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Faraday induction and the current carriers in a circuitTimothy
H. Boyer Citation: American Journal of Physics 83, 263 (2015); doi:
10.1119/1.4901191 View online: http://dx.doi.org/10.1119/1.4901191
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Faraday induction and the current carriers in a circuit
Timothy H. BoyerDepartment of Physics, City College of the City
University of New York, New York, New York 10031
(Received 16 August 2014; accepted 27 October 2014)
This article treats Faraday induction from an untraditional,
particle-based point of view. The
electromagnetic fields of Faraday induction can be calculated
explicitly from approximate
point-charge fields derived from the LienardWiechert
expressions, or from the DarwinLagrangian. Thus the electric fields
of electrostatics, the magnetic fields of magnetostatics, and
the
electric fields of Faraday induction can all be regarded as
arising from charged particles. Some
aspects of electromagnetic induction are explored for a
hypothetical circuit consisting of point
charges that move frictionlessly in a circular orbit. For a
small number of particles in the circuit
(or for non-interacting particles), the induced electromagnetic
fields depend upon the mass and
charge of the current carriers while energy is transferred to
the kinetic energy of the particles.
However, for an interacting multiparticle circuit, the mutual
electromagnetic interactions between
the particles dominate the behavior so that the induced electric
field cancels the inducing force per
unit charge, the mass and charge of the individual current
carriers become irrelevant, and energy
goes into magnetic energy. VC 2015 American Association of
Physics Teachers.
[http://dx.doi.org/10.1119/1.4901191]
I. INTRODUCTION
When students are asked what causes the electric field in
aparallel-plate capacitor, the response involves charges on
thecapacitor plates. Also, students say that the magnetic field ina
solenoid is due to the currents in the solenoid winding. Butwhen
asked for the cause of the Faraday induction field in asolenoid
with changing currents, the usual student responseis that the
induction field is due to a changing magneticfieldnot that it is
due to the acceleration fields of thecharges in the solenoid
winding. The student view reflectswhat is emphasized in the
standard electromagnetism text-books.1,2 Indeed, although Darwin3
computed inductionfields from accelerating charges in the 1930s,
today it is rareto have a physicist report that the Faraday
induction fieldarises from the acceleration of charges.4,5
In this article, we wish to broaden the perspective onFaraday
induction by reviewing some aspects of the particlepoint of view.
We treat the induction fields as arising from theelectromagnetic
fields of point charges as derived from theLienardWiechert
expressions or from the DarwinLagrangian. First, we mention the
LienardWiechert6 formtaken by the electric and magnetic fields of a
point charge ingeneral motion. Then we turn to the
low-velocity-small-dis-tance approximation derived in the 1940
textbook by Pageand Adams.7 This approximate form for the
electromagneticfields is the same as that obtained from the
DarwinLagrangian8 of 1920. Here in the present article, the
approxi-mated fields are used to treat Faraday induction in detail
for ahypothetical circuit consisting of point charged particles
mov-ing frictionlessly on a circular ring. This circuit provides
arough approximation to that of a thin wire that is bent into
acircular loop. For small numbers of particles (or for
noninter-acting particles), we see that the magnitudes of the
massesand charges of the charge carriers are important, that
theinduced electric fields can be small, and that energy goes
intothe mechanical kinetic energy of the charge carriers.However,
for large numbers of interacting charges, the mutualinteractions
make the magnitudes of the masses and chargesunimportant, the
induced electric fields balance the inducingfields, and the energy
goes into magnetic energy of the circuit.
