Phys Educ 22 (19871 Prlnted In the UK
Explaining electromagnetic
induction:
a critical re-examination
The clinical value of history in physics
J Roche
The study of the historical evolution of a theory or
explanation in physics canhelptoimprove
our
presentunderstanding of it.Supersededtheories
sometimeslinger on in modern physics literature.
Technical terms devised more than a century ago,
perhaps, may now be misleading or inappropriate.
Well established conventional explanations or auxiliary constructions may have becomeso familiar that
they
may
occasionally
be
mistaken
for
correct
physical explanations. Students whofail to recognise
thehistoricaloriginsandpresentstatus
of such
concepts may find themselves unable tofit them into
a coherent picture, and may become perplexed
or
even discouraged as a result. History can often help
to clarify such problems and it may even suggest a
solution.
As a case history illustrating this approach,I shall
examine briefly the more qualitative aspectsof some
of theexplanations which leading physicists have
offeredforelectromagneticinductionduringthe
past one hundred and fifty years or so. I shall use
the term ‘correct’ physical explanation to mean an
explanation which is inclose agreement withexperimental evidence and with modern fundamental
theoryandalsoone
which is logically soundand
accurately worded. I am not, of course, suggesting
that correct explanations are physically true in any
unqualified or final sense, nor do I supposethat
conventional explanations or falsified theories have
no further useful role in physics.
The term ‘electromotive force’
A Volta (1745-1827) in 1802 introduced the term
‘electric moving force’ to describe the prime move
responsible for driving an electric current in a ‘perpetualmotion’ in a‘perfect circle of conductors’
(Volta1800).Todaythederivativeterm‘electromotive force’ means the potential difference across
that prime mover.
0031-9120/87/020091+03$2.50~1987
IOPPublishing Ltd
It has frequently been pointed out that this usage
is now unfortunate, sincephysics today distinguishes
sharplybetweena
‘force’ anda‘potentialdifference’. The conflict between the apparent meaning
and the intended meaning hereis unlikely to resolve
itself, such is the suggestive power of language.
Sooner or later a change in nomenclature may be
needed and ‘electromotive potential difference’ is a
possible candidate.However,thecommonabbreviation EMF lessenstheproblemconsiderably
by
eliminating explicit referencetotheterm‘force’,
and I shall continue to employ itin thepresent
article.
It was shown by G Kirchhoff (1824-87) in 1849
that localisedEMFS generally set up auxiliary electrostatic forces, by means of surface charges, in order
toestablishauniformcurrentaroundthe
circuit
(Kirchhoff 1879 pp49-55, 1 5 1 4 ) . Since these additional fields areconservative,thesum
of thenet
potential differences around the whole circuit
will be
exactly equaltothesum
of the PDS acrossthe
localised underlying EMFS only. This is the substance
of Kirchhoff’s second network law (Kirchhoff 1879
~~15-16).
In particular, if there is an accumulator, a generator, or a coil in acircuit,thesurfacecharges
in
these devices will set up electrostatic fields which so
mask the drivingEMFS within them that the resultant
internalpotentialdifferenceacrosseach(thatis,
the work-doneon unit charge passing through it) will
John J Roche is a member ofLinacre College,
Oxford, where he tutors in historyof science and
physics. He is Secretary of The Institute of
Physics History of Physics Group. He has
obtained aBSc, and MSc, an MAand a DPhil and
was previouslyHead of Physics Department at
Strathmore College, Nairobi. Hischiefresearch
interest liesin the application ofthe history of
physics to the clarification of concepts in presentday physics. He hasverious publicationsin
critical physicsand in the pure history
science.
of
91
be exactly equal to the
ance only.
PD
across its internal resist-
Theoretical contrasts
Modern macroscopic electromagnetism, or ‘classical
electromagnetism’ to give it its more usual but misleading description, owes its experimental discoveries, its mathematical formulation and its interpretationtoagreatmanyinvestigatorsextendingfrom
H C Oersted (1777-1851) to A Einstein (18791955).
