
11.1 Introduction
- In this chapter we return to the complete set of four Maxwell equations that we saw at the
beginning of this course. So far, we have been studying Maxwell's equations in bits and pieces; it
is time to add one last piece and put them all together. We will then have the complete and
correct story of classical electromagnetism.
- The following is a brief review of the puzzles associated with electromagnetic fields, which can
change in any way over time. Gauss’s law always applies to both dynamic and static fields. Since
ሬԦ through any closed surface is always zero. Faraday's
there are no magnetic charges, the flux of ‫ܤ‬
law, ‫ܧ × ׏‬ሬԦ = െ
ሬԦ
డ஻
డ௧
ఫԦ
ሬԦ = െ ,
, is also generally true. According to Ampere’s law, we said that ܿ ଶ ‫ܤ × ׏‬
ఢబ
but the correct general equation has a new part, called the displacement current, which was
discovered by J. C. Maxwell.
- We will also derive a law of the electromagnetic energy and their local conservation, known as
the Poynting theorem, and the concept of electromagnetic waves. Finally, we will see the general
solution of the Maxwell equations and understand the complete realm of classical physics.
11.2 Maxwell’s equations and their empirical basis
(Sections 16-1, 16-2, and 16-6 of Reitz / Sections 9.1-9.3 of Purcell)
- Maxwell-Ampere’s law
1.
Until Maxwell's work, the known laws of electricity and magnetism were those we have
studied so far. In particular, the equation for the magnetic field of steady currents,
Ampere’s circuital law, was known as
ሬԦ = ߤ଴ ‫ܬ‬Ԧ
‫ܤ×׏‬
2.
Maxwell began by considering these known laws and expressing them as differential
equations, although the ‫ ׏‬notation was not yet Invented.
3.
If one takes the divergence of this equation, the left-hand side will be zero, because
the divergence of a curl is always zero.
ሬԦ൯ = 0 = ߤ଴ (‫ ڄ ׏‬ଔԦ)
‫ ڄ ׏‬൫‫ܤ × ׏‬
4.
So, this equation requires that the divergence of ଔԦ also be zero. But if the divergence
of ଔԦ is zero, then the total flux of current out of any closed surface is also zero. This
certainly cannot in general be zero, because we know that the charges can be moved
from one place to another.
5.
We have the equation that expresses the very fundamental law that electric charge is
conserved - any flow of charge must come from some supply/source.
߲ߩ
‫ܬ ڄ ׏‬Ԧ = െ
߲‫ݐ‬
6.
Maxwell proposed that this problem could be solved by adding the term related to
డாሬԦ
డ௧
to the right-hand side of the equation of Ampere’s law. According to Gauss’ law, the
charge density is proportional to the divergence of the electric field; ‫ܧ ڄ ׏‬ሬԦ = ߩ/ߝ଴ .
߲‫ܧ‬ሬԦ
ሬԦ = ߤ଴ ‫ܬ‬Ԧ + ߤ଴ ߝ଴
‫ܤ×׏‬
߲‫ݐ‬
߲‫ܧ‬ሬԦ
߲ߩ
ሬԦ൯ = ߤ଴ ቆ‫ ڄ ׏‬ଔԦ + ߝ଴ ‫ ڄ ׏‬ቇ = ߤ଴ ൬‫ ڄ ׏‬ଔԦ + ൰ = 0
֜ ‫ ڄ ׏‬൫‫ܤ × ׏‬
߲‫ݐ‬
߲‫ݐ‬
7.
ሬԦ
డா
This is the Maxwell-Ampere equation, and the term, ‫ܬ‬Ԧ஽ = ߝ଴ , is called the
డ௧
displacement current. Of course, in a macroscopic dielectric, the displacement current
ሬԦ
ሬԦ
డ௧
డ௧
డா
డ஽
is given by ‫ܬ‬Ԧ஽ = ߝ = .
- How the displacement current works
1.
We shall see that Maxwell's little addition to the Ampere’s circuital equation has farreaching consequences. We can touch on only a couple of them in this chapter.
2.
As our first example, we may consider a small parallel-plate capacitor
A.
Assume that the capacitor is being charged by a constant current ‫ܫ‬.
B.
If Ampere’s circuital law is applied to the contour ‫ ܥ‬and the surface ܵଵ , we find
ሬԦ ‫݈݀ ڄ‬Ԧ = ර ൫‫ܤ × ׏‬
ሬԦ൯ ‫݊ ڄ‬ො݀ܽ = ර ߤ଴ ‫ܬ‬Ԧ ‫݊ ڄ‬ො݀ܽ = ߤ଴ ‫ܫ‬
ර ‫ܤ‬
஼
ௌభ
ௌభ
C.
Here, the direction of ݊ො is set to be the same as the direction of the current ‫ܫ‬.
D.
On the other hand, if Ampere’s circuital law is applied to the same contour ‫ ܥ‬and
the surface ܵଶ , we get the controversial result
ሬԦ ‫݈݀ ڄ‬Ԧ = ර ൫‫ܤ × ׏‬
ሬԦ൯ ‫݊ ڄ‬ො݀ܽ = ර ሬ0Ԧ ‫݊ ڄ‬ො݀ܽ = 0
ර ‫ܤ‬
஼
E.
ௌమ
ௌమ
Again, the displacement current solves the problem. If the electric field can be
ignored outside the capacitor (an ideal capacitor), the electric field between the
parallel plates, in terms of the charge density (ߪ) on the right plate, is given by
ߝ଴ ‫ܧ‬ሬԦ = െߪ݊ො
F.
If the Maxwell-Ampere equation is then applied to the contour ‫ ܥ‬and the surface
ܵଶ , we get
ሬԦ ‫݈݀ ڄ‬Ԧ = ර ൫‫ܤ × ׏‬
ሬԦ൯ ‫݊ ڄ‬ො݀ܽ = ර ቆߤ଴ ‫ܬ‬Ԧ + ߤ଴ ߝ଴
ර ‫ܤ‬
஼
= ර ߤ଴ ߝ଴
ௌమ
G.
3.
ௌమ
ௌమ
߲‫ܧ‬ሬԦ
ቇ ‫݊ ڄ‬ො݀ܽ
߲‫ݐ‬
߲‫ܧ‬ሬԦ
߲
߲
߲ܳ
‫݊ ڄ‬ො݀ܽ = ߤ଴ ර ߝ଴ ‫ܧ‬ሬԦ ‫݊ ڄ‬ො݀ܽ = െߤ଴ ර ߪ݀ܽ = െߤ଴
= ߤ଴ ‫ܫ‬
߲‫ݐ‬
߲‫ ݐ‬ௌమ
߲‫ ݐ‬ௌమ
߲‫ݐ‬
This result is generally correct, as shown in the figure below.
As the second example, consider a spherically symmetric radial distribution of current.
A.
Let’s assume that the radial current has the same magnitude in all directions.
B.
Let the total charge inside any radius ‫ ݎ‬be ܳ(‫)ݎ‬. If the radial current density at
C.
the same radius is ‫)ݎ(ܬ‬, the charge conservation law requires
߲ܳ(‫)ݎ‬
= െ4ߨ‫ ݎ‬ଶ ‫)ݎ(ܬ‬
߲‫ݐ‬
We now ask about the magnetic field produced by the currents in this situation.
Suppose we draw some loop Ȟ on a sphere of radius ‫ݎ‬, as shown in the figure
above. There is some current through this loop, so we might expect to find a
magnetic field circulating in the direction shown.
