Electromagnetic Induction
by R. R. BIRSS
Imperial College, London
SUMMARY
Existing rules for calculating induced e.m.f. are examined and a nonrelativistic expression is developed for the force on a charged particle moving
in the vicinity of a time-varying but stationary source of magnetic field.
This expression is used to predict the e.m.f. induced in three basic electromagnetic devices. The controversial experiment in which a long cylindrical
permanent magnet is rotated about its axis of symmetry is discussed, together
with a new experiment of this type which suggests that, although the field may
be regarded as rotating, this does not, in itself, provide any indication as to
the location of the seat of the e.m.f. in closed circuits.
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1. INTRODUCTION
Faraday summarized the results of his experiments on magneto-electric
induction only in the form of a qualitative rule. The familiar mathematical
formulation
a@
e.m.f. = E . dl = - - ,
(1)
at
where
@ = J J B . dS,
was given by Neumann, a contemporary of Faraday, and was based principally
on an analysis of the work of Ampkre. The line integral in eqn. (1) is usually
evaluated around the closed electrical circuit itself whereas @ is the integral of B
over any surface, S, bounded by the path of integration.
Over one hundred years after its initial pronouncement, the interpretation
of (1) is often still attended by uncertainty and controversy. This unsatisfactory
situation has arisen chiefly because the flux-cutting rule (Carter 1954)
$
E=vxB,
(2)
gives a rival formulation but also because of the so-called electromagnetic
paradoxes. There is a temptation to ignore the apparent discrepancies and
to rely on direct experiment. Usually, this is the solution adopted by those
who teach the basic principles of electromagnetism-for it is at an elementary
level that the problem of reconciling (1) with (2) is most acute. A possible
single expression for E which can be interpreted unambiguously is presented
in Section 2 where the consequences of replacing a fixed path of integration by
a variable one are examined. Some electromagnetic paradoxes are discussed
in Section 3.
2. THEFORCE ON A CHARGED PARTICLE
2.1. Introduction
The difficulty of reconciling eqns. (1) and (2) may best be appreciated by
considering two basic electromagnetic devices, the iron-cored transformer and
the homopolar generator known as Faraday’s disc. Equation (1) may be
successfully applied to the iron-cored transformer but its application to Faraday’s
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287
disc is complicated by the fact that there is no unique linear path around which
the line integral of E may be calculated and hence there is no precise boundary
to the circuit. Thus eqn. (1) may only be applied to a homopolar machine by
the introduction of an arbitrary linear boundary along a particular path. Conversely, the flux-cutting rule may readily be used to explain the operation of
Faraday's disc but not that of an iron-cored transformer, in which it may be
supposed that all the flux is sensibly confined to the iron.
I n the special case of a closed linear rigid circuit and a constant magnetic
field produced by sources which are uninfluenced by the presence of the closed
circuit, then the flux through the circuit can only change if elements of the
circuit cut lines of force, so that (1) may be derived immediately from (2),
v being the velocity of an element of the circuit relative to the field sources.
Consequently, it is often maintained that (1) is a fundamental physical law
whilst (2) is merely a convenient computational rule. There is, however, a
strong physical reason for retaining the flux-cutting rule since it may be deduced
directly from the equation
F=Q(E +V x B)
(3)
which gives the force on an individual charge q moving with a uniform velocity
This equation has been
conclusively verified experimentally, at least for velocities considerably less than
the velocity of light, and it serves to explain in more fundamental terms not
only induced e.m.f's but also the forces exerted on conductors carrying current
in magnetic fields. A further advantage of (3) and of the flux-cutting rule is
that they predict the force on an individual charge at any point rather than the
integrated value round a closed circuit. Thus (2) enables the e.m.f. between
any two points in the circuit to be calculated and, when the total e.m.f. is concentrated in one element of the circuit, it predicts the seat of the e.m.f. It
might therefore be thought that (1) should be employed to calculate induced
e.m.f's whilst (2) should be used to provide a fundamental understanding of
the physics of the process. The apparatus which, perhaps, best illustrates the
fallacy of this contention is the high-energy particle accelerator known as the
betatron.
