Electrodynamics of moving media - Minkowski covariant formulation - Radiation due to oscillating dipole in vacuum - Field of moving charge in bounded region and Cerenkov radiatio

Kalafus, R. M.

Pyati, V. P.

Tai, C. T.

English

NASA Technical Reports Server

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THE UNIVERSITY OF MICHIGAN 
COLLEGE OF ENGINEERING 
DEPARTMENT OF ~LECTRICAL ENGINEERING 
Radiotion laboratory 
Hard copy (HC) / /  0 ,am 1 Soveinlxr 19G5 
Microfiche (MF) 
ff 653 July 65 
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(NASA CR OR TMX OR AD NUMBER) 
435 
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93 
(CATEGORY1 
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TABLE OF CONTENTS 
INTRODUCTION 
INDEFIN'ITE AND DEFINITE FORMS OF MAXWELL - 
lMINKOWSKl EQUATIONS 
RADIATION DUE TO AN OSCILLATING DIPOLE OVER 
MEDIUM 
A LOSSLESS SEMI-INFNTE ILIOVING DIELECTXC 
FIELD OF A MOVING CHARGE IN A BOUKDED REGION 
CERENKOV PHENOMENA IN A WAVEGLTIDE 
PUBLICATIONS AND PAPERS PRESENTED AT 
CQrJFERENC ES 
FUTURE WORK 
REFERENCES 
2 
4 
7 
12 
13 
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I 
INTRODUCTION 
Several problems are currently being investigated under this grant. The 
details a re  described in Sections II, m and IV. A s  a result of this grant, the 
Radiation Laboratory of The University of Michigan is able to support, in a part- 
time basis, two graduate students working in this area. The work by V. Pyati on 
the radiation due to an oscillating dipole over a lossless semi-infinite moving 
dielectric medium has been supported in part by Grant NGR-23-005-107. This 
study provides the thesis material for M r .  Pytai's Ph. D dissertation and will soon 
be published as a technical report. The study on the field of a moving charge in a 
bounded region has yielded some interesting results which may have some close 
bearing on the understanding of tile excitation of Zereirirov Tadiiitioii. Tiie fwiidatiixi 
of electrodynamics of media is further consolidated by a thorough investigation of tlie 
EHPMv formulation based upon Born's equations. It is hoped that this will end the 
lack of comprehension that has resulted from considering the EHPMv fornmlation 
as a new and independent formulation of the electrodynamics of moving nzdia. 
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Il 
DEFINITE AND INDEFINITE FORhB OF MAXWELL-MINKOWSKl EQUATIONS 
Much work has been published recently about the different fornlulations of 
electrodynamics of moving media. The problem was practically resolved in a 
previous study (Tai, 1964) however much confusion remains as can be seen in the 
articles appearing in recent issues of the 'Proceedings of the Institute of Electrical 
and Electronics Engineers (Szablya, 1965; Uia ,  1965; Penfield and Szablya, 1065 and 
Tai and Szablya, 1965). In summarizing these articles, it appears tlxit some of the 
authors still do not appreciate Minkowski's great work, while others still think 
the EIIPMv foriiiulation is an  independent formulation. 
we have re-examined the definite and indefinite forms of Maxwell-Minkowski equations 
bwec! slpm f?~m's qat ions .  
The contents of the paper are outlined below. 
To clarify the inatter further 
Thc tmtitr~~cat is cmt&cd a pqxr  by Tsi (196%) . 
The relativistic transformation between the polarization aid map&ization 
veotors in two inertial systems is given by 
where 
wp=,=. ( E + i i x P )  
.1 = Y "  1 0 (l-p)-@ 0 Cl-$,-@ . 0 
l o  
p * v / c  . 
0 ' J  
The velocity of separation between the two inertial systems is assuined to be in the 
z-directioii. -rnvs eqimtions (Born, 1910) are obtained by solving (2.1) aucl (2.2) 
for 
system. 
and % and substituting them into the Maxwell equatioiis defined in tlie wipriiiicd 
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This yields 
If we define 
(2 .3)  
