ELECTROMAGNETIC MOMENTUM AND ELECTRON INERTIA IN A …€¦ · The electromagnetic field at a point...

621.3.011:538.3Monograph No. 150

Sept. 1955

ELECTROMAGNETIC MOMENTUM AND ELECTRON INERTIA IN A CURRENTCIRCUIT

By Professor E. G. CULLWICK, O.B.E., M.A., D.Sc, F.R.S.E., Member.(The paper was first received 12th April, and in revised form 13th June, 1955.

September, 1955.)SUMMARY

In the second volume of his "Treatise on Electricity and Magnetism"Clerk Maxwell developed the theory of electric current-circuits fromgeneral dynamical principles, and discussed the experimental effectswhich should occur if an electric current is a true motion of somesubstance possessing inertia. Since none of these effects had at thattime been observed, Maxwell developed his general electromagnetictheory on the assumption that they do not exist, or at least that theyproduce no sensible effect.

It is now known, however, that an electric current in a conductorconsists of moving electrons, and the inertia effects which were dis-cussed by Maxwell have been observed experimentally. They areextremely small, and have not been brought within the scope of electro-magnetic theory. A conduction current is usually assumed to be dueto the drifting along the conductor, with a very small mean velocity,of all the available conduction electrons, so that the kinetic energy ofthe electrons due to this motion is negligible in comparison with themagnetic energy of the current. Electron-inertia effects in currentcircuits have therefore been accepted as something outside classicalelectromagnetic theory—a position which is illogical if, as is usual,we identify the kinetic and magnetic energies of a free electron.

It is shown in the paper that it is possible to identify the kineticenergy of the conduction electrons in a current circuit with the magneticenergy of the current, so that electron-inertia effects can be includedin the general electromagnetic scheme. In consequence, a currentcircuit can be said to possess an electromagnetic mass whose motion,when current flows, entails electromagnetic momentum. This momen-tum accounts for the known effects of electron inertia and also for theforce on the end wire of a long rectangular circuit.

The relativistic form of the theory indicates the possibility thatelectromagnetic laws may depart from the classical form, becomingnon-linear in circuits where a high inductance per unit length ofconductor is combined with a current greater than is usually foundin practice.

The inadequacy of classical theory also extends to the known electro-magnetic properties of superconductors, and the present hypothesissuggests the possibility of a unified theory in which there would be nonecessity to distinguish between a superconductor and a perfectconductor.

LIST OF SYMBOLS

(Rationalized M.K.S. units)a = 1-7 x 10-3//L.

A,A = Vector potential, webers/m.B,B— Magnetic flux density, webers/m2.

c = Velocity of light in vacuo, m/sec.D, D = Electric flux density, coulombs/m2.

e = Electronic charge (a negative quantity), coulombs.E,E = Electric field intensity, volts/m.

Ve = E.M.F., volts.F = Mechanical force, newtons.

H,H= Magnetic field intensity, AT/m./, / = Electric current, amp./, / = Electric current density, amp/m2.

/ / was published as an INSTITUTION MONOGRAPH in

Total self-inductance of a current circuit, henrys.Self-inductance per unit length of a 2-wire transmission

line, henrys/m.Length of circuit, m.Mass, kg.Rest mass of an electron, kg.Electromagnetic mass of a current circuit, kg.Numbers of turns; effective number of conduction

electrons per unit length of a conductor.Electromagnetic momentum per unit volume of the

field.Total electromagnetic momentum.Electric charge, coulombs.Radius, radius vector, m.Resistivity, ohm-m.Resistance, ohms.Poynting vector.Time, sec.Mutual kinetic energy, current and conductor.Energy, joules.Velocity, m/s.Co-ordinates.Angles.

Correspondence on Monographs is invited for consideration with a view topublication.

Professor Cullwick is Watson-Watt Professor of Electrical Engineering, St.Andrews University (Queen's College, Dundee).

L

/ =m, M =

rriQ =

Mo, M =N =

p,p =

Ptotai ~q, Q =R,r=

r =R =

S, S =t =

Tme =U, W =

u, v, w =x, z =cc,d =

e0 = Primary electric constant,* 8-854 x 10~12.(JL0 = Primary magnetic constant,* 1-257 X 10~6.p = Charge of effective conduction electrons per unit length

of conductor (a negative quantity), coulombs/m.O = Magnetic flux linkage, weber-turns.ifj = Scalar potential function.

(1) INTRODUCTIONIt is a common practice in electromagnetic theory to regard

the magnetic energy of a current circuit as electrokinetic, and tocompare the expression \Ll2 with the kinetic energy of a movingmass, %mv2. It is the purpose of the paper to show that themagnetic energy of a current circuit can be identified with thekinetic energy of the mass-equivalent of the total electromagneticenergy of the conduction electrons. The concept of electro-magnetic momentum in a current circuit will then be used todetermine the force on the end wire of a long rectangular circuit,and to bring the known effects of electron inertia in a circuitwithin the scope of electromagnetic theory.

(2) ELECTROMAGNETIC MASS OF A MOVING CHARGEDPARTICLE

Fig. 1 shows a small positive charge, q, moving in free spacewith a constant velocity v in a straight line. The electromagneticfield at a point P(r, a) consists of a radial electric field of intensityE and flux density D = €0E, together with a magnetic field whose

• The idea that "free space" possesses the physical properties of permittivity andpermeability is based on the concept of a material medium or aether, and is clearlyincompatible with the hypothesis of this paper.

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160 CULLWICK: ELECTROMAGNETIC MOMENTUM AND ELECTRON INERTIA IN A CURRENT CIRCUIT

Enx H, Dn*B

Fig. 1.—Moving charge: electromagnetic momentum and energy flux.

lines of force are circles concentric with the path of the charge.At P the magnetic field is vertically out of the paper, withintensity H and flux density B = /XQ//.

If the charged particle is spherical when at rest, and thevelocity v is very small compared with c, we have

and

or

E =

H= v x D

sin

(0

. . . . (2)

where En is the component of E perpendicular to the velocity.The flux density is

..i v F

(3)

If v is comparable with c the particle suffers the Lorentz con-traction, and becomes an oblate spheroid whose polar axis is(1 — v2/c2yi2 times its equatorial axis. The electric field at Pis still radial, but has the value

1 ?E =

This has components:

normal to v: En =

parallel to v: Ev =

3/2

47T€Q (jS2*2 + z2)3/i

q px(S2*2 + 22)3/2

• (4)

(5)

(6)

where /? = (1 — v2\c2)~x\2. The magnetic field is still given byH= v X D, B= v x Efc2, or

H = ve0En, B = (7)

According to classical electromagnetic theory we also have,at the point P,

(a) a flux of electromagnetic energy given by the Poynting vectorS = E x H (8)

which is the rate at which energy passes through unit area normalto S, and

(b) an electromagnetic momentum

per unit volume of the field.

Both S and p are shown in Fig. 1. They are directed inwardstowards the path of the charge, at an angle TT/2 — a to thedirection of motion.

