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Radiation Due to an Oscillating Dipole over a Lossless Semi-Infinite Moving Dielectric Medium, Study notes of Electromagnetism and Electromagnetic Fields Theory

A technical report on the radiation due to an oscillating dipole over a lossless semi-infinite moving dielectric medium. It covers topics such as electrodynamics of moving media, reflection and refraction of a plane electromagnetic wave, and oscillating dipole over a moving dielectric medium. The report includes mathematical equations, diagrams, and numerical results. The report was prepared for the National Aeronautics and Space Administration and was administered through the Office of Research Administration at the University of Michigan.

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THE
UNIVERSITY
OF
MICHIGAN
COLLEGE
OF
ENGINEERING
DEPARTMENT
OF
ELECTRICAL
ENGINEERING
Radiation La
bototory
Rudiution
Due
to
an
Oscillating
Dipole
Over
u
Lossless
Semi-lnfinite
Moving
Dielectric
Medium
by
VITTA1
P.
PYATI
,
r-
r[
.
s
February
1966
-
Grant
NGR-23-005-107
National Aeronautics and Space Administration
Langley Research Center
Langley Station
Hampton, Virginia
23365
Contract
Mth
:
Administered
through:
OFFICE
OF
RESEARCH ADMINISTRATION ANN
ARBOR
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Download Radiation Due to an Oscillating Dipole over a Lossless Semi-Infinite Moving Dielectric Medium and more Study notes Electromagnetism and Electromagnetic Fields Theory in PDF only on Docsity!

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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

b y

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

I *

. (^) 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

.

I *'

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.

i ii

T H E U N I V E R S I T Y O F M I C H I G A N

7322-2-T

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

iv

J

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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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

  1. 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
  2. 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
  3. 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

  1. 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

vi

.

r' (^8)

l

3

T H E U N I V E R S I T Y O F M I C H I G A N 7322-2-T

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.

4

  1. 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:

  1. Reflection and refraction of a plane electromagnetic wave at the boundary of a moving dielectric medium.

8

i^ #

W '

T H E U N I V E R S I T Y O F M I C H I G A N 7322-2-T

CHAPTER II

ELECTRODYNAMICS OF MOVING MEDIA

  1. 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.

3

T H E U N I V E R S I T Y O F M I C H I G A N 7322-2-T

  1. 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.

  1. 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

.

T H E U N I V E R S I T Y O F M I C H I G A N 7322-2-T

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

7

T H E U N I V E R S I T Y O F M I C H I G A N 7322-2-T

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

  1. 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

9

T H E U N I V E R S I T Y O F M I C H I G A N 7322-2-T

Reflected

Transmitted

FIG. 2: PLANE WAVE INCIDENCE ON A MOVING MEDIUM

T H E U N I V E R S I T Y O F M I C H I G A N

7 322 -2 -T

.

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

  • (^) Y

c

T H E U N I V E R S I T Y O F M I C H I G A N 7322-2-T

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.