RL 896 A DIGEST OF HERTZ'S ELECTROMAGNETISM Chen-To Tai and John H. Bryant June 1993 RL-896 = RL-896

A Digest of Hertz's Theory of Electromagnetism Chen-To Tai and John H. Bryant Radiation Laboratory Department of Electrical Engineering and Computer Science The University of Michigan Ann Arbor, MI 48109-2122 Abstract An attempt is being made to recast a masterpiece in the theory of electromagnetism by the great scientist in modern notations and to supply the derivations of several important equations in the work. By invoking the radiation condition in his formulation, an amendment has been made to clarify a criticism addressed to that theory. Also, upon removing the constraint placed on the current distribution in the original theory we have demonstrated a complete union of Hertz's theory of electromagnetism with Maxwell's theory. The distinct and independent features of Hertz's theory have been emphasized. The importance of this theoretical work by Hertz and its significance appear to have not been fully recognized. 1

Introduction The work under discussion was published by Hertz in 1884 [1], twenty years after the publication of Maxwell's theory [2], and three years before the start of his renowned experimental verification of Maxwell's theory, a monumental work in the history of science. The paper has been examined by several authors [3, 4], but the importance of this theoretical work and its significance appear to have not been fully recognized. The purpose of the present study is to recast the entire work in modern notations, to fill in some detailed steps not found in Hertz's original paper, and most important of all, to deduce some new information which can be extracted from his theory. For the convenience of modern readers, all quantities are now defined in MKS system. Gibb's notations of vector analysis are used throughout. The Formulation According to Hertz, the following premises are adopted in his theory: 1. The principle of the unity of electric force and that of the unity of magnetic force. The two forces which Hertz referred to correspond to the electric and the magnetic fields in modern nomenclature. 2. The principle of the conservation of energy, that of the action and reaction as applied to systems of closed circuits; that of the superposition of electric and magnetic actions, and lastly, the well-known laws of the magnetic and electromagnetic actions of closed currents and of magnets. There are two sets of equations in the original work. The first set consists of equations representing Ampere's law and Faraday's law for fields produced by electric current. For a steady current flow with electric current density J, which is a function of time Ampere's law states VxH=Je (1) then V J =O (2) 2

In free space (air) V. (poH) = 0, (3) so we can introduce an electric vector potential function, denoted by A, such that ~oH = V x A, (4) then, in view of (1), we obtain V x V x A, = /oJe. By assuming V Ae = 0, (5) the previous equation can be changed to V2Ae = -oJe. (6) By defining Ae = PoIe, (7) (4) becomes H =-V x Ae = V x Ie, (8) Ho and (6) can be written in the form V2Ie = -J. (9) The quantity Ie will be designated as the electric current potential. It has the dimension of [Ie] = [Ampere]. Hertz invoked the induction law that the time rate of change of Ie would yield an electric field given by E -p t(10) 3

In an interpretation of the origin of (10), it appears that this induction law is a consequence of Faraday's law, namely, OH V x E =-o (11) With H given by (8), (11) becomes Vx(E + =0. (12) A particular solution of the above equation is E = —ot ' (13) which is the same as (10). Equations (8) and (13) are two basic equations used by Hertz. A second set of equations were introduced by Hertz based on the concept of magnetic current, which was interpreted by him as the rate of the change of magnetization, i.e., aM Jm = -o at. (14) The negative sign was introduced in (14) so that the magnetic Ampere's law would have the same form as the electric Ampere's law. Namely, Vx E=Jm. (15) We designate (15) as the 'magnetic Ampere's law' a term not used by Hertz. For the electric field produced by a magnetic current V. (EoE) = 0. (16) A magnetic vector potential, denoted by Am, can thus be introduced such that CoE = V x Am. (17) Substituting (17) into (15), we obtain V x V x Am = oJm. (18) 4

