A postulational approach to electromagnetism

A POSTULATIONAL APPROACH TO ELECTROMAGNETISM BY

PARRY MOON i AND DOMINA

EBERLE SPENCER 2

" T h e first time t h a t a French reader opens Maxwell's book, a feeling of m a l a i s e and even distrust at first mingles with his admiration. Only after long acquaintance and much work does this feeling vanish. For some eminent minds, the feeling never disappears. "The English scientist does not try to construct a single, definite, well-ordered structure: he seems rather to raise a great number of provisional and independenl houses, among which communication is difficult and sometimes impossible." H . Poincar~ ;~

"To a mathematician of the school of Laplace and Amp8re, it would be absurd to give two distinct theoretical explanations of the same phenomenon, and to mai~tai~ that these two explanations are simultaneously valid. "Io a physicist of the school of Thomson or Maxwell, there is no contradiction in representing the same phenomenon by two different models. Moreover, the complication thus introduced into science never shocks the Englishman: for him it adds the charm of variety . . . . Thus in English theories we find those inconsistencies, those incoherencies, those contradictions which we are forced to judge severely because we seek a rational system, whereas the author has sought to present only a work of imagination." P. Duhem ~

"We have still in front of us the important task of deriving Maxwell's equations in such a way as will not contradict the atomic theory of electricity . . . when it Ethe task] is completed we shall find that, by one of those ironies of history, we shall have to a large extent returned to the views of Maxwell's predecessors and contemporaries, which he unwisely rejected." A. O'Rahilly ~ 1. I N T R O D U C T I O N

A few definitions, chosen more or less at r a n d o m from the literature of electromagnetism, are as follows : (a) The electric field strength E at a given point P is equal to the force per unit charge which acts on a charged particle placed at P. (b) The voltage V between two points a and b in an electric circuit is equal to the difference in potential between the two points: Yab =

~a-

~b.

Department of Electrical Engineering, Massachusetts Institute of Technology, Cambridge, Mass. Department of Mathematics, University of Connecticut, Storrs, Conn. 3 H. POINCARf~, "Electricit~ et optique," Paris, Carre et Naud, 1901, p. III. 4 p. Duhem, "La th~orie physique," Paris, Chevalier & Riviere, 1906, p. 129. 5 A. O'Rahilly, "Electromagneties," London, Longmans, Green & Co., 1938, p. 180. 293

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(c) The magnetomotive force around a closed path is equal to the total conduction current linking t h a t path :

MMF

= f J.dA.

Are these s t a t e m e n t s true or false? Most students and some professors would u n d o u b t e d l y say " t r u e " ; but actually, all three statements are false in general and apply only in certain special cases. Does this n o t indicate a weakness in the c u s t o m a r y development of electromagnetic theory? Maxwell (1) e himself was aware t h a t his treatise did not constitute a logical whole, yet he was never able to remedy the defect. Subsequent investigators have added further complications to Maxwell's theory but have not eliminated the looseness and ambiguity. Electromagnetic theory is a peculiar subject. T h e peculiarity resides not so m u c h in the stratification--superposed layers of electrostatics, magnetostatics, steady currents, and time-varying fields--as in the failure t h a t has a t t e n d e d all a t t e m p t s to weld these layers into a logical whole. T h e lowest layer, electrostatics, defines certain concepts, such as E, D, ¢, in a way t h a t is generally satisfactory only for the static case. Yet the a t t e m p t is made to force these specialized definitions into the higher strata, with ad hoc modifications when necessary. T h e student, in looking t h r o u g h his textbooks on electromagnetics, can find general definitions only with difficulty, if at all ; and even the most advanced treatises fail to present a rigorously logical development of the subject. In the following paragraphs, we examine the possibilities of formulating classical electromagnetism in a logical manner. This outline might be used, for instance, in writing an advanced textbook. Much of the material is obvious to anyone who has t h o u g h t about the subject, b u t the inconsistencies and redundancies still existing in modern books support the belief t h a t the present s t u d y is needed. In the final section of the paper, we compare the postulational approach for the classical theory with t h a t for the new electrodynamics (2). 2. MAXWELL'S EQUATIONS

T h e essence of classical electromagnetism is contained in the four equations of Maxwell: 0D curl H = J + -~-, curl E =

0B Or'

""

div B = 0, div D = o. 6 T h e boldface numbers in parentheses refer to the references appended to this paper.