The classical interpretation of eqn. (8) is that, as the positionof the charged particle changes, the electromagnetic energy at

stationary points in the medium or aether is maintained atappropriate values by the flow, with velocity c, of energy in themedium. There is, however, an alternative viewpoint whichmerits investigation. Since the velocity of the charge is constantthere is no change in the total energy of the system, and for anobserver moving with the particle there is no flow of electro-magnetic energy at all. Tf, therefore, the electromagnetic energyis regarded as belonging to the particle, rather than to themedium, it is reasonable to suppose that when the particlemoves it takes the energy with it, with its own velocity v.

If we define the electromagnetic mass of the particle, Me, bythe relation (for v < c)

\Mev2 = magnetic energy

it is easy to show that

M =6TTR

(10)

(ID

where R is the radius of the charged sphere. Now let me be theelectromagnetic mass per unit volume of the field, and supposethis mass to move, with the charge, with velocity v. We thenhave

so thatHB = ^V2 V2

• • • (12)C2

the last expression arising from eqn. (7). The momentum of me

is clearly,

mjv = . . . . (13)

or p=DnxB (14)

which is the component of the classical electromagnetic momen-tum, D x B, in the direction of the motion of the chargedparticle.

If we similarly resolve the Poynting vector S, we find theflux of energy in the direction of motion to be

H2

v = EltH=— or Se = Ett . . (15)

We next suppose that this represents the motion of electro-magnetic energy with velocity v, moving with the charge. Itsvolume density will be

£ H2

rj — zi —V €0V

2

(16)

This is obviously not the magnetic energy, \HB, for it is fargreater. But if we use the mass-energy equivalence we find adirect re lat ionship between these two energy densities. T h emass of the energy U is

U

~c2EJK

c2 • • • (17)

the electromagnet ic mass as defined by eqn. (12).

We therefore see that

(a) The magnetic energy of the moving charged particle can becompletely identified with the kinetic energy of the mass of themoving energy U.

(b) The electromagnetic momentum in the direction of motion ismerely the momentum of the mass of the energy U.

It should, however, be noted that eqns. (12)—(17) cannot beregarded as providing an accurate microscopic account of theenergy of the field. The expression for me given by eqn. (12)


CULLWICK: ELECTROMAGNETIC MOMENTUM AND ELECTRON INERTIA IN A CURRENT CIRCUIT 161

does not possess spherical symmetry about the particle, butsymmetry about the direction of motion, so that it cannot bean accurate detailed description of the energy-mass distributionof a stationary spherical charge. The quantities me, p, Sv and Umust therefore be regarded as functions which possess physicalidentity only when integrated throughout the complete volumeof the field.

Since we assume that v is small compared with c, we neglectin the first instance the mass of the kinetic energy. This involvesa relativistic modification, which will be considered later. Ifthe electromagnetic mass Me is the only mass possessed by theparticle (e.g. if the particle is an electron and we assume that itsmass is entirely electromagnetic), then clearly the energy U isthe total mass-energy of the particle, or the energy which wouldbe liberated if the particle disintegrated into electromagneticradiation.

If a charged sphere is stationary or moving with a velocityv < c, and if the charge is uniformly distributed on the sphere,the energy of its electric field is easily found to be

W =STT€0R

which has a mass-equivalent

877-i?

(18)

(19)

Thus Ms = 3MJ4, so that if an electron is to be regarded as aspherical aggregate of charge whose mass is entirely electro-magnetic, it is necessary to postulate additional electromagneticenergy of amount

q2

( 2 0 )

It is well known that a stable spherical electron cannot existwithout internal forces in addition to those of classical theory,and eqn. (20) may be taken to represent the energy of the non-classical, and at present unknown, forces which keep the electronfrom exploding.

If we identify the mass, Me, given by eqn. (11) with the restmass m0 of an electron, and then add the mass, mm, of themagnetic energy \Mev

2, we obtain the total mass of the movingparticle

D2\

m = mn + m,^ = M<, + Mo [ —; 1

( v2

1+2*. . . (21)

This is a first approximation, with v < c, to the relativisticrelation

/••* /y i 0 \ __ i fy fyY\

(3) TWO MOVING CHARGED PARTICLES(3.1) Energy Fluxes

Fig. 2 shows two particles with charges qx and q2 movingwith velocities vx and v2, respectively, in paths which are notnecessarily co-planar. At a point P the electric field is made upof components Ex from qx and E2 from q2, and the magneticfield similarly has components Hx = vx x Dx, H2 = v2 x D2.The resultant magnetic intensity is

H= Hx+ H2, where H2 = H2 + H2 + 2HX . H2 (23)

The magnetic energy in the field has a volume density

2HX.H2) . (24)

Fig. 2.—Two moving charges.

and since the mutual energy IX-QHX . H2 is symmetrical in Hx andH2, we shall assume that equal portions of it are associated witheach charge.

We therefore suppose that each charge carries its share of themutual energy, so that using the relation given in eqn. (15) weattribute to qx, in addition to the flux of its self-energy, a flux ofenergy with velocity vx,

mX (25)

and to q2 a flux of energy with velocity v2

D2). Ht _ (E2 x / / , ) . y2

€0V2 €0V2

If vx and v2 are parallel or anti-parallel, these reduce to thesimple forms

Sm\ = En\ X Hi, Sm2 = En2 X Hx . . . (27)

where EnX and En2 are the components of Ex and E2 perpendicularto the velocities.

In addition, the charge qx carries a self-energy flux SsX= EnX x Hx with velocity vx, and q2 carries a self-energy flux5̂ 2 = En2 x H2 with velocity v2. So the total energy fluxcarried by each charge is

= Ss2 Sm2 = Em2 n2

• . (28)

where H = Hx + H2, the resultant magnetic intensity. Theseenergy fluxes are assumed to exist entirely independently of eachother. Even if vx and v2 have different values, or are in differentdirections, the two energy streams are assumed not to interfere.

If the two charges are moving with the same velocity, theirindividual energy fluxes can be combined into one, of value

S=(EnX+En2)xH=EnxH (29)

VOL. 103, PART C.

and since this combination can be continued indefinitely foradditional charges moving with the same velocity, it followsthat eqn. (29) applies to any rigid configuration of charges inuniform rectilinear motion, and that eqn. (28) is valid for thetwo lines of moving charge in two parallel conductors.

(3.2) The Effective Mass of a Conduction ElectronExperiments on electron inertia in closed conducting circuits

(see Section 11.2) have shown that the ratio e/m for a conductionelectron in a current-carrying conductor is approximately thesame as that obtained in experiments on low-energy cathoderays, i.e. the effective mass of a conduction electron is approxi-mately the same as the rest mass of a free electron.

According to Section 3.1, however, the mass of the particle q{is a function of the mutual energy of qx and q2 as well as of itsself-energy. Let us examine the probable order of magnitudeof this effect for a conduction electron.