By assuming V.Am =0, (19) (18) reduces to V2Am = -oJm. (20) Let us define a magnetic current potential, denoted by Im, such that Am Im =. (21) Then (17) can be written in the form E = V x I. (22) The dimension of the magnetic current potential is [Im] = [volt]. The induction theorem used by Hertz for the magnetic field produced by a magnetic current is aI, H = E t (23) We can give an interpretation of the origin of (23), by considering a 'magnetic Faraday's law' in the form aE V x H = o t. (24) Substituting (22) into (24-) we obtain V x -e at )0. (25) A particular solution of (25) is dIm H = t atm H- so Bt ' which is the same as (23). We must call attention to the fact that the term o aE in (24) should not be viewed as the displacement current in Maxwell's dslee 5

theory because the fields E and H described by (22) and (23) are produced by magnetic current. There is no electric current involved in this set. Equations (22) and (23) are the dual set of (8) and (10). Let us summarize the two basic sets of equations used by Hertz to develop this theory of electromagnetism to be disclosed shortly. They are: H V x I, (26) E = - oI (27) E = V x Im (28) H e= ot, (29) where Ie and Im denote, respectively, the electric current potential and the magnetic current potential which are solutions of the equations: V2I, = -Je (30) V2I = -Jm. (31) The explicit expressions of Ie and Im will be discussed later. A most remarkable feature of Hertz's theory is the casting of the electric field given by (27) into a form of (28) by an equivalent magnetic current potential, denoted herein by I'1. Thus let -P = V x I. (32) The prime on I'1 is merely a notation, not a differential sign. This is the most important step in his theory. It invokes the concept of the interaction or coupling of two otherwise separate systems, the consequence of which shows that Ampe;re's law, both electrical and magnetic individually, is not sufficient to describe electromagnetic phenomena. The end result of his theory is the natural appearance of the electric displacement current in the electric system and the magnetic displacement current in the magnetic system. This logic and its development are quite different from that of Maxwell. In certain 6

aspects, it is richer in its physical content. Hertz did not use the word 'equivalent' to introduce I'1. This terminology is our suggestion. The meaning of the subscript 'ml' will be evident later. By taking the curl of (32) we obtain -otV X I = V x V x I. (33) By assuming V. I' =0, (34) we can reduce (33) to the form V2I1 = - toV x Ie. (35) Now we introduce a new function Ie1 such that a I'=- VxIe, (36) and substitute it into (35). We obtain V2Il = — Ie. (37) In (37), we treat Ie as a known function which is a solution of (30) and Ie1 as an unknown function to be determined from (37). Once Il is so determined we can find I', based on (36), i.e., Im1 =- -o^oV X Ie. (38) The magnetic field due to this equivalent magnetic current potential can be cast in the form of (29). Denoting this magnetic field by H1, we obtain aIml a2 H1 = at- = -ot2 -V X Iel. (39) In the words of Hertz this is the corrected part of the magnetic field which has to be added to the magnetic field due to Ie alone. The total magnetic field is now given by H2 = H+H1 =V x [I- C tIel, ' (40) ]2a 7

where c = 1/(/oIo)2 denotes the velocity of light in empty space. It is interesting to observe that in his original work this constant appears very early in his formulation because his quantities are defined in the absolute electric and magnetic units. The change of the electric current potential as seen in (40) also changes the corresponding electric field. Thus, E2= E + E1 =-1 - - Ic2 - Ie, (41) The iteration process can be continued on indefinitely to yield a series for the total electric current potential, denoted by Pe, given by a2 1 a2'n ] Pe l e t2I...(-n 2 n 0.... (42) The relation between the successive electric current potentials is governed by the equation V2Ie = -Ie(n-1). (43) It is understood that Ie(-l) = Je (44) IeO = Ie. (45) The total electromagnetic field due to Pe is then given by He = V x Pe (46) aPe Ee = - O, (47) By means of the same technique, it is obvious that the fields produced by the magnetic current can be developed in a similar manner. In fact, by considering (28) and (29) as the dual of (26) and (27), we can deduce the following key equations: Pm -= Im- 2 Or Iml*'*( — ) n C2 *Imn+... (48) 8