(1)

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1)o not the equations themselves constitute a sufficient basis for electromagnetisnl, without any further definitions or postulates? Lindsay and Margenau (3) say, " | n a very real sense, therefore, these equations m a y be said to constitute a definition of E and H . " But this cannot be true. least two ways:

Maxwell's equations are incomplete in at

(i) T h e y do not completely define the five field vectors E, D, B, H, J. (ii) T h e y do not specify the force on a charged particle. It is well known t h a t the specification of the curl or the divergence does not completely specify a vector. So even if one could assume t h a t the quantities on the right side of Eq. 1 are known, he still would not have definitions of H, E, B, and D. Additional equations m a y be introduced : D = ~E,

B = #H,

where ~, ~, and ¢ are scalar constants.

J = aE,

(2)

Substitution into Eq. 1 gives

curl H = ~E + ~ $ ' 1 OH curl E = - g 0--t-'

(la)

div H = 0, div E = p/~. E q u a t i o n la applies only to a linear, homogeneous, isotropic, time-invariAnd even in this special case, Maxwell's equations do not constitute a complete basis for electromagnetism, since t h e y do not specify a m e t h o d for determining ~, #, a nor do t h e y give any information about the force on charged particles. We conclude, therefore, t h a t in a logical t r e a t m e n t of electromagnetism, Maxwell's equations m u s t be preceded by definitions.

ant medium.

3. FU~DAMS~TALS

The usual concepts of mechanics, including length, mass, time, and As a basis for any formulation of electromagnetism, we m u s t also have definitions of electric charge and current. These concepts will be defined operationally (4). T h a t is, a definition will give either

force are assumed to be known.

(a) A description of how the q u a n t i t y can be measured experimentally, or

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(b) An equation t h a t relates the new concept to concepts previously defined. As the f u n d a m e n t a l concept of electromagnetism, we take the

electric charge (2. F r e q u e n t reference will be made to forces between charges, but such s t a t e m e n t s should not be construed as indicating a microscopic theory. We shall deal t h r o u g h o u t with aggregates of charge, never with single electrons.

I. Definition of Charge A particle m a y be acted on by forces of mechanical origin--by pushes and pulls c o m m u n i c a t e d by rods or strings, by air currents, by inertial and frictional effects. It can also be acted on by gravitational forces. If all these forces on a dielectric particle are eliminated and still a force remains, we say t h a t it is caused by electric charges• By means of a torsion balance, Coulomb showed t h a t the magnitude of the electrostatic force between two charges is

Q1Q2 F = K

r2 ,

(3)

where K is a constant t h a t depends on the units and on the medium. E q u a t i o n 3 m a y be used as an operational definition of the concept,

electric charge Q. T a k e three particles with charges Q1, Q2, Q3. balance and keeping the distance r fixed, we find

F'

K Q~Q3 r2

F"

Using a torsion

K Q2Q3

,

r2

Q2 is adjusted so t h a t F' = F". T h e n Q1 = Q2 = Q. ure the force between Q1 and (22 :

F=K

We now meas-

Q'r.2

or

O = (F/K)~r.

(3a)

Equation 3a defines charge in terms of the known concepts of force and distance.

I L Definition of Charge Density Charge density p is defined by the equation (4) V-~0

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T h e limit is, of course, a m a t h e m a t i c a l fiction. Physically, the v o l u m e is imagined to be reduced to a small but macroscopic size which still contains a large n u m b e r of electrons (say 10~°). T h e function p(uL u'-', u a, t) is considered to be continuous and with continuous first deriwitiv('s at o r d i n a r y points.

III. Definition of Current Density Current density J is defined as J = p+v+ + p_v_,

where the subscripts refer to the sign of the charge. trons move, as in a metallic conductor, J = p_v_

=-

:vlO,,lv-,

<5) If only the elect5")

vvhere

N = n u m b e r of electrons per unit volume, I Q~ J = m a g n i t u d e of electronic charge. ]'he velocities in Eqs. 5 and 5a are ordinarily measured with respect to the conductor. But here, as t h r o u g h o u t classical electromagnetism, velocity is ambiguous.

IV. Definition of Current The c u r r e n t t h r o u g h a surface S is defined as

x = fJ.dA.

16)

V. Definition of Permittivity Consider a parallel-plate capacitor connected to a b a t t e r y . Measure t h e charge on one of the plates, first with v a c u u m dielectric, seco n d l y with a given dielectric. T h e ratio of the two values of Q is called the relative p e r m i t t i v i t y (~/~0) of the given dielectric, or =

~o(Q/Qo),

~7 )

where ~0 is an a r b i t r a r y c o n s t a n t which depends only on the s y s t e m of units. E q u a t i o n 7 defines a p r o p e r t y of a material, and this p r o p e r t y is called permittivity.