II. POINT-CHARGE FIELDS
A. Point-charge fields for general motion
Although students are familiar with the Coulomb electricfield of
a point charge, many do not study electromagnetismto the point that
they see the full retarded point-charge fieldsfrom the
LienardWiechert potentials6
E r; t e1 v2e=c2
n ve=c 1 n ve=c 3jr rej2
" #tret
ec2
n f n ve=c aeg1 n ve=c 3jr rej
" #tret
; (1)
and
Br; t ntret Er; t; (2)
where the unit vector n r re=jr rej, and where theposition ret,
velocity vet, and acceleration aet of thecharge e must be evaluated
at the retarded time tret such thatcjt tretj jr retretj. These
field expressions, togetherwith Newtons second law for the Lorentz
force, give thecausal interactions between point charges.9 The only
thingmissing from this formulation of classical electrodynamics
isthe possible existence of a homogeneous solution ofMaxwells
equations (such as, for example, a plane wave)that might interact
with the charged particles. However, elec-tromagnetic induction
fields are distinct from homogeneousradiation fields, and
therefore, we expect that the fields ofFaraday induction can be
treated as having their origin fromcharged particle motions. The
exploration of this point-charge point of view in connection with
Faraday induction isthe subject of the present article.
B. Low-velocity-small-distance approximation
withoutretardation
Since the electric field of Faraday induction is distinctfrom
the Coulomb field of a stationary charge, we expect
263 Am. J. Phys. 83 (3), March 2015 http://aapt.org/ajp VC 2015
American Association of Physics Teachers 263
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this induction field to involve the additional velocity-
andacceleration-dependent terms of Eq. (1). The use of the
fullexpressions involving retardation in Eqs. (1) and (2) can be
aformidable task. For the radiation fields, which fall off as1=r at
large distances, the presence of retardation cannot beavoided as
the signal travels from the source charge to thedistant field
point. However, for field points that are nearpoint charges that
are moving at low velocities, it is possibleto derive from Eqs. (1)
and (2) approximate expressionsfor the electric and magnetic fields
that involve no retarda-tion. The task of approximation is not
trivial and is carriedout in the textbook Electrodynamics by Page
and Adams7
giving
Ej r;t ejrrj jrrjj3
1v2j2c2
32
vj rrj cjrrjj
2" #
ej2c2
ajjrrjj
aj rrj rrj
jrrjj3
" #O 1=c3
;
(3)
and
Bj r; t ejvjc
r rj jr rjj3
O 1=c3
; (4)
where in Eq. (3) the quantity aj refers to the acceleration
ofthe jth particle. These approximate expressions are powerseries
in 1=c, where c is the speed of light in vacuum. Theapproximate
fields in Eqs. (3) and (4) can be quite useful;they were used by
Page and Adams10 to discuss action andreaction between moving
charges, and by the current authorwhen interested in
Lorentz-transformation properties ofenergy and momentum11 and
questions of mass-energyequivalence.12 Here, the approximate
expressions for thefields are exactly what is needed to understand
Faradayinduction from a particle point of view.
The approximate fields in Eqs. (3) and (4) also correspondto
those that arise from the Darwin Lagrangian8
L XiNi1
mic2 1 v
2i
2c2 v
2i
28c4
! 1
2
XiNi1
Xj6i
eiejjri rjj
12
XiNi1
Xj 6i
eiej2c2
vi vjjri rjj
vi ri rj vj ri rj
jri rjj3
" #
XiNi1
eiUext ri; t XiNi1
eivic Aext ri; t ; (5)
where the last line includes the scalar potential Uext andvector
potential Aext associated with the external electromag-netic
fields. The Darwin Lagrangian omits radiation butexpresses
accurately the interaction of charged particles
through order 1=c2. The Darwin Lagrangian continues toappear in
advanced textbooks,8 but the approximate expres-sions (3) and (4)
seem to have disappeared from the con-sciousness of most
contemporary physicists. The Lagrangianequations of motion from the
Darwin Lagrangian can berewritten in the form of Newtons second law
dp=dt dmcv=dt F with c 1 v2=c21=2. In this Newtonianform, we
have
d
dt
mivi
1v2i =c2 1=2" #
ddt
mi 1v2i2c2
vi
eiE eivicB
ei Eext ri; t Xj 6i
Ej ri; t
eivic Bext ri; t
Xj 6i
Bj ri; t
;
(6)
with the Lorentz force on the ith particle arising from
theexternal electromagnetic fields and from the electromag-netic
fields of the other particles. The electromagnetic fieldsdue to the
jth particle are given through order v2=c2 byexactly the
approximate expressions appearing in Eqs. (3)and (4).