Modernfundamentaltheory
is oftencalledthe
‘Maxwell-Lorentz theory’ for short and these were
indeed the two most important contributors to the
mathematicalsynthesisandmoderninterpretation
of thetheory.Nevertheless,thename
is a little
unfairtoFaraday,amongstothers,and
it is also
somewhat misleading, since the theory has evolved
considerablyduringthepresentcentury.Itnow
includes features which are incompatible with some
of the explanations of Maxwell and Lorentz, and
also with some of thenotions of Faraday.It is
important to recognise what these are.
Both Maxwell and Lorentz thoughtof the fields as
specialstates of amaterialaether(Maxwell
1873
vol. 2 pp384 and 438, Lorentz 1909 pp2, 11 and 13).
Following the demise of the aether in the 1920s, the
fields again became, as they had been for Faraday,
conditions existing in their own right, rather than
states or disturbances of some underlying medium
(Faraday 1855
vol.
3 pp447-52). This was not
altogether a return to the theory of Faraday, however,sinceelectrostatic,magneticandradiation
fields werenowthought
of as beingpropagated
outwards with the velocity of light from elementary
electriccharges, in accordance with thetheory of
Hertz and Lorentz (Hertz
1893 pp118,122-4and
137,Lorentz 1935-9 vol. 2 pp228-9). The field at
any point is not, therefore, homogenised, as it was
with Faraday and Maxwell, but it is the resultant of
myriads of differentially delayed contributions from
each of the source charges (Lorentz 1909 pp19-20
and 240). Furthermore, again in contrast with Faraday, and in accordance with the earlier principle of
superposition, the static electric and magnetic fields
can pass through any substance whatever, without
attenuation, and all cases of apparent shielding or
deflection of flux are. due to the induction of opposing fields.
These ideas, although they are now virtually unchallenged, and are more than
60 years old, have
not yet penetratedintoeverycorner
of electromagnetism.
The nature of lines of force
For Faraday (1838 ~ 4 0 9 ,185.5 p437), J Poynting
92
(1920 p270) and J J Thomson (1891 pp149-50), lines
of electricandmagneticforcewere
self-existing
discrete physical units, with longitudinal tension and
lateral repulsion. Lines of force could maintain their
separate identities under various sorts of deformation and alteration in theirsources. In this theory
also,oneline
of forcecouldactdirectlyupon
another,andcharges,currentsandmagnetswere
often thought of as terminations or as carriers of the
lines, or even as mere passive appendages to them.
Inthis view chargesandcurrentswerenotspontaneously active in their own right but were carried
along by the motion of the lines.
In modernfundamentalelecromagnetictheory,
lines of force are still real in the restricted or weaker
sense that they do represent a sequence of possible
or latent forces in the field. However, there is no
tension along the lines or repulsion between them
(Lorentz 1909 pp20 and 240). Electric and magnetic
fields can act only on charges, not on other
fields.
The field at each point along a lineis independently
serviced by thesourcecharges,
it represents, in
general, a different force at each point, and a
different delay time, and there seems to be no reason
to attribute an intrinsic physical unity to any line.
Faraday himself (1855 p369) was fully awarethat
posterity might adopt a more restricted interpretation of his lines of force.Theyremainfor
most
an
indispensable
tool
in the visualization and
explanation of electromagnetism.
Do the fields translate and rotate?
In Faraday’stheorymagneticlines
of forceare
convected
along
by moving
a
coil or magnet
(Faraday 1855 p335),theymoveoutwardsfroma
growing current and inwards to a decreasing current
(Faraday 1838 pp68-9), buttheyremainatrest
whenasymmetricalmagnetspinsabout
itsaxis
(Faraday 1855 pp3367).
In modernelectromagnetictheorytheradiation
fields arethought of asmoving,sincetheyare
propagated through space. In the case of a steadily
translatingchargedbody,
or currentbearingcoil,
it wouldseemcorrectto
say thatthefieldsare
convected along by this motion since linear electromagnetic momentum then appears
in the fields. It
also seems correct to say that
all constant properties
of the field, including latent structures such as the
lines of force, are carried along by this motion.