First suppose that the component of magnetic field, H2, dueto sources other than qx is uniform. Then it is clear that the

6



total volume integral of H\. H2 is zero. Consider two pointsdiametrically opposite on a circular line of force of H{: thevalues of Hl. H2 at these two points are equal and oppositeand their contributions to the volume integral cancel. Thus theelectromagnetic mass of an electron moving in a uniform externalmagnetic field is the same as that of an isolated electron.

Next consider the orders of magnitude of H\ and Ht. H2 atpoints in the vicinity of a spherical electron. The portion of theelectromagnetic mass of a charged sphere contained within aconcentric sphere of radius r is

m, = 6v \R

Let r = 100OR. Then 99-9% of the electromagnetic mass iscontained in the concentric sphere. The magnetic flux-densityat a radius r on the equatorial plane of the moving electron,taking R = 2 x 10~15 m and r = 2 x 10~12 m, is

B{ = = (4 x 10"3)v weber/metre2

Suppose a second electron to be moving with the same velocityin a parallel path, with a common equatorial plane, at a distanceof 10~10 m (the order of magnitude of the diameter of an atom)from qv Then its magnetic field at the same point as before isabout B2= (1-6 x 10~6)vweber/m2, so that B2lBlB2 =H\\HJI2 ~ 2500.

Nearer to q{ this ratio will be greater. It is therefore evidentthat the effective electromagnetic mass of a conduction electronwill not differ, according to this hypothesis, appreciably fromthat of a free electron.

(4) APPLICATION TO A LONG RECTANGULAR CIRCUIT

(4.1) General ConsiderationsFig. 3 shows a long rectangular circuit, ABCD, which carries a

constant current /. The sides AB, BC and CD are connected

B I

C D

Fig. 3.—Long rectangular circuit.

rigidly together, but the end wire DA makes contact with therest of the circuit by means of mercury cups, so that the forceon this member can be measured. Then the force on this endwire can be calculated from a knowledge of the magnetic field,at points on DA, due solely to the current in the three sides AB,BC and CD, and calculated from the law of Biot and Savartfor the magnetic field of a current element:

(30)

where l8s is the current element and r is the radius vector fromthe element to the point where the field is 8H. The force on acurrent element I'8s' at the point is then F = /*0/'(oV x 8H),which is not, in general, equal to the force on I8s due to themagnetic field of the element I'8s'. So the resultant force onthe three-sided portion AB, BC, CD is not that which would becalculated from the magnetic field of the other portion, DA,alone, acting on the current in the three sides. Since the forceon each side is perpendicular to the wire, the resultant force onthe three sides is merely that on BC. But the magnetic field atBC due to DA alone, as calculated from eqn. (30), is very small

and approaches zero as the length of the circuit increases. Thus,if the circuit is very long the force on the end wire DA is dueentirely to the magnetic field of the two long sides, but theforces on these long sides are not due to the magnetic field ofthe current in DA alone.

If, as in Maxwell's original theory, the electromagnetic fieldis regarded as a physical condition in a material medium oraether, the reaction of the force on the end wire is considered tobe borne by the medium in the vicinity and transmitted bystresses in the medium to the other end wire. It is, in fact, onlysince the concept of a material medium has fallen into obsoles-cence that problems of this kind have caused discussion,1"7 forthere appears to be nothing to take the place of the medium as anagent for transmitting the force. We shall show that such anagent is provided by the momentum of moving energy.

First let us calculate the force on the end wire by the usualmethod. Assume the length of the circuit to be great in com-parison with its width, and let the self-inductance per unit length(two wires) at points remote from the ends be Lo. Suppose thewire DA to move a small distance 8x under the action of theforce F, thus increasing the length of the circuit by 8x. Assumethat the current / is maintained constant during the displacement.Then the magnetic flux Unking the current increases by 8<I>= Lol8x, so that an e.m.f. Ve = — SO/8/ is induced, and inorder to keep the current constant additional energy must besupplied, from the source, of amount

8W = (-Ve)l8t = LQI28x

The magnetic energy of the circuit increases by an amount%L0I

28x, and since the energy supplied must be equal to themechanical work done by the force F plus the increment inmagnetic energy, we have

= F8x "I• • (31)

or, if L is the total inductance of the circuit

(32)

(4.2) The Flow of Electromagnetic MomentumIn order to interpret the force F given by eqn. (32) in terms of

energy momentum, we distinguish the components of electricfield arising from the positive and negative charges in the con-ductors. We shall simplify the problem by assuming that theparallel wires have the same uniform section and negligibleresistance, so that the electric field between them, except near theends, is perpendicular to the current flow and the potentialdifference between them is constant. The end wire then providesa resistance "load" on the long transmission line.

Fig. 4(a) shows a portion of the circuit remote from theends, and Fig. A{b) is a cross-sectional view. We consider eachconductor to contain the same quantity of stationary positivecharge in its structure per unit length. The current in conductor 1is provided by the motion, with a mean velocity vu of conductionelectrons whose charge per unit length is />,, and the current inconductor 2 is provided by the motion, with mean velocity v2,of conduction electrons whose charge per unit length is p2. Thedirection of motion is, of course, opposite to that of the current,and />! and p2 are very nearly equal. Clearly pvvi = p2v2 — I.

Then at any point P we recognize four components of electricfield, all perpendicular to the wires, namely

E\ from the conduction electrons in conductor 1.E2 from the conduction electrons in conductor 2.£3 from the stationary positive charge in conductor 1.£4 from the stationary positive charge in conductor 2.



and this is deflected at the end wire at the rate

(a)

Fig. 4.—Energy fluxes in long rectangular circuit.(a) Energy streams and electric field components.(6) Electric and magnetic field components.

These components are in the direction shown in Fig. 4(a). Thecomponents of the magnetic field intensity, arising from themotion of the sources of Ev and E2, are shown in Fig. 4(6).

The positive charges, being stationary, carry no energy fluxes.The conduction electrons in conductor 1 carry an energy fluxwith velocity v\'.

Sx = Exx (Hi + H£

or Sx = \E{ x Hi\ + \E{ x H2\ = Ey(Hx + H2cos 0) (33)

directed towards the end DA.TT

But E, = —-, so that

(H2 + HXH2 cos 0)_ v " l . (34)

Since the velocity of this energy flux is vu its volume density isSJvx and its mass per unit volume is

m, = v{c2

(35)

Its kinetic energy is therefore

\m<o\ = f^(H2 + HxH2 cos 0) . . . (36)

Similarly, the conduction electrons in conductor 2 carry anenergy flux with velocity v2:

Hj + H^cose€QV2

away from the end DA, and its kinetic energy is

- ^ ( H 2 + H x H z c o s 8 ) . . . . ( 3 8 )

Thus the total kinetic energy of the moving energy fluxes, perunit volume, is

(39)

i.e. the magnetic energy density.Consider the change of electromagnetic momentum which

occurs at the end wire DA. The momentum of the Sx stream,per unit volume, is

HxH2cos$)jvx . . . (40)

o&e) . . . . (41)

per unit area of an infinite plane, remote from the ends, per-pendicular to the side wires. So the total loss of momentum persecond is

jj (Hf + HXH2 cos 6)ds . . . (42)

the integration being over the infinite plane. Similarly, the S2stream is given momentum at the rate

\H2 cos 6)ds . . . (43)

so that the total rate of change of momentum at the end wire is

• L 0 I 2 . . . . ( 4 4 )

where Lo is the inductance of the circuit per unit length of thetransmission line.