III. ELECTROMAGNETIC INDUCTION
Electromagnetic induction was discovered by MichaelFaraday, not
as a motion-dependent modification ofCoulombs law, but rather in
terms of emfs producingcurrents in circuits. This circuit-based
orientation remainsthe way that electromagnetic induction is
discussed in text-books today. The emf in a circuit is the closed
line integralaround the circuit of the force per unit charge f
acting on thecharges of the circuit: emf
f dr. Faradays emfs were
associated with changing magnetic fluxes, and Faradays lawof
electromagnetic induction in a circuit is given by
emfF 1
c
dUdt
; (7)
where U is the magnetic flux through the circuit.As correctly
emphasized in some textbooks,13 electromag-
netic induction in a circuit can arise in two distinct
aspects.The motional emf in a circuit that is moving through
anunchanging magnetic field can be regarded as arising fromthe
magnetic Lorentz force acting on the mobile charges ofthe moving
circuit. On the other hand, when the circuit is sta-tionary in
space but the current in the circuit is changing,new electric
fields arise. These new electric fields can causean emf in an
adjacent circuit (mutual inductance) or in theoriginal circuit
itself (self-inductance); the new electric fieldsare precisely
those appearing due to the motions of thecharges of the circuit as
given in Eq. (1), or, through orderv2=c2, as given in Eq. (3). It
is these electric fields that arethe subject of our discussion of
Faraday induction.
It should be noted that for steady-state currents in a
multi-particle circuit with large numbers of charges where
thecharge density and current density are time-independent, allthe
complicating motion-dependent terms in Eqs. (1)(2) or(3)(4) beyond
the first leading term in 1=c actually cancel,so that the
electromagnetic fields can be calculated simplyusing Coulombs law
and the Biot-Savart Law.14 However,for time-varying charge
densities and/or current densities,the motion-dependent terms in
Eqs. (1)(2) or (3)(4) do notcancel and indeed provide the Faraday
induction fields.
If an external emf, emfext, is present in a continuous cir-cuit
with a self-inductance L and resistance R, the current iin the
circuit is given by the differential equation
264 Am. J. Phys., Vol. 83, No. 3, March 2015 Timothy H. Boyer
264
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emfext Ldi
dt iR; (8)
where the term L di=dt corresponds to the negative of
theFaraday-induced emf associated with the changing current inthe
circuit. Here, in traditional electromagnetic theory,
theself-inductance L of a rigid circuit is a
time-independentquantity that depends only upon the geometry of the
circuit.The energy balance for the circuit is found by
multiplyingEq. (8) by the current i
emfext i d
dt
1
2Li2
i2R; (9)
corresponding to a power emfext i delivered by the exter-nal emf
going into the time-rate-of-change of magneticenergy 1=2Li2 stored
in the inductor and the power i2Rlost in the resistor.