Not every apparent motion of a property is a true
physical motion, however. The phase velocity in a
waveguide, when it is greater than the velocity of
light, is an obvious example of this. The motion of
the Faraday lines, inwardsor outwards, in the neighbourhood of a changing primary current is also an
example of an apparent or phase velocity. Thisis not
because the velocity of these lines is greater than
that of light-indeed it can have any value up to the
velocity of light-but because we can identify the
sameline or group of linesatdifferenttimes
in
the near regionby convention only, since thereis no
physical reason for any such identification.
In modern electromagnetic theory thereis angular
momentum in the field of a rotating current distribution or magnet only when thereis asymmetry about
the axis of rotation. When thereis perfect symmetry
we seem to be obliged to agreewith Faraday, therefore, that the magneticfield and its lines of force do
not rotate.
Qualitative and quantitative laws
M Faraday (1781-1867) discovered in 1831, in rapid
succession, the inductionof a current in a stationary
secondary helix by changing the currentin a stationary primary helix, the induction
of a current in a
stationary helix by plunging a magnet into it, or by
moving a current-bearing circuit close to i t , and the
induction of a current in a spiral by moving it close
to the poleof a magnet. Thelast experiment also led
Faraday to discover that a current
is induced in a
wire which moves so as to 'cut the magnetic curves'
(Faraday 1838 pp344). I shall call thesethefirst,
.'
Y
".
1
*--
I
Figure 1 Faraday's
electromagnetic induction
93
second and third kindsof electromagnetic induction,
respectively. Figure 1 is a reproduction of Faraday’s
diagrams of his apparatus.
Faraday did not study electromagnetic induction
quantitatively at this time, nor did he ever formulate
equationsforit.Insteadheestablished
in great
detail, and with experimental rigour. the qualitative
structure of the phenomenon, includingmuch of the
information now expressedby Lenz’s law and by the
generatorrule.Hedidprovidesomequantitative
hints, however.
H E Lenz (1804-65), of St Petersburg, was not
satisfied with Faraday’s statement of the rule relating themotion of a coil tothedirection
of the
current induced in it. His restatementof the law is as
follows (Lenz 1834 p485).
‘If a metallic conductor in the vicinity of a Galvanic current, or a magnet, is set in motion, then a
Galvanic current is induced in it [i.e in the ‘metallic
conductor’] which has a direction such that it would
bringabout it in theconductor, [if] motionless,a
movementcontrarytothat[actually]impartedto
it . . ..‘
Lenz’s ownstatement
is not explicit aboutthe
importance of relative motion, but subsequent versions of the law took this into consideration.
Lenz did not feel that there was any obscurity in
Faraday’streatment of directions in primaryand
secondary induction, so he did not extend his law to
coverit.Subsequently,however,
Lenz’s law was
generalised by Maxwell to cover all cases of electromagnetic induction by reinterpreting it to mean that
the induced EMF in a circuit acts in such a direction
as to oppose the change in the number of lines of
magnetic flux linking it (Maxwell 1873 vol. 2 p176).
Thevariousrestatementsandreformulations
of
Lenz’s qualitative law which are now current represent an interesting exampleof how a law can change
its meaning fundamentally andyet retain its original
name.Suchaprocess
is surelyunavoidable in a
developingsubject.Ambiguitycan
best beconfronted here, perhapsby pointing out the difference
betweenLenz’sownstatementandsubsequent
re-statements of the law, such as that of Maxwell.
F E Neumann of Konigsberg (1798-1895) in 1845
and 1847, in a purely theoretical analysis guided by
the known properties of electromagnetic induction
and also by Ampere’s electrodynamics, postulated
and derived preliminary versionsof the quantitative
laws now written as E=Blv and E=-dWdt, for the
EMF induced in a moving wire and in a closed circuit,
respectively (Neumann 184.5 pp2-3 and 69-77, 1847
pp3-4). The modern physical interpretation of these
equations, although partly due to Neumann himself,
largely derive from Faraday. Maxwell and Lorentz.
