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On the Advanced Solutions ofMaxwell's EquationsByron T. Darling aa Département de Physique, Université Laval, Ste-Foy, QuébecG1K 7P4, CanadaPublished online: 03 Dec 2010.

To cite this article: Byron T. Darling (1986) On the Advanced Solutions of Maxwell's Equations,Optica Acta: International Journal of Optics, 33:11, 1397-1404, DOI: 10.1080/713821896

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OPTICA ACTA, 1986, VOL. 33, NO. 11, 1397-1404

On the advanced solutions of Maxwell's equations

BYRON T. DARLINGtD6partement de Physique, Universit6 Laval, Ste-Foy,Qu6bec GlK 7P4, Canada

(Received 4 July 1985; revision received September 1986)

Abstract. In this paper we show how the results of two preceding papers on theretarded solutions of Maxwell's equations may be transcribed to advancedsolutions. We also give the results of the retarded surface integral for the advancedsolution, and in the n J = 0 case, also for the retarded solution, and the half sumand half difference of the advanced and retarded solutions.

1. IntroductionIn a previous paper [1] (henceforth referred to as paper 1), we derived an identity

using retarded Green's functions, and we applied it there, and in a following paper[2] (henceforth referred to as paper 2), to the retarded solutions of Maxwell'sequations. The main objective of paper was to derive an identity, based on thevector wave operator W= curl curl + (1/c 2 )82, that is satisfied by any time-dependentvector function, whose first and second partial derivatives are continuous through-out the region A interior to, and on a closed surface S.

The derivation used a time-dependent free-space retarded dyadic Green'sfunction F, where

wr=I(T)(v) (1.1)

and I is the unit dyadic, =t-t', vx'-x. The equations in this section arenumbered to correspond to those in paper 1.

The identity may be written (in equivalent ways (1.8a)-(1.8d))

fs F= A(x')F(x',t') - f(WF)' Dd 4 x+V' A(V F)gd 4x, (1.8b)

where g = 6(s)/47rv (s t t' + v/c) is the retarded Green's function of the scalar waveequation

[(1/C2)02 -V2]g= (t) (v),

where D=Ig and v is the magnitude of v. The symbol JsF has the definition

FV' fs FgdtdS-V'x f x FgdtdS- f x (Vx F)gdtdS. (1.9)~~~~~s s s

The quantities x', t'; x, t denote respectively the field and source points and times.The function 1A(X) is unity if x is in region A, and is zero elsewhere. If the integrationover t from - oo to + oo is carried out, the result is expressed by means of the usual

t Present address: Department of Physics, University of Florida, Gainesville, Florida32611, U.S.A.

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brackets [ ] indicating quantities evaluated at the retarded time, denoted asequation (9 a) in paper 1.

In the above definition and identity the operator V' has been commuted outside ofthe integral signal, which is possible if the surface is of finite extent. The advantage indoing this will be mentioned later.

In order to use this identity with Maxwell's equations for the field E, H in ahomogeneous isotropic medium where a charge-current density distribution p, Jexists, it is first necessary to eliminate, in turn H and E to obtain the inhomogeneousvector wave equations

WE= -0tJ (1.11 a)

WH=VxJ. (1.11 b)

When these are used in the identity to replace WE and WH in their respectiveequations, and when V E is replaced by p/e, the identities for E and H may be written

lA(X')E(x', t') = E (x', t') + E (1.14)s

1A(x')H(x ,t')= Hr(x',t')+ fix JgdtdS+ fH (1.15)s s

where

E= - V'4~'- 'A

HA =(1/#)V' x Ar,

and (/), A' are the retarded vector potentials due to sources in region A:

4r=(1/ ) { pgd4xA

Ar=#J f Jgd4x.A

The fields Er, H' could be written in these familiar forms in virtue of commutatingthe operator V' to the outside of the integral sign. Even if V' may not be so commuted,expressing the fields Er, HA in this form serves as a useful mnemonic (for V' may justbe commuted back where it belongs). These fields bear the subscript (instead of A)in paper 1, and were changed in paper 2.

