Kalafus, R. M.

Pyati, V. P.

Tai, C. T.

English

http://deepblue.lib.umich.edu/bitstream/2027.42/7867/5/bad2613.0001.001.pdfhttp://deepblue.lib.umich.edu/bitstream/2027.42/7867/4/bad2613.0001.001.tx

Pyati, Vittal P.

Kalafus, Rudolph Morton

Deep Blue Documents at the University of Michigan

Topics in electrodynamics of moving media : final report : 1 May 1965-1 May 1966

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T H E  U N I V E R S I T Y -  O F  M I C H I G A N  
7322-1-F 
Topice in Electrodynamics of Moving Media 
Final Report 
1 May 1965 - 1 May 1966 
BY 
C. T. Taf, V. Pyati and R. M.Kahfu8 
mt NGR-23-005-107 
Prepared for 
I -  
o 
I -  
* 
THE U N I V E R S I T Y  O F  M I C H I G A N  
7 322 -1 -F 
TABLE OF CONTENTS 
I 
II 
m 
Iv 
V 
VI 
VII 
vm 
INTRODUCTION 
RADIATION DUE TO AN OSCILLATING 
MOVING DIELECTRIC MEDIUM 
DEFINITE AND INDEFINITE FORMS OF 
MAXWELL -MINKOWSKI EQUATIONS 
ELECTRODYNAMICS OF MOVING, 
CONDUCTlNG MEDIA 
CERENKOV RADIATION TOPICS 
DRIFTING, MAGNETO-IONIC MEDIUM 
-FIRST ORDER THEORY 
DIPOLE OVER A LOSSLESS SEMI-INFMITE 
THE TIME-DEPENDENT DYADIC GREEN'S 
FUNCTION FOR A MOVING ISOTROPIC 
MEDIUM 
REFERENCES 
Page 
1 
2 
3 
4 
8 
14 
18 
21 
iii 
I .  
c 
T H E  U N I V E R S I T Y  O F  M I C H I G A N  
7322-1 -F 
I 
INTRODUCTION 
In preparing the final report, we have given some technical outline of those 
works which have not yet appeared in a technical report or been submitted for 
publication. Work performed on the project that has been published is indicated 
in the references. The latter includes a detailed study of the problem of a small 
antenna over a moving dielectric medium (Section 2), and a paper concerning the 
present views on the theory of electrodynamics of moving media (Section 3). In 
Section 4 are discussed some of the complexities introduced into the theory by 
extending it to conducting media. 
which involve the Cerenkov radiation effect. 
drifting, magneto-ionic plasma is extended to finding the characteristic indices of 
refraction for some configurations of plasma parameters. The last section deals 
with the formulation of the time-dependent Green's function for moving, lossless 
media. 
Several applications are presented in Section 5 
In Section 6, earlier work on a 
1 
T H E  U N I V E R S I T Y  O F  M I C H I G A N  
7322-1-F 
11 
RADIATION DUE TO AN OSCILLATING DIPOLE OVER A 
LOSSLESS SEMI-INFINITE MOVING DIELECTRIC MEDIUM 
A thesis bearing the above title was completed by Pyati (1966) duri 
period, and copies of this were distributed to NASA on March 25, 1966. 
paper describing this work was presented a t  the Washington URSI meeting 
April 19 - 21, 1966. It is expected that a written paper will soon be sub] 
publication in a technical journal. The main contribution resulting from 
is the exact determination of the effects of the motion on the radiation pat 
Hertzian antenna. 
2 
. 
T H E  U N I V E R S I T Y  O F  M I C H I G A N  
7322-1 -F 
rn 
DEFINITE AND INDEFINITE FORMS OF MAXWELL-MINKOWSKI EQUATIONS 
The basic equations which govern the field vectors in a moving medium are 
LOW well understood. 
me all equivalent. 
Minkowski. 
nedia was presented at the International Convention of the Institute of Electrical 
md Electronics Engineers (Tai, 1966). 
he Langley Research Center. 
recent controversies since the publication of the book by Fano, Chu and Adler (1960) 
These equations may be presented in various forms but they 
All of them can be derived from the original theory of 
A paper concerning the present views on electrodynamics of moving 
Twenty copies of this paper were sent to 
It is hoped that this work has clarified some of the 
- T H E  U N I V E R S I T Y  O F  M I C H I G A N  
7322-1 -F 
rv 
ELECTRODYNAMICS OF MOVING, CONDUCTING MEDIA 
4 . 1  Ohm's Law 
Upon the introduction of finite conductivity into the study of isotropic moving 