94
Qualitative explanations and models
AccordingtoFaraday’spreferredexplanation,
all
forms of electromagnetic induction are due to lines
of magneticforcecuttingacrossaclosed
or open
circuit and releasing their power into the
wire (Faraday 1855 pp344-5). There was complete symmetry
for Faraday between a moving wire cutting stationary lines of force and moving linesof force cutting a
stationarywire.Furthermore,lines
of forcewere
alsosupposedtomoveoutwardsfromagrowing
primary current and to cut across a secondary
circuit, thereby inducing a current.
J C Maxwell (1831-79) published in 1861
(Maxwell 1965 pp451-513) description
a
of an
elaboratehydrodynamicmodel,
which he offered
as apossiblemechanicalexplanationforelectromagnetic
induction
for
and
many other
electromagnetic phenomena as well. In the model
magnetic lines of forcewererepresented
by line
vortices in theether,and
all forms of electromagneticinductionwereduetothedifferential
rotation of neighbouringvorticesactinguponthe
rotating
particles
which separated
them.
The
resultingthrustupontheseparticlesconstituted
the
induced
electromotive
force.
Subsequently
Maxwell gaveupparticularmechanicalmodels;
heretained,however,ageneralmechanical
view
of electromagnetism.Mechanicalexplanations
of
electromagneticinductiondonotappearto
have
survived the decay of aether theories.
Anonmagneticexplanation
of electromagnetic
induction grew up on the Continent from
183.5 onwards, due to the theoretical investigationsof Gauss
(1863-1933, vol. 5 pp616-7), Neumann (1848 pp116
and171),Weber (1852 pp 511, 518-9 and 526-9)
and many others, all of whom were inspired by the
electrodynamics of Ampere. According to this view
the magnetic force was simply a modification introduced by motion into the electric force between two
charges(Ampere
1823pp28&90).
Themagnetic
forcewas.
in reality. an ‘electrodynamic’force.
Furthermore, all forms of electromagnetic induction
were special cases of this electrodynamic force. In
particular,
the
electromagnetic
induction
which
appears in a wire which is moved in the neighbourhood of a magnet or current-bearing conductor, is
due to an electric intensity which acts along the wire
and which is produced by the interaction between
the electric charges in the wire and those constituting thecurrent or magnet.Thistraditiondidnot
concern itself with the manner of communication of
force between separated charges, nor i tdid
introduce
Faraday’s concept of field. In the present century,
however,theelectrodynamictradition
led tothe
belief that the E M F induced in a moving wire is due
to electric field acting along that wire.
In 1892 H A Lorentz of Leiden (1853-1928) first
trons which constitute a growing current in a primary coil produce both the induced electricfield in the
secondary and primary and the changing magnetic
flux linking both. The changing magnetic field here
does not cause the induced electric field, both are
joint effects of moving charges in the primary. Both
fields, of course, are correlated by the familiar equations E = - a p i a t or equivalently by the Maxwell
equation BxE=-aBlat. In a somewhat similar sense
a stretching force applied to a spring simultaneously
causes a tension and an extension in the spring, and
both effects are correlated by Hooke’s well known
law, 7‘=xe.
Theincongruity
of statingthatthechanging
magnetic flux ‘causes’theinduced
EMF emerges
most clearly, perhaps, in the case of a large closed
coil linkingatoroid.Hereagrowing
flux in the
toroidcrossesa
small partonly of theimaginary
surface of the coil; nevertheless an electric field is
induced in theperiphery of thelatter.Fromthe
Electromagnetic induction of the first kind
viewpoint of modern Lorentzian electromagnetism,
the changing currentin the toroid generatesboth the
Towards a coherent explanation It is still frequentchangingmagnetic field within it andtheinduced
ly stated in many textbooks, following Faraday, that
thechangingmagnetic
flux throughastationary
electric field outside it (and inside it, also).