D.
But we are already in difficulty. A different choice of Ȟ would allow us to conclude
that its direction is exactly opposite to that shown. So how can there be any
ሬԦ around the currents?
circulation of ‫ܤ‬
E.
ሬԦ depends not only on
We are saved by Maxwell's equation. The circulation of ‫ܤ‬
the total current through Ȟ but also on the rate of change with time of the electric
flux through it. It must be that these two parts just cancel.
ொ(௥)
F.
From Gauss’ law, the electric field at the radius ‫ ݎ‬must be
G.
Since Gauss’ law is already correct in dynamics, the rate of change is then
ସగఢబ ௥ మ
.
߲‫ܧ‬
1 ߲ܳ(‫)ݎ‬
‫)ݎ(ܬ‬
=
=െ
߲‫ ݐ‬4ߨ߳଴ ‫ ݎ‬ଶ ߲‫ݐ‬
߳଴
H.
ሬԦ is always zero; There is
Therefore, the two source terms cancel and the curl of ‫ܤ‬
no magnetic field in this case.
4.
From our discussion so far of Maxwell's new term, you may have the impression that
it doesn't add much. It is true that if we just consider the forth equation by itself,
nothing particularly new comes out. But Maxwell's small change, when combined with
the other equations, does indeed produce much that is new and important.
- Maxwell’s equations and All of classical physics
1.
James Clerk Maxwell (1831-1879) indeed brought together all of the laws of electricity
and magnetism and made one complete and beautiful theory; we got all the pieces of
the classical electromagnetism.
2.
We may ask whether the mathematical equations are true or false. This is answered by
doing experiments, and countless experiments have confirmed Maxwell's equations. In
this light, it is apparent that they cannot be proved; however, the applicability to any
situation can be verified.
3.
In the following table, we have all that was known of fundamental classical physics,
that is, the physics that was known by 1905 (excluding the general relativity). Here it is
all in one table. With these equations we can understand the complete realm of classical
physics.
ߩ
‫ܧ ڄ ׏‬ሬԦ =
‫ۓ‬
ߝ଴
ۖ
߲‫ܧ‬ሬԦ
‫ܬ‬Ԧ
ۖ ଶ
ሬԦ െ
ܿ ‫ܤ×׏‬
=
߲‫ߝ ݐ‬଴
‫۔‬
ሬԦ = 0
‫׏‬
‫ڄ‬
‫ܤ‬
ۖ
ሬԦ
߲‫ܤ‬
ۖ
ሬԦ +
‫׏‬
×
‫ܧ‬
=0
‫ە‬
߲‫ݐ‬
߲ߩ
+ ‫ܬ ڄ ׏‬Ԧ = 0
߲‫ݐ‬
‫ܨ‬Ԧ =
4.
Maxwellᇱ s Equations
microscopic
ሬԦ
‫ߩ= ܦڄ׏ ۓ‬
ሬԦ
߲‫ܦ‬
ۖ
ۖ‫ܪ × ׏‬
ሬԦ െ
= ‫ܬ‬Ԧ
߲‫ݐ‬
ሬԦ = 0
‫ܤڄ׏ ۔‬
ۖ
ሬԦ
߲‫ܤ‬
ۖ
ሬԦ
‫ ܧ × ׏ە‬+ ߲‫ = ݐ‬0
macroscopic
Charge conservation
ሬԦ൯
‫ܨ‬Ԧ = ‫ݍ‬൫‫ܧ‬ሬԦ + ‫ݒ‬Ԧ × ‫ܤ‬
Lorentz force
݀
݉‫ݒ‬Ԧ
݀‫݌‬Ԧ
= ቆ
ቇ Law of motion
݀‫ ݐ݀ ݐ‬ඥ1 െ ‫ ݒ‬ଶ /ܿ ଶ
‫ܨ‬Ԧ = െ‫ܩ‬
݉ଵ ݉ଶ
‫ݎ‬Ƹ
‫ݎ‬ଶ
Gravitation
In one small table we have all the fundamental laws of classical physics-even with room
to write them out in words and with some redundancy. We have climbed a great peak.
We are on the top of Mount K2; We are nearly ready for Mount Everest, which is
quantum mechanics. We have climbed the peak of a "Great Divide," and now we can
go down the other side.
- Boundary conditions
1.
As we have seen a lot of times, typical physical problems in the macroscopic view take
the form of boundary value problems with respect to the interface of piecewise
homogeneous media.
2.
So, we need to consider the boundary conditions of the fields with respect to the
Maxwell equations.
3.
The four equations provide four boundary conditions. For simplicity, the divergence
equations and the curl equations in the Maxwell equations can be considered
separately.
4.
The divergence equations may be considered using a pillbox-shaped surface involving
the interface as illustrated below.
A.
Here, the flux of a vector through the side of the pillbox can be ignored as the
ሬԦ ‫݊ ڄ‬ො݀ܽ ՜ 0
height ݄ becomes infinitesimal, e.g., ‫׬‬ௌ ‫ܤ‬
య
B.
C.
5.
Gauss’ law
ሬԦ = ߩ
‫ܦڄ׏‬
֜
(‫ܦ‬ଵ௡ െ ‫ܦ‬ଶ௡ ) = ߪ
Magnetic Gauss’ law
ሬԦ = 0
‫ܤڄ׏‬
֜
(‫ܤ‬ଵ௡ െ ‫ܤ‬ଶ௡ ) = 0
The curl equations may be considered using a rectangular path involving the interface
as illustrated below.
A.
Here, the flux of a vector through the area of the rectangular can be ignored as
the height ݄ = ݄ଵ + ݄ଶ becomes infinitesimal, e.g., ‫׬‬ௌ
B.
6.
ሬԦ
డ஻
డ௧
‫݊ ڄ‬ො݀ܽ ՜ 0.
Maxwell-Faraday equation
ሬԦ
߲‫ܤ‬
‫ܧ × ׏‬ሬԦ +
= 0 ֜ (‫ܧ‬ଵ௧ െ ‫ܧ‬ଶ௧ ) = 0
߲‫ݐ‬
C. Maxwell-Ampere equation
ሬԦ
߲‫ܦ‬
ሬԦ െ
ሬԦ
‫ܪ×׏‬
= ‫ܬ‬Ԧ ֜ (‫ܪ‬ଵ௧ െ ‫ܪ‬ଶ௧ ) = ‫ܭ‬
߲‫ݐ‬
Depending on the form of the permittivity/permeability, the Drude model, Lorentz
model, Debye model, and so on, the detailed expressions of the boundary conditions
can be different. However, the boundary conditions represented above are as valid as
the Maxwell equations.
7.
The boundary conditions for the scalar and vector potentials follow those of the electric
and magnetic fields,
- One note on the relation between the electric field (the magnetic field) and the displacement (the
magnetic induction)
1.
This topic is beyond the scope of the class this semester, and I am not going to discuss
it in the class. But, if you are interested in it, you may read Chapter 19 of the textbook
of Reitz or Chapter 7 of the book by Jackson.
2.
Briefly speaking, for a non-dispersive medium in which the permittivity/permeability is
a constant (real or complex), the relation between the electric/magnetic field and the
displacement vector/magnetic induction is localized in time so that
ሬԦ(‫ݎ‬Ԧ, ‫ݎ(ߝ = )ݐ‬Ԧ)‫ܧ‬ሬԦ (‫ݎ‬Ԧ, ‫)ݐ‬
‫ܦ‬
3.