Then (2.1) and (2.2) can be written in the form 
a V x ( E  -p VxGc)=S + -/J (K +ac) 
c o  a t 0  c 
a Ox(E  + ; x i 5  ) = J + - ( E  E + P )  , 
C C a t o c  c 
(2.5) 
(2.7) 
(2.8) 
Eqwtioils (2.7) and (2.8) are identical to the two equations derived by Chu (Fano, Chu 
d Adler, 1960), using a kinematic method. The undesirable feature of titat deri- 
vation is 'that it postulates the so-called magnetic charge model. As we see here, 
such a fictitious model is not necessary. We can trace the origin of the field 
quantities in a precise manner from the relativistic formulation. Other indefinite 
forms of Maxwell's equations can be derived accordingly in a siiidar manner. This 
work sliows the precise relations between the field vectors in various foriiiulatioiis 
wing only Minkowski's covariant forixdation of electrodynamics of moving iiiedia 
within the framework of the special tlimry of relativity. 
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RADIATION DUE TO AN OSCILLATING DIPOLE OVER A LOSSLZSS 
SEm-INFINITE MOVLNG DIELECTL~IC MEDIUnl 
The solution to two boundary value problems in the electrodynamics of 
moving media is nearing coiqletion. In both problems, ,as shown in Fig. 3-1, 
the upper half space ( z > 0) is free space and the lower half space (z < 0 ) is 
filled by a lossless dielectric which is inoviug with a uniforiii velocity v in tlie 
y -dir ection. 
Y 
FIG 3-1: GEOMETRY OF THE PROBLEM 
The primary problem involves the determination of the radiation field due to 
an oscillating (IIertzian) dipole located in a vacuum. A secondary problem, the 
reflection and refraction of a plane electromagnetic wave incident on the moving 
dielectric, is considered as a preliminary to the more difficult problem above. The 
latter is a generalization of the work reported by Tai (1964) in the sense that the 
plane of incidence is not restricted to the yz-plane 
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Reflection-Xefraction Problem 
The Maxwell-Minkowski equations in the moving medium, assuming e-id 
variation, are given by 
where 
C 1 -n 
1-3 
l-n2p2 
a =  
(3.1) 
(3 .2)  
(3.3) 
(3.4) 
(3.5) 
Plane wave solutions of the above equations which represent the transmitted wave 
can be constructed thus: 
iK 6 E = ( E  E oxJ oy’ - 
where 
E E E are independent of the coordinates oxJ oy’ 02 
6 = x sin et cos f +y sin et sin #-z cos O~ 
(3.6) 
and K, the propagation constant satisfies the dispersion relation 
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K2sin20tcos2f+- 1 (KsinOtsing+012) 2 +$cos 2 Ot-a.k 2 = 0 . 
a 
The amplitudes must satisfy the relation 
1 KsinetcosftE + - (Ksinetsin$dttwL?)E -KcosOtE = 0. ox a OY oz 
The i i m p t i c  field is given by 
Alternatively, one may start with the magnetic field 
(3 .7)  
(3.10) 
and proceed in a similar manner. The rest of the sulalysis is quite straightfonvard 
a d  the salient results axe: 
1. Snell's law is modified, thus 
2. The reflected and the transmitted waves possess components not 
present in the incident wave, except when $d= ~112, and the phenomenon 
of Brewster's angle is absent. 
3. W~en f=n/2, the results are quite similar to the non-moving case. 
Dipole Problem 
(3. 11) 
Two cases corresponding to a vertical dipole or  a horizontal dipole (in the 
direction of velocity) are treated in detail. In each case, the problem is formulated 
in  terms of the vector and scalar potentials amropriate to each region shown in 
Fig. 3-1. Formal solution is obtained by applying the boundary conditions at the 
interface z = 0. Asymptotic expansion of the Fourier integrals is obtained by 
using the sadclle point metlid. The results will appear in a report now under pre- 
paration. 
Sonze allied problems for future research are; 1) magnetic dipofe over moviig 
mediwLl, 2) complementary problem in which the sources are locatcd in the iiioving 
medium, and 3) extensiom to lossy (conducting) media. 