Although the energy analysis for Eq. (9) seems natural,the
differential equation (8) presents some unusual aspects ifwe
consider the circuit from the particle point of view. If attime t
0, the constant external emf (emfext) is applied tothe circuit and
the current is zero (i0 0) then the Faradayinduced emf (emfF L
di=dt) must exactly cancel emfextso that emfext Ldi=dt 0 at time t
0. Phrased in termsof forces per unit charge applied to the
circuit, the Faradayinduced electric field EF must exactly cancel
the externalforce per unit charge fext associated with the external
emf.Indeed, if the resistance R of the circuit becomes vanish-ingly
small (R ! 0) then this canceling balance of theFaraday electric
field EF against the force per unit chargefext associated with the
external emf holds at all times, andyet the current increases at a
constant rate following di=dt emfext=L: But if the net force per
unit charge goes to zero,why do the charges accelerate so as to
produce a changingcurrent di=dt? After all, in classical mechanics
it is the re-sultant force FR on a particle that determines the
particlesacceleration a; that is, FR ma. Thus, we expect that if
theresultant force on a particle is zero, then the particle does
notaccelerate. However, electromagnetism involves someaspects that
are different from what is familiar in nonrelativ-istic mechanics,
and electromagnetic circuit theory involvessome approximations that
go unmentioned in the textbooks.In this article, we wish to explore
these differences andunmentioned approximations by using a particle
model inconnection with Faraday induction. We will note the
approx-imations involved in Eq. (8) that lead to the troubling
appa-rent contradiction with Newtons second law.
IV. FARADAY INDUCTION IN A SIMPLEHYPOTHETICAL CIRCUIT
A. Model for a detailed discussion
Here, we would like to explore Faraday induction in somedetail
for the simplest possible circuit, in hopes of obtainingsome
physical insight. Accordingly, we will discuss a hypo-thetical
circuit consisting of N equally-spaced particles ofmass m and
charge e, which are constrained by centripetalforces to move in a
circular orbit of radius R in the xy-plane,centered on the origin.
A balancing negative charge to makethe circuit electrically neutral
can be thought of as a uniformline charge in the orbit, or as a
single compensating chargeat the center of the orbit; the choice
does not influence the
analysis to follow. The system can be thought of as consist-ing
of charged beads sliding on a frictionless ring. There isno
frictional force and hence no resistance R in the model.The model
is intended as a rough approximation to a circularloop of wire of
small cross section.15
We now imagine that a constant external force per unitcharge
fext is applied in a circular pattern in the tangential /direction,
fext /fext, to all the charges of the ring. Oneneed not specify the
source of fext, but one example wouldbe an axially symmetric
magnetic field applied perpendicularto the plane of the circular
orbit in the z direction, increas-ing in magnitude at a constant
rate. The external emf aroundthe circular orbit is
emfext
fext dr 2pRfext: (10)
The external force per unit charge fext places a tangentialforce
Fi ei/ifext on the ith particle located at ri. TheFaraday
inductance of the charged-particle system is deter-mined by the
response of all the particles ei in the circularorbit.
B. One-particle model for a circuit
1. Motion of the charged particle
We start with the case when there is only one charged par-ticle
of mass m and charge e in the circular orbit. In thiscase, the
tangential acceleration a/ of the particle arisesfrom the
(tangential) force of only the external force per unitcharge fext,
since the centripetal forces of constraint are allradial forces.
From Eq. (6), written for a single particle andwith dmcv=dt mc3a/
where c 1 v2=c21=2, wehave
a/ efextmc3
; (11)
where fext is the magnitude of the tangential force per
unitcharge due to the external emf (emfext) at the position of
thecharge e.
2. Magnetic field of the charged particle
The magnetic field Be at the center of the circular orbitdue to
the accelerating charge e is given by Eq. (4)
Be 0; t kev
cR2; (12)
where the velocity v is increasing since the external force
perunit charge fext gives a positive charge e a positive
accelera-tion in the / direction. This magnetic field Be produced
bythe orbiting charge e is increasing in the z direction, which
isin the opposite direction from the increasing external mag-netic
field that could have created fext and emfext in Eq. (10).