ሬԦ(‫ݎ‬Ԧ, ‫ݎ(ߤ = )ݐ‬Ԧ)‫ܪ‬
ሬԦ(‫ݎ‬Ԧ, ‫)ݐ‬
‫ܤ‬
and
However, if the medium is dispersive (ߝ = ߝ(߱)), we have to consider the nonlocality in
time of the relation: e.g.,
ஶ
ሬԦ(‫ݎ‬Ԧ, ‫ߝ = )ݐ‬଴ ቈ‫ܧ‬ሬԦ (‫ݎ‬Ԧ, ‫ )ݐ‬+ න ݀߬‫ܧ‬ሬԦ (‫ݎ‬Ԧ, ‫ ݐ‬െ ߬) ‫)߬(ܩ‬቉
‫ܦ‬
૙
4.
The integral term represents the response of the medium against the external input
stimuli of ‫ܧ‬ሬԦ at the time of ‫ ݐ‬െ ߬.
5.
The initial time of ߬ = 0 in the integral interval indicates the causality, in which only
the input at the past ‫ ݐ‬ᇱ = ‫ ݐ‬െ ߬ < ‫ ݐ‬can affect the response at the present ‫ݐ‬.
11.2 Electromagnetic energy and Poynting theorem (Section 16-3 of Reitz / Sections 9.6 of Purcell)
- Poynting theorem: energy conservation law of electromagnetism
1.
We now want to state the conservation of energy for electromagnetism quantitatively.
2.
First, let us consider the rate of work exerted on a charge in a unit volume. Since the
magnetic Lorentz force on a charged particle is always perpendicular to the velocity of
the particle, the rate of work done on the charge per unit volume is
3.
‫ܨ‬Ԧ ‫ݒ ڄ‬Ԧ ‫ݍ‬
‫ݒݍ‬Ԧ
ሬԦ൯ ‫ݒ ڄ‬Ԧ =
= ൫‫ܧ‬ሬԦ + ‫ݒ‬Ԧ × ‫ܤ‬
‫ܧ ڄ‬ሬԦ = ‫ܬ‬Ԧ ‫ܧ ڄ‬ሬԦ
ܸ
ܸ
ܸ
Second, we need to evaluate this rate, ‫ܬ‬Ԧ ‫ܧ ڄ‬ሬԦ , employing Maxwell’s equations.
A.
ሬԦ
ሬԦ = ߤ଴ ‫ܬ‬Ԧ + ߝ଴ ߤ଴ డா,
Using Maxwell-Ampere’s equation, ‫ܤ × ׏‬
డ௧
‫ܬ‬Ԧ ‫ܧ ڄ‬ሬԦ = ‫ܧ‬ሬԦ ‫ ڄ‬ቆ
=
߲‫ܧ‬ሬԦ
1
ሬԦ െ ߝ଴ ቇ
‫ܤ×׏‬
߲‫ݐ‬
ߤ଴
1
߲‫ܧ‬ሬԦ
1
߲ 1
ሬԦ൯ െ ߝ଴ ‫ܧ‬ሬԦ ‫ڄ‬
ሬԦ൯ െ ൬ ߝ଴ ‫ܧ‬ሬԦ ‫ܧ ڄ‬ሬԦ ൰
‫ܧ‬ሬԦ ‫ ڄ‬൫‫ܤ × ׏‬
= ‫ܧ‬ሬԦ ‫ ڄ‬൫‫ܤ × ׏‬
߲‫ߤ ݐ‬଴
ߤ଴
߲‫ ݐ‬2
B.
ሬԦ × ‫ܧ‬ሬԦ ൯ = ‫ܧ‬ሬԦ ‫ ڄ‬൫‫ܤ × ׏‬
ሬԦ൯ െ ‫ܤ‬
ሬԦ ‫ ڄ‬൫‫ܧ × ׏‬ሬԦ ൯,
Using the vector identity, ‫ ڄ ׏‬൫‫ܤ‬
1
1
߲ 1
ሬԦ × ‫ܧ‬ሬԦ ൯ + ‫ܤ‬
ሬԦ ‫ ڄ‬൫‫ܧ × ׏‬ሬԦ ൯ െ ൬ ߝ଴ ‫ܧ‬ሬԦ ‫ܧ ڄ‬ሬԦ ൰
‫ ڄ ׏‬൫‫ܤ‬
ߤ଴
ߤ଴
߲‫ ݐ‬2
1
߲ 1
ሬԦ൯ + ‫ܤ‬
ሬԦ ‫ ڄ‬൫‫ܧ × ׏‬ሬԦ ൯ െ ൬ ߝ଴ ‫ܧ‬ሬԦ ‫ܧ ڄ‬ሬԦ ൰
= െ‫ ڄ ׏‬൫‫ܧ‬ሬԦ × ‫ܪ‬
ߤ଴
߲‫ ݐ‬2
‫ܬ‬Ԧ ‫ܧ ڄ‬ሬԦ =
ሬԦ
C.
డ஻
Using Maxwell-Faraday’s equation, ‫ܧ × ׏‬ሬԦ = െ ,
D.
ሬԦ ߲ 1
1
߲‫ܤ‬
ሬԦ ‫ڄ‬
‫ܤ‬
െ ൬ ߝ ‫ܧ‬ሬԦ ‫ܧ ڄ‬ሬԦ ൰
߲‫ ݐ߲ ݐ‬2 ଴
ߤ଴
߲ 1
1
ሬԦ൯ െ ൬ ߝ଴ ‫ܧ‬ሬԦ ‫ܧ ڄ‬ሬԦ + ߤ଴ ‫ܪ‬
ሬԦ ‫ܪ ڄ‬
ሬԦ൰
= െ‫ ڄ ׏‬൫‫ܧ‬ሬԦ × ‫ܪ‬
߲‫ ݐ‬2
2
ሬԦ = ߤ଴ ‫ܪ‬
ሬԦ has been used.
‫ܤ‬
E.
We can now define a scalar and a vector:
డ௧
ሬԦ൯ െ
‫ܬ‬Ԧ ‫ܧ ڄ‬ሬԦ = െ‫ ڄ ׏‬൫‫ܧ‬ሬԦ × ‫ܪ‬
4.
1
1
ሬԦ ‫ܪ ڄ‬
ሬԦ and ܵԦ = ‫ܧ‬ሬԦ × ‫ܪ‬
ሬԦ
‫ߝ ؠ ݑ‬଴ ‫ܧ‬ሬԦ ‫ܧ ڄ‬ሬԦ + ߤ଴ ‫ܪ‬
2
2
߲‫ݑ‬
= െ‫ܵ ڄ ׏‬Ԧ െ ‫ܬ‬Ԧ ‫ܧ ڄ‬ሬԦ
֜
߲‫ݐ‬
The derived equation is known as Poynting’s theorem.
- Local conservation principle
1.
The Poynting theorem evaluates the rate of change of the local quantity ‫ݑ‬, since the
ሬԦ = ‫ܪ‬
ሬԦ(‫ݎ‬Ԧ, ‫)ݐ‬, and ‫ݎ(ݑ = ݑ‬Ԧ, ‫)ݐ‬.
electric and magnetic fields are local functions: ‫ܧ‬ሬԦ = ‫ܧ‬ሬԦ (‫ݎ‬Ԧ, ‫)ݐ‬, ‫ܪ‬
2.
Considering that ‫ܬ‬Ԧ ‫ܧ ڄ‬ሬԦ is the mechanical energy dissipation rate to a charge distribution,
‫ ݑ‬should be the energy density in space. Furthermore, in analogy to the charge
conservation (
3.