I n the simplest form of betatron, electrons are constrained to describe a
circular path, between the poles of an electromagnet, by a radial electromagnetic
force v x B. For a constant value of B the trajectories of the electrons may
thus be satisfactorily predicted from eqn. (3), but in the normal operation of
the machine the electrons are accelerated to high energies by a tangential force
produced by a steady rate of decrease of the magnetic induction, B. I n general,
each electron is therefore subject to two forces, a radial force due to the existence
of B and a tangential force due to its diminution. The tangential force cannot
be reconciled with (3)-which is therefore seen to hold only for constant B-but
it may be deduced, albeit somewhat artificially, from (1). Although the existence
of the tangential force emphasizes the inadequacy of (3), and hence of (2),the
existence of the radial force-even for time-invariant B-does not invalidate (l),
for eqn. (1)predicts only the value of the tangential component of e.m.f. integrated
around the ' circuit '. Thus although (1) provides the correct value for
$ E .dl it fails to predict the total force acting on an electron. Even for a
closed rigid linear circuit of conducting material (which may be referred to for
v relative to the (time-invariant) sources of E and B.
288
R. R. Birss
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convenience as a ‘ wire loop ’) eqn. (1) is not completely satisfactory. A
material circuit is to some extent simpler for it permits a unique resolution of
electromagnetic forces into e.m.f’s and ponderomotive forces but, although the
circuit of integration need not necessarily coincide with the wire loop, the
e.m.f’s which result from the presence of transverse electric fields (the Hall
effect) cannot be readily determined. For solid bodies of conducting material,
or for charged particles in oacuo, the applicability of eqn. (1) is, of course, even
more limited except in cases which are simplified by considerations of symmetry.
It is therefore desirable to investigate the possibility of developing eqn. (1) to
deduce an expression for the electric field at any point (i,e. the force on an
individual charge) rather than the integrated value round a closed path. Such
an expression would facilitate a more complete understanding of the trajectories,
in oacuo, of charged particles in varying magnetic fields and of magnetoelectric induction in ‘ solid ’ bodies.
2.2. Expressions for E
In terms of the magnetic induction, B, eqn. (1) may be written
$E
. dl = -aat
11..
dS .
(4)
Since div B = 0, it is permissible to put B = curl A so that (4)may be transformed
by Stokes’ theorem to give
$
(5)
where the variation with time of the vector A and of the shape and position of
the path of integration both contribute to the rate of change of $A . dl. It is
shown in the Appendix that eqn. (5) may be written in the form
where v is the velocity of any point on the path of integration.
when written in the alternative form
f
Equation (6),
(7)
illustrates a separation of the e.m.f. into a variational or transformer term and
a motional or flux-cutting term. It should be emphasized, however, that this
separation is only a unique one relative to the coordinate system to which the
velocities, magnetic inductions and the position and configuration of the circuit
are referred.
T o deduce, from (6),the value of E at any point, it must be remembered that
any acyclic function may be added to the integrands without affecting the value
of the line integrals around any closed path. E may therefore be written in
the form
E=-*+vxB-grad(++$)
at
(8)
where 4 is the electrostatic potential, which exists even in the absence of changes
in the magnetic field and the circuit, and t,h is an unknown scalar function.
Similarly, the function A is not completely defined by the equation B =curl A,
since any acyclic function may be added to A without altering the value of the
Electromagnetic Induction
289
curl. However, in view of the occurrence of the unknown function $
,I no
additional restriction is imposed by postulating that div A=O so that A is
thereby completely defined and may be identified with the Maxwell vector
potential.