Paper underlined the fact that although equations (1.14) and (1.15) aresolutions of their respective vector wave equations, they are not necessarily solutionsof Maxwell's equations. It also noted that the formulae of Baker and Copson [3], andof Jones [4], can be obtained directly from (1.14) and (1.15) and that these also are notnecessarily solutions either of Maxwell's equations or of the vector wave equations.The final conclusion was stated as a theorem (paper 1 , p. 743)-that the necessary andsufficient condition, for any of these formulations to constitute a solution ofMaxwell's equations in the interior region of a closed surface, is that the Maxwellequations are satisfied on the boundary surface.

Paper 2 showed, firstly, that even though (1.14) and (1.15) for E and H are asolution of Maxwell's equations (subject to the above mentioned theorem) for thedistribution of charge-current p, J in region A, the retarded fields E , H' are a

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Advanced solutions of Maxwell's equations

solution of Maxwell's equations in region A for a different charge-current distri-bution if the normal component of J does not vanish on S. That article examined indetail the curious situation where the retarded fields calculated from the retardedpotentials on p and J dto nol satisfy the Maxwell equations for p and J, yet the fieldsE and H of (1.14) and (1.15) do.

Secondly, paper 2 broadened the conditions to include the case where S may bean infinite surface which separates space into two regions but where the sources are offinite extent. This was based on an all too brief discussion of the asymptoticbehaviour of the retarded solution. One of the objectives of the present paper is toprovide an adequate discussion of this behaviour for both the retarded and advancedsolutions.

Thirdly, paper 2 (p. 102) showed that the so-called Huygens' principle, or better,the principle of continuation of solutions [5], which states that the surfaceintegrals of equations (1.14) and (1.15) for the electromagnetic field E, H give theeffect on one side due to the sources on the other side of S, is true only if there is nocurrent flowing across S. A theorem covering the case when there is a current acrossS was presented (paper 2, p. 102).

In the present paper, in addition to the results and objectives mentioned in theabstract, we extend the foregoing results to the advanced solutions.

2. Advanced solutionsWe may follow all the transformations for deriving the identity equations (8),

(8a) (8d) of paper [] but using the advanced (,reen's functions in place of theretarded ones. We will indicate the advanced function by means of the symbol placed above the corresponding retarded function. Thus, =t-t'-v/c, D=Ik,where g=J5(v)/47rv satisfies the same wave equation as g. The advanced dadicGreen's function r=+ C, where

= c2VV'(§l (s)/47rv) (1)

satisfies equation (1) of paper 1. A calculation performed on U similar to that at thetop of p. 737 of paper I ields

02C= C2VV', (2)

and it is only necessary to put the symbol on D andg from equations (6) to (9) of thatpaper.

Equation (9) was a definition of the notation appearing on its left-hand side whichwe shall now denote by affixing an 'r' at the top of the integral sign. Thus ( F denotesthe right-hand side of equation (9), paper I whilst .j F will denote the sameexpression with g replaced by g. Similarly, in equation (9 a) the superscript 'a' on [ ]"denotes the contained quantity has been evaluated at the advanced time t = t'+ v/c.Corresponding changes are made in equations (8 a)-(8 d).

In paper 1, § 4, the same changes are made, and in addition, we add a superscript 'a'on Bat, AA, and Ea in equation (13a). Thus equation (14): of paper would be

t Note the typographical error in equation (12 a) of paper 1 where it should read d3x in thelast integration.

I In paper I the equation following equation (14) should have appeared with the last twoterms interchanged to agree with the order in definition (9) and to make sense of the statementson p. 7 43 . Some typographical errors also appear on p. 742: Vg should replace g in the n Jterm of the third equation and, below, a brace should be inserted in the second line of 12.

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written

1A(x')E(x',t') = E(x', t')+ | E (3)S

and equation (15) would be written

l^(')H(x',t) =Ha(x,,t,)_ + ,jaIA(X )H(x t =HA(x, t')+ A fx JgdtdS+ H. (4)

All the other expressions are valid with the mentioned changes in notation. Thetheorem on p. 743 remains unchanged.

With reference to paper 2 [2]. if in addition to the changes in notation alreadymentioned, we affix a superscript 'a' on pa, pI, J and J then all of the results of thatpaper may be transcribed to the advanced solutions. Except for these changes ofnotation, the theorem on p. 102 of paper 2 remains the same.