(4.3) The Force on the End WireSuppose the end wire to move a very small distance Sx, thus

lengthening the circuit by the same amount. The work doneby the change of momentum is then L0I

28x, and this is preciselythe amount of additional energy which must be provided by thesource if the current is to be kept constant. We then have:

Work done by change of momentum = Mechanical work done+ the increment in mag-

netic energy,

Lol28x = F8xor

andas before.

F=iL0/2 (45)

We shall see later that the electromagnetic momentum can beidentified with the momentum of the conduction electrons, soaccording to our hypothesis the force on the end wire in a verylong rectangular circuit is entirely due to electron inertia. Theforce between the two side wires cannot, of course, be explainedin this way, except in a short region near the corners of thecircuit. The force between two long parallel current-carryingconductors may be regarded as arising from a very smallunbalance of the mutual forces between the electric charges inthe wires, some of which are moving and some of which arestationary.

It is evident that this hypothesis of moving energy-mass leadsdirectly to the conclusion that Newton's third law (equality ofaction and reaction) does not apply to the mutual forces of twoindividual current elements unless the elements are parallel toeach other. The hypothesis thus gives a physical justificationto the accepted classical form for the mutual force of two currentelements. Ampere's law of force between two current elements,upon which Weber's electromagnetic theory was based, obeysNewton's third law whatever the direction of the current elementsmay be. Ampere's theory thus treats the problem from theprinciples of statics rather than of dynamics, and for this reasontheories based upon it fail to lead to electromagnetic radiationor the equivalence of mass and energy.

(4.4) The Resultant Flux of Energy

The resultant energy flux towards the end wire DA is,vectorially,

S = S{ + S2 = (Ex + EJ x H . . . (46)



Since each side wire contains equal amounts of stationarypositive charge and the wires have the same section, it followsthat

x H}.dS=0 (47)

for we may divide this integral into two equal and oppositehalves by a plane half way between the side wires and per-pendicular to the plane of the circuit. The resultant total rateof energy flow towards the end wire is therefore

P = J J{(£i + E2 + £3 + E4) x H} . dS = f J (£ X H) . dS

. . . . (48)

That is, it is the same as that given by the classical Poyntingvector E x H. This energy is converted into heat in the endwire.

Any change in the resultant electromagnetic field at a point iseffected in accord with classical electromagnetic theory, withthe velocity of an electromagnetic wave, which in free space is c.For example, consider Fig. 5, which shows conditions near a

' d t

Fig. 5.—Energy fluxes when the current is changing.

straight conductor which carries a changing current 1. Themain energy flux, Sv = En x H, travels with velocity v parallelto the wire in the direction of electron flow. If the currentincreases at the rate di\dt, an induced electric field E{ = — dA/dtis produced parallel to the wire in a direction opposite to thatof the current. The classical Poynting vector Sc = E-t x H isthen directed radially out from the wire, and represents a flowof electromagnetic energy moving outwards with velocity c.This increases the classical field energy \{HB + ED), and maytherefore be considered as providing the increment in the kineticenergy of the accelerating main energy flux. If the currentdecreases, Et and Sc reverse, and the main energy flux decelerates.

(5) THE ELECTROMAGNETIC MASS AND CONDUCTIONCHARGE OF A CURRENT CIRCUIT

If we attempt to apply the above detailed analysis of movingenergy streams, whose energy density at a point is specified, tocircuits of any shape it is clear that we shall encounter greatdifficulties; for each current element contributes its own com-ponent of energy flux, parallel to itself, at a given point, andeach of these components must be considered to exist inde-pendently. We shall therefore proceed on the assumption that,in general, the total kinetic energy of the moving energy mass isequal to the magnetic energy of the circuit \LP-.

If the mean velocity of the conduction electrons in a completecurrent circuit can be taken to have the same value, v, at all partsof the circuit, we can then introduce the concept of the electro-

magnetic mass of the circuit, which we may call Mo when v < c.We define Mo by the relation

2

. . . (49)

L being the inductance of the complete circuit. The totalelectromagnetic momentum is then

P total = M0V = LI2IV (50)

If p is the charge of the conduction electrons comprising thecurrent per unit length of wire, I = pv and

M0 = Lpi (51)

We now take the radical step of identifying Mo with the totalmass of the conduction electrons whose motion comprises thecurrent. We have already noted that the identification of themagnetic energy of a free electron with its kinetic energy isgenerally accepted, so it is no more than a logical extension ofthis idea to postulate the same for the electrons in a completecurrent circuit. Indeed, a failure to do so seems to leave anuntidy inconsistency in electromagnetism, however minute thepractical consequences of this inconsistency may be. By makingthis identification, moreover, we can bring the known effects ofelectron inertia in a circuit into the general electromagneticscheme—a unification which seems highly desirable.

Let the charge p consist of N conduction electrons, per unitlength of wire, each of mass m0, so that Ne = p, where e ( < 0)is the electronic charge. If / is the total length of the wire, thetotal momentum of the conduction electrons is Nlmov, andequating this to Mov gives

o —plm0

• • (52)

So from eqn. (51) we obtain

and

We also have v — -—— iJ^r = (—}-7 . . .p \M 0 \moj I

(54)

(55)

(56)

v and / being in opposite directions, since e is negative. Thismay be put in the form

_, . a ,. , Mean momentum of a currentMagnetic flux-linkages per , ,. ., , , .,. , ., - • r TII = electron, aiviaea oy tne

unit length of wire, U\l e l e c t r o n i c c h a r g e } m ^ e

. . . . (57)

(6) THE CONDUCTION CHARGE AND ELECTRON FLOWFrom the last Section we see that, if the identity of Mo with

iV7ra0 is to be valid, the mean velocity of the conduction electrons,assumed here to be small compared with c, must be determinedboth by the current and the geometry of the circuit. The chargep is also determined, or quantized, by the geometry of the circuit.This may at first sight appear to be a difficult condition. Acurrent in a wire is often taken to be due to the motion of allthe conduction electrons in the atomic structure (of the order of1023 per cubic centimetre for copper) drifting along the wirewith a very low velocity which depends solely upon the resultant



electric field in the conductor. This is not, however, a necessaryinterpretation of the theory of electronic conduction. Calcula-tions8 based on the Fermi-Dirac statistics, on the assumption ofone conduction electron per atom, give the r.m.s. velocity of afree electron in copper as about 1-6 x 106m/sec. This meansthat individual electrons have velocities somewhere within arange which extends far below and far above this figure. If weresolve the velocity of every electron along the wire, when nocurrent is flowing, there will be equal streams of electrons inopposite directions, with a great range of velocities in each stream.When a current flows, the above hypothesis requires that itmust be provided by the requisite number of electrons whoselongitudinal velocities are as close as possible to the value givenby eqn. (56). This condition is satisfied if the opposing streamsof electrons possessing this particular mean velocity are nolonger equal, so that the difference in their charge per unitlength is equal to p. Provided that this theory does not lead toabsurd results, such as a value for p greater than the total chargeavailable, there seems to be no a priori objection to it.