2.3. Three basic electromagnetic devices
Equation (8) may be readily used to explain the operation of the three
electromagnetic devices discussed in sub-section 2.1 provided it is remembered
that E is the force on an individual charge and that v must be interpreted as
the velocity of an individual charge relative to a set of coordinate axes in which
the sources of B are stationary. If this set of axes is stationary in the laboratory
frame of reference, then the operation of the betatron shows that the term in
is not present, since the radial and tengential forces on each electron are observed
to be given exactly by v x B and - aA/at respectively.
In an iron-cored transformer, e.m.f's may arise from both the v x B and the
- a A / a t terms. For example, consider a set of cylindrical polar coordinates,
I , 8, z, in which a long cylindrical iron core, of radius a, has the x-axis as its
axis of symmetry. The vector potential, A , is everywhere perpendicular to
both the radius I and the z-axis. In terms of the alternating flux density Bt
in the iron core, the magnitude of A is given by A,= ijrBc for r,ca (i.e. within
the core) and A = &a2Bi/rfor r a a . The e.m.f. developed across the terminals
of a single-turn coil of radius u', concentric with the core, is thus
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+
where @g is the magnetic flux in the iron. When a current flows in this coil an
e.m.f. is developed across the turn and perpendicular to the z-axis : its magnitude
is v x B,, where v is the velocity possessed by individual charges by virtue of
the current flowing in the coil and B,, is the small leakage flux density outside
the iron at a'. This e.m.f. gives rise to a compensating transverse separation
of static charge and hence to the Hall effect.
Since Faraday's disc operates in a constant magnetic field, the observed
e.m.f. arises only from the v x B term. The use of eqn. (8) rather than (6)
or (7) obviates the difficulty, already mentioned, that there is no precise boundary
to the circuit and hence no unique linear path of integration. It shows, however,
since v is the velocity of any point on the path of integration, that, if (6) or (7)
is to be used, the motional term should be calculated by considering the path
of integration to be completed by a radius vector rotating with the disc and by
a line (of constantly increasing length) drawn around the periphery of the disc
from this radius vector to the sliding contact. The variational term,
--= -
- . dl,
(9)
at
is, of course, zero since, as indicated by the partial derivatives, it is evaluated
by considering the path of integration to be stationary. It may be mentioned
at this point that the change in length of the peripheral segment of the path of
integration need cause no difficulty since the derivation of (6) and (7) applies
to a circuit which is being deformed both by flexure and extension. Nor does
the mere existence of the sliding contact cause any difficulty per se.
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290
R. R.
Birss
3. SOMEELECTROMAGNETIC PARADOXES
3.1. Stationary magnetic-field sources
It was stated in section 2.2. that eqn. (8) gives the force on an individual
charge if v is interpreted as the velocity of the charge relative to a set of coordinate axes in which the sources of B are stationary, and it is thereby assumed
that none of the sources of B move relative to each other. If this is true, and
if the sources of B are also stationary in the laboratory frame of reference, the
force on an individual charge may be written in the form
aA
E=--+vxB,
at
where, for simplicity, the electrostatic term - grad 4 has been omitted. Under
such circumstances the only paradoxes which can arise are those which result
from employing (1) or (2) or both instead of (10). A typical example, which
also illustrates the danger of attaching too much importance to the concept of
flux-linkages, is afforded by the experiments of Blonde1 (1914). I n these
experiments a coil of wire is unwound in a steady magnetic field so that the
number of turns linking magnetic flux is progressively decreased. Despite the
fact that the flux-linkages are progressively decreased, no e.m.f. is generated
unless the connection, from the axis of the coil to the last turn, rotates with the
coil. This result is, of course, to be expected from eqn. (lo), although believers
in the dictum that the e.m.f. is equal to the rate of diminution of flux-linkages
may draw some comfort from the fact that it is only when a ‘ solid ’ body is used to
complete the circuit or when the number of ‘ turns ’ changes progressively but
discontinuously that the expected e.m.f. is not obtained. Nor are these the
only b2tes noires, for paradoxical results can also arise from a procedure known
as ‘ substitution of circuit ’. A typical example is that of an ordinary tubular
sliding resistor situated in a magnetic field. Again the number of flux-linkages
can be changed without inducing an e.m.f. but only by substitution of circuit.