media, one is immediately beset by the existence of two apparently different forms 
of Ohm's law: 
1 
= ya ( E + + X X )  (Weyl, 1922) 
C 
and 
where 
1 
t2' = (T ? 7 (E + B x B) (Sommerfeld, 1952, and Cullwick, 1959) 
C Y 
7 is the conduction current density 
a the rest-frame conductivity of the medium 
C 
1 
V = v$ is the velocity of the medium 
P = v/c 
( 2) Schlomka (1950) claimed that Jc is correct on the basis of an electron- 
theoretic model. 
to the relative velocity of two bodies as seen by a third observer. 
H i s  crucial step, however, involves attaching a physical meaning 
This is dubious 
4 
c 
T H E  U N I V E R S I T Y  O F  M I C H I G A N  I 
showed that the decomposition of current density into conduction and convection 
terms is not unique. 
volume per unit time, 
An expression was derived for the Joule heat loss per unit 
that is independent of the decomposition, using elementary relativistic thermoa 
dynamics. 
decomposition may be used, and the choice of Ohm's law is immaterial. 
4 .2  Characterization of Charge Densities 
Since both field and energy terms are uniquely specified, either 
In extending the work of Tai (1965a) on point sources in isotropic, moving 
media to conducting media, the characterization of charge density requires careful 
interpretation. Denoting quantities measured in the rest system of the medium by 
primes, the total current density J is made up of convection and conduction terms 
in general: 
2 
1 1 1  1 1  -
J = p  E +<T 
S 
1 1 
where p is the density of the charges introduced into the medium with velocity Ti . 
The quantity ps cannot be considered to be the total charge density, for this leads 
to inconsistencies; there must be an additional partial charge density caused by the 
presence of the source charge density ps . 
viewed in the stationary (unprimed) system has two convection currents, one due 
S ? 
? 
Furthermore, the current density 
to the motion of the source charge i p s  3 , and one due to the motion of the medium 
? 
(YP, 7) . 
L 5 
T H E  U N I V E R S I T Y  O F  M I C H I G A N  
7322-1-F 
1 1 ? ? 
where p = p - p s  , and p is the total charge density. 
the introduction of source charges has no effect on Ohm's law, as would be 
required. 
special case where p 
With this interpretation, r 
1 
The charge density y p is determined completely by p s  , and in  the r 
is a point charge 
S 
then 
? 
" p r  
The scalar potential 9 for the time-independent case (measured in the stationary 
system) must satisfy the following differential equation: 
where 
and 
2 n = p  E / p  E 
0 0  
1 1 
E andp are the permittivity and permeability, measured in the 
rest-frame of the medium. 
6 
I -  
* 
. 
T H E  U N I V E R S I T Y  O F  M I C H I G A N  
7322-1-F 
A partial solution has been found in closed form for the scalar potential, that 
s, a solution found by ignoring the second term on the right of the equation. By 
?ourier transform analysis, the partial scalar potential function 9 can be shown 1 
0 be: 
where 
1 1  
and it is assumed that n B > 1 . 
f i l  exhibits an exponential decay for large z . 
the radiation properties are determined by the vector potential A . 
can be shown to depend only on the partial solution P1 , so that it is unnecessary to 
The surface described by I a1 z = r is a cone, and 
In the rest system of the medium, 
This in turn 
1 
evaluate the total scalar potential 9 . 
section 5.3 . 
The results are identical with those of 
7 
T H E  U N I V E R S I T Y  O F  M I C H I G A N  
'7322-1-F 
V 
CERENKOV RADIATION TOPICS 
5.1 Unbounded, Lossless Media 
The well-known Cerenkov effect, arising from charged particles travelling 
with velocities exceeding the speed of light in the medium, has many interesting 
properties. 
energy densities in the calculation of the fields, while the assumption of a 