circuit ‘causes’ the induced electromotive force. This We must disagree with Faraday, therefore, that
magnetismassuch‘evolves’electricity
in the first
statement is not compatible with modern Lorentzian
electromagnetism. In the latter theory the induced
kind of electromagnetic induction. Havingsaid that,
E M F in a stationary coil in all cases is caused by an
however, there is no doubt that the changing magneinducedelectric
field
which
is producedsimultic flux linkage is usually the most convenientway of
taneously with themagnetic field by themoving
inferringtheexistenceof,and
of calculating,the
source electrons. In particular, the accelerated elecinduced E M F in a coil. Furthermore, there are more
proposedatheory
of electromagneticinduction
which drew heavily bothonContinentaland
on
British traditions (Lorentz 1935-9 vol. 2 pp253 and
256). Lorentz accepted the existenceof qualitatively
distinct electric and magnetic fields, but they were
produced by and propagated outwards from electric
charges. According to Lorentz, the first and second
kinds of electromagnetic induction were both produced by induced electric fields, but not the third.
Lorentz recognised that the convected electrons
in a wire moving at right angles to a magnetic field
will be interpreted by the field as an electric current
andthey will, therefore,experienceatransverse
force which will act along the wire. On Lorentz’s
theory, therefore, the motional electromotive force
is a simple magnetic force (Lorentz
1909 p5). It is
caused directly by the local magnetic field and not by
any induced electric field.
A
Figure2 The inducedelectric
field in the neighbourhoodof B
growing primarycurrent
n
v
95
complex situations, suchas the induction of currents
in a transformer, where it is partly correct, and very
convenient to say that the changing flux ‘causes’ the
induced EMF.
Lines offorce The numberof lines of force passing
throughanycircuitlinked
by amagnetic field is
nondenumerablyinfinite,justlikethenumber
of
points on a surface. It does not make strict physical
sense,therefore,to
say thatthenumber
of lines
increaseswhenthe
flux increases. If, however,
following Faraday (1855 p349) and Maxwell (1965
vol. 1 pp160-1) a discrete selectionof lines is made by
conventionwhosesurfacedensity
is chosentobe
proportionaltothe
local magnetic field strength,
then the increase in flux will, of course, be approximately proportional to the increase
in number of
theselines.Thisproportioncannotbeexact,
in
general, since whole numbers of lines are involved.
When
Faraday’s
geometrical
quantification
is
imposeduponthemagnetic
field produced by a
changing primary current, the number of lines irl its
neighbourhood will appear to multiply or decrease
andthepatternappearstoexpandoutwards
or
contract inwardswith a velocity which depends upcn
the rate of change of the current. A simple calcula
tion shows that the induced E M F in the secondary is
not given by the rule E= Blv,where v is the apparent
velocity of the lines. This is not surprising, since we
have seen earlier that the
velocity of the Faraday
lines in this instance is an apparent or phase velocity
onlyandnota
physical velocity.Althoughthe
growth of a primary current and the movement of a
magnetcanboth
giverise toanincrease
in the
conventional number of lines in a secondary circuit,
the two situations are quite distinct physically. since
the rule E=Blv does apply to the latter.
I have found figures 2 and 3 particularlyhelpful in
explainingelectromagneticinduction
in terms of
induced
electric
fields.
Figure
2 represent
the
electric field pattern
established
by growing
a
current in a coil. The accelerating electrons react
upon each other producing a retarding electric field
andtherefore,a
back E M F in the coil itself.The
induced electric field outside the coil is responsible
for currents in any local secondary circuits. These
electric fields disappear when the current becomes
steady and reverse when the current decreases. The
magnetic field has been excluded from the diagram
in order to simplify it, but also to draw attention to
the immediate cause of electromagnetic induction.
The second kind
Traditionally,
the
induced
electric
field which
appears in the neighbourhood of a moving magnet
or current-bearing circuit, is deduced from the E M F
96
induced in a moving wire by appealing to the symmetry of relative motion. When the observer moves
at the same speedas the wire, the magnetic force on
the conduction electrons necessarily disappears and
the observed force which they still experience must
be attributedtoanelectricfield,sinceonlyan
electric field acts on charges at rest.