Em = V x Pm (49) aPm Hm = e, (50) at The relation between the successive magnetic current potentials is given by V2Imn = -Im(n-l) (51) with Im(-1) = Jm (52) and Imo = Im. (53) Based on (42) and (43), Hertz showed that Pe satisfies the Helmholtz wave equation. The proof is as follows. By taking the Laplacian of (42) we obtain ~~ 1 o 02' V,2p, = V2 (- 1) 2 anIen. (54) n=0 Ot2I Since V2Ieo = -Je, (55) and V2Ien = -Ie(n-1), n > 1. (56) (54) can be written in the form 1 02 oo 2(n-1) VP e = - -Je- E(-1),) e(n-1) c2 — 1 c2(n ) t2(_1)_ 1 02 oo 1 02n - Je + EZ(-1) _- -- O Ien = - J~ -t c20t2 + -) c22n 5t2nIe, 1 02Pe - Je + c2 at2 9

or 1 O2Pe V2P - t = -J (57) which is the vector Helmholtz equation. It should be emphasized that the wave equation for Pe as stated by (57) was derived under the condition that V * Je = 0 and its derivation does not depend on the explicit expressions for Ie. Only (43) is needed. By eliminating Pe between (46) and (47), with the aid of (57), one finds 0H, Vx Ee = -o (58) V x H = Je +e, OE (59) at These are two of the differential equations found in Maxwell's theory now derived from Hertz's theory under the constraint V- J = 0, hence V (coE) = 0. One important feature of Hertz's theory is the natural appearance of the displacement current co2 in (59). The constraint on Je can be removed. But before we do that, let us complete the discussion of Hertz's work, particularly in regard to his solutions for Ien for n > 0. Solutions for Pe The series expansion for Pe derived by Hertz as given by (42) contain the potential functions,,en. An expression for Ien was found in the original paper without showing its derivation. We attempt to give a derivation of that expression using a procedure that is presumably similar to Hertz's original scheme. We start with Ieo, which is a solution of the equation V2Ieo -Ie(-) = -Je. (60) One particular solution of Ieo in free space is given by Ieo(R) = J J (R'),dV' (61) I~(R,- J f R - R/[ 10

or simply 1 fJd leo = -|- dr, which will be written in the form 471 Ieo = W | JJKo(r)d~, with Ko(r) = 1/r. To find Il1 which is a solution of V2Ie = -Ieo, (62) (63) (64) we let Iel= / Je, Ki (r)dr. 4r (65) Substituting (63) and (65) into (64), we see that satisfy the equation the function Kl(r) must or V2K(r) = -KI(r) = -1/r, 1 d { 2dKIl 1 r2 dr dr - r (66) The particular solution for K1 is KI'(r) = -r/2. Following the same procedure progressively, one finds (67) en Je'n()d I~,, /,~rd. (68) with Kn(r) = (-l)n2"-1/(2n)!. (69) 11

hence Pe=)n 02n Pe = -I,, n=O O(t)2n I r2n-l 2nJe d. (70) 47r n=O (2n)! a(ct)2" This is the expression for Pe found in Hertz's original paper, although he used the component form of Pe in stating his result. It can be proved by means of (68) that V Ien = 0 when V Je = 0. This is a verification of the postulate stated by (34) and passed on to Ie1 and subsequently to Ien. According to the wave equation for Pe stated by (57), there are, mathematically, two independent solutions given by [P Ir Je(t- r/c) dT (71) [Pe]r r, (71) [Pea = e(72) in free space. The function [Pe]r is designated as the retarded potential and [Pe]a as the advanced potential. The Taylor series expansions of J(t - r/c) and Je(t + r/c) are given by ae r 2 O2ac) J(t- r/c) = Je + r(ct) + 2 (ct)2 + (73) J(t + r/c) = [Je- r(ct) + 2 d(ct)2 (74) Thus, Hertz's expression for Pe as stated by (70) is the arithematic mean of [Pe]r and [Pe],, i.e., Pe= {[Pe]r + [Pe]a) (75) 2 This relation was first pointed out by Havas [3], after commenting on a paper by Zatkis [4] who misinterpreted Hertz's expression for Pe. According to Havas, Hertz's solution for P, is not acceptable from the physical point 12