dispersionless medium results in infinite energy. 
wave has the form of a cone (see Fig. 1); fields are zero ahead of the cone, and 
behind the cone behave as 
The assumption of a point charge as a source results in infinite 
For a point charge, the shock 
in the rest  system of the medium, or  
1 
Jlal- 
in the rest system of the charge. 
For a line charge moving perpendicular to its axis with velocity v , while 
the wedge angle Q is the same as before (see Fig. l), there is no wake. 
the fields are confined to a small region around the shock wave, and vanish in  
front and behind. 
media. 
That is, 
This was worked out for the case of lossless, non-dispersive 
a 
- 
. 
~ 
I 
. 
- 
In front: 
THE U N I V E R S I T Y  O F  M I C H I G A N  
7322 -1 -F 
Shock wave travels with velocity 
8 / c/n, where 
C - < v  n 
Charge travels 
with velocity v 
X 
sin (Y = l /nP A 
r = O  - - CI 
6 
FIG. 1: CERENKOV SHOCK WAVE GEOMETRY 
9 
T H E  U N I V E R S I T Y  O F  M I C H I G A N  
7322-1-F 
5.2 Bounded Media 
The introduction of perfectly conducting boundaries causes some interesting 
effects. 
spectrum discrete, instead of continuous. 
boundary is moving or stationary if it is perfectly conducting. 
An observer in the rest system of the medium finds the radiation 
It is immaterial whether or not the 
For the case of a point charge moving along the axis of a perfectly con- 
ducting cylinder of radius "b" having an appropriate medium (n /3 > I), the solution 
has three regions, indicated in Fig. 2 .  In regions I and II, the solution is the Sam( 
as for the unbounded case; in region III the fields have the behavior 
4 7  where r = + y , and j is the m-th zero of J (x), the Bessel function. 
Work on the energy distribution was found to have been discussed in detail by 
Russian authors (Bolotovskii, 1961). 
m 0 
For the corresponding two-dimensional case, that of a line charge moving 
between two parallel conducting plates, the fields are still restricted to the shock 
wave front which forms the zig-zag pattern shown in Fig. 3. 
spectrum seen by an observer in the rest-frame of the medium is again discrete, 
the frequencies excited depending on the eigen-values of the parallel-plate geometrq 
The frequency 
10 
. 
T H E  U N I V E R S I T Y  O F  M I C H I G A N  
7322-1-F 
/ / / I / / / / / / / /  
r = b  
III Waveguide Mode Region 
r = O  - 
I / / / / / / / / / / / / / / / / / / / / / / / /  
cr=m 
FIG. 2: SHOCK WAVE GEOMETRY, CYLINDRICAL BOUNDARY 
FIG. 3: SHOCK WAVE GEOMETRY, PARALLEL PLATE BOUNDARY 
11 
T H E  U N I V E R S I T Y  O F  M I C H I G A N  
7322-1-F 
5.3 Unbounded Conducting Media 
The case of a point charge moving in a conducting medium was considered 
As in the program, tying in with the general study of conductive, moving media. 
one would expect, the introduction of conductivity in the medium does not change 
the shape of the shock wave. The fields ahead of the shock wave are zero, and 
behind are attenuated in comparison with the lossless case. 
the vector potential becomes: 
The behavior of 
1 1  
a p v  where 
1 
Note that when the conductivity o vanishes, p vanishes, and the behavior becomes 
that of the lossless caae above. 
behavior to be expected from the introduction of losses. 
The exponentially decaying factor is the type of 
5 .4  Dispersion 
When dispersion is considered, a convenient quantity to utilize is the amount 
In symbols, we 
1 
of radiated energy passing through a small cylinder of length dz . 
are  interested in the quantity 
1 
1 - -  dw - 2 n r  I m S 1  dt( 
r 1 dz 
12 
. 
T H E  U N I V E R S I T Y  O F  M I C H I G A N  
7322-1-F 
t 
where S is the Poynting vector component in the radial direction, from the point 
of view of the medium rest system. The expression for the lossless case is, in 
M. K. S. units (cf. Tamm (1939) ) 
r 
1 1 
1 2  
8n 
d w q  