Some authors seem to feel that the electric
field
thus inferred by ‘merely’ moving the framework of
theobservercannotbetruly‘real’.But
such an
attitude of mindwould imply that all other properties which appear when a framework of reference
is moved, such as momentum and kinetic energy,
arenottrulyrealeither.
If specialrelativity has
taught us anything, it is surely that we must take
relationalpropertiesseriously.Furthermore.the
inducedelectric field which appears in the neighbourhood of amovingmagnet
or current-bearing
solenoid can be deduced directly from the motion
of the source electrons in modern electromagnetic
theory (Lorentz 1909 p19).
Figure 3 illustrates the induced electric field pattern established by a magnet when it is translated
along its axis. The electric field intensity is zero on
themagneticequator.Whenthemotion
of the
magnet reverses the fields reverse their directions.
Theydisappear,
of course, when themagnet
is
brought to rest.
Modern motional EMF
Lorentz’s explanation of electromagnetic induction
in a moving wire as a simple magnetic force on the
convected electrons has gained ground rapidly during thepresentcentury.but
it hasnotyetfully
displaced the old electrodynahic tradition according
to which themotional E M F is dueto an induced
electric field. It is not my purpose here to decide in
favour of any theory, butsimply to point out that
Lorentz’sexplanation is theonlyonecompatible
with modernfundamentalelectromagnetismand
with electron physics.
It follows from this that the use of the symbol E .
which commonly represents the electric field intensity, can be very misleading if it is used to represent
the force perunit charge in a wire cutting a magnetic
field. This symbol implies that a force
will act as a
chargeatrest,
which is notgenerallytrue
in the
present case.
Some modern authors, while accepting the theory
of Lorentz here, nevertheless state that the motional
E M F is due to an ‘apparent‘ or ’equivalent’ or ‘effective’ electric field. Thisusagemay,perhaps,be
justified in certain circumstances but the fictional or
conventionalcharacter of this field needstobe
emphasised in teaching, since it can very easily be
lost sight of.
Faraday’s explanation of the motional E M F is so
vivid and well-established that it is easy to assume
that a magneticEMF is induced in a moving wire only
when it ‘cuts’ the lines of force. All that is required
in the theory of Lorentz, however, is that the moving wire be located in anappropriatelydirected
magnetic field. Whether or not the lines of force are
cut by the moving wire is irrelevant. However, when
the magnetic lines of force move there is invariably
an induced electric field accompanying them. as we
have seen. When a wire moves at the same speed as
the magnetic lines, or, equivalently, at the speed of
the conductor or magnet which produces them. the
induced magnetic force on each conduction charge
is
exactly balanced by theinducedelectricforce.
In
this case the relative velocity of the wire and magnetic lines of force is,in fact. directly proportional to
the resultant induced force on each convected conductionchargeand
it is thereforea
significant
quantity.
It also follows from Lorentz‘s theory that, were
the magnetic lines of force to rotate with the cylindrical magnet mentioned earlier, the electronsin the
magnet which are convected along by this motion
wouldexperiencenoresultantforce,sincethe
magnetic and induced electric forces on them would
balance. The observed electrostatic charge on the
magnetwouldthen
fail to appear.Thisfurther
confirmsFaraday’stheory
of stationarymagnetic
lines of force around a rotating magnet.
It has frequently occasioned surprise that the most
general
statement
of the law of Faraday
and
Neumann,E=-dWdt,shouldapply
to all three
kinds of electromagneticinduction.despitetheir
considerable differences. Thisis indeed a remarkable
example of the coherance of the laws of mathematicalphysics. I t should be noted, however, that the
samequantitative law is abletohandle physically
distinct cases here precisely because i t is a quantitative correlation rather than a causal law.