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IV 
FIELD OF A MOVING CHARGE IN A BOUNDED REGION: 
CERXNKOV PIIENObIENA IN A iVAV!:GUIDE 
4. 1 Introduction 
The original analytical work on Cerenkov radiation (Frank and Tamm 1937, 
Tanim, 1939) treated the problem of a point charge moving in an infinite medium. 
Their niethd assumed a point charge moving with velocity vz^  with respect to tlie 
stationary medium, giving a current density of 
- 
J = q V 6 (XI 6(y) 6(~-vt) . 
Nag and Sayied (1956) approached the problem from a moving medium point of 
view, using the I\"linkowski theory. That is, they considered the problem of a 
statinnary pl2;?pge 1?7_gyinv mpdiiim, a i d  & ~ ~ c f ~ r i c &  t!ie p~gi_?lfi?lg fields. --D --------- - _--a- --- 
The problem discussed here is the case of a charge moving along the axis 
of a cylindrical medium bounded by a conducting shell. This differs from the 
case above in that there are reflections, and the radiation is enclosed. All three 
cases discussed above assume a point charge and constant velocity, which result 
in infinite energy density. The assumption of constant velocity is cle'wly rutreal 
since the radiation must result in a loss of kinetic energy of the charge, and there 
are no sources of energy available to maintain the velociw in the problem. A 
third difficulty is the assumption of constant permeability and permittivity: all  
real materials exhibit dispersion, a variation of these parameters with frequenoy. 
This assumption results in infinite energies, but the character of tlie fields 
remains intact, 
d 
The problem will be presented in the following manner. First, Tanim's 
method is used to derive an expression for the vector potential; then the moving 
medium approach is used to yield the same expression. 
the integrals and interpret the results. 
Finally, we solve the 
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4.2 Solution of tlx Vector Potential - Tanim's Metlid 
Here the problem is characterized as a point charge q moving dong the 
= -vz . The vector po@ntial in the ME3 systerii satisfies A z-axis with velocity 
the differential equatiun 
- 2p q v  S(X)d(y)6(Z+vt) 
n2 a 2 i  
c2 at2 
- -  -- 
where 
n 2 Z 1 - 1 .  E 
poco 
Casting this in cyli&rical coordinates, applying the bouic^.xy condition 
that n x E  = 0 at r=b, and applying the Fourier transform ethod, it can be 
where 
CY = r/b 
u =  %qT 
V 
z + v t  
7 =  
b J n w  
4.3 Solution for the Vector Potential - i'\loving Medium Method 
Here the charge is assumed stationary in a iiiediurn moving with velocity 
A - v = vz . Due to the constitutive relation 
where the quantities are defined in Section III, the scalar potential satisfies the 
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By Fourier analysis, this has the solution 
Tlie relation between the vector pteiltial in the primed system and the scalar 
potential in the unprimed (stationary) system is 
Using the coordinate transformation 
It cafl be shown that the expression for the vector potential is the same as for 
Section 4.2, where, 
The primed system here corresponds to the unprimed systeril in Section 4.2. 
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4.4 Evduation of the Integral and Interpretation 
The integral in the expression for the vector poteiitial c a i  be put in a n~orc 
tractable form by using the method of contow iiitcgation. If the propc;- contour 
is chosen, the integral along the 
illfinite quarter-circle will vanish, 
and 
0 kim 
f(u)du ? (residues) 
where the plus sign applies when the contour 
is closed in the upper quarter-circle, and the 
minus sign below. The Hankel functioiis have a Ixmcli poiiit at u=O, but there 
is no difficulty here. The Hankel functions have a logaritlmic belmvior near the 
origin, and the integral exists as the lower limit approaches zero. 