3. Induced electric field from Faradays law
Associated with this changing magnetic field Be, createdby the
orbiting charge e, there should be an induced electricfield Eer; t
according to Faradays law. Thus averagingover the circular motion
of the charge, we expect an averageinduced tangential electric
field hEe/ri at a distance r from
265 Am. J. Phys., Vol. 83, No. 3, March 2015 Timothy H. Boyer
265
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-
the center of the circular orbit (where r R so that themagnetic
field Be has approximately the value B0; t at thecenter) given from
Eq. (12) by
2prhEe/ r i emfe 1
c
dUedt
1c
d
dtBe 0; t pr2
1c
d
dte
v
cR2pr2
1
ce
a/cR2
pr2;
(13)
since dv=dt a/: Using Eq. (11), the average tangentialelectric
field follows from Eq. (13) as
hEe/ r; t i /e2rfext
2mc2c3R2: (14)
This derivation of the Faraday induction field corresponds tothe
familiar textbook approach.
4. Induced electric field from the approximate
point-chargefields
We will now show that this induced average tangentialelectric
field hEe/r; ti is exactly the average electric field
due to the charge e obtained by use of the approximate elec-tric
field expression given in Eq. (3). Thus we assume thatthe charge e
is located momentarily at re xR cos /eyR sin /e, and we average the
electric field Eer; t due to eover the phase /e. Since the entire
situation is axially sym-metric when averaged over /e, we may take
the field pointalong the x-axis at r xr, and later generalize to
cylindricalcoordinates. The velocity fields given in the first line
ofEq. (3) point from the charge e to the field point. Also,
thevelocity fields are even if the sign of the velocity ve
ischanged to ve. Thus the velocity fields when averaged overthe
circular orbit can point only in the radial direction.
Theacceleration fields arising from the centripetal accelerationof
the charge will also point in the radial direction. Since weare
interested in the average tangential component of thefield Ee, we
need to average over only the tangential acceler-ation terms in the
second line of Eq. (3). If the field point isclose to the center of
the circular orbit so that r R, thenwe may expand in powers of r=R;
we retain only the first-
order terms, giving jxr rej1 R11 xr re=R2 andjxr rej3 R31 3xr
re=R2. Then the average tan-gential component of the electric field
due to the charge ecan be written as
hEe/ xr; t i e
2c2ae/
jxr rej ae/ xr re xr re
jxr rej3
" #* +
e2c2
ae/R
1 xr reR2
ae/ xr re xr re
R31 3xr re
R2
: (15)
Now we average over the phase /e with re xR cos /e yR sin /e and
ae/ ae/x sin /e y cos /e. We note thathae/i 0; ae/ re 0, and hae/x
rei yae/R=2 hae/ xrei. After averaging and retaining termsthrough
order r/R, Eq. (15) becomes
hEe/xr; ti yea/r
2c2R2; (16)
which is in agreement with our earlier results in Eqs. (13)and
(14). Thus indeed the electric field of Faraday inductionin this
case arises from the acceleration of the charged cur-rent carrier
of the circuit.
5. Limit on the induced electric field
We are now in a position to comment on the averageresponse of
our one-particle circuit to the applied externalemf. If the source
of emfext is a changing magnetic field,then this situation
corresponds to the traditional example fordiamagnetism within
classical electromagnetism.16 For thisone-particle example, the
response depends crucially uponthe mass m and charge e of the
particle. When the mass m islarge, the acceleration of the charge
is small; therefore theinduced tangential electric field Ee/ in Eq.
(14) is small.This large-mass situation is what is usually assumed
inexamples of charged rings responding to external emfs.17 On
the other hand, if we try to increase the induced
electromag-netic field Ee/ by making the mass m small, we encounter
afundamental limit of electromagnetic theory. The allowedmass m is
limited below by considerations involving theclassical radius of
the electron rcl e2=mc2. Classicalelectromagnetic theory is valid
only for distances large com-pared to the classical radius of the
electron. Thus in ourexample where the radius R of the orbit is a
crucial parame-ter, we must have R rcl. This means we require the
massm e2=Rc2 and so e2=mc2R 1. Combining this limitwith r=R < 1
and 1 < c leads to a limit on the magnitude ofthe induced
electric field in Eq. (14)
hEe/r; ti fext for r < R: (17)
The induced electric field of a one-particle circuit is
smallcompared to the external force per unit charge associatedwith
the external emf.