It has often been stated. correctly in my opinion.
that
although
electromagnetic
induction
of the
secondandthirdkindscanbededucedfromthe
magnetic force on a moving charge. the first kind
of electromagneticinductioncannotbe
similarly
deducedfrommoreprimitiveexperimental
laws,
Arguments which claim to achieve such a deduction
often rely on an appeal to mathematical coherence
among all cases. However plausiblesuch an assumption might be it does not constitute a demonstrative
proof.
A special case: the inductor voltage
What is the nature and origin of the ‘inductor voltage’ V!-=Ldfidt which leads the current by f cycle
in an A C circuit? I t is not the induced EMF in the coil,
E;= - Ldfidt since the latter lags behind the current
by cycle. It is not the voltage ‘applied‘ to thecoil in
general,either.sincetheresultantvoltagewould
then be zero, and no voltage would be available to
drive a current through the internal resistanceof the
inductor, which we will supposetobe significant.
Finally the ‘inductor voltage’ V Lis not the PD across
its owninternalresistance. since thelatter always
differs in magnitude and phase from the former.
I shall examine this topic carefully, since students
find it difficult and also becauseit is a prime example
a
A
Figure3 Theinduced
electric field in the
neighhourhood of a
moving magnet
A
v
97
of themanner
in which physics sometimesreinterprets the systemit deals with in order to make it
more amenable to mathematical treatment.
ApplyingKirchhoff'ssecondnetwork
law toa
simple series A(' circuit we obtain. E,+ E,= V,.+ VR.
where E, and E , represent the instantaneous values
of thesupplyandinduced
EMFS. respectively, and
V( and V R represent the PIX acrossthecapacitors
andacross all of theresistorsaroundthecircuit,
respectively.The
physical interpretation of this
equation, in accordance with the way it was discussed at the start of this article. is that the algebraic
sum of the potential differences across the underlying localised electromotive forces, in the generator
and in the inductors respectively, is exactly equal to
the algebraic sum of the net internal PIX across each
element of the circuit (including the generator and
inductors).
Strictly speaking. there are four voltages
associated with any inductor (figure 4); the induced E M F ,
E,=- Ldlidt: the masking electrostatic potentialdifferenceduetosurfacecharges
on thecoil;the
resultant of thesetwovoltages,
which is thenet
internal
R,/. where R, is the internal resistance
of the coil; and lastly the external or voltmeter I'D, V
across the coil, which is not equal to the net internal
PD.
I'D.
Theexternal r w can be calculated by applying
Kirchhoff's law to the voltmeter circuit, adopting the
same clockwise convention as in the main circuit
-E,=V-RII.
It follows that
V=Ldlldt+RII
The external P V across a pure capacitor or resistor
is equal to its internal I'D in each case. because the
latter voltages are conservative. It should be noted
that neither the external PD nor the electrostatic rr)
of the inductor appear in the Kirchhoff equation for
the main circuit.
When the latter equation is translated into a phasor diagram (figure 5 ) it becomes
or
IB,+B,~'=V,.?+V,?.
where E, etc. are peak values. Such diagramswill be
found in old text books (e.g. Joubert 1806 ~ 3 8 1 ) .
The
phasor
diagram
IS not mathematical
a
description of the A ( ' circuit: i t is rather a rotating
auxiliary construction whose projectiono n a suitable
axis
such
asAOB
is numericallyequaltothe
instantaneous values of the circuit variables. Time
differencesbetweenthephases
of voltagesand
currents in the circuit are encoded in the diagram a s
angular differences.
Consider next the following rearrangementof
+Vl2%a
the Kirchhoff equation.E,=(-Ei)+V,
manoeuvre which is reminiscent of 'd'Alembert's
principle'. This invites the conventional interpretation that (-E,)=+L d//dt=V, is not an induced I . M F
but a receivedvoltage on apar with V ( and V/<.
Clearly. V , is in antiphase with the true induced! ; M P .
The supply voltageis then the onlyE M F in the circuit.
It wouldseemthattherealinductorhasbeen
notionally replaced by an 'equivalent' inductor with
aconservativevoltageacrossit,
like that of the
capacitor. but opposite to the latter in almost every
respect in its effect upon the current. Furthermore.
the resultant internalPD across the equivalent inductor is now V , + R , I , which is exactly equal to the
external or voltmeter PD.