(7) TWO PRACTICAL CASES{a) A concentric cable, of inductance 5 X 10~4 henry/km.

The inductance per metre length of single conductor is

— = 2-5 x 10~7 henry/metre;

em0

1-6 x 10~19

iFll X 10-31= - 1 - 7 6 X 1011

so that\L

— )- ; /= —4-4 x 104/metres/secmoj I

where / is the current in amperes. The moving charge per unitlength is

p = - = — 2-27 x 10~5 coulomb/metre

and Mo = Lp2 = 2-57 X 10~13 kilogramme/kilometre.

(b) A toroidal coil, of ring radius R= 10""1 m, mean turnradius r = 2 X 10~2m, with N = 1 000 turns and carryinga current of 5 amp.

The inductance is L ~ fx0r2N2/2R = 8TT X 10~4 henry, and

the internal flux density is about 10~2 weber/m2 or 100 gauss.The length of wire is / = inrN = 40TT metres, so

v=s (U)^l= - (3-52 X 106)/\moj I

= 1-76 x 107 metres/second = 0-0585c

p = j/v = — 2-82 x 10-7 coulomb/metre

Mo = Lp2 = 2-012 x 10~16kilogrammeThe effective number of moving conduction electrons per

metre of wire isN= ple= 1-76 x 1012

Suppose the toroid to be wound with two layers of 500 turnseach, with wire of diameter lmm. Then the volume of theconductor per metre length is 7r/4cm3 and the effective numberof conduction electrons per cubic centimetre is 2-24 x 1012.This is extremely small in comparison with the total availablenumber of conduction electrons, which is of the order of 1023 percubic centimetre.

(8) SELF-INDUCED E.M.F.It is evident that the e.m.f. induced in a circuit when the

current changes, according to this hypothesis, is merely aneffect of electron inertia. If the conductor has no longitudinalfreedom of movement this e.m.f., in the direction of the current /,is given by

re~ ^dt~ I

which from eqns. (53) and (54) gives

dv

pv

4 U • . (58)

or, if the velocity v is not the same at all parts of the circuit,

. . . . (59)e Jdt

The equivalent self-induced electric-field intensity, of classicaltheory, in the conductor is evidently

so that

e dt

dv

(60)

= - nto=r (61)

which may be compared with eqn. (57).The classical law of conduction, / = E\r, where / i s the current

density and r the resistivity, states that the current density is, atevery instant, proportional to the resultant electric-field intensity.That is, E is regarded as the cause of the current, whereas bythe present hypothesis a current continues to flow in a short-circuited circuit because of the inertia of the conduction electrons.Classical theory requires this self-induced E because electroninertia is considered to be negligible or absent. The averageforce opposing the motion of the conduction electrons is —Jrper unit charge. The electrons are decelerated by this resistanceto motion, and in classical theory their deceleration induces anelectric field E which cancels the decelerating force. Theresultant force acting on the electron, regarded as a particlewithout inertia, is therefore zero. So if we use our presenthypothesis in a theory of conduction, we must not include aself-induced electric-field intensity.

The linking magnetic flux, if v is single-valued, is

= Ll= -

$> = b A .dl

. • (62)

. . (63)

or, in general,

But since

where A is the magnetic vector potential at the axis of the wire,we can put

A = ( ^ \ v + gradtf; . . . . ( 6 4 )

beingwhen A is due to the current in the conductor alone,an arbitrary scalar potential function.

(9) VARIATION OF v IN A CONDUCTOR OF FINITECROSS-SECTION*

We have tacitly assumed that the conductor has a negligiblysmall cross-section, so that the electron velocity given by eqn. (56)

• The case of a finite conductor is treated more rigorously in the Appendix.



applies to all parts of the section. Actually v must vary across aconductor of finite section, for from eqn. (64) we have

€ 6curli; = —curl /4 = — B

m0 m0(65)

where B is the flux density inside the conductor. From eqn. (60)this is seen to be consistent with the classical equation curl E= - B.

(10) RELATTVISTIC EXPRESSIONS FOR THE MAGNETICENERGY AND THE SELF-INDUCED E.M.F.

So far we have assumed that the velocity v of the effectiveconduction electrons is sufficiently small in comparison with c tojustify the neglect of any variation, with velocity, of the electro-magnetic mass. In the example of the toroidal coil, however,we obtained an electron velocity of about c/17. It is thusevident that, in circuits of high L\l with high currents, the varia-tion of mass may be significant.

If v is not negligible in comparison with c the kinetic or mag-netic energy is not $M0v

2 but

Um=(M-MlP)c* (66)

where M is the electromagnetic mass of the moving energy andM/j8 is its rest mass, not necessarily equal to Mo. For simplicitywe assume that the circuit is such that v is the same over itsentire length, for otherwise we shall encounter great complexity.We require Um as a function of the current, and proceed on theassumption that the relation between electron momentum andflux linkage, eqn. (57), remains unchanged. Since the electronicmass is now m = jSw0, this requires

^ - f r , • • (67)

so that the number of conduction electrons per unit length isN = f$N0, where p0 and No are the low-velocity values.

The electromagnetic mass is then

M = Nlm = j82M0 .

I ILWe also have v = - = — ~\ — = — I\l-^-,

LM

(68)

• (69)

whence v2 = _ ^ 1 + ^ 2 ) . • • • (70)

If L/2 = MQC2, then / = molcleL > 1 -7 x 10~3//L. Denote thelatter quantity by a. Then

(a) if / < a, v = - /*/

(6) if / > a, v -> c

From eqn. (70) we obtain

£ / 2 \

• • • (71)

M-Qc>) ™and from eqns. (66) and (68)

Um = (73)

This reduces to \Ll2 if / < a, but approaches L/2 if / > a.The self-induced e.m.f., Ve, is given by VJ = — dUJdt,

whence

")

which reduces to — L-r- if / < a, and approaches —2L — if / > a.at at

For the toroidal coil (Section 7), l\L = 5 x 104 and a > 85 amp.This is greatly in excess of the current-carrying capacity of thewire. Our theory thus becomes non-linear under conditionswhich are outside the range of normal practice, and this departurefrom classical electromagnetism must also, of course, apply to theforces experienced by current-carrying conductors.

