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THEUNIVERSITY OF MICHIGAN COLLEGE OF ENGINEERING DEPARTMENT OF ELECTRICAL ENGINEERING Radiation L abototory
Rudiution Due to an Oscillating Dipole Over u Lossless Semi-lnfinite Moving Dielectric Medium
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VITTA1 P. PYATI ,
r- r[. s
February 1966 -
Grant NGR-23-005-
National Aeronautics and Space Administration Langley Research Center Langley Station Hampton, Virginia 23365
Contract M t h :
Administered through:
O F F I C E O F R E S E A R C H A D M I N I S T R A T I O N A N N A R B O R
b r'^ T H E^ U N I V E R S I T Y7322-2-T^ O F^ , ~ I C H I G A N
RADIATION DUE TO AN OSCILLATING DIPOLE
OVER A^ LOSSLESS SEMI-INFINITE MOVING DIELECTRIC MEDIUM
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. (^) by Vittal P. Pyati
February 1966
Report No. 7322-2-T on
Grant NGR -23-005-
Prepared for NATIONAL AERONAUTICS AND SPACE ADMINISTRATION NASA-LANGLEY RESEARCH CENTER LANGLEY STATION HAMPTON, VIRGINIA
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T H E U N I V E R S I T Y d~ M I C H I G A N 7322-2-T
A CKNOWLEDGE ME NT
The author is indebted to Professor Chen-To Tai for suggesting the problem and invaluable guidance and encouragement throughout the inves - tigation. Thanks are also due to the members of the Thesis committee for their helpful suggestions. A portion of this investigation w a s conducted while the author was with the Radio Astronomy Observatory of The Univer- sity of Michigan (NsG-572) and he is grateful to the Director, Professor Fred T. Haddock, for his support. The author wishes to thank Professor
Ralph E. HkU, H e a d of the Radiation Laboratory for his encouragement.
In addition, he wiehes to eqxess his sincere appreciation to Mrs. Claire
F. White and M i s s Madelyn L. Hudkins for typing the manuscript.
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TABLE OF CONTENTS
FOREWORD
A CKNQWLEDGEMENT LIST OF ILLUSTRATmNS CHAPTER I I1 ELECTRODYNAMICS O F MOVING MEDIA
INTRODUCTION AND STATEMENT OF THE PROBLEM
2.1 The Lorentz Transformation 2.2 Maxwell-Minkowski Equations
2. 3 The Method of Potentials for Moving Media III REFLECTION AND REFRACTION OF A PLANE ELECTRO- MAGNETIC WAVE AT THE BOUNDARY OF A MOVING DIELECTRIC MEDIUM 3.1 Geometry of the Problem 3.2 Plane Waves in Moving Media 3. 3 The Modified Snell's Law 3.4 Electric Field Perpendicular to the Plane of Incidence 3.5 Electric Field Parallel to the Plane of Incidence 3.6 Perpendicular Incidence 3.7 Summary OSCILLATING DIPOLE OVER A MOVING DIELECTRIC MEDIUM 4.1 Introduction 4.2 Vertical Dipole
IV
4.2.1 Fourier Integral Method 4.2.2 Method of Weyl 4.2.3 Approximation of the Integrals; Asymptotic Forms 4.2.4 Numerical Results
4. 3.1 Fourier Integral Method 4.3.2 Approximation of the Integrals; Asymptotic Forms 4.3.3 Numerical Results
4.3 Horizontal Dipole in the Direction of the Velocity
V CONCLUSIONS
REFERENCES
APPENDIX A: POINT CHARGE IN MOVING MEDIA;
CERENKOV RADIATION
ABSTRACT
Page ii iii V 1 3 3 4 6 9 9 9
40 40 40 40 49 55 77 84 84 87 93 101 103
105 108
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LIST OF ILLUSTRATIONS (CONT'D)
Fig. No. Page
IN THE AIR IN THE Y Z PLANE FOR A VERTICAL DIPOLE R n = 2 , h=O.5X 80
- pd IN THE XZ PLANE FOR A VERTICAL DIPOLE FOR n = 0.5, h = O 81 IN THE AIR IN THE XZ PLANE FOR A VERTICAL DIPOLE n = 0.5 , h = 0.25X^82
- I^ I N^ THE AIR^ IN^ THE^ Y Z^ PLANE FOR A VERTICAL DIPOLE F$R n=O.5, h=O.25X 83 IN THE XZ PLANE FOR A HORIZONTAL DIPOLE FOR n = 2, 94 IN THE AIR IN THE YZ PLANE FOR A HORIZONTAL DIPOLE R n = 2 , h=O.25X 95
- IIN THE AIR IN THE Y Z PLANE FOR A X~ORIZONTALDIPOLE F $ R n = 2 , h=O.5X 96
25. E^ IN^ THE XZ PLANE FOR^ A^ HORIZONTAL DIPOLE FOR^ n^ =O.^ 5,
b A0 97
- b61 IN THE YZ PLANE FOR A HORIZONTAL DIPOLE FOR n =0.5, h = O 98 IN THE AIR I N THE XZ PLANE FOR A HORIZONTAL DIPOLE R n = 0.5, h=O.25X 99 IN THE AIR IN THE YZ PLANE FOR A HORIZONTAL DIPOLE
n = 0.5 , h = 0.25X 100
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CHAPTER I INTRODUCTION AND STATEMENT OF THE PROBLEM
In the last decade, interest in the study of the electrodynamics of moving
media has increased considerably. Based on Minkowski's theory, Nag and
Sayiedl have presented an alternate derivation of Frank and Tamm's2 formula
for Cerenkov radiation. Boundary value problems involving stationary charges
and one or more moving dielectric media have been considered by Sayied3,
Zelby and others. While Frank5 has analyzed the problem of an oscillating
dipole in miform motion, the complementary problem, in which the medium is
in uniform motion and the source and the observer at rest, has been indepen-
dently solved by Tai^6^ and^ Lee^ and^ Papas^7^.^ The present work^ is^ concerned with the following boundary value problem.