డఘ
డ௧
= െ‫ܬ ڄ ׏‬Ԧ), ܵԦ should be the flow of ‫ݑ‬.
Consequently, Poynting's theorem states that the rate of energy transfer from a region
of space is equal to the rate of work done on the charge distribution in that region
plus the energy flux leaving that region. If energy goes away from a region, it is because
it flows away through the boundaries of that region: the local conservation principle.
4.
The relativistic validity of the local conservation law
A.
In general physics, you might have considered the total energy conservation of a
closed system or in the whole world. However, there is a difficulty with such a
"world-wide" conservation law in the theory of relativity.
B.
For example, suppose that there is some energy ܷଵ in some region (1) while there
is some energy ܷଶ in another region (2) some distance away.
C.
For "world-wide" conservation of the kind described, it is necessary that the energy
lost from ܷଵ should appear “simultaneously” in ܷଶ .
D.
But, the concept of "simultaneous moments" at distant points is one which is not
equivalent in different systems. Two events that are simultaneous in one system
are not simultaneous for another system with a different reference frame and there
would be some moments when the energy was not conserved.
E.
There seems to be no way to make the law of charge conservation relativistically
invariant without making it a "local" conservation law.
F.
It turns out that energy conservation should be a local process. In fact, Maxwell’s
equations satisfy the Lorentz relativistic invariance (this will be discussed in the
next semester) and the energy conservation in electromagnetism is already “local”.
- Examples of energy flow
1.
Plane electromagnetic wave: A light wave
A.
In a light wave, we have an electric field and a magnetic field at right angles to
each other and to the direction of the Poynting vector.
1
ሬԦ = ‫ܧ‬ሬԦ × ‫ܤ‬
ሬԦ = ߝ଴ ܿ ଶ ‫ܧ‬ሬԦ × ‫ܤ‬
ሬԦ
ܵԦ = ‫ܧ‬ሬԦ × ‫ܪ‬
ߤ଴
B.
ሬԦห from the Maxwell-Faraday equation, and thus the flow of
For light, ห‫ܧ‬ሬԦ ห = ܿห‫ܤ‬
energy per unit area per second is
ሬԦห = ߝ଴ ܿ‫ ܧ‬ଶ and ‫ۄܵۃ‬ୟ୴ୣ୰ୟ୥ୣ = ߝ଴ ܿ‫ ܧۃ‬ଶ ‫ۄ‬ୟ୴ୣ
หܵԦห = ߝ଴ ܿ ଶ ห‫ܧ‬ሬԦ × ‫ܤ‬
C.
When we have a light beam, there is an energy density in space given by
‫=ݑ‬
D.
1
1
1
1
‫ܧ‬ଶ
ሬԦ ‫ܤ ڄ‬
ሬԦ = ߝ଴ ‫ ܧ‬ଶ + ߝ଴ ܿ ଶ ቆ ቇ = ߝ଴ ‫ ܧ‬ଶ
ߝ଴ ‫ܧ‬ሬԦ ‫ܧ ڄ‬ሬԦ + ߝ଴ ܿ ଶ ‫ܤ‬
ܿଶ
2
2
2
2
Now the wave travels at the speed ܿ, so we should think that the energy that
goes through a square meter in a second is ܿ times the amount of energy in one
cubic meter. So, we would say that
‫ۄܵۃ‬ୟ୴ୣ = ܿ‫ۄݑۃ‬ୟ୴ୣ = ߝ଴ ܿ‫ ܧۃ‬ଶ ‫ۄ‬ୟ୴ୣ
E.
This is the same as the flow of energy per unit area per second obtained above.
F.
Here, time averages, ‫ۄ ۃ‬ୟ୴ୣ , have been used because the electric and magnetic
field and the Poynting vector are oscillating in time. In fact, the Poynting theorem
has to be discussed in further details with respect to time average, which is topics
of the classical electrodynamics of a graduate level course (Chapter 6 of Jackson’s
Classical Electrodynamics, 3rd edition).
G.
Note that ܿ = (ߝ଴ ߤ଴ )ିଵ/ଶ is indeed the speed of light, which will be discussed in
detail in the next section.
2.
Slowly charging planar capacitor
A.
If the plates have a radius ܽ and a separation ݄, the total energy due to the
B.
stored electric field between the plates is
ߝ଴
ܷ = ‫ = ܸݑ‬ቀ ‫ ܧ‬ଶ ቁ (ߨܽଶ ݄)
2
When the capacitor is being charged, the volume between the plates is receiving
energy at the rate of
C.
ܷ݀
݀‫ܧ‬
= ߝ଴ ߨܽଶ ݄‫ܧ‬
݀‫ݐ‬
݀‫ݐ‬
There must be a flow of energy into this volume from somewhere. Of course, you
might think that it comes in through the charging wires, but it does not.
D.
Poynting’s theorem gives the answer. Using Maxwell-Ampere’s law, we found that
E.
the magnetic field at the edge of the capacitor in free space (‫ܬ‬Ԧ = 0) is given by
߲‫ܧ‬ሬԦ
݀‫ܧ‬
ܽ ݀‫ܧ‬
ሬԦ = ߤ଴ ‫ܬ‬Ԧ + ߤ଴ ߝ଴
‫ܤ×׏‬
֜ 2ߨܽܿ ଶ ‫ܽߨ = ܤ‬ଶ
֜ ‫ =ܤ‬ଶ
߲‫ݐ‬
݀‫ݐ‬
2ܿ ݀‫ݐ‬
ሬԦ must be parallel to the plates. So there is
‫ܧ‬ሬԦ is perpendicular to the plates; ‫ܧ‬ሬԦ × ‫ܤ‬
an energy flow that comes in all around the edges from the space surrounding
the capacitor, as shown in the figure above.
F.
ሬԦห = ଵ ߝ଴ ܽ‫ ܧ‬ௗா .
The area of the surface is 2ߨ݄ܽ, and หܵԦห = ߝ଴ ܿ ଶ ห‫ܧ‬ሬԦ × ‫ܤ‬
G.
So, the same result of the total flux of energy is obtained again:
ଶ
ௗ௧
1
݀‫ܧ‬
݀‫ܷ݀ ܧ‬
ර ܵԦ ‫݊ ڄ‬ො݀ܽ = 2ߨ݄ܽ ‫ߝ ڄ‬଴ ܽ‫ܧ‬
. = ߝ଴ ߨܽଶ ݄‫ܧ‬
=
2
݀‫ݐ‬
݀‫ݐ‬
݀‫ݐ‬
ௌ
H.
When the charges are far away, there is a weak but enormously spread-out field
that surrounds the capacitor. Then, as the charges come together, the field gets
stronger nearer to the capacitor. So, the field energy which is way out moves
toward the capacitor and eventually ends up between the plates.
3.
A resistance wire carrying a current (skipped in class)
A.
Since the wire has resistance, there is an electric field to drive the current: ‫ܬ‬Ԧ = ߪ‫ܧ‬ሬԦ .
B.
The resistance also causes a potential drop along the wire, so there is also an
electric field just outside the wire, parallel to the surface.
C.
According to Biot-Savart’s law, the current generates a magnetic field which goes
around the wire because of the current.
D.
The electric and magnetic fields are at right angles; there is a Poynting vector
directed radially inward, as shown in the figure below.
E.