The conventional phasor diagram(figure 6) shows
that
E,=(P/.-V<)+P/(
Figure4 Four voltages associatedwith an inductor: the
induced I v t . the electrostatic PI). the resultant internal1'1)
and the external or voltmeter
f w
and
&L(V/,-v(.)2+v/<~.
Summing up. therefore, it would appear that the
real circuit has been replaced by a mathematically
simpler equivalent circuit and the induced E M F E , in
the inductor by a conventional or fictional conservative voltage VL=- E,. This sort of substitution is, of
course, common practice in physics.
Acknowledgments
I wish to express my gratitude to Dr H G Schneider,
Mrs U Clark and Dr A Best for their assistance in
translating some difficult German passages, and to
Dr B Minakovic and Dr N Doe of the Department
of Engineering Science. Oxford. for programming
the graphics.
98
A
“L
-
B
Figure 5 The ‘physical’ phasor diagram for a series AC
circuit
Figure6 The conventional phasor diagram f o r an A (
circuit
References
-1873 A Treatise on Electricity and Magnetism 2 vols
Ampere A-M Annee 1823 Memoire de I’Academie Royule
des Sciencer de I’lnstitut de Frunce vi (Paris 1827)
pp175-388.
Faraday M 1835,1844 and1855 Experimental Researches
in Electricity 3 vols (London)
Gauss C F 1863-1933 Werke 12 vols (Leipzig and Berlin)
Hertz H 1893 Elecwic Waves (London)
Joubert J . Foster G and Atkinson E 1896 Elementary
Treatise on Electricity and Magnetism (London)
Kirchhoff G 1892 Gesammelte Abhandlungen (Leipzig)
Lenz E 1834 Annalen der Physik und Chemievol. 21
pp483-94
Lorentz H A 1935-9 Collected Papers 9 vols (The Hague)
1909 The Theory of Electrons (Leipzig)
Maxwell J C 1965 Scient$c Papers (New York: Dover)
Queries in physics
A623 (from QlP70) (No polarising filters, yet the
Schlieren pattern above the shadowof my soldering
iron warned me thatit was about to setfire to the
curtain. How are heat shadows possible without
polarisation?) A correspondent writes: heat
shadows of the kind referred to are nothing to do
with polarisation. They are causedby variations in
refractive index. It is instructive to place a point
source of light a few metres away from a Bunsen
burner or other heat source. and to
view the effect
on a screen several metres on the other side.
(I have
produced dramatic imagesof convection currents in
liquids this way,using a transparent tank and a
(Oxford)
Neumann F E 1845 Physikalische Ahhandlungen der
Koniglichen Akademie der Wissenschufien 211 Berlin
1-87: ibid. (1847) 1-72
-1848Journal de Mathematiques Pureset Appliquees
13 113-78
Poynting J H 1920 Collected ScientiJic Papers (Cambridge)
Swenson L 1972 The Ethereal Aether (Austin and London)
Thomson J J 1891 Philosophical Maguzine 31 pp149-71
Volta A 1800 Phil.Trans. 90 pp403-3(1
Weber W 1852 (ed. R Taylor) Scientific Memoirs vol. 5
pp489-529
The authorwill be happy to provide more detailed
references for those interested.
heated wire.) If the point sourceis exchanged for an
extended source the shadows from the various parts
of it overlap so that little or no pattern is seen on the
screen. Thisis a nice illustrationof why (tiny) stars
twinkle in an atmospherewhich otherwise seems not
to be displaying turbulence.
The above item was selected from
Q l P . a thriceyearly broadsheet which includes answers. Itis
available on subscription at a rate
of f2.75 (f4
overseas airmail,f3.50 surface mail) from the
Editor, MrW H Jarvis, Salewheel House.
Ribchester. Preston PR3 3XU.All correspondence
concerning this feature should be addressed to
Mr Jarvis.
99