of view, which is certainly true. However, the 'shortcoming' can be readily remedied. In fact, the concept of retarded potential was introduced by Hertz several years later [5]. If he had pursued the matter further that shortcoming could have been easily removed. In any event, there are two approaches by which Hertz's result can be properly modified to yield the correct answer. The missing criterion is the radiation condition which Hertz never mentioned in his paper. If we add to Hertz's expression for Pe a term denoted by P,, which is a solution of the homogeneous wave equation V2p- _2 at2 = 0, (76) then the radiation condition can be satisfied by a proper choice of Po. The desired function is obviously given by Po= {[Pe]r- [Pe]a} (77) Since both [Pe]r and [Pe]a satisfy (57), their difference certainly satisfies (76). Let the resultant potential function be denoted by PeT, then PeT = Pe + Po I 1 -= {[Pe]r + [Pe]a} + - {[Pe]r - [Pe]a} = [Pelr, (78) which is the desired answer. Another approach to find PeT is more complicated but it follows closely Hertz's original analysis with a modification. Returning to the differential equation for leo, namely, V2Ieo = -Ie(-l) = -Je, (79) we observe that the general solution for Ieo is Ieo = - Je - C+ dr, (80) 4,x r 13

where c, is an arbitrary constant (function of time). In Hertz's treatment he let c, = 0. Now if we choose c, such that 'J, CoJe = (t) (81) then 1 r Je OJe] Ie~ = 4-Je - + -at) dr. (82) Ieo: 47r r 9(ct)J (82) For Iel we let Iel 1 /JeII() + (ct)I;(r). (83) The functions Ki(r) and KI are solution of the equations V2K (r) = -1/r (84) V2K;(r) = -1. (85) The particular solutions are I (r) - -r/2! Ki(r) =-r2/3! hence I = 47r 2! 3! (ct) (86) Similarly, 2 = r J4! 5! (ct) (87) Using these modified expressions for Ien in (42), now representing PeT, we find 1 = r2n-1 dJ)n PeT 4 = n! r(ct d' (88) 14

which is the same as [Peir. It is seen that the new expression contains terms with odd derivatives of the current function, these terms represent precisely the mean value of the difference of the retarded and the advanced potentials. Of course, this approach is guided by our anticipation that the resulting potential function should represent a retarded potential. The method, however, is essientially based on Hertz's formulation with a proper modification. Non-solenoidal Distribution of Je So far we have been dealing with Hertz's theory under the constraint V - Je = 0. For non-solenoidal current distribution the equation of continuity reads: V.Je = a. (89) In view of (59) we must have V (6,E) = P, (90) which corresponds to Gauss law for time varying charge. With this removal of the original constraint we have obtained the complete system of Maxwell's equations from Hertz's theory of electromagnetism based on an independent method quite distinct from Maxwell's path. As far as the function for PeT or [Pe]r is concerned, the expression represented by (71), or its equivalent (88), is still valid, but V * PeT is no longer vanishing. It can be shown that VPeT = 1- pt -a /)d (91) 47T r 9t This part of the theory was investigated thoroughly by Hertz later in 1889 [5]. The consolidation of Maxwell's theory could have been established by Hertz theoretically in 1884 if he had pursued a little further based on his own model. The modification of Hertz's theory for the magnetic current model can be executed in the same manner. The most convenient derivation is to apply the duality principle by replacing (Je, pe, PeT, Ee, He,, 0o, o) in the electric model by (Jm, 0, PmT, -Hm, Em Eo, 6P.) in the magnetic model. 15