Obviously, there is a flow of energy into the wire all around, equal to the energy
being lost in the wire as ‫ܬ‬Ԧ ‫ܧ ڄ‬ሬԦ . Notably, Poynting’s theorem successfully predicts
that the electrons are getting their energy to generate heat from the field outside.
- Field momentum
1.
Just as the field has energy, it will have a certain momentum per unit volume.
2.
An approach similar to that used in the derivation of Poynting theorem from Maxwell’s
equations allows for a rigorous determination of the linear momentum of the
electromagnetic field and raises the local conservation of momentum and the key
quantity that describes the flow of energy and momentum in spacetime, known as
Maxwell’s stress tenor or the electromagnetic stress–energy tensor.
3.
However, the details of this topic are somewhat beyond the scope of this class. If you
are interested, please refer to the additional reading material, a one-page summary of
the linear momentum of the field and the Maxwell stress tensor.
4.
Just providing the conclusion, the linear momentum of the electromagnetic field in a
volume is given by the Poynting vector as we expect:
ܵԦ
1
ሬԦ݀‫ = ݒ‬න
ሬԦ݀‫ݒ‬
‫݌‬Ԧ୤୧ୣ୪ୢ = න ଶ ݀‫ = ݒ‬න ߝ଴ ‫ܧ‬ሬԦ × ‫ܤ‬
‫ܧ‬ሬԦ × ‫ܪ‬
ଶ
ܿ
ܿ
௏
௏
௏
5.
For a planewave in a volume in free space, the magnitude of the linear momentum is
‫ܧ‬ሬԦ
1
1
1
‫ܧ‬
ሬԦห݀‫ = ݒ‬න ߝ଴ ‫ܧ‬ሬԦ ‫ = ݒ݀ ڄ‬න ൬ ‫ܦ‬
ሬԦ ‫ܧ ڄ‬ሬԦ + ‫ܤ‬
ሬԦ ‫ܪ ڄ‬
ሬԦ൰ ݀‫= ݒ‬
‫ = ݌‬න ߝ଴ ห‫ܧ‬ሬԦ × ‫ܤ‬
ܿ
ܿ ௏ 2
2
ܿ
௏
௏
6.
Here, ‫ ܧ‬on the right-hand side is the total energy of the planewave. This result is the
relativistic relation between the energy and momentum of a photon: ‫ܿ݌ = ܧ‬.
11.3 Electromagnetic waves (Sections 16-4, 16-5, and 16-7 of Reitz / Sections 9.4 and 9.5 of Purcell)
- Wave equations of electric and magnetic fields in free space
1.
2.
In free space, where there is no charges and currents, Maxwell’s equations are
ሬԦ
߲‫ܧ‬ሬԦ
߲‫ܤ‬
ሬԦ = 0, ‫ܧ × ׏‬ሬԦ +
ሬԦ െ ߝ଴ ߤ଴
‫ܧ ڄ ׏‬ሬԦ = ‫ܤ ڄ ׏‬
= 0, and ‫ܤ × ׏‬
=0
߲‫ݐ‬
߲‫ݐ‬
Taking the curl of the Maxwell-Faraday and Maxwell-Ampere equations, we can have
two separate equations with respect to the electric and magnetic fields, respectively.
ሬԦ
߲ ଶ ‫ܧ‬ሬԦ
߲‫ܤ‬
߲
ሬԦ = ‫ܧ × ׏ × ׏‬ሬԦ + ߝ଴ ߤ଴
‫ܧ × ׏ × ׏‬ሬԦ + ‫× ׏‬
= ‫ܧ × ׏ × ׏‬ሬԦ + ‫ܤ × ׏‬
=0
߲‫ݐ‬
߲‫ ݐ‬ଶ
߲‫ݐ‬
ሬԦ
߲
߲ଶ‫ܤ‬
߲‫ܧ‬ሬԦ
ሬԦ െ ߝ଴ ߤ଴ ‫× ׏‬
ሬԦ െ ߝ଴ ߤ଴ ‫ܧ × ׏‬ሬԦ = ‫ܤ × ׏ × ׏‬
ሬԦ + ߝ଴ ߤ଴
=‫ܤ×׏×׏‬
=0
‫ܤ×׏×׏‬
߲‫ݐ‬
߲‫ ݐ‬ଶ
߲‫ݐ‬
ሬԦ = 0 and ‫ܣ × ׏ × ׏‬Ԧ = ‫׏‬൫‫ܣ ڄ ׏‬Ԧ൯ െ ‫׏‬ଶ ‫ܣ‬Ԧ, we have
3. Using ‫ܧ ڄ ׏‬ሬԦ = ‫ܤ ڄ ׏‬
߲ ଶ ‫ܧ‬ሬԦ
=0
߲‫ ݐ‬ଶ
ሬԦ
߲ଶ‫ܤ‬
ሬԦ െ ߝ଴ ߤ଴
=0
‫׏‬ଶ ‫ܤ‬
߲‫ ݐ‬ଶ
Truly, the electric and magnetic fields in free space satisfy the wave-equation in three
‫׏‬ଶ ‫ܧ‬ሬԦ െ ߝ଴ ߤ଴
4.
dimensions, which predicts the existence of the electromagnetic wave and its
propagation even without any supporting medium.
5.
Here, we can define the propagation speed of the electromagnetic wave in free space:
ܿ = (ߝ଴ ߤ଴ )ିଵ/ଶ , which is a constant called the speed of light, as you already know.
6.
Further details of the electromagnetic wave and its behavior, including the boundary
condition problems, resonance, propagation, dispersion, light-matter interaction and so
on, will be discussed in the next semester.
- Wave equations of scalar and vector potentials (General EM wave-equation) and Lorenz gauge
1.
Previously, we derived the wave equation of the electric and magnetic field in free
space. In case including charges and currents, the wave equations are complicated.
2.
However, it is possible to derive the wave equations of the scalar and vector potentials,
which are equivalent to Maxwell’s equations in terms of electric and magnetic fields.
3.
Determination of the scalar and vector potentials
A.
ሬԦ = 0, we determine the vector potential as
From ‫ܤ ڄ ׏‬
ሬԦ = ‫ܣ × ׏‬Ԧ
‫ܤ‬
B.
Faraday’s law gives
ሬԦ
߲‫ܤ‬
߲‫ܣ‬Ԧ
= െ‫× ׏‬
߲‫ݐ‬
߲‫ݐ‬
߲‫ܣ‬Ԧ
߲‫ܣ‬Ԧ
‫ܧ‬ሬԦ +
= െ‫ܧ ֜ ߮׏‬ሬԦ = െ‫ ߮׏‬െ
߲‫ݐ‬
߲‫ݐ‬
‫ܧ × ׏‬ሬԦ = െ
֜ ‫ × ׏‬ቆ‫ܧ‬ሬԦ +
4.
߲‫ܣ‬Ԧ
ቇ=0 ֜
߲‫ݐ‬
Wave equation and Lorenz gauge
A.
Combining the vector and scalar potentials with the other two equations, the
B.