A more satisfactory explanation is provided by eqn. (10).
3.2. Moving magnetic-field sources
If, relative to the laboratory frame of reference, all the sources of B are
moving with the same constant (linear) velocity, u, which is small compared
with the velocity of light, then the force on an individual charge is
aA
E = - -+(v-U)X
B,
(11)
at
where v is the velocity of the charge in the laboratory frame of reference and
a A / a t is calculated relative to axes which move with the sources of B. An
interesting paradox which arises with a linearly moving magnet is Herring’s
experiment (1908) in which a magnet is forced between a metal spring clip,
the two halves of which are joined electrically only by an external circuit.
From eqn. (11) it may be seen that, since a A / a t = O , the value of E at any
point depends only on the relative velocity of the circuit and the magnet, and
the e.m.f. obtained is therefore the same as if the clip and circuit were in motion
and the magnet at rest. Integration of E from one side of the clip to the other
around the circuit gives no resultant e.m.f. (This may be most easily seen for
the special case in which the clip is at the extreme end of the magnet. If the
e.m.f. induced in a circuit in which the clip is bridged by a wire sliding over
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Electromagnetic Induction
29 1
the end face of the magnet is compared with the rate of change of flux through
that circuit it may be seen that the net e.m.f. around the circuit bridged by the
magnet is zero.)
If two or more sources of B are moving with different velocities then a
number of terms of the same form as (11) must be used, summed over all the
sources of B. For example if two similar magnetic poles are symmetrically
situated near a conductor so that instantaneously they lie on a straight line
through the conductor and perpendicular to it, whilst moving at right angles
to this line and to the conductor in opposite directions, then the e.m.f. obtained
is double that appropriate to a single pole although the conductor itself is in
zero magnetic field. This result is often stated to be paradoxical. Similarly
a summation of eqn. (11) may be applied to any number of linearly moving
sources of B although its application is seldom straightforward when materials
are present which act as magnetic shields without themselves being primary
sources of B.
3 . 3 . Rotating magnetic-field sources
More complicated-and more interesting-cases arise when the sources of
B are rotating in the laboratory frame of reference ;for example, a long cylindrical
bar magnet rotating about its axis. Again the - aA/% term in eqn. (8) is zero
and a stationary charge should experience a force from the v x B term or, more
exactly, from a Z v x B term where the summation extends over the elementary
dipoles forming the magnet (each moving with a linear instantaneous velocity v).
However, there is no experimental evidence, as there is with a stationary magnet,
that the term -grad t,4 is zero and there is thus the possibility that such a term
may be introduced by the rotation of the magnet. It is not therefore clear whether
or not a stationary charge in the vicinity of the magnet will be acted upon by a
force. For example, in the modified Faraday's disc arrangement shown in
fig. 1, when the disc D spins in the field of the magnet M a current flows through
M
=2
Fig. 1.
the circuit AG,B which is completed by means of sliding contacts on the
periphery and the axle of the disc. If, alternatively, the disc is stationary
whilst the magnet rotates about the common axis EB no current is registered
by GI. However, this does not prove that E is everywhere zero; for Xv x B - grad $
integrates to zero around the complete circuit whatever the value of t,4. Further,
although a galvanometer G, connected to brushes E and F bearing on the
magnet as indicated shows a deflection, this could be interpreted as being due
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R. R. Birss
either to a field of force (E) surrounding the magnet or to one within the magnet
itself. These difficulties most usually present themselves in terms of a conceived motion of the magnetic field surrounding the magnet. This point of
view originated with Faraday, who was accustomed to thinking in terms of
lines or tubes of magnetic force and who, in his Diary of July 14, 1851, asked:
When the magnet is still and the wire is moving, it seems
“ 11345
unlikely that the current should be generated anywhere else than in the
moving wire; for its motion or quiescence makes all the difference. But
then, when the magnet is moving, where is the current then generated?