n@> 1 dz 
where the integration is over only those (positive) frequencies for which n @ > 1 . 
For conducting media one would not expect this quantity to be independent of radius , 
because of losses. Indeed, for frequencies high enough that w > > 
, the integrand is modified only by the exponential factor o(n2  8 and w > >  1 
E 
The modifying factor is still tractable when the second condition on w is removed, 
but there has not yet been found a convenient factor good for all frequencies. 
T H E  U N I V E R S I T Y  O F  M I C H I G A N  
7322-1-F 
VI 
DRIFTING, MAGNETO-IONIC MEDIUM - FIRST ORDER THEORY 
Tai (196513) applied Minkowski's theory of moving media to the case of a 
drifting, magneto-ionic plasma, where the drift velocity v is small compared to 
the velocity of light in vacuum c. This involves an eighth-order set of equations 
for the index of refraction n, given by 
det = 0 . 
where the components of x are the following: 
2 
A xx = ( n  - l ) ( l - n P Z - j Z ) + ( l - n P ) X  z 
A = - j ( n  - l ) Y  
A = ( l - n P  - j Z ) n P x - j n P  Y - j ( l - n P Z ) Y Y  
A 
A = ( n  - l ) ( l - n P  - j Z ) + ( l - n P ) x  
A = ( l - n P  - j Z ) n P Z + j n P  Y + j ( l - n P ) Y  
A = - j ( n 2 - 1 ) y  + n ~  X x 
A = j ( n 2 - 1 ) Y  + n ~  x 
A = - ( l - n P ) ( l - n P Z - j Z ) - j n P x Y  + j n p  Y + X  
2 
Xy z 
XZ z Y Z  
z 
= j (n2 - 1) Y 
YX 
2 
YY z z 
YZ z x z  z x  
zx Y 
ZY X Y 
zz z Y Y X  
14 
. 
T H E  U N I V E R S I T Y  O F  
7322-1-F 
M I C H I G A N  
where 
V = drift velocity of the medium 
n 
w = plasma frequency 
P 
W 
W 
C z = -  
w = collision frequency 
C 
m w  W e 
w = gyromagnetic frequency . m 
For the stationary magneto-ionic plasma, there are four possible values of the 
index of refraction: two llordinary” waves and two llextraordinary” waves. When 
the medium moves as well, there are in general eight values, or modes, possible. 
Making use of the assumption that v < < c a the index of refraction was 
found for a variety of orientations of magnetic field, drift velocity, and direction 
of propagation. These are presented below for a wave of the €orm 
n 
j w  (t --z) 
C e 
15 
T H E  U N I V E R S I T Y  O F  M I C H I G A N  
7322-1-F 
1. 
2. 
Propagation, drift, magnetic field parallel: P = YG, 8= P; 
X (Y - j Z) 
1 + Y - j Z  2 
( 1 + Y - j  Z) 1 , 2  -- 
n 
X (Y+ j Z) 
3 , 4 - -  1 - Y - j z  2 n 
(1 - Y - j Z) 
Drift and magnetic field parallel, transverse to propagation: 
x ( 1 -  j z - x) \ 
2 Y - ( 1  - j  Z) (1  - j Z - X )  
n 
(no change from the stationary medium case) 
16 
T H E  U N I V E R S I T Y  O F  M I C H I G A N  
7322-1-F 
3. Propagation, drift, and magnetic field mutually orthogonal: 
H = Y S  , E = S $  
The effect of drift is a shift of the order of v /  c in the ordinary and extraordinary 
wave indices, and the appearance of totally new modes. 
correspond to very slow waves having velocities on the order of the drift velocity. 
These are rapidly attenuated for most frequencies and plasma parameters 
encountered. 
of a boundary-value problem. 
pessimistic results on this matter for the all-parallel case. 
These new modes 
The extent to which these modes can be excited would be the subject 
M. Epstein (1962) reported some initially 
T H E  U N I V E R S I T Y  O F  M I C H I G A N  
7322-1-F 
THE TIME -DEPENDENT DYADIC GREEN'S FUNCTION 
FOR A MOVING ISOTROPIC MEDIUM 
The dyadic Green's function for a harmonically oscillating field in a moving 
isotropic medium was  previously derived by means of an operational method 
(Tai, 1965a). 
for a particular frequency w : 
There it was shown that the associated scalar Green's function is, 
Case I: n S <  1 
where 2 2 112 R = ( p  + a t )  a 
p 2 = ( x - x J  2 + ( Y - Y J  2 
5 = z - z  
0 
S 
2 2  c 
2 -2 n -1) 1 - 8  , ( 
( I - n  B )  2 2  
a =  
I - n  B 
' I  
v is the velocity of the medium 