As will be noted in the next Section, Maxwell developedhis general electromagnetic theory on the assumption that thecarriers of current in a conductor have no inertia. A study ofChapters 5-9 of the fourth part of his Treatise will show howclassical theory is deeply rooted in the general principles ofdynamics, and Maxwell certainly regarded magnetic energy askinetic. He called the vector potential A the electrokineticmomentum, but without attempting to associate with it anyparticular velocity. Now it is impossible to conceive the ideasof kinetic energy and momentum without accepting the conceptof moving mass, which in Maxwell's day was uncomplicated bythe equivalence of mass and energy and the variation of masswith velocity. It should not therefore surprise us if our presenthypothesis throws doubt on the complete validity of Maxwell'sequations.

(11) ELECTRON-INERTIA EFFECTS

(11.1) History and Present Theoretical PositionIn his "Treatise on Electricity and Magnetism" (Part IV,

Chapter VI), Clerk Maxwell discussed three types of experimentaleffect which should exist if an electric current in a conductor is atrue motion of some substance having inertia.

(a) If a circular coil is freely suspended by an axial thread withits axis vertical, any change in the current flowing in it should beaccompanied by a rotation of the coil.

(b) A coil carrying current should exhibit gyroscopic effects and,if Ampere's hypothesis that ferromagnetism is due to atomic currentsis correct, the same should apply to a magnet.

(c) When a rapidly rotating coil, which is part of an unenergizedclosed conducting circuit, is suddenly stopped, the inertia of thecurrent carriers should cause a momentary displacement of elec-tricity (i.e. a current) through the circuit.

Maxwell stated that no such phenomena had ever beenobserved, and apparently performed an experiment to test (b),but without a positive result. He showed that all three effectsdepend on the possible existence in the expression for the kineticenergy of a moving current-carrying conductor of a term involvingthe product of the velocity of the conductor and the velocity ofthe electricity relative to it.concluded:

We have thus three methods of detecting the existence of the termsof the form Tme, none of which have hitherto led to any positiveresult. I have pointed them out with the greater care because itappears to me important that we should attain the greatest amountof certitude within our reach on a point bearing so strongly on thetrue nature of electricity.

He therefore developed his electromagnetic theory on theassumption that such effects do not exist, or at least that theyproduce no sensible effect, and they cannot be deduced from hisfundamental equations of the electromagnetic field.

Nevertheless, all three types of effect have now been experi-mentally observed.9 The first successful experiments on thegyromagnetic effect were those of Barnett10 in 1915, who suc-ceeded in magnetizing an iron rod by rotating it. The converseeffect, the production of rotation by magnetization, was observedby Einstein and de Haas11-13 (1915 and 1916). A conclusionfrom these and later experiments is that the magnetic momentof a ferromagnetic atom must be due to spinning electronsrather than to orbital electrons. In 1916 Tolman and Stewart14

He called this term Tme, and



succeeded in detecting effect (c). Furthermore, their experi-ments proved that the carriers of electricity in a conductioncurrent have a negative charge, and the ratio ejm for a con-duction electron was found to be approximately the same asthat for a free electron. Effect (a) was sought by Sir OliverLodge,16 but was not detected until 1930, when experiments byBarnett17 gave positive and theoretically consistent results.Barnett used alternating current, tuning the frequency so thatmechanical resonance occurred, and measured the torque byremoving the oscillations by means of an equal and oppositetorque which could be calculated.

In the past treatment of electron-inertia effects it has beenusual to take the momentum and kinetic energy of the conductionelectrons as extra-electromagnetic phenomena, p being taken asequal to the charge of the total number of conduction electronsavailable, so that the mean velocity v becomes very small. For agiven current, so that pv is constant, the kinetic energy of theelectrons is proportional to v, and this older viewpoint gives akinetic energy which is extremely small in comparison with themagnetic energy of the circuit. It has therefore previously beenheld that it is not possible to identify the two energies, and thetotal energy in a stationary circuit due to the current has beentaken as being equal to the magnetic energy plus a very smallcorrection for the kinetic energy of the conduction electrons.18

According to our hypothesis, however, the magnetic energy ofthe circuit and the kinetic energy of the conduction electronsare the same thing, and in a stationary current circuit the energydue to the current flow is exactly equal to either. It also followsfrom this theory that these very small manifestations of electroninertia in a current circuit are a necessary consequence of themomenta of the energy fluxes. We shall apply our hypothesisto the experiment of Tolman and Stewart [effect (c)] and to theBarnett effect (a). Experiments on the gyromagnetic effect havebeen carried out on iron rods, which are outside the scope of thepaper.

(11.2) The Experiment of Tolman and Stewart: Production of anInertia! Current

A rapidly rotating coil is connected, by sliding contacts, in astationary circuit which includes a ballistic galvanometer. Thecircuit contains no source of e.m.f., so that when the coil is atrest or rotating uniformly there is no current. When it issuddenly stopped the momentum of the conduction electronscarries them on and the galvanometer registers the chargedisplaced around the circuit.

Consider the coil to be rotating uniformly, with no current,with peripheral velocity w. As before, let there be N conductionelectrons per unit length of wire, so that p = Ne. The momentumof these electrons, due to the rotation, is Nmwl where m is themass of an electron. If / is the current which would have thesame energy-stream momentum, we have

LI2jv = Lph = Nmwl

Nmwl mwl ._„I = Pv = —TT- = — . . . . (75)

and

andLp Le

The quantity of electricity displaced when the rotation is stoppedwill be the same as that displaced when a current of this value isdissipated in a circuit of inductance L and resistance R. Since

diin such a case L-j- + Ri = 0,

atato

\ di•7

R\ idt = 0

so that

em

wl(76)

This is the same relation as used by Tolman and Stewart, andtheir experiment gave results consistent with a value of elmapproximately equal to that for a slowly moving free electron.More recent experiments by Kettering and Scott19 have confirmedthis identity to a considerable degree of accuracy.

(11.3) The Barnett Effect

The experiment of Barnett,17 in 1930, confirmed that whenthe current in a circuit changes, the conductor experiences avery small longitudinal force. Suppose the circuit includes ahelical coil free to rotate about its axis, which is vertical. Ifthe coil is stationary and carries a current /, the angular momen-tum of the energy stream is Movr, where r is the mean radius ofthe coil and Mo is the electromagnetic mass of the coil. Letthe mass of the wire be Mc; then when the current is stoppedthe electromagnetic angular momentum Movr is converted intomechanical angular momentum Mcwr, where w is the finalperipheral velocity of the wire. Thus

w = w? (77)

Let us apply this to the case of a solenoid of the same axiallength, turn area and number of turns as the toroid previouslyconsidered. For a rough calculation of the order of magnitudeof the effect we may take the inductance, and hence MQ, tobe the same as that of the toroid. The mass of the wire will beabout 0-9kg, so that, if a current of 5 amp is stopped, the finalperipheral velocity of the wire will be

2 x lO"16 X 1-76 x 107 A in Qw ~ — ~ 4 x 10~y metre/second

in the same direction as the original electron flow, and opposedto the direction of the original current.

(11.4) The Self-Induced E.M.F.Since classical electromagnetic theory is based on Maxwell's

assumption that the kinetic energy of a moving current circuitis independent of the velocity product vw, and since the existenceof these electron-inertia effects shows that this is not rigorouslytrue, it follows that the fundamental law of self-induction, whena circuit is not constrained by external mechanical forces, isno more than a very close approximation.