I n the wire across which the curves, that may be supposed to move with
the magnet, are passing? Or in the magnet, which may be supposed to
be moving (as the wire did) whilst the curves are considered still? Do the
lines of force revolve with the magnet or do they not ? ”
If a motional e.m.f. is conceived to arise from motion relative to lines of
magnetic force rather than motion relative to the field sources themselves, then
the concept is only a useful one in so far as the motion of the lines of force may
be determined for a given motion of the field sources. The connection between
eqn. (8) and the concept of motion relative to lines of force is an interesting one.
If a magnetic-field source is stationary in the laboratory frame of reference,
the lines of magnetic force must also be assumed to be stationary since it is
known from experiment that z,h= 0. If the field source moves with a constant
velocity v which is small compared with the velocity of light, then, because of.
the equivalence of inertial frames of reference, it may be supposed that I,$ is
again zero and hence the lines of force must be assumed to move with the field
sources. A rotating field source, however, is not stationary in an inertial
frame of reference and therefore the principle of equivalence cannot be invoked
to deduce that z,h must be zero. A finite value of I,$ does not, of course, alter the
e.m.f. induced in a closed circuit, for -grad I,$ integrates to zero around any
closed path, but it may affect the location of the seat of the e.m.f.
For example, in the arrangement shown in fig. 1, a rotation of the magnet M
results in a deflection of the galvanometer G, which, if I
)
= 0, arises wholly in
the circuit EG,F. If + # O the observed e.m.f. is no longer generated wholly
in this circuit but partly in the magnet itself and, for a certain finite value of z,h,
the seat of the e.m.f. is wholly in the magnet. Thus, if the e.m.f. is conceived
to arise from motion relative to lines of force, the lines of force may be assumed
to be either stationary or rotating. Both assumptions are widely held although,
as yet, the hypothesis that the lines of force rotate at a speed which is different
from that at which the magnet rotates has found no adherents.
Similar considerations arise for the stationary circuit AG,B of fig. 1, in which
no e.m.f. is generated by a rotation of the magnet M. Although it may be
argued that it is inconsistent to concede a variation in the place where the
e.m.f. is generated since the relative motion of the magnet and the disc is the
same as in the conventional Faraday’s disc experiment, it must again be remembered that rotation, unlike translation, is absolute and not relative.
3.4. A meaningless question ?
The quotation from Faraday’s Diary given above poses two questions, viz.
(u) When the magnet is rotated where is the e.m.f. generated?
(b) Do the lines of force rotate with the magnet or do they not?
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I t is obvious that Faraday considered these questions as being two alternative
but equivalent formulations of the same problem. This view was shared by
Steinmetz, Barnett (1912, 1913), Kennard (1913, 1916) and Pegram (1917)
who also believed, with Faraday, that the lines of force do not rotate and that
an e.m.f. is induced in the rotating magnet. Weber-to whom the term unipolar
induction is due-believed that they did rotate, a view which was shared by
Cullwick (1938), Sir Oliver Lodge (although this has been subsequently
disputed), Andrew Gray, Cramp (1933) and Cramp and Norgrove (1936). A
recent paper by Hammond (1954) promoted further discussion ; Professor
Cullwick and Dr. Wilkinson maintained that the e.m.f. was generated in the
stationary leads whereas Hammond said that he still inclined to the view that
the seat of the e.m.f. was in the rotating magnet.
Whatever the true state of affairs, it had long been known (Preston 1885)
that the integrated value of E around the complete circuit was the same under
either hypothesis and it was not therefore expected that experiments on complete
circuits would settle the issue. In such circumstances it was not long before
it was maintained that the question of whether the field rotates with the magnet
or not is fundamentally a meaningless one. This point of view was put forward
by PoincarC (1900), Abraham, Kennard and Fiippl and found its most recent
and most vehement protagonist in Professor G. W. 0. Howe (1934, 1936 a, b,
1950).