The presence of Jo(u) in the denominator poses a problem, since the Gcssel 
function has its zeros on the real axis. In order to obtain physically realistic 
results corresponding to the retarded potential, a small iiiiagklary part is iiitro- 
duced hito 11, negative for the first bracketed term in the integral, positive for 
the second. This has the effect of lowering or raising the coiitour below or above 
the real axis. Thus some contours will enclose these zeros - some will not. 
The integrals along the imaginary axis become Laplace transforms of the 
modified Bessel functions, which are known . 
The resultmg expression has three distinct regions, shown in the followiiig 
Fig. 4-1. In regions I and I[ the solution is the same a s  in the unbounded case. 
c y =  1 
D[I 
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A 
-/.l q v z  A(r, z, t)= 
4;7bd?JCi 
0 7 < a  
* 
where urn denotes the nith zero of Jo(u). Since J1(um)z N o ( ~ l l )  for all m, the 
last sum is approximately 
sin 7 u Jo(aum) 
--1 2 m 
111- A 
and since 
u r n a n (  m -  i )  , 4 
this is appraximately 
" h i s  can be seen to be periodic in T with a period of 8 . 
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V 
I'UTI,IJCATIONS AND PAPERS PRESENTED AT CONFERENCES 
Tai, C. T. "Comments on the Lorentz Force, 
Tai, C. T. , "Xadiation in Moving ILiedia, ' ' Presented at The Ohio State 
PROC. IEEE, 5 3 ,  1145 
(August 1965). 
University Syniposiwn on Antennas (July 1965). 
Tai, C. T . ,  "The Dyadic Green's Function for a Xoving Isotropic Xedium, ' '  
IEXE Trans. G A P ,  AP-13, 322-323 (March 1'365). 
Tai, C .  T. , "Huygen's Principle in a Moving Isotropic Iiomogeneous and Linear 
Xediuiu, ' I  Appl. Optics, 4,1347-1349 (October 1965). 
12 
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VI 
P U N S  FOR FUTURE XOi<K 
The constitutive relationship of a iiioving conductive medium will be 
investigated. We hope to resolve the question '%%ether o r  not a moving 
conductive medium is anisotropic'' . There is evidence that the previous 
discussions on this subject by Soiiimerfeld and Cullwick are not completely 
satisfactory. 
13 
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REFERENCES 
Born, AI.  (1310), "A Derivation of tine Fundamental Equatioiis for Electro- 
magnetic Processes in hloving Bodies froiii the Viewpoilit of 
Electron Theory, " X ! I a t l ~  Ann., 68, pp. 526-551 
Fano, R. M., I .  J. Chu, and R. B. Adler (1960), Electromagnetic Fields, Xnergy 
and Forces, J. Wiley and Sons, Inc. , New York. 
Frank, I. and I. Tamm (1937), "Coherent Visible Radiation of Fast Electrons 
Passing Through Matter, I '  C. R. Acad. Sci. USSR, l4, Xo. 3, pp. 103-114. 
Nag, B. D. and A. M. Sayied (1956), "Electrodynamics of Moving Media and the 
Theory of the Cerenkov Effect, 
Penfield, P and J F Szablya (19G5), ""lie Lorentz Force, 
Szablya, J. F. (1965), the Lorentz Force, 'I  Proc. IEEE, 53, p. 418. 
Tai, C. T. (1064), "A Study of Xlectrodynamics of Moving Media, 
Tai, C. T. (1965a), "Reflection and Refraction of a Plane Electromagnetic Wave 
Proc. Roy. Soc. , 235A, pp. 544-551. 
h o c .  IEEE, 53- p.  1134. 
h o c .  IELE, 52, 
pp. 685-689. 
at the Boundary of a Semi-infinite Moving Medium, ' I  Presented at 
Spring URSI Meeting, Washington, D. C. April. 
r\iIovhg Illedia, 
volunie to R. W. P. King. 
Tai, C. T. and J .  F. Szablya (1965), "Cominents on the Lorentz Force, ' I  J?roc. IEEE, 
53, p. 1145. 
T a m ,  I. (l939), "Radiation Emitted by Uniformly Moving Electram, I '  J. Phys., I, 
NO. 5-6, pp. 439-454 
Unz, H. (19G5), "On the Induction Laws in Electromagnetic Theory, 
Tai, C. T. (19G5b) "Definite and Indefu~te Forms of h13x\vcllfs Equations for 
Submi t ted  for i i ture publication in a coninicniorativc 
Proc. XELL, 
-J 53 p.546. 
14 


Electrodynamics of moving media First semiannual report, 1 May - 1 Nov. 1965

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