Consider the freely suspended solenoid, initially stationaryand carrying a steady current. If the circuit is short-circuitedthe energy momentum will be transferred to the coil as a whole.If the total mass of the wire, Mc, is taken as including the electro-magnetic mass Mo, and Mo is taken as constant, the totalkinetic energy during the acceleration of the coil is given by

Kinetic energy = %(MC — M0)w2 + \MQ{\v + v)2

. . (78)

the middle term being Maxwell's Tme. The total momentum isconstant, so that

anddw

zTtdv

(79)



Then since there are no external forces, we have (13) INERTIAL CURRENT IN A PERFECT CONDUCTORIf a perfectly conducting ring, without current, is rotating

Rate of increase of kinetic energy + Rate of energy conversion w i t h a p e r i p h e r a l velocity u and is then stopped, a steady "super-into heat = 0, and if V, is the self-induced e.m.f., it follows that c u r r e n t » s h o u i d b e p r oduced, since the conduction charge p

should continue to move with velocity u. This current should(80) therefore be

'"~}-UK (85)

dv JI

whence from eqn. (79),_ dl Mov dw

Ve~ ~ ~dt~~^~~di

ldt Mc

from eqn. (51). If, however, the circuit is constrained by externalforces in such a way that the conductors have no longitudinalfreedom of movement, the induced e.m.f. will suffer nomodification.

For the given solenoid, MQjMc ~ 2-3 x 10"16. The originalmagnetic energy of the stationary solenoid is \LI2 = 10~2TT joule,and the final energy of the rotating coil, assuming a perfectlyfree suspension and no frictional loss, is %Mcw

2 = 7-2x 10~18 joule. Except for this extremely small amount ofenergy, all the original magnetic energy is converted into heatin the conductor.

(11.5) The Force on the Conductor

The longitudinal force on the conductor, when the currentchanges, is —Mod(v + w)ldt, but since w < v

p dt dt

dl= — — -r per unit length of wire

e dt• (82)

If the resistance of the solenoid is 2-72 ohms, the maximumvalue of —dl\dt when a current of 5 amp is short-circuited isIRjL = 5-4 X 103amp/sec. The maximum value of thelongitudinal force is therefore about 3 x 10~8 newton/m, or3 x 10~5 dyne per centimetre of wire.

(12) THE E.M.F. INDUCED BY THE LONGITUDINALACCELERATION OF A CONDUCTOR

Whenever the longitudinal velocity of a conductor changes,there is a very small induced e.m.f., as shown in Section 11.4.If the longitudinal acceleration is du/dt, in order to prevent aninduced current the electromagnetic mass Mo must be given thesame acceleration, and this requires an impressed force, per unitcharge, equivalent to an electric field of intensity (Molpl)dujdt.The impressed e.m.f. required to prevent the induced current istherefore V'e = iMQjp)dufdt, and the e.m.f. induced in the wireby the acceleration is equal and opposite to this, i.e.

_Moau =

* p dt

„„ /mQ \duo)d7~~{Tl)dl • (83)

In M.K.S. units

1 0 - 1 2 / ^ volts (84)

in the direction of the acceleration. This e.m.f. is additional tothat given by the classical laws of electromagnetic induction.

producing a Unking flux

(86)

The mean flux density through the ring, if R is the mean radiusand to the angular velocity, is

( 8 7>

It is of interest to note that this is the same as the flux densityinside a rotating superconducting sphere as deduced from theLondon electrodynamic theory of superconductivity.20 By theLondon theory, however, the current and field should be inde-pendent of the previous deceleration, i.e. if a ring with nocurrent is rotating with constant speed at a normal temperatureand is then supercooled, the supercooling alone should generatethe supercurrent. This curious result does not follow from ourtheory. Although the expected effect is very small, it should bepossible to determine the truth by experiment.

(14) THE ENERGY REQUIRED TO ESTABLISH A CURRENTIN A COIL ROTATING AT CONSTANT SPEED

Consider a circular coil rotating about its axis with constantangular velocity, so that the linear velocity of the conductor is u,taken positive in the direction of the current (Fig. 6). Then thetotal kinetic energy due to the current flow is

Wo[(V - u)2 - u2] = iLI2 - Mouv . . (88)

Fig. 6.—Rotating current-loop.

In order to keep the velocity of the wire constant during thegrowth of the current, a mechanical force must be exerted onthe wire of value M^dv/dt in the direction of v, or —M^dvjdtin the direction of /. Thus, in raising the velocity of the electro-magnetic mass M o relative to the wire from zero to v, mechanicalwork must be done by the driving motor at the rate —Moudvldt,and the total mechanical work required is —MQuv. The remainderof the required energy given by eqn. (88) must be provided bythe source of e.m.f. causing the current. Thus the energyrequired to establish the current, apart from ohmic loss, is\Ll2 provided electrically, and —Mouv = — uI\/(LMQ), providedmechanically (both v and / are here taken as positive quantities).The latter term becomes positive if u is in the opposite directionto the current. The e.m.f. of self-induction is —Ldl/dt and isindependent of u.

(15) THE LONDON THEORY OF SUPERCONDUCTIVITYAt present the macroscopic theory of superconductivity,

originally due to H. and F. London,21 is an ad hoc modificationof classical theory, an essential difference being that account is



taken of the inertia of the "superconducting" electrons. Classicaltheory, as we have noted, neglects electron inertia. The Londontheory has some points of similarity with the hypothesis presentedin this paper. For example, it leads to the conclusion that"supercurrents" are quantized in a macroscopic way, for aconductor as a whole,22 and that for a large superconductor thekinetic-energy density of the superconducting electrons is equalto their magnetic-energy density.23 However, these two energiesare regarded as being distinct and additive, and not, as we havepostulated, different aspects of the same thing.

It is difficult to reconcile the London equations with some ofthe basic relations of electromagnetic theory. For example, afundamental equation of the theory is our eqn. (64), but withreversed sign.21 Now the vector potential A, where curl A = B,is parallel, not anti-parallel, to a normal current in a straightconductor, so that by the London theory the induced electricfield when a supercurrent changes is quite different from thatinduced by the same rate of change of a similar normal current.From the form of the London equations it would appear thatsuch peculiar and rather incredible features of the theory maybe related to the inclusion of both electron inertia and theequivalent self-induced electric field of classical non-inertialtheory, contrary to our conclusion in Section (8). If our presenthypothesis should lead to an alternative theory of super-conductivity, one would expect it to cover both superconductionand normal conduction, with superconductors appearing merelyas perfect conductors (r = 0), but further discussion of thisinteresting problem is clearly beyond the scope of the paper.

(16) REFERENCES(1) MATHUR, S. B. L.: "Biot-Savart Law and Newton's Third

Law of Motion," Philosophical Magazine (7th Series),1941,32, p. 171.

(2) DUNTON, W. F.: "Validity of Laws of Electrodynamics,"Nature, 1937, 140, p. 245.

