ELECTROMAGNETIC FIELD IN DISPERSIVE MEDIA G. P. Novikova and V. L. Kon'kov

UDC 5 3 8 . 3

As known, the electromagnetic field created by moving charges is described by the Maxwell equations rotH=-rot E =

1 c)D 4r~ --+--,o~; c c)t c

--

divB----0;

(i)

1 0 Ba--~; div D = 4=p, C-

where p and v are the density and velocity vector of the electric charges, which we shall consider as given functions of the coordinates and time. In the case of nonmagnetic media the relation A

(2)

o=~.e

is added to equations (i)t the dielectric permittivity ~ in dispersive media having an operator meaning. It is commonly assumed that if the vectors D and ~ are expanded into a Fourier integral with respect to time, relation (2) assumes the form D~ = E(~)~, where E(~) is the value of the dielectric permittivity for an electromagnetic wave of frequency m; D m and Rm are the Fourier components of the vectors D and E. This statement is equivalent to the assumption that vector D is related to vector E by the relation [I] t

O ( r , t) = E ( r , t) + j f ( t - - : ) E ( r ,

z) d:

--03

and r

is determined by the equality e(~) = i + f(~).

Introducing the electromagnetic potentials ~ and A~ and expanding them as functions of the coordinates into triple Fourier integrals, for example,

j

OO

% (r) -

1

(2=)a/2 % (K) e --OO

.~r

dr,

from the equations we find for the scalar potential + (r, t) = ~

tlr, at'

P (r', t) R

--0o

1$ f~CO

J ~ (,,,)

--oo

t~

T

1/2 co i~(U--l) ( )e ,

R

=

r-- f'.

(3)

If we assume that the dielectric permittivity r does not depend on the frequency ~, we find at once that the frequency integral J is given by the half-sum of the retarded and advanced 6-functions: jr_

~

t--t'--

R .I!2 .~_~ t - - t ' - F C

) (

R ml]2 r

)I

From inherent causality considerations, the advanced solution is usually discarded and only the retarded potential is retained:

Gorki Physicotechnical Scientific-Research Institute, Lobachevskii Gorki State University. Translated from Izvestiya Vysshikh Uchehnykh Zavedenii, Fizika, No. 4, pp. 113-115, April, 1979. Original article submitted April 6, 1978. 0038-5697/79/2204-0441507.50

9 1979 Plenum Publishing Corporation

441

(

I

p r', t-- --R~lt2 (r. t) = -~ ~ d r ' R c 1

)

In the presence of dispersion, i.e., when e depends on ~, it is no longer possible in (3) to separate out the retarded potential in Chls form. From the physical point of view, this is associated with the fact that in a dispersive medium each harmonic has its retardation time and the frequency integral in this case is not a 6-function. Taking into consideration that E is an even function ofw and retaining exp (i~/c)Rc ~/2(~), which in the absence of dispersion corresponds to the retarded potential, the frequency integral J in the case when the ~(~) function has the "stepped" form 9=

{~;

mx.<.o~<~,~

~; out,de the Inte~a!

(4)

indicated

becomes 1 J = --(sin e~ qt

w h e r e qa a n d qa a r e tegral

given

1 ~lqt--sillm2ql)+--(sin~2q ~: q2

by qi " t'

The given example clearly is not a 8-function.

1 ~ - - sin ~ q:) + - - ~ ( q l ) , cx

-- t + ( R / c ) c ~ / a i ,

shows that

i - l,

in the case

2.

when r d e p e n d s

on u t h e

frequency

in-

Let us, however, expand, in (3), ~(~, R) - (i/E(m)) cos (~R/c)E1/a(~) as a function of frequency into a Fourier integral with respect to the varlable q. The frequency integral J is then represented in the form r

dq e~q (R) [~ (t' - - t + q) + ~ if' - t -- q)l,

J ---

(5)

0

w h e r e @q(R) is the Fourier component of @(~, R), whereby, e~q(R)=~

t--

C

R

in the absence of dispersion,

.

If we substitute this expression for J in (3) the first term in the square brackets will represent the retarded, while the second the advanced, field. However, in the general case, the retardation and advancement time q varies from 0 to | In particular, the value q = 0 corresponds practically to the instantaneous propagation of the field, i.e., to action at a distance. In order to clarify the meaning of the latter, let us consider the example when p = Po(r) cos wot. In this case we have (r. 0 = ~

1 r po(r') ~ R ~1~ (~) J d r ' - ~ - - cos c cos ~o t.

Hence it follows that the field at the time t is determined by the source at that same moment of time which, too, corresponds to action at a distance. It is necessary to underline that we arrive at the same results if in (5) we take the half-sum of the retarded and advanced 6-functions. The value q - m, on the contrary, corresponds to the case when the retardation and advancement time of the field is infinite. It is possible that this is associated with the assumption on the homogeneity and infinity of the medium. It would appear that if in (5) we retain only the first term, it is necessary to take the integral with respect to q within the boundaries from qmin = R/vmax to qmax " R/vmin, correspondigw to the retarded harmonics with the phase velocities Vma x = c/E~/amin and Vmln = c/g ~! m ~ . However, this kind of potential does not satisfy the initial equations.

442

Consequently, in dispersive media it is impossible to separate out the retarded potential by the conventional Fourier method. This fact leads to some difficulties. Thus, for instance, it is kno~1 that in the theory of the Vavilov--Cerenkov effect the inequality BE ~/2 > 1 (fl is the ratio of the particle velocity to the velocity of light) is an important condition. This condition is usually fulfilled in some finite spectral interval m: < ~ < m2 in which e depends weakly on frequency. And if, in accordance with this, we assume that the e(~) dependence has the form (4), the field created by the moving "superlight" charge is given by [2]

where r is the polar radius and y2 = B2e2 _ i. At large distances from the z axis, the cylindrical functions can be replaced by their asymptotic values and as a result we obtain F~ (r,t)=---

d . sin ~ u 2 - - T ) ;

~-

v

Thus, the field of a "superlight" charge is expressed in terms of known Fresnel functions [3] and it is nonvanishing in the whole space [4]. Naturally there arises the problem: How does the field whose phase velocity in the chosen spectral interval is less than the velocity of the moving charge outstrip the latter? It should alw be mentioned that if the E(m) dependence has the previous form (4) but the condition Be ~/2 > i is fulfilled in the whole spectral interval, in this case, too, the field of the "superlight" electron is nonvanishing in the whole space. This is evidently associated with the fact that in the presence of dispersion the potential being considered is not retarded. Possible ways of overcoming these difficulties will be discussed by us elsewhere. LITERATURE CITED .

2. .

4.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media, Pergamon. D. D. Ivanenko and A. A. Sokolov, Classical Field Theory [in Russian], GITTL, MoscowLeningrad (1951). E. Jahnke and F. Emde, Tables of Functions with Formulae and Curves, Dover (1945). S. Datta Majumdar and G. Pu Sastry, Z. Naturforsch., ~, 28a (1973).

443