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- Radiation due to an oscillating (Hertzian) dipole over a iossiess semi-infinite moving dielectric medium. Here lossless means zero conductivity and the dipole source is assumed to be
located in free space or vacuum which is stationary w i t h respect to an observer
in whose frame of reference all the fields will be determined. The problem may be regarded as an extension of Sommerfeld's8 dipole problem to moving media. The object of this study is, first to develop techniques of formulation of boundary value problems in moving media, and then to apply these techniques to the above problem to ascertain the extent to which the radiation patterns are
modified due to the motion of the dielectric medium.
It may be recalled that Weyl' developed a method by which Sommerfeld's solution for a dipole over a flat earth could be interpreted as a b-e of plane waves reflected and refracted by the earth at various angles of incidence. In
order to give such a physical interpretation to the present problem, it is neces-
s a r y that we extend F r d l ' s results to moving me'dia, nameky:
- Reflection and refraction of a plane electromagnetic wave at the boundary of a moving dielectric medium.
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CHAPTER II
ELECTRODYNAMICS OF MOVING MEDIA
- 1 The Imentz Transformation
Consider two coordinate systems as shown in Fig. 1, in which the y and y
axes coincide and the system S' is mbving tKith a uniform v d o d t y v in the y-direction
with respect to S. For the case when the two origins 0 and 0 ' coincide at the
Z'
S'
V
Y
V
! L
Z'
Y'
FIG. 1: THE COORDINATE SYSTEMS
instant t = t' = 0, the equations of transformation of the space-time coordinates
from one system into another are given by
I
y' = y(y-vt) y = y(y'+vt')
x' = x, z' = z x = x', z = z '
t' = y(t- g x ) t^ =^ y(t'+Ex')
The above is known as the Lorentz or the Lorentz-Minkowski transformation. The various constants appearing above are given by
c = (Po€o) -'= velocity of light in free space or vacuum
eo = permittivity of free space = ( 3 6 a ~ l O ~ ) - ~ farads/m.
p, = permeability of free space = 4 ~ x 1 0 - ~henries/m
p = v/c.
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- 2 Maxwell-Minkowski Equations Consider an isotropic homogeneous lossless (zero conductivity) medium
moving uniformly with a velocity v in some direction. Without loss of generality,
we can choose this to be the y-direction and orient the axes as in Fig. 1. Now,
according to the theory of relativity, Maxwell equations must have the same form in all inertial frames of reference, that is, they must be covariant under the Lorentz transformation (2.1). Therefore in the unprimed o r laboratory system, we have aB V X E = - --at a s V x H = J + -- - (^) at (2.3)
and by attaching primes, we get Maxwell equations in the primed system, for instance, (2.2) becomes
It may be noted that the divergence equations follow from the curl equations and the equation of continuity; hence do not yield any new information. To formulate a problem completely the constitutive relations must be known. These can be derived in the following manner. In the primed system where an observer is at rest with the medium, we have
- D' = EE' - (^) (2. 7)
- B ' = p€J' (2.8) where E: and p are the permittivity and the permeability of the medium in the primed system. Now, according to Minkowski's theory, which is based upon the special theory of relativity, the fields in the two systems transform according to the following scheme..
T H E U N I V E R S I T Y O F M I C H I G A N
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of this type of formulation a modified version of Nag and Sayied's method for Cerenkov radiation is presented in Appendix A.