(3) CLAYTON, A. E.: Discussion on the above, ibid., p. 246.(4) DUNTON, W. F., and DRYSDALE, C. V.: "A Comprehensive

Fundamental Electrical Formula," ibid., 1939,143, p. 601.(5) HOWE, G. W. O.: "The Application of Newton's Third

Law to an Electric Circuit," Wireless Engineer, 1945, 22,p. 521.

(6) HOWE, G. W. O.: "Mechanical Force on the Short Sideof a Long Rectangular Circuit," ibid., 1952, 29, p. 83.

(7) MOULLIN, E. B.: Discussion on the above, ibid., p. 193.(8) MOTT, N. F., and JONES, H.: "The Theory of the Properties

of Metals and Alloys" (University Press, Oxford, 1936),p. 268 and Chapter 5.

(9) BARNETT, S. J.: "Gyromagnetic and Electron-InertiaEffects," Reviews of Modem Physics, 1935, 7, p. 129.(For a comprehensive review.)

(10) BARNETT, S. J.: "Magnetization by Rotation," PhysicalReview, 1915, 6, p. 239.

(11) EINSTEIN, A., and DE HAAS, W. J.: "ExperimentellerNachweis der Ampereschen Molekularstrome," Verhand-lungen der Deutschen Physikalischen Gesellschaft, 1915,17, p. 152.

(12) EINSTEIN, A.: "Ein einfaches Experiment zum Nachweis derAmpereschen Molekularstrome," ibid., 1916, 18, p. 173.

(13) DE HAAS, W. J.: "Weitere Versuche iiber die Realitatder Ampereschen Molekularstrome," ibid., p. 423.

(14) TOLMAN, R. C, and STEWART, T. D.: "The ElectromotiveForce produced by the Acceleration of Metals," PhysicalReview, 1916, 8, p. 97.

(15) TOLMAN, R. C, and STEWART, T. D.: "The Mass of theElectric Carrier in Copper, Silver and Aluminium," ibid.,1917, 9, p. 164.

(16) LODGE, O. J.: "Modern Views of Electricity" (Macmillan,London, 1892), Second edition, p. 97.

(17) BARNETT, S. J.: "A New Electron-Inertia Effect and theDetermination of mje for the Free Electrons in Copper,"Philosophical Magazine (7th Series), 1931,12, p. 349.

(18) LORENTZ, H. A.: "The Motion of Electricity in Metals,"Journal of the Institute of Metals, 1925, 33, p. 265. Asummary appeared in Engineering, 1925, 119, p. 625.

(19) KETTERING, C. F., and SCOTT, G. G.: "Inertia of the Carrierof Electricity in Copper and Aluminium," PhysicalReview, 1944, 66, p. 257.

(20) LONDON, F.: "Superfluids. Volume 1: Macroscopic Theoryof Superconductivity," (Wiley, New York, 1950),p. 82.

(21) LONDON, F., and LONDON, H.: "Electromagnetic Equationsof the Supraconductor," Proceedings of the Royal Society,A, 1935, 149, p. 71.

(22) LONDON, F.: "Superfluids, Volume 1," pp. 2 and 3.(23) LONDON, F.: ibid., p. 66.(24) LIVENS, G. H.: "The Theory of Electricity" (University

Press, Cambridge, 1918), p. 553.(25) LIVENS, G. H.: ibid., p. 555.(26) SLEPIAN, J.: "Energy Flow in Electric Systems—the V(

Energy-Flow Postulate," Transactions of the AmericanI.E.E., 1942, 61, p. 835.

(27) CARTER, G. W.: "The Electromagnetic Field in its Engi-neering Uses" (Longmans, London, 1954).

(28) MAXWELL, J. C : "A Treatise on Electricity and Mag-netism" (University Press, Oxford; Third Edition, 1892).Vol. H, Part IV, p. 634, eqn. (16), p. 322.

(17) APPENDIX*In this Monograph the conventional idea of field energy

has been used, and the concept of a component energy flux ofthe Poynting form was introduced. This led to eqn. (64) for therelation between A and v for a filamentary circuit, and inobtaining eqn. (65) it was tacitly assumed that eqn. (64) isapplicable to a conductor of finite section. To justify this it isnecessary to express the total magnetic energy of the current ina way which attributes it to the interior of the conductor andnot throughout the whole of the magnetic field.

If T is the magnetic energy and W the total electromagneticenergy within any given volume which encloses the circuit,24

dTIt

dWIt

{E . J)dr (89)

where / is the total current density, including the displacementcurrent. It is assumed that the current is quasi-steady, so thatradiation of energy can be neglected. Following Livens25 we

7)Aput E = — — — grad <j>, where (f> is the electric scalar potential,so that dt

{E.J)dr=-

~ • j)dr + J J J (<f> div J)dr - J J <f>JndS. . . . (90)

where /„ is the component of J normal to the boundary surface ofthe volume. Since div / = 0 we then obtain

• The Appendix was received 18th November. 1955.


170 CULLWICK; ELECTROMAGNETIC MOMENTUM AND ELECTRON INERTIA IN A CURRENT CIRCUIT

The term <f>Jn represents an energy flux through the surface.*If it be denoted by Sn it follows that

• • • (92)

and thus T = dr\j. dA • • (93)f

The displacement current is negligible inside the conductor,so if the integration is confined to the volume of the conductorwe may take / in eqn. (93) as consisting entirely of conductioncurrent. Furthermore, since there is no radiation the magneticfield of the external displacement current can also be neglected.The normal component of conduction current Jn can exist onlywhen the surface charges, and therefore the external electricfield, are changing and the outward normal component of thedisplacement current from the surface is equal to Jn. Thus Snrepresents the outward flow of energy through the surfacenecessary to provide the energy increase in the external electricfield. Since E inside the conductor is negligible compared withthe external field, we have a consistent scheme in which, to avery close approximation, all the magnetic or kinetic energy is

• Compare the energy flux discussed by Slepian" and Carter.27

f Since A is proportional to J, this result is the same as that given by Maxwell28

as an alternative to the usual expression which assigns the clectrokinetic energy tothe magnetic field. He adopted the latter as being more in accord with his fundamentalhypothesis of a medium or aether.

confined within the conductor and all the electric or potentialenergy is outside it.

Since / is entirely conduction current within the conductorwe have

/ = enev (94)

where ne is the number of effective conduction electrons per unitvolume. The kinetic energy density is $mnev

2 = %J(mle)v,m being the effective electromagnetic mass per electron. So ifthe magnetic energy of the current is to be equal to the kineticenergy of the effective conduction electrons we must have

/ . « -»,(=).

or, from eqn. (94),

fo

the solution of which is clearly

(95)

(96)

Eqn. (64), with the arbitrary scalar potential function iff taken aszero, is therefore valid for conductors of finite section, andeqn. (65) is also valid. Furthermore, these relations are notrestricted to a circuit with a single-valued current and apply, forinstance, to the current in a long transmission line.


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