Maxwell equations can be equivalently converted as
߲
ߩ
‫ۓ‬
‫ܧ ڄ ׏‬ሬԦ = ‫׏‬ଶ ߮ + ൫‫ܣ ڄ ׏‬Ԧ൯ = െ
ۖ
߲‫ݐ‬
ߝ଴
ଶ Ԧ
ሬԦ
1 ߲‫ܧ‬
1߲ ‫ܣ‬
1 ߲߮
‫ܬ‬Ԧ
‫۔‬
ଶ Ԧ
Ԧ+
ሬԦ െ
=
‫׏‬
‫ܣ‬
െ
െ
‫׏‬
൬‫׏‬
‫ڄ‬
‫ܣ‬
൰
=
െ
ۖ‫ܤ × ׏‬
ܿ ଶ ߲‫ݐ‬
ܿ ଶ ߲‫ ݐ‬ଶ
ܿ ଶ ߲‫ݐ‬
ߝ଴ ܿ ଶ
‫ە‬
These equations look complicated at glance. However, we can simplify them
further using the freedom to choose arbitrarily the divergence of ‫ܣ‬Ԧ as
1 ߲߮
߲ఓ ‫ܣ‬ఓ = ߲ఓ ‫ܣ‬ఓ = ‫ܣ ڄ ׏‬Ԧ + ଶ
=0
ܿ ߲‫ݐ‬
C.
This choice of gauge is known as the Lorenz gauge after Ludvig Lorenz (18291891), a Danish physicist, whose name is often confused with Hendrik Lorentz.
D.
E.
Using the Lorenz gauge condition, the Maxwell equations become
1 ߲ ଶ ‫ܣ‬Ԧ
‫ܬ‬Ԧ
1 ߲ଶ߮
ߩ
‫׏‬ଶ ‫ܣ‬Ԧ െ ଶ ଶ = െ
and ‫׏‬ଶ ߮ െ ଶ ଶ = െ
ଶ
ܿ ߲‫ݐ‬
ߝ଴ ܿ
ܿ ߲‫ݐ‬
ߝ଴
What a beautiful set of equations! The equations for ‫ܣ‬Ԧ and for ߮ are nicely
separated but of the same form.
5.
Using the four-vector potential, current, and differential operator, which will be
discussed in the discussion of the special relativity next semester, a single equation can
describe everything in the classical electrodynamics as
ǹ‫ܣ‬ఓ = ߲ఓ ߲ఓ ‫ܣ‬ఓ = ‫ܬ‬ఓ
6.
We now have another form of the electromagnetic laws exactly equivalent to Maxwell's
equations, and in many situations they are much simpler to handle. In the next section,
we will study the general solution of the simplified Maxwell’s equations.
- Note on gauge transformation
1.
Setting the term ‫ܣ ڄ ׏‬Ԧ is called choosing a gauge.
2.
The condition, ‫ܣ ڄ ׏‬Ԧ = െ మ
ଵ డథ
௖ డ௧
, is called the Lorenz gauge. The Coulomb gauge, ‫ܣ ڄ ׏‬Ԧ =
0, makes the equations of statics somewhat simpler: ‫׏‬ଶ ߮ = െߩ/ߝ଴ and ‫׏‬ଶ ‫ܣ‬Ԧ = െߤ଴ ‫ܬ‬Ԧ.
3.
Changing ‫ܣ‬Ԧ by adding ‫ ;߰׏‬is called a gauge transformation.
4.
ሬԦ - that is, without
We can allow ‫ܣ‬Ԧ to be changed without affecting the fields ‫ܧ‬ሬԦ and ‫ܤ‬
changing the classical physics - if we always change ‫ܣ‬Ԧ and ߮, together by the rules:
߲߰
‫ܣ‬Ԧᇱ = ‫ܣ‬Ԧ + ‫ ߰׏‬and ߮ ᇱ = ߮ െ
߲‫ݐ‬
5.
ଵ డమ ట
When ‫׏‬ଶ ߰ െ మ
௖
డ௧ మ
‫ܣ ڄ ׏‬Ԧᇱ +
= 0, the new potentials still preserve the Lorenz condition as
1 ߲߶ ᇱ
1 ߲߶
1 ߲ଶ߰
1 ߲߶
= ‫ܣ ڄ ׏‬Ԧ + ଶ
+ ‫׏‬ଶ ߰ െ ଶ ଶ = ‫ܣ ڄ ׏‬Ԧ + ଶ
=0
ଶ
ܿ ߲‫ݐ‬
ܿ ߲‫ݐ‬
ܿ ߲‫ݐ‬
ܿ ߲‫ݐ‬
- Particular solution of the wave equation
1.
1D planewaves in free space
A.
B.
As we have discussed, the wave equation in free space is given by
1 ߲ ଶ ߰(‫ݎ‬Ԧ, ‫)ݐ‬
‫׏‬ଶ ߰(‫ݎ‬Ԧ, ‫ )ݐ‬െ ଶ
=0
߲‫ ݐ‬ଶ
ܿ
If ݂(‫ݎ‬Ԧ, ‫ )ݐ‬is homogeneous except a certain axis (here, let us set it ‫ݔ‬-axis), we have
one-dimensional wave equation as
C.
߲ ଶ ߰(‫ݔ‬, ‫ )ݐ‬1 ߲ ଶ ߰(‫ݔ‬, ‫)ݐ‬
െ ଶ
=0
߲‫ ݔ‬ଶ
߲‫ ݐ‬ଶ
ܿ
From general physics, you may already know that the general solution of the 1D
wave equation is given as
߰(‫ݔ‬, ‫ ݔ(݂ = )ݐ‬െ ܿ‫ )ݐ‬+ ݃(‫ ݔ‬+ ܿ‫)ݐ‬
D.
This general solution leads to planewaves, homogeneous over the yz plane, and
its first and second solutions correspond to the waves propagating to +‫ ݔ‬and െ‫ݔ‬
directions, respectively.
2.
3D spherical waves from a point source
A.
Now let us consider the fields or the potentials outgoing from a point source at
the origin:
B.
1 ߲ ଶ ߰(‫ݎ‬Ԧ, ‫)ݐ‬
= ‫)ݎ(ߜ)ݐ(ݏ‬
ܿ ଶ ߲‫ ݐ‬ଶ
Considering the isotropic symmetry, the wave equation becomes
C.
1 ߲ଶ
1 ߲ ଶ ߰(‫ݎ‬, ‫)ݐ‬
൫‫ݎ(߰ݎ‬, ‫)ݐ‬൯ െ ଶ
= ‫)ݎ(ߜ)ݐ(ݏ‬
ଶ
‫ݎ߲ ݎ‬
߲‫ ݐ‬ଶ
ܿ
߲ଶ
1 ߲ଶ
֜
൫‫ݎ(߰ݎ‬,
‫)ݐ‬൯
െ
൫‫ݎ(߰ݎ‬, ‫)ݐ‬൯ = ‫)ݎ(ߜ)ݐ(ݏݎ‬
߲‫ ݎ‬ଶ
ܿ ଶ ߲‫ ݐ‬ଶ
Except ‫ = ݎ‬0 , the outgoing solution, that propagates to the +‫ ݎ‬direction, is
‫׏‬ଶ ߰(‫ݎ‬Ԧ, ‫ )ݐ‬െ
obviously given by
D.
݂(‫ ݎ‬െ ܿ‫ ݐ(݂ )ݐ‬െ ‫ݎ‬/ܿ)
=
… (1)
‫ݎ‬
‫ݎ‬
This is the solution of the outgoing spherical wave.
E.
As the retardation, െ‫ݎ‬/ܿ, can be neglected and ߰ becomes the solution Just like
F.
a Coulomb field for a charge at the origin that vanes with time
݂(‫)ݐ‬
߰(‫ݎ‬, ‫= )ݐ‬
as ‫ ݎ‬՜ 0 … (2)
‫ݎ‬
As we go very close to the origin, the 1/‫ ݎ‬dependence of ߰ causes the space
‫ݔ(߰ݎ‬, ‫ ݎ(݂ = )ݐ‬െ ܿ‫)ݐ‬
֜
߰(‫ݎ‬, ‫= )ݐ‬
derivatives to become dominant in the 3D wave equation, but the time derivatives
keep their same values and can be neglected;
G.
