faraday induction and the current carriers in a circuit

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 View Table of Contents: http://scitation.aip.org/content/aapt/journal/ajp/83/3?ver=pdfcov Published by the American Association of Physics Teachers Articles you may be interested in The electromagnetic momentum of static charge-current distributions Am. J. Phys. 82, 869 (2014); 10.1119/1.4879539 Mutual inductance between piecewise-linear loops Am. J. Phys. 81, 829 (2013); 10.1119/1.4818278 Teaching Faradays law of electromagnetic induction in an introductory physics course Am. J. Phys. 74, 337 (2006); 10.1119/1.2180283 Challenges to Faradays flux rule Am. J. Phys. 72, 1478 (2004); 10.1119/1.1789163 The Flexible Faraday Cage Phys. Teach. 42, 181 (2004); 10.1119/1.1664386

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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

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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



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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



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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

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