However, we must examine the argument sketched at the bottom of p. 101 ofpaper 2 more closely. The assertion was that , Er, over any portion S' of a sphericalsurface centred on the origin, yields zero in the limit when the radius of the spheretends to infinity. The additional assertion now is that , Ea over any portion S' of aspherical surface centred on the origin, vanishes as the radius of the sphere tends toinfinity: in other words Ea , which has the asymptotic form of an incoming wave atinfinity, makes the advanced surface integral vanish asymptotically. To avoidcarrying along the advanced notation we give the details for the retarded case, andthen mention how the advanced case differs.

First we calculate an expression for V'g. We obtain

V'g = ['(s)/47rvc- (s)/47rv 2 ]V'v. (5)

Now

1/v 1/x + x' cos (')/x 2 x'2/2x 2 = 1/x + O(1/x 2 )

and

V' _ v/v x'/x- + O(1/x 2 ),

where *=x/x is a unit vector in the direction x, and O(1/x 2 ) denotes a quantity oforder /x2 . Since S' is centred on the origin = A.

Substituting in equation (5) we obtain

V'g - (s)/4icxc + O(1 /x2). (6)

Let u = t - x/c, then the asymptotic form of E (or Hr) will be

Er - f(u, fi)/x + O(1I/x2), (7)

where iA f = 0 as x - x, and = . The dependence of f on fi allows for the possibleangular dependence of E.

The first term of our surface integral fs E, i.e. A, A * Erg dt dS tends to zero asx- o, since E'g- A fo(s)/47Ex2 which tens to zero because n f- 0. We may writedS=x2 dQ where d is an element of solid angle.

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Advanced solution o Maxwell's equations

The second term of the surface integral will be asymptotic to

-V' x f x ErgdtdS= -V'x f x (f/x)gdt dSs' . s'

= I(f x f/x) x V'gdtdSs,

f x ( x f/x)0,(s)/47rxc dt dS

- X f x ( x auf/x) b(s)/47rxc dt dS (8)fSI

because the terms of O(1/x 2 ) in V'g vanish in the limit x-eoo, and where we havemade an integration by parts on t in the last step.

The terms of O(1/x 2 ) in Er vanish as x-o, so the third and last term min thesurface integral is

Ax (Vx E')gdtdS= - f x (Vxf/x)gdtdS.

But

V x (f/X)- -fi x f/xc+O(1/x 2).

Thus the third term is asymptotic to

fi x ( x f/x)5(s)/4rxc dt dS. (9)fS'

This cancels with the result for the second term. Consequently,

IET 0. (10)

In the case of the advanced solution we have a change in sign of the x/c term in u,and also in the argument of the delta-function in g. This change in sign of x/c in theargument of the delta-function changes the sign of the asymptotic form of the secondterm of the surface integral; whereas the change in sign of x/c in u changes the sign inthe asymptotic form of the third term. Thus

x f x /xgdtdS- + f x ( x aO./x)dtdS

and

- f Ax (V x Ea)gfdtdS= - f Ax(Ax /uf/x)gdtdS.S' S'

These cancel with each other, whilst thefirst term of the surface integral, as before, isasymptotically zero. Hence we have

as required for the proof of the theorem on p. 102 of paper 2, when S' is an infinitesurface.

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Note that, in paper 2, we considered the surface S, to be a mathematical (i.e. anon-material) surface. The E, H or Ea, H were the respective retarded andadvanced solutions of Maxwell's equations throughout all space for the entirecharge-current distribution assumed to be finite in extent. Now, if S is a finite closedsurface whose interior we denote by A and if E, H is any solution (for p, J) ofMaxwell's equations in the interior of and on the surface S, we know from theidentity (8) of paper 1, that

T E= lA(x')E(x', t)-E (x', t') (12)

and

T H = lA(x')H(x', t')- Hr(x', t')- f Ax JgdtdS. (13)S A f~~~~~~S

Similar expressions are valid if we replace the superscript 'r' by 'a', and'g'by' ,'.In particular, we replace E, H by Er, Hr or Ea, Ha (which are solutions of Maxwell's

equations for p; J inside S). On one hand, if we calculate the retarded surface integralof E we obtain equation (18) of paper 2. On the other hand, if we calculate theretarded surface integral of E' we may write

EA = 1 A(x')EA(X, t) - E(X, t')- E S(x, t) (14)fE s(

with

E(x', t') =-V' A (Ps/e)g d4 x-_ ,.Yf J sgd 4 x,A A

and where

P5s(x', t') = (1 /c2)01 f ' Jg dt dSS

Js(x', t') =- V' J Jg dt dS.J s

If the surface S is chosen where - J =0, then the last term in equation (14) isnon-existent, and we obtain a number of interesting relationships.