1 ‫)ݐ(ݏ‬
… (3)
4ߨ ‫ݎ‬
Comparing (1), (2), and (3), we can finally reach out the Coulomb potential-like
H.
solution regarding the retardation time as
1 ‫ ݐ(ݏ‬െ ‫ݎ‬/ܿ)
߰(‫ݎ‬, ‫ = )ݐ‬െ
‫ݎ‬
4ߨ
Of course, in order to get the rigorous proof of this solution, you need to study
‫׏‬ଶ ߰(‫ݎ‬Ԧ, ‫)ݎ(ߜ)ݐ(ݏ = )ݐ‬
֜
߰(‫ݎ‬, ‫ )ݐ‬՜ െ
ଵ డమ
the Green function of the wave equation operator, ‫׏‬ଶ െ మ
௖ డ௧ మ
, which you may
encounter in the class of mathematical physics or graduate level electromagnetism.
If you are interested in it, refer to the additional reading material of Green’s
function of the inhomogeneous wave equation or Chapter 6 of Jackson’s book.
11.4 General solution of Maxwell’s equations (Section 16-7 of Reitz)
- Linear superposition of the contributions of different point sources
1.
We have found the solution for a point source.
2.
For an arbitrary density distribution of source ‫ݎ(ݏ‬Ԧ ᇱ , ‫ )ݐ‬in time and space, we can simply
make the solution up of the sum of many point sources, one for each volume element
݀‫ ݒ‬ᇱ , and each with the source strength ‫ݎ(ݏ‬Ԧ ᇱ , ‫ ݒ݀)ݐ‬ᇱ :
݀߰(‫ݎ‬Ԧ, ‫ = )ݐ‬െ
3.
1 ‫ݎ(ݏ‬Ԧᇱ , ‫ ݐ‬െ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |/ܿ) ᇱ
݀‫ݒ‬
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
4ߨ
Since the wave equations (equivalently Maxwell equations) are linear, the resultant field
is the superposition of the fields from all of such source elements:
߰(‫ݎ‬Ԧ, ‫ = )ݐ‬െ
1
‫ݎ(ݏ‬Ԧ ᇱ , ‫ ݐ‬െ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |/ܿ) ᇱ
න
݀‫ݒ‬
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
4ߨ ୟ୪୪
ୱ୮ୟୡୣ
4.
That is, the field at ‫ݎ‬Ԧ at the time ‫ ݐ‬is the sum of all the spherical waves which leave
the source elements at ‫ݎ‬Ԧ ᇱ at the times ‫ ݐ‬െ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |/ܿ. The figure below illustrates the
case of a charge moving in an arbitrary way.
5.
Employing this general solution of the 3D wave equation to the master equations for
the scalar and vector potentials, we have
߮(‫ݎ‬Ԧ, ‫= )ݐ‬
1
ߩ(‫ݎ‬Ԧ ᇱ , ‫ ݐ‬െ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |/ܿ) ᇱ
1
‫ܬ‬Ԧ(‫ݎ‬Ԧ ᇱ , ‫ ݐ‬െ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |/ܿ) ᇱ
Ԧ(‫ݎ‬Ԧ, ‫= )ݐ‬
න
݀‫ݒ‬
and
‫ܣ‬
න
݀‫ݒ‬
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
4ߨߝ଴ ୟ୪୪
4ߨߝ଴ ܿ ଶ ୟ୪୪
ୱ୮ୟୡୣ
6.
ୱ୮ୟୡୣ
The electric and magnetic fields are then found by differentiating the potentials.
- Everything of the classical electromagnetism
1.
In the following table is the complete theory of electricity and magnetism, and of light;
a complete description of the fields produced by any moving charges; and more.
Maxwellᇱ s Equations
ߩ
ߝ଴
ሬԦ
߲‫ܤ‬
=0
‫ܧ × ׏‬ሬԦ +
߲‫ݐ‬
‫ܧ ڄ ׏‬ሬԦ =
ሬԦ = 0
‫ܤڄ׏‬
ሬԦ െ
ܿଶ‫ܤ × ׏‬
߲‫ܧ‬ሬԦ
‫ܬ‬Ԧ
=
߲‫ߝ ݐ‬଴
Their solutions
߲‫ܣ‬Ԧ
߲‫ݐ‬
ߩ(‫ݎ‬Ԧ ᇱ , ‫ ݐ‬െ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |/ܿ) ᇱ
݀‫ݒ‬
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
ሬԦ = ‫ܣ × ׏‬Ԧ
‫ܤ‬
‫ܧ‬ሬԦ = െ‫ ߮׏‬െ
߮(‫ݎ‬Ԧ, ‫= )ݐ‬
1
න
4ߨߝ଴ ୟ୪୪
ୱ୮ୟୡୣ
2.
‫ܣ‬Ԧ(‫ݎ‬Ԧ, ‫= )ݐ‬
1
‫ܬ‬Ԧ(‫ݎ‬Ԧ ᇱ , ‫ ݐ‬െ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |/ܿ) ᇱ
න
݀‫ݒ‬
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
4ߨߝ଴ ܿ ଶ ୟ୪୪
This is the goal of this lecture, and we finally get it!
ୱ୮ୟୡୣ
(HW #11)
Due: 20240612
1. Consider a piece of wire with a resistance of ܴ as shown in the figure below. The wire has a
circular cross-section with a radius of ܽ and its length is ݈. In the wire, a constant current ‫ ܫ‬is
supported. Due to the resistance, heat is generated in the wire. Determine the Poynting vector right
on the surface of the wire, which generates the heat. Show that the energy being lost in the wire is
equal to the energy flowing into the wire: ‫׬‬஺ ܵԦ ‫݊ ڄ‬ො݀ܽ = ‫ ܫ‬ଶ ܴ.
2. The Lorentz force per unit volume on a region of free space containing charges and currents is
expressed as
‫ܨ‬Ԧ
ሬԦ
= ߩ‫ܧ‬ሬԦ + ‫ܬ‬Ԧ × ‫ܤ‬
ܸ
Using the Maxwell equations and vector identities, show that
݂Ԧ =
݂Ԧ = െ
ሬԦ
߲ ‫ܧ‬ሬԦ × ‫ܪ‬
1
1
ሬԦ ‫ܪ ڄ ׏‬
ሬԦ െ ߤ଴ ‫׏‬൫‫ܪ‬
ሬԦ ‫ܪ ڄ‬
ሬԦ ൯ + ߤ଴ ൫‫ܪ‬
ሬԦ ‫׏ ڄ‬൯‫ܪ‬
ሬԦ
ቆ ଶ ቇ + ߝ଴ ‫ܧ‬ሬԦ ‫ܧ ڄ ׏‬ሬԦ െ ߝ଴ ‫׏‬൫‫ܧ‬ሬԦ ‫ܧ ڄ‬ሬԦ ൯ + ߝ଴ ൫‫ܧ‬ሬԦ ‫׏ ڄ‬൯‫ܧ‬ሬԦ + ߤ଴ ‫ܪ‬
߲‫ݐ‬
ܿ
2
2
From this result, we call the term
ሬԦ
ாሬԦ ×ு
௖మ
as the linear momentum density of the electromagnetic field.