18 
. 
T H E  U N I V E R S I T Y  O F  M I C H I G A N  
7322-1-F 
Case 2: n 8 > 1 (Cerenkov condition) 
E .e p r 1 
47r Ra J 
where 
The time-dependent case has recently been treated by Compton (1966). 
There he derived the following expression for Gt "2 : 
where 
and 7 = t - t  . 
0 
19 
T H E  U N I V E R S I T Y  O F  M I C H I G A N  
7322-1-F 
It can be shown that the complicated analysis involved in Compton's work can 
be avoided by taking the Fourier integral of the harmonic solution g (n I R )) 
with respect to the angular frequency w . 
will be submitted for publication in the near future. 
( w  0 
A note comparing these two derivations 
20 
. 
. 
T H E  U N I V E R S I T Y  O F  M I C H I G A N  
7322-1 -F 
REFERENCES 
Bolotovskii, B. M., (March-April 19621, "Theory of Cerenkov Radiation (m), 
Sov. Phys. Uspek., 4 No. 5, pp. 781-811. 1 
Compton, R. T., Jr., (1966), "The Time-dependent Dyadic Green's Function for 
a Moving Medium, " presented at URSI meeting, Washington, D. C. 
April 19-21, 1966. 
Cullwick, E. G. (1959), Electromagnetism and Relativity, (Longmans) p. 92. 
Epstein, M., (Sept. 19621, "Effect of Fluid Motion on Electromagnetic Wave 
Propagation", IRE Trans., AP-10, No. 5, pp. 645-646. -
Fano, R. M., Chu, L. J., and Adler, R. B. (1960), Electromagnetic Fields, Energy 
and Forces, (Wiley, N.Y.), pp. 376-503. 
Pyati, V. (Feb. 1966) "Radiation Due to an Oscillating Dipole Over a Lossless 
Semi-Infinite Moving Dielectric Medium, 
Laboratory Report No. 7322-2-T. 
University of Michigan Radiation 
Schlomka, T. (1950), W a s  Ohmsche Gesetz bei Bewegten Koerpern," A d e n  der  
Physik, 443-444, NO. 8 -9, pp. 246-252. 
Sommerfeld, A. (1952), Electrodynamics (Academic Press Inc., New York, N. Y.) 
p. 282. 
Tai, C. T. (March 1965a), "The Dyadic Green's Function for a Moving Isotropic 
Medium," IEEE Trans. AP-13, No. 2, pp. 322-323. 
Tai, C. T. (March 1965b), "Electrodynamics of Moving Anisotropic Media: The 
First-Order Theory," Rad. Sci. J. of Res. NBS, % No. 3. 
Tai, C. T. (1966), "Present Views on Electrodynamics of Moving Media", 
presented at the 1966 International Convention of IEEE, to be published 
in the convention records. 
21 
T H E  U N I V E R S I T Y  O F  M I C H I G A N  
7322-1-F 
Tamm, I. , (1939), "Radiation Emitted by Uniformly Moving Electrons," 
J. Of Phys. - 1, NO. 5-6, pp. 439-454. 
Weyl, H. (1922), Space-Time-Matter, (Dover Press), p. 195. 
22 
t 
8 
3 
_/ ' . (. 
- , ,. 
9 HE L'Ni'YrERSITY O F  MICHIGAN 
THE RADIATION LABORATORY 
A N N  ARBOR AREA CODE 313 
761-0500 
31 August 1966 
DEPARTMENT OF ELECTRICAL ENGINEERING 
2 0 1  C A T H E R I N E  STREET 
A N N  A R B O R .  MICHIGAN 
a 
MEMO TO Recipients of "Topics in Electrodynarm 'cs of Moving 
Media, The UniVeratty of Michigan Radiatian 
Laborabbry Report 7322-1-F (NGR-23-005-107) 
Dated May 1966 - Final Report - Page 17 
FROM Professor Chen To 28t 
SUBJECT Addendum to Report 7322-1-F 
A question has been raised concerning the appearance of new 
propagation modes procfuced by the motion of the plasma, pointing ouk 
that the Lomtz transformation of modes in a etatfcmary medium to a 
moving medium should not result in new ones. using ths concept of 
p W e  invariance, i. e. treating 6, j+) zm a *vector, where 
- w -  k =  -11 
C 
in any system of reference, it can be shown that to the first order in f!, 
t€ie index of refraction transforms in a one-to-one faahion as follow= 
This casts doubt on the physical significaxme of the apparently new modes, 
and suggests that the first-order a,ssumptioxuii may give misleading results. 
This problem needs further consideration. 
REH/C~W 
, 


Topics in electrodynamics of moving media Final report, 1 May 1965 - 1 May 1966

Topics in electrodynamics of moving media Final report, 1 May 1965 - 1 May 1966

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