In Retrospect Several authors in the past have reviewed the paper under discussion, but it is our opinion that the value of this masterpiece was not fully recognized. It is remarkable that an alternative method was available to derive Maxwell's equations based on a quite different approach. Even though a constraint was originally imposed on the method, the fact that a retarded potential formula can be extracted from the formulation demonstrates convincingly the power of his method. As has been shown, by removing that constraint and invoking the equation of continuity and the Gauss law, the complete system of Maxwell's equations evolve from Hertz's theory. For the magnetic current model, since there is no magnetic charge, the condition V. Jm = 0 is not a constraint. It is a mathematical statement of a physical law. The theory for the magnetic current model is, therefore, complete by itself provided that a homogeneous solution is added to the original magnetic current potential function. This portion of Hertz's theory appears to be not covered in Maxwell's theory. The physical insight of Hertz's work seems to be not well appreciated in the past. In a comment by the renowned physicist Max Planck [6], Hertz's derivation of Maxwell's equations is considered to be peculiar. We feel that the derivation was very brilliant, logical, and not peculiar at all. Hertz's modest conclusion of his own theory "... if the choice rests only between the usual systems of electronmagnetics and Maxwell's, the latter is certainly to be preferred..." might have casted a negative image in the eyes of later scholars. It may be of interest to remark that forty years ago one of the present authors (C.T.T.) considered a method [7] to extend Rayleigh's theory of diffraction of electromagnetic waves by small bodies [8]. In that method, we expand E and H for a monochromatic oscillating field into two series in the form E = Z(ik)nEn (92) n=O H = E(ik)nHn, (93) n=o where k = ) According to Maxwell's theory, in free space, V x E = ikZ0H, (94) 16

V x H=-ik z Zo = 2 (95) By substituting (92) and (93) into (94) and (95) and equating the terms of the same power of (ik)n, we obtain, for n > 1, V x (ZoH,) = -En —1 (96) V x En = ZoHn-1. (97) In particular, for n = 1 V x (ZoH)=-Eo (98) V x E1 = ZoHo. (99) In a later paper by Stevenson [9] the solution for En, Hn for n > 1 are found systematically by an interative method based on a known pair of solution for Eo and Ho. The similarity of the approaches between the works of Stephenson/Tai and that of Hertz is quite evident. However, Hertz's theory is more profound and authorative. Both of these authors were not aware of Hertz's iterative method. In fact, even Lord Rayleigh did not quote Hertz's theory relating a quasi-static field with a dynamic field or a wave theory. The scientific community had done a great injustice in not fully recognizing the value of this work which probably had prompted Hertz to search vigorously for the experimental evidence in verifying Maxwell's theory, now confirmed theoretically by an independent approach. The authors gratefully acknowledge the support from Dr. Fawwaz T. Ulaby, Director of the Radiation Laboratory at the University of Michigan for this work. The technical assistance of Mr. Jim Ryan is very much appreciated. 17

References [1] Hertz, Heinrich "On the relations between Maxwell's fundamental electromagnetic equations and the fundamental equations of the opposing electromagnetics," Wiedemann's Annalen, Vol. 23, pp. 84-103, 1884, English translation by D.E. Jones and G.A. Schott in Miscellaneous Papers by Heinrich Hertz, MacMillan and Co., London, 1896. [2] Maxwell, James Clerk, "A dynamical theory of electromagnetic field," Scientific Papes, Vol. 1, pp. 526-597, Dover Publications, New York; original paper published in London Phil. Trans. Soc., Vol. 155, 450, 1864. [3] Havas, Peter.,"A note on Hertz's 'derivation' of Maxwell's equations," Am. J. Phys., Vol. 34, 667-669, 1966. [4] Zatzkis, Henry, "Hertz's derivation of Maxwell's equations," Am. J. Phys., Vol. 33, 898-904, 1965. [5] Hertz, Heinrich, "The forces of electric oscillations, treated according to Maxwell's theory," Wiedemann's Annalen, Vol. 36, 1, 1889, English Translation by D.E. Jones in Electric Waves by Heinrich Hertz, MacMillan and Co., London, 1896, also Dover Edition, 1962. [6] Planck, Max in James Clark Maxwell, A Commemoration Volume 1831 -1931, pp. 60-62, Cambridge University Press, Cambridge, England, 1931. [7] Tai, C.T., "Quasi-static solution for diffraction of a plane electromagnetic wave by a small oblate spheroid," IRE Trans., PGAP-1, 13-36, 1952. [8] Rayleigh, Lord, "On the incidence of aeriel and electric waves upon small obstacles in the form of ellipsoids or elliptic cylinders, and on the passage of electric waves through a circular aperture in a conducting screen," Phil. Mag., Vol. 44, 28-52, 1897. [9] Stevenson, A.F., "Solution of electromagnetic scattering problems as power series in the ratio (Dimension of Scatter)/Wavelength," J. Appl. Phys., Vol. 24, No. 9, 1134-1142, 1953. 18