6. Energy balance
We also note that the power delivered by the externalforce per
unit charge goes into kinetic energy of the orbitingparticle. Thus
if we take the Newtons-second-law equationgiving Eq. (11) and
multiply by the speed v of the particle,we have
266 Am. J. Phys., Vol. 83, No. 3, March 2015 Timothy H. Boyer
266
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d
dtmcv v
d
dtmcc2
mc3a/v efextv; (18)
so that the power efextv delivered to the charge e by the
exter-nal force goes into kinetic energy of the particle.
The situation of a one-particle circuit can be summar-ized as
follows. For the one-particle circuit, the inducedelectric field is
small compared to the external force perunit charge and depends
explicitly upon the particlesmass and charge, while the energy
transferred by theexternal field goes into kinetic energy of the
one chargedparticle. Clearly, this is not the situation that we
usuallyassociate with electromagnetic induction for
circuitproblems.
C. Multiparticle model for a circuit
1. Motion of the charged particles
In order to make contact with the usual discussion ofFaraday
inductance in a circuit, we must go to the situationof many
particles, each of charge e and mass m. However,if we take the
one-particle circuit above and simply super-impose the fields
corresponding to N equally spacedcharges while maintaining the
acceleration appropriate forthe single-particle case, we arrive at
a completely falseresult. If we take Eq. (14) for the
Faraday-induced averageelectric field hEe/r; ti due to a single
particle of charge eand mass m accelerating in the external force
per unitcharge fext and then simply multiply by the number
ofcharges, we have a result that is linear in N and
increaseswithout bound. Thus, merely extrapolating from the
one-particle circuit suggests that the Faraday induced
electricfield arising from many charges might far exceed
theinducing force per unit charge fext.
To obtain a correct understanding of the physics, wemust include
the mutual interactions between the accelerat-ing charges of the
circuit. With these mutual interactions,the force on any charge in
the circular orbit is not just theexternal force efext due to the
original external force perunit charge, and the acceleration of any
charge is not givenby a/ efext=mc3. Now the force on any charge is
a sumof the force due to the original external force per unitcharge
plus the forces due to the fields of all the othercharged particles
in the circular orbit as given in Eq. (6).The magnetic force eivi
B=c is simply a deflection anddoes not contribute to the tangential
acceleration of thecharge ei. Thus the equation of motion for the
ith particlebecomes
d
dtmicivi /
mic3i ai/ mic3i Rd2/idt2
/i ei fext ri Xj6i
ejri rj jri rjj3
Xj 6i
ej2c2
(
ajjri rjj
aj ri rj ri rj jri rjj3
" #); (19)
where it is understood that the factor ci 1 v2i =c21=2
should be expanded through second order in vi=c so as to be
consistent with the remaining terms arising from the
approxi-mate field expression (3).
Since the particles are equally spaced around the circularorbit
and all have the same charge e and mass m, the situa-tion is
axially symmetric. The equation of motion for everycharge takes the
same form, and the angular acceleration ofeach charge is the same:
d2/i=dt
2 d2/=dt2. For simplicityof calculation, we will take the Nth
particle along the x-axisso that /N 0; rN xR, and /N y. The other
particlesare located at rj xR cos2pj=N yR sin2pj=N, corre-sponding
to an angle /j 2pj=N for j 1; 2;;N 1. Thetangential acceleration of
the jth particle is given byaj/ d2/=dt2xR sin2pj=N yR cos2pj=N.
Bysymmetry, it is clear that the electrostatic fields, the
velocityfields, and the centripetal acceleration fields of the
other par-ticles cannot contribute to the tangential electric field
at par-ticle N. The equation of motion for the
tangentialacceleration for each charge in the circular orbit is the
sameas that for the Nth particle, which from Eq. (19) is
mc3Rd2/dt2
efext XN1j1
e2
2c2y aj/
jxR rjj
8