The most cogent expressions of the belief that the question is in reality
meaningless are due to Howe. He pointed out that it would be meaningless
to ask whether the temperature field of a hot body rotated with it or whether
the uniform distribution of illumination from a lamp rotated with it. Further,
he maintained that the concept of a rotating field implies that the lines of force
have an individuality, that one can be distinguished from the next: but lines
of force are mathematical abstractions, their density is decided only by convention,
I ‘ One is not justified in regarding them as physical entities like bristles that
one can chalk-mark and thus trace their movement”.
Against attaching too much physical significance to lines of force Howe argued
that although, under certain circumstances, e.m.f.’s may be generated, and the
magnitudes of these e.m.f.’s may be proportional to the number of lines of force
cut, this does not mean that the e.m.f.’s are due to the cutting of physical lines.
These, and similar, arguments have been advanced so frequently and with such
persistence that it is of considerable interest to consider them in more detail.
T h e analogy between a magnetic field and, for example, a temperature field
is a doubtful one since it is not possible to deal purely with a magnetic field
isolated from associated electric effects. T h e possibility of an electrical effect
associated with the rotation of a magnet is neglected in the analogy with a
temperature field, yet such an effect can be made a criterion of rotation. The
claim that the concept of a rotating field implies the individuality or distinguishability of lines of force overlooks the elementary distinction between a necessary
and a sufficient condition. Following the path of an individual line of force
might well be sufficient to define the rotation of a magnetic field; the impossibility of this procedure does not, however, render invalid the concept of field
rotation since alternative criteria may exist which permit a unique determination
CP
T
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R. R. Birss
of whether the field is rotating or not. Furthermore, such criteria need depend
in no way on the possibility of tracing the movement of individual lines of force.
Such a criterion would, for example, be to define the rotation of a field in terms
of the force on a stationary charge. T h e argument against attaching too much
physical significance to lines of force cannot be treated too seriously for, as with
many other physical concepts, their justification is only on the grounds of convenience. Whether the cutting of lines and the e.m.f. are cause and effect or
not they certainly co-exist and mathematically this is the same thing. There
is much point in Professor Cramp’s argument (1938) that relative motion
between a conductor and the lines of force confers physical reality to the lines,
in other words a physical explanation of the e.m.f. involves the physical existence
of the lines of force. Relative motion between a conductor and a structureless
medium is a ‘ contradiction in terms-an abrogation of essential consistency ’,
it is ‘ operationally meaningless ’.
As well as advancing counter-arguments to the charge that it is a meaningless
question t o ask whether the field rotates with the magnet or not, it is also possible
to perform an experiment which proves that the question is not meaningless
and which also has some bearing on whether the field should, in fact, be regarded
as rotating or not. This experiment is discussed in section 3.5. It is
therefore desirable to consider whether questions (a) and ( b ) are, in reality,
alternative formulations of the same question. T h e location of the seat of the
e.m.f. is a question of fact which, unfortunately, cannot be answered by experiments on closed circuits. Measuring the force on an isolated charged particle
would, of course, be sufficient to answer question (a). Such experiments-for
example rotation of the magnetic poles in the betatron experiment described in
section 2.1.-have,
however, not yet been performed. I t may therefore
be concluded that ( a ) is a question of fact which has not yet been decided by
experiment but to which a definite answer exists. Question ( b ) is not a question
of fact but of convention: if a motional e.m.f. is conceived to arise from motion
relative to lines of magnetic force then a set of rules which governs the motion
of the lines of force must be formulated so as not to contradict any existing
experimental results. It follows that an answer to (a) in conjunction with all
the other available experimental evidence, must either automatically answer (b)
or invalidate altogether the concept of motion relative to lines of force. It is,
however, possible to devise an experiment which will provide an answer to ( b )
that must be regarded as valid until it is verified or invalidated by a decision
on ( a ) , or until it is invalidated by further experiments to decide (b). Such an
experiment is described in section 3.5 but it should be emphasized that a
decision as to whether or not the field may be regarded as rotating does not, in
itself, provide any indication as to the location of the seat of the e.m.f.