VI. Definition of Permeability Consider a toroidal core m a d e of t h e material to be tested. A uniform winding on the core is c o n n e c t e d to a sinusoidal electric source of

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c o n s t a n t frequency. T h e relative p e r m e a b i l i t y (~/~0) of the core m a y be defined as the ratio of the magnetizing c u r r e n t s obtained with the material core and with a v a c u u m core, or U = U0(Z/I0).

(S)

E q u a t i o n 8 defines a p r o p e r t y of a material, this p r o p e r t y being called the permeability.

I. Postulate of Retardation It is a well-known experimental fact t h a t electromagnetic disturbances do n o t arrive i n s t a n t a n e o u s l y at a d i s t a n t point b u t are retarded b y a time interval equal to l/c. H e r e c is a c o n s t a n t t h a t d e p e n d s on the m e d i u m . In a v a c u u m , c = 2.99792 X 108 na sec -1. In dissipative media, the form of a t r a n s i e n t m a y be so c o m p l e t e l y distorted t h a t the concepts of phase velocity and group velocity are practically meaningless. Nevertheless, we shall p o s t u l a t e t h a t all time relations obtained experimentally at v e r y small distances can be e m p l o y e d at large distances merely b y replacing t b y (t - I/c). TABLE I.--Fundamentals. I. Definition of charge:

O = (F/K)¢r. II. Definition of charge density: O--*0

,

"

III. Definition of current density: J = p+v+ + 0-v-. IV. Definition of current:

I = ~J.d~. V. Definition of permittivity:

e = eo(Q/Qo) for parallel-plate capacitor. VI. Definition of permeability: u = uo(I/Io)

for toroidal coil on a.c.

I. Postulate of retardation. A s u m m a r y of this section is given in T a b l e I. T h e definitions of this section are a necessary p r e l i m i n a r y to a n y formulation of classical electromagnetism, either of the t y p e of Section 4 or t h a t of Section 5.

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4. D E F I N I T I O N S OF E A N D B

Consider first the problem of finding a definition of E that will apply in both electrostatics and electrodynamics. The usual electrostatic definition is E = F/Q. (9) But this equation is inadequate for general use : (a) In free space, the operational experiment (4) associated with Eq. 9 is easily imagined and can even be performed. But in a solid dielectric, the operational significance of the definition becomes questionable. (b) Equation 9 does not limit the magnitude Q of the test charge, though a large test charge will distort the original field that is to be measured. This difficulty is eliminated by the modification,

E = lira ( F ) Q--*0

~)

(9a) "

But obviously the limiting process cannot be taken literally, since the minimum possible charge is that of an electron. With either the microscopic or the macroscopic interpretation, Eq. 9a involves troublesome explanations. (c) The usual equation for the force on a moving charge is F = QEE +

v

x B],

(10)

and this equation may be employed as a general definition of E. Lorentz considered v as the velocity of the test charge with respect to the aether. But since the aether has been abandoned, v seems to be the velocity of Q with respect to the magnetic field, If B is produced by several current-carrying conductors moving with respect to each other, however, it is not clear (at least, without further explanation) what v is.

Similar difficulties are encountered in attempting to define (5) the magnetic flux density B. We may imagine a current element I ds, placed at point P where B is to be measured. The force on the element is d F = I ( d s X B), (11) and the equation defines B. All the weaknesses of the previous definitions appear, plus the added troubles associated with a fictitious, isolated current element. In particular, Eq. 11 raises the question of the absolute velocity of the charges which constitute the current. Evidently, an explanation is needed about this velocity--is it with respect to the aether, with respect

300

PARRY MOON AND DO:MINA EBERLE SPENCER TABLE I I . - - O u t l i n e o f E l e c t r o m a g n e t i c T h e o r y .

DEFINITIONS (1) Definition of 9:

(2) Definition of A :

(3) Definition of E : 0A E = - grad 9

Ot"

(4) Definition of B : B = curl A. (5) Definition of D : D = eE, where e = e0(Q/Q0) for a parallel-plate capacitor. (6) Definition of H : H = B/u,

where u = u o ( I / I o ) for a toroid excited with sinusoidal a.c.

POSTULATES (1) C o n s e r v a t i o n of c h a r g e : div J -

00 ot"

(2) Force on c h a r g e : F = Q[-E+v)<

B~,

where v is the v e l o c i t y of Q with respect to the~magnetic field.