ଵ
ሬԦ ‫ܪ ڄ ׏‬
ሬԦ is trivial: ‫ܪ ڄ ׏‬
ሬԦ = ‫ܤ ڄ ׏‬
ሬԦ = 0.
Hint: In fact, the term ‫ܪ‬
ఓ
బ
3. Consider a medium in which ߩ = 0, ‫ܬ‬Ԧ = 0, ߤ = ߤ଴ , but where the polarization ܲሬԦ is a given
function of position and time: ܲሬԦ = ܲሬԦ (‫ݎ‬Ԧ, ‫)ݐ‬. Show that the Maxwell equations are correctly obtained
from a single vector function ܼԦ satisfying the following equations:
1 ߲ ଶ ܼԦ
ܲሬԦ
‫׏‬ଶ ܼԦ െ ଶ ଶ = െ
ܿ ߲‫ݐ‬
ߝ଴
1
1
μܼԦ
ሬԦ െ ܲሬԦ and ‫ܤ‬
ሬԦ = ‫× ׏‬
‫ܧ‬ሬԦ = ‫ × ׏ × ׏‬Z
ଶ
ߝ଴
ܿ
߲‫ݐ‬
Here, the vector ܼԦ is called the Hertz vector (https://en.wikipedia.org/wiki/Hertz_vector).
4. (“Very” optional) We have studied that the Maxwell equations are equivalent to the wave
equations of the scalar and vector potentials as
1 ߲ ଶ ‫ܣ‬Ԧ
‫ܬ‬Ԧ
1 ߲ ଶ߮
ߩ
ଶ
‫׏‬ଶ ‫ܣ‬Ԧ െ ଶ ଶ = െ
and
‫׏‬
߮
െ
=െ
ଶ
ଶ
ଶ
ܿ ߲‫ݐ‬
ߝ଴ ܿ
ܿ ߲‫ݐ‬
ߝ଴
Also, we saw that the solutions to the wave equations are given as
1
ߩ(‫ݎ‬Ԧ ᇱ , ‫ ݐ‬െ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |/ܿ) ᇱ
1
‫ܬ‬Ԧ(‫ݎ‬Ԧ ᇱ , ‫ ݐ‬െ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |/ܿ) ᇱ
߮(‫ݎ‬Ԧ, ‫= )ݐ‬
න
݀‫ ݒ‬and ‫ܣ‬Ԧ(‫ݎ‬Ԧ, ‫= )ݐ‬
න
݀‫ݒ‬
ᇱ
ଶ
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ |
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
ୟ୪୪
4ߨߝ଴ ୟ୪୪
4ߨߝ଴ ܿ
ୱ୮ୟୡୣ
ୱ୮ୟୡୣ
However, unlike Coulomb's law, these integrals are difficult to calculate.
Now, let us consider one of the simplest cases, a moving point charge ‫ݍ‬. If the trajectory of the
charge is given as a function of time by ‫ݎ‬Ԧ௦ (‫ ݐ‬ᇱ ), the charge and current densities are as follows
ߩ(‫ݎ‬Ԧ ᇱ , ‫ ݐ‬ᇱ ) = ‫ߜݍ‬൫‫ݎ‬Ԧ ᇱ െ ‫ݎ‬Ԧ௦ (‫ ݐ‬ᇱ )൯ and ‫ܬ‬Ԧ(‫ݎ‬Ԧ ᇱ , ‫ ݐ‬ᇱ ) = ‫ݒݍ‬Ԧ௦ (‫ ݐ‬ᇱ )ߜ൫‫ݎ‬Ԧ ᇱ െ ‫ݎ‬Ԧ௦ (‫ ݐ‬ᇱ )൯
Then, the potentials are represented as
߮(‫ݎ‬Ԧ, ‫= )ݐ‬
1
‫ܣ‬Ԧ(‫ݎ‬Ԧ, ‫= )ݐ‬
න
4ߨߝ଴ ܿ ଶ
Here, ‫ݐ‬௥ = ‫ ݐ‬െ
1
න
4ߨߝ଴
‫ݒݍ‬Ԧ௦ ൬‫ ݐ‬െ
‫ ߜݍ‬ቆ‫ݎ‬Ԧ ᇱ െ ‫ݎ‬Ԧ௦ ൬‫ ݐ‬െ
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
ܿ ൰ቇ
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
݀‫ ݒ‬ᇱ =
‫ߜݍ‬൫‫ݎ‬Ԧ ᇱ െ ‫ݎ‬Ԧ௦ (‫ݐ‬௥ )൯ ᇱ
1
න
݀‫ݒ‬
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
4ߨߝ଴ ୟ୪୪
ୱ୮ୟୡୣ
ᇱ|
ᇱ|
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ
ᇱ
ܿ ൰ ߜ ቆ‫ݎ‬Ԧ െ ‫ݎ‬Ԧ௦ ൬‫ ݐ‬െ
ܿ ൰ቇ
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
݀‫ ݒ‬ᇱ =
‫ݒݍ‬Ԧ௦ (‫ݐ‬௥ )ߜ൫‫ݎ‬Ԧ ᇱ െ ‫ݎ‬Ԧ௦ (‫ݐ‬௥ )൯ ᇱ
1
න
݀‫ݒ‬
|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ ᇱ |
4ߨߝ଴ ܿ ଶ
ห௥Ԧି௥Ԧ ᇲ ห
௖
.
Show that evaluation of the integrals gives
߮(‫ݎ‬Ԧ, ‫= )ݐ‬
1
‫ݍ‬
1
‫ݍ‬
ቌ
ቍ=
൥
൩
4ߨߝ଴ ቀ1 െ ݊ො (‫ߚ ڄ ) ݐ‬Ԧ (‫) ݐ‬ቁ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧ (‫|) ݐ‬
4ߨߝ଴ ൫1 െ ݊ො௦ ‫ߚ ڄ‬Ԧ௦ ൯|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ௦ |
௦
‫ܣ‬Ԧ(‫ݎ‬Ԧ, ‫= )ݐ‬
௥
௦
௥
௦
௥
௧ୀ௧ೝ
1
‫ߚݍ‬Ԧ௦ (‫ݐ‬௥ )
1
‫ߚݍ‬Ԧ௦
ቌ
ቍ=
൥
൩
4ߨߝ଴ ܿ ቀ1 െ ݊ො (‫ߚ ڄ ) ݐ‬Ԧ (‫) ݐ‬ቁ |‫ݎ‬Ԧ െ ‫ݎ‬Ԧ (‫|) ݐ‬
4ߨߝ଴ ܿ ൫1 െ ݊ො௦ ‫ߚ ڄ‬Ԧ௦ ൯|‫ݎ‬Ԧ െ ‫ݎ‬Ԧ௦ |
௦
௥
௦
௥
௦
௥
௧ୀ௧ೝ
ሬԦ (௧ )
൫௥Ԧି௥Ԧ (௧ )൯
௩
where ߚԦ௦ (‫ݐ‬௥ ) = ೞ ௖ ೝ and ݊ො௦ (‫ݐ‬௥ ) = |௥Ԧି௥Ԧೞ (௧ೝ)| .
ೞ
ೝ
These are known as the Liénard–Wiechert potentials.
(https://en.wikipedia.org/wiki/Li%C3%A9nard%E2%80%93Wiechert_potential)
This problem is in fact beyond the scope of this class. So, don't worry if you can't solve it right away
or don't understand the answer (refer to Section 21-1 of Reitz or the wikipedia page above) at this
stage. Just give it a try. And you can always come to my office if you have any questions.