3.5. A new experiment
An experiment has recently been performed with the apparatus illustrated
diagrammatically in fig. 2. M and M’ are two hollow cylinders of Alcomax
magnetized with adjacent surfaces of opposite polarity. Y is a soft iron yoke, of
square cross section. D and D’ are copper discs, A is an annular brass disc
insulated from D‘C which is a copper rod. T h e two systems (D’DC) and
(AB, MYM’) can both be rotated independently of each other by means of a
motor and a driving arrangement on the end C (which have been omitted for
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295
clarity). This driving system is surrounded by an electrostatic screen and it
is not in electrical contact with the rest of the apparatus. The wire AB is
insulated from the yoke Y and the electrical circuit, which is completed by the
galvanometer G, is D’DBA.
When the disc rotates and the yoke is stationary it is found that an e.m.f.
is obtained proportional to the speed of rotation, whilst with the yoke rotating
and the disc stationary an equal and opposite e.m.f. is obtained. When both
disc and yoke rotate together no galvanometer deflection is observed. It may
be assumed that the magnetic field is confined to the yoke and the pole-pieces
MM’, except in the region of the air gap. A quantitative analysis of the experimental data confirmed that the effect of the stray field could, in fact, be neglected.
It may be noted that the results obtained with this apparatus are in complete
agreement with eqn. (8) because the - aA/& term vanishes since it is calculated
relative to coordinate axes which rotate with the yoke and pole-pieces.
If it is asked what bearing the results of this experiment have on question (b)
of section 3.4, a plausible answer would be to say that, since an e.m.f.
can be induced when the disc D is stationary in a magnetic field which is undergoing no change either in magnitude or direction and when the connecting
circuit AB is rotating in zero magnetic field, then the magnetic field between
the pole-pieces M and M’ must be regarded as rotating. However, since it
could be argued that this e.m.f. is a variational one which arises from a change
in the flux linking the complete circuit, it is necessary to interpret the experimental data with reference to the rules which must be complied with if observed
electromagnetic effects are to be correctly explained by a consideration of lines
of magnetic force.
These rules have been given by Bewley (1952) and may be explained by
reference to eqn. (7). They state that, in a closed circuit, the motional e.m.f.
is to be calculated by interpreting v as the velocity of any point of the circuit
relative to the lines of force and by assuming that v and 6 are instantaneously
constant, whilst the variational e.m.f. is to be calculated by assuming that the
circuit is instantaneously stationary relative to the lines of force and that only B
varies. For a closed linear circuit these rules are ambiguous only in so far as
the motion of the lines of force is uncertain, but the introduction of a pluridimensional body complicates the evaluation of both motional and variational
components since the circuit (or path of integration) need not necessarily be the
same for both components. However, a certain simplification is afforded by
the fact that, if two particular paths of integration are postulated, these paths
must depend only on the relative disposition and motion of the various parts
of the actual circuit and in no way on whether or not the magnet is rotating.
For example, however the circuit D‘DBA of fig. 2 is considered to be completed through the disc D, it must be completed in the same way as is the
circuit AG,B of fig. 1. When either the circuit AG,B or the disc D rotates, the
observed e.m.f. is a motional one whether or not the magnet M rotates and
whether or not, if it does, the field rotates with it. The $(v x B) . dl term must
therefore be calculated around a path of integration which includes, as described
in section 2.3, a radius vector rotating with the disc and a peripheral segment
of constantly increasing length, Since the variational component is absent,
the -a@/& term must be calculated relative to a circuit which is completed
by a straight line from A to the axis of D and which thus does not link any
296
R. R. Birss
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magnetic flux. Making the same assumptions for the apparatus shown in
fig. 2, it follows that the experimental results described above may only be
Fig. 2.
interpreted by assuming that the lines of force rotate with the magnet. This
does not, in itself, provide any indication as to the location of the seat of the
e.m.f.