DEDUCTIONS (1) F r o m Definition 4, d i v B = 0. (2) F r o m Definitions 3 a n d 4, 0B curl E = - - -

Ot"

(3) F r o m Definitions 1, 2, 3, and P o s t u l a t e 1, d i v D = 0, (4) A n d

V2~ + p / ~ = Eu 02~

Ot~ _•

(5) F r o m Definitions 2, 3, 4, 0D curl I-I = J + -~-, (6) A n d 0~A • A+uJ

= ~u Ot~,

IJ. I;. I.

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to the wire, or with respect to the magnetic field ? One m a y introduce the Einstein special theory of relativity, which is based on a naive faith in the infallibility of Maxwell's equations. But if an electrodynamic is itself based on the fundamental principle that absolute velocity is meaningless (as was done by Ampere, Gauss, Ritz, and others), then no special relativity crutch is needed (2). In this section, we have considered the possibility of a logical development of electromagnetism beginning with a minimum number of definitions, to be followed by Maxwell's equations. The preliminary definitions seem to offer almost insuperable difficulties. In view of this result, it is perhaps better to abandon the attempts made in this section and to t r y an entirely different approach. A postulational development based on scalar and vector potentials is outlined in the following sections. 5. FURTHER DEFINITIONS AND ]POSTULATES

Starting with the preliminary definitions of Table I, we proceed to the logical development of electronlagnetism summarized in Table I I. There are six definitions, two postulates, and a set of deductions. The first definition specifies the scalar potential in terms of a distribution of charge p. The integral is taken over all space, and the charge density Ep] = p ( u , , u"-, u ~, t - r / c ) is retarded in accordance with Postulate I. A similar definition gives the vector potential A. The square brackets indicate retardation, according to the usual Lorenz notation. The scalar constants k and k' depend on the medium and on the system of units. Note t h a t the foregoing definitions are general, in as much as they are not limited to electrostatics or magnetostatics. Unfortunately, however, t h e y are limited to l i n e a r media, since only with a linear system can we use superposition or integration. Also, c is questionable unless the medium is linear and lossless. There seems to be a widespread misconception as to the universality of electromagnetic equations. Even if one assumes t h a t the Maxwell equations in differential form are universally true, he can do practically nothing with them until he introduces further definitions or manipulations that limit generality. Thus there seems to be little disadvantage in restricting ourselves from the start to media that are linear, homogeneous, isotropic, time-invariant, and lossless. We next define the vector E in terms of the potentials (Definition 3, Table II). Similarly, the vector B is defined in terms of the vector potential (Definition 4). Definitions 5 and 6 allow the evaluation of k and k' in Definitions 1 and 2. In the rationalized inks system, k = 1/47r~,

k' = #/4~r.

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Only t w o postulates are n e e d e d : conservation of charge, and the e q u a t i o n for force on a moving charge (Table II). 6. DEDUCTIONS

F r o m the six definitions and t w o postulates (plus the material in T a b l e I), one should be able to d e d u c e all of electromagnetic theory. Using Definition 4 and a familiar v e c t o r identity, we obtain d i v B -- div c u r i a

= 0

or

d i v B = O,

(12)

which is one of Maxwell's equations. F r o m Definition 3 and a n o t h e r v e c t o r identity, 0

curl E = -- curl grad 9 -- ~ (curl A). B u t curl A = B and curl grad ~o = 0, so 0B

curl E = -- 0-7"

(13)

F r o m Green's t h e o r e m (6) and Definition 1, we o b t a i n the i m p o r t a n t differential equation, 02~o (14)

V"-~ + p/e = ~# Ot--~.

Thus, 02~o

div grad ~ + p/e = e# Ot~, and b y Definition 3, - dive

+o/e

=

divA+

~u~-

.

(15)

Using P o s t u l a t e 1 and Definitions 1 and 2, div A = ~ ------

f__ d i v r[ J ] d ~ ) = - - -

f I a[p] d~

4~r J

r

Ot

O~ e/Z ~ - .

T h u s the b r a c k e t in Eq. 15 is zero, and div D = p.

(17)

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F r o m Green's theorem and Definition 2, we obtain the partial differential equation (7),

O~-A . A + u J = ~u ot-~..