- 3 The Method of Potentials for Moving Media The method developed here is due to Tai6. The Maxwell equations assuming e-iwt variation a r e V x - E =iwg V x _ H = J -iw_D. Making use of the constitutive (2. 15) and (2.16) we obtain (V+iwQ)xg =iwp g - _ H - (V+iwa)x_H= - i w E g. - _E + J. Applying the transformation
we get
V x - H 1 = - i w E a,. - El+e i w R y J- Now introduce the vector potential A1 such that p q. l l l =^ vx^ a-1 - -^ * A or
(2. 17) (2. 18)
.
. (2.19) (2.20)
(2.22) (2.23)
Substituting in (2.22), we obtain
-1E^ = i w a - l .= A-1^ - V g^ 1 (2.25) where $d1 is the scalar potential. Substituting for (^) - El and H -1 in (2.23), we obtain
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The vector operator on the left band side can be expanded in rectangular coor-
dinates thuf
va=x-+y^ n a^ ~a -^ +z- h^ a
ax aay az
Now define the gauge condition V. A = -ik2a -1 w I, so that (2.26) becomes
To integrate (2.29) in an infinite domain we introduce the scalar Green's function
G which satisfies the equation
where q refers to source point.
Two distinct cases depQrding upon the sign of a exist.
Case 1: a > o o r v < c/n The solution can be obtained by dimensional scaling a@ .ika@Rl
G1 = 4 r R
where R1, the modified distance is given by 2
~ a s e 2 :a < O o r v > c / n In this case, we have a two-dimensional Klein-Gordon equation and the
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CHAPTER III
REFLECTION AND REFRACTION OF A PLANE ELECTROMAGNETIC
WAVE AT THE BOUNDARY OF A MOVING DIELECTRIC MEDIUM
3. 1 Geometrv of the (^) ~~ Problem~ A s shown in Fig. 2, the region z < 0 is filled by a medium, with a per- meability p , and a permittivity E, moving uniformly in the y-direction with a velocity v. The region z > 0 is free space bo, eo) and is stationary. A plane electromagnetic wave traveling in free space in an arbitrary direc- tion is incident upon the interface; as a result there will be a reflected wave and a transmitted wave. azimuthal angles being measured from the x-axis.
Let the orientation of the three waves be as in Fig. 2 , the
- 2 Plane Waves in Moving Media In order to represent the transmitted field, we need plane wave solutions in the moving medium. for e-iwt variations are given by
The Maxwell equations in the absence of sources and
VxE=- i w g ( 3. 1 ) VxH_ = -iwD_ (3.2) V * B = O ,- V * D = O (3.3) Making use of the constitutive relations (2.15) and (2.16) we obtain
The plane wave solutions which we a r e seeking can be represented thus i K K i K K
E = E e H = H e
(3.4) (3.5)
(3.6) (3.7) where the first factor denotes complex amplitude, K the propagation constant and
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Reflected
Transmitted
FIG. 2: PLANE WAVE INCIDENCE ON A MOVING MEDIUM
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Eliminating one variable at a time between (3.14) and (3.15), we find that gf and H_" satisfy the modified vector wave equation
(3.17)
Expanding the left hand side according to (2.27) and making use of (3.16), we
find and H_" satisfy
(Va * V)E"- + a k^2 E_" = 0
(Va.V) H_"+ak2 HIf - = 0
These equations separate into three scalar equations in rectangular coordinates of the form
[i2+: 5 +-+A? 2 I+!/=o
where I+!/ stands for one of the components. Plane wave soluti
given by i(klx +$y -k3z) e provided kl, %, k3 (^) rl satisfy the characteristic equation 2 k kl t +^ k i -ak2^ =^0
ns of the above a r e
Now, we are ready to construct plane wave solutions in the moving medium starting
with either the electric or the magnetic field. Thus setting
we get
b
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The amplitudes in (3.23) must satisfy the relation (3.16), so that klE" ox +$E'' OY-k3E" 02 = 0. Comparing (3.24) with (3.9), we get from the phase functions
K sin Otcos ft = kl
K sin Opingt = (k2-ws2)
K COS et = kg
(3.26a) (3.26b) (3. substituting for kl, %, k3 in (3.22) we get the following dispersion relation for K
K 2 sin 2 et (^) , a 1 (K sinet singt+ua) 2 + K 2 cos 2 et-ak2 = o (3.27)
Solving for K /ko , the refractive index of the moving medium, we get
where CY is the angle between the direction of propagation and the velocity of the
medium, (cos a = sin 0 sinp ) This expression checks with that of Papas" (see
page 231, Eq. 61 ). We also note that the amplitudes in (3.9) will have to satis- fy the following relation
t t --
K sinOtcOS$tE (^) OX +- a^1 (KshOtsingt+wQ)EOY -KCOSOtE oz = O (3.29)
The magnetic field H- can be obtained from (3.4). Making use of (3.27) and (3.29), one can show that the H- field thus obtained satisfies (3.5). From this it follows that the divergence relations in (3.3) are also satisfied. We conclude this section by summarizing the method of construction of plane wave solutions in moving media.