Define

EA =(1/2)(E - E) (15)

and

E = (1/2)(E,+ E). (16)

Then:

|E =2EA -E (17)

' Es =EA A (18)s~~~~~~~~~~~~(8

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{EA-=EA 0 (19)fs A; A-[

E = - E (20)JS

where the first column of values is for x' interior to A, and the second is for x' exteriorto A.

A similar set of equations may be written for the advanced surface integral.If we take for S a spherical surface centred on the origin, and if we substitute Ea

for E in equation (12), then it is evident that as the radius of the sphere tends toinfinity the retarded surface integral of the advanced solution does not vanish.Likewise, the advanced surface integral of the retarded solution does not vanish.

On the other hand, if we substitute Er into equation (12), and let the radius of Stend to infinity, the region A tends to infinity, so all the sources are interior to S, andE= E',, and the retarded surface integral over the entire sphere of the retardedsolution vanishes. Similarly for the advanced surface integral with the advancedsolution. These are, of course, not as strong as the proof we gave above which appliedto any portion of the spherical surface.

The important point we want to emphasize is that, if we are treating the retardedsurface integral of the retarded solution, and if the sources are of finite extent, then ifthe surface S consists of several finite portions, with some portions extending toinfinity (screens), only these need to be taken into account because of the asymptoticproperties proved above.

3. Some observationsFor monochromatic standing wave solutions in a dissipationless cavity [6, 7], in

the case where there is no current to the walls of the cavity, the solution may beexpressed in terms of E+, and the integral equation in terms of E-. We shall,however, not pursue the details of this.

The conditions of continuity mentioned in the introduction under which thebasic identity was derived may be relaxed to allow point charged particles(electrons), since we may always consider them mathematically as a limit of asmoothly spread out charge distribution (the Lienard-Wiechert potentials areusually derived in this way) where the identity is applicable. We may than apply ourresults to the classical motion of a single electron. If the entire trajectory of theelectron is contained within a closed surface S then equation (19) is valid for eitherthe retarded or advanced surface integral.

Dirac, in an article 'Classical theory of radiating electrons' [8], suggested that theradiation field produced by the electron is merely

-2Ek, -2H-.

But we have just noted that for any point outside a surface containing the radiatingelectron, the retarded (or advanced) surface integral of this radiation field yields zero,consequently for such a radiation field there would be no Huygens' principle [2, 5].

AcknowledgmentsI am pleased to thank Dr Real Tremblay, directeur du department de physique

de l'Universit& Laval and Dr Charles F. Hooper, Jr., Chairman of the Department of

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Physics of the University of Florida, for placing the facilities of their laboratories atmy disposal, and Professor Pierre Ramond for his helpful arrangements.

Je tiens a souligner ma reconnaissance pour le soutien qui m'a kt accoud. par leLaboratoire de Physique et Mol&culaire, en particulier son directeur le Dr MarcelBaril, de l'Universit6 Laval.

References[I] DARLING, B. T., 1983, Optica Acta, 30, 733.[2] DARLING, B. T., 1984, Optica Acta, 31, 97.[3] BAKER, B. B., and CopsoN, E. T., 1950, The Mathematical Theory of Huygens' Principle

(Oxford University Press).[4] JoNEs, D. S., 1964, The Theory of Electromagnetism (New York: Pergamon Press), p. 43.[5] LARMOR, J., 1903, Proc. London math. oc., 1, 1.[6] IMBEAU, J. A., DARLING, B. T., and AMIOT, P., 1983, J. Can. Physique, 61, 571.[7] I)ARLING, B. T., 1986, Optica Acta, 33, 1405.[8] DiRAc, P. A. M., 1938, Proc. R. Soc., Lond., A, 167, 148.

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