(18)

Thus,

O'-'A grad div A - curl curl A + u J = ~ / ~ - -

Ot"- '

and from Definitions 3 and 4, 0E grad div A - curl B + # J = -- e u - ~ -- E/~grad (O~/0t). ButdivA

=-

0~ eu~,so 0D curl H = J + or-"

(19)

Equations 12, 13, 17, 19 are Maxwell's equations, which are here derived from the definitions and postulates of Tables I and II, without use of Section 4. All of classical electromagnetic theory is obtainable directly from Maxwell's equations, so the whole subject rests on the postulation system of Tables I and II. In specific applications, the vectors E, D, B, t t are usually obtained from the potentials ~ and A. These potentials m a y be evaluated from Definitions 1 and 2 or from the partial differential expressions, Eqs. 14 and 18. The vector quantities are then found by applying Definitions 3, 4, 5, and 6. 7. THE NEW ELECTRODYNAMICS

An alternative to Maxwell's theory is based on the work of Amp+re, Gauss, and Ritz. In this form of electrodynanlics, the physically measurable quantities force and charge are emphasized, and the fictitious magnetic field is entirely abandoned. The result is a logical system with only a few definitions and postulates, free from the ambiguities of the Maxwell theory. TABLE I II.--The New Electrodynamics. I. Definition of charge: Q = (F/K)~r. l I. Definition of charge density:

I| I. Definition of permittivity: =

~o(Q/Qo).

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I. Postulate of retardation. II. Postulate of force (2) Q2

ak~E~" -- aa 4r~c~.r dt v(t -- r / c ) -- ar -4~re Or

r

Table III lists the entire set of definitions and postulates needed for the new electrodynamics (z). Postulate II expresses the force per unit charge, on a macroscopic charge Q2, caused by a charge Q1 which m a y have a n y velocity with respect to Q~ and a n y acceleration. For a continuous distribution of charge, Q~ = p d ~

and the total force is obtained by integration. The new electrodynamics m a y be expressed as a field theory if desired. The only field vector is ~, where =

lim (F/Q2). Q-*o

(20)

The usual quantities, E, D, B, H, J and ~ are eliminated. With stationary charges (electrostatics), ~ become identical with the familiar E and a scalar potential m a y be introduced : ~

=-

grad ¢.

(21)

With moving charges, a vector potential is also needed. Evidently this form of electrodynamics is simpler and more closely related to the measurable quantities t h a n is the Maxwell theory. T h e new formulation is also free from the outstanding defects of the Maxwell theory. As mentioned previously, one of the troubles with classical electromagnetism is the ambiguity with respect to velocity. In the pre-Maxwellian work of Galileo, Newton, Ampfire, and Gauss, r e l a t i v e velocities were used as a m a t t e r of course. But F a r a d a y and Maxwell introduced the questionable idea of fields in a stationary aether, t h u s apparently employing the idea of a b s o l u t e velocity. The concepts of current and current density are tainted by this vagueness. Even Definitions 1 and 2 of Table II are under suspicion, for they assume t h a t we can always tell whether charges are classified as 0 or J. Physically, there are n o t two distinct quantities here b u t only one; and whether a given charge distribution is specified as p or J depends on what physical velocity is arbitrarily considered to be zero. The force between two charges depends on the velocity of one with respect to the other, not primarily on their velocities with respect to the aether, the laboratory,

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or the observers. Two parallel streams of electrons moving at the same velocity exert no more force on each other than they would if they were stationary in the laboratory. Since the new electrodynamics employs only relative velocities, it does not encounter these difficulties nor does it need to be patched up by the introduction of a special theory of relativity. To summarize, we feel that all special cases of electromagnetism, such as the stationary and quasi-stationary states, may well be derived from Maxwell's equations. But before these equations are stated, logical development requires that the vectors be defined. Even when the investigator carefully avoids the familiar fairy tales dealing with fictitious cavities and "wormholes" and the pushing of charged particles through solid but frictionless media, he still encounters trouble in defining the concepts (Section 4). The least objectionable approach that we could find to classical electromagnetics is summarized in Table I I. And even that approach has trouble with the concept of velocity and with the generality of the results. Another possibility is the new electrodynamics, which offers interesting possibilities that merit further study. REFERENCES

(1) J. C. MAXWELL,"Treatise on Electricity and Magnetism," Oxford University Press, 1892. (2) PARRY MOON AND D. E. SPENCER, "A New Electrodynamics," JovR. FRAN~;L1NINST., Vol. 257, p. 369 (1954). (3) R. B. LINDSAYANt) H. MARCENAU,"Foundations of Physics," New York, John Wiley & Sons, 1936, p. 306. (4) P. W. BRII)GMAN,"The Logic of Modern Physics," New York, The Macmillan Co., 1932. (5) J. V. HVGHES, "The Definitions of Magnetic Flux Density and Field Intensity," A m. J. Phys., Vol. 21, p. 89 (1953). (6) J. A. STRATTON,"Electromagnetic Theory," New York. McGraw-Hill Book Co., 1941. p. 166. (7) PARRYMOONAND D. E. SPENCER, "The Meaning of the Vector Laplacian," JotrR. FRaXKLIN INST., Vol. 256, p. 551 (1953).