It may, of course, be objected that proving the field rotates with the unsymmetrical magnet shown in fig. 2 does not prove that it would rotate with a
completely symmetrical magnet-nor
indeed does it. However, even for a
symmetrical rotating magnet, relativistic electromagnetic theory (Cullwick
1954) confirms that the field may be regarded as rotating and assigns the seat
of the e.m.f. to the stationary conductors. T o pursue this further would,
however, be beyond the scope of the present article for the treatment is controversial and one in which the permitted limits of scientific heterodoxy are soon
reached.
ACKNOWLEDGMENT
T h e experiment described in section 3.5. was performed while the
author was a member of the Physics Department of Nottingham University
and he is indebted to Professor L. F. Bates for providing the facilities for the
construction of the apparatus and for the experimental work.
REFERENCES
BARNETT,
S. J., 1912, Phys. Rev., 35, 324; 1913, Ibid., 2, 323.
BEWLEY,
L. V., 1952, Flux Linkages and Electromagnetic Induction, p. 26 (New York:
MacMillan).
BLONDEL,
A., 1914, Comptes Rendus, 1959, 674.
G. W., 1954, The Electromagnetic Field, p. 149 (London: Longmans).
CARTER,
CRAMP,W., 1933, Electrician, 111, 471; 1938, Ibid., 121, 371.
E. H., 1936, Journal I.E.E., 78, 481.
CRAMP,W., and NORGROVE,
CULLWICK,
E. G., 1938, Electrician, 121, 286 and 467; 1954, Proc. I.E.E. (Part I), 101, 162.
HAMMOND,
P., 1954, Proc. I.E.E. (Part I), 101, 147.
C., 1908, Trans. Amer. I.E.E., 27, 1341.
HERRING,
297
Electromagnetic Induction
HOWE,G. W. O., 1934, Electrician, 113, 329; 1936a, Ibid., 117, 191,493 and 625; 1936 b,
Journal I.E.E., 79, 345; 1950, Proc. I.E.E. (Part I), 97, 129.
KENNARD,
E. H., 1913, Phys. Rev., 1, 355; 1916, Ibid., 7, 399.
PEGRAM,
G. B., 1917, Phys. Rev.,10, 591.
PO IN CAR^, H., 1900, L’,&lairage $lectrique, 23, 46.
PRESTON,
S. T., 1885, Phil. Mag., 19, 131 and 215.
APPEND I X
If the shape and position of the path of integration do not change with time,
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then
2at $ A . dl = $ $ . dl .
I n general, however, an additional term is present
due to variations in the shape and position of the path of integration. This
term may be evaluated by considering a finite section PQ of the path of
integration which moves to the position P’Q’ after a short interval of time dt.
If A is considered to be constant over the interval dt then the change due to
the motion of the curve alone is
If v is the velocity of any point on the path of integration, the first two terms
on the right hand side of this equation represent dt times the difference between
the values of A . v at Q and at P, whilst the third term may be expressed, by
Stokes’ theorem, as a surface integral over elements of area dt(dl x v). Thus
1I:A.dl-/>.dl=dt
dl . g r a d ( A . v ) - d t
The complete rate of change of
a~
%SPA . dl =
IP[at+
Q
aA
1:
A . dl is thus
grad (A . v) -v x curl A
be written in the form $E
(vxcurlA). d l .
.dl=f[
1.
dl and eqn. (5) may therefore
-:.
+v x B]
dl
.
The Author :
Dr. R. R. Birss is at present engaged at Imperial College upon research in solid state
physics with particular emphasis on ferromagnetic materials. He is also interested in
special relativity and the electrodynamics of moving bodies.