Table Of ContentELECTROMAGNETIC FIELDS
Theory, worked examples and problems
ELECTROMAGNETIC FIELDS
Theory, worked examples and problems
Ruth V. Buckley
Principal Lecturer,
School of Electrical Engineering,
Leeds Polytechnic
© Ruth V. Buckley 1981
All rights reserved. No part of this publication may be reproduced or
transmitted, in any form or by any means, without permission.
First published 1981 by
THE MACMILLAN PRESS LTD
London and Basingstoke
Associated companies in Delhi Dublin
Hong Kong Johannesburg Lagos Melbourne
New York Singapore and Tokyo
ISBN 978-0-333-30664-2 ISBN 978-1-349-05792-4 (eBook)
DOI 10.1007/978-1-349-05792-4
This book is sold subject to the standard conditions of the Net Book Agreement.
The paperback edition of this book is sold subject to the condition that it shall
not. by way of trade or otherwise, be lent. resold, hired out. or otherwise
circulated without the publisher's prior consent in any form of binding or cover
other than that in which it is published and without a similar condition including
this condition being imposed on the subsequent purchaser.
CONTENTS
Preface vi
1. ELECTRON BALLISTICS 1
2. ELECTROSTATIC FIELD THEORY 25
3. ELECTROMAGNETIC THEORY 50
4. PRINCIPLES OF ELECTROMAGNETIC INDUCTION 84
5. ELECTROMAGNETIC WAVES 108
6. PRINCIPLES OF GUIDED WAVES 135
Appendix Stokes' Theorem 161
163
Bibliography
PREFACE
Although most electrical engineering curricula include a study of
network theory as well as electric and magnetic field theory, it is
usually recognised that on most courses much of this subject mate
rial can and is separated out into different syllabus contents.
This book on electromagnetic theory provides a chapter on elec
tron ballistics separate from the field theory; the following
chapters concentrate on electrostatics and electromagnetics under
steady state conditions, with a chapter on time varying fields,
Maxwell's equations and their relations to each media property and
finally a chapter on waveguide developments.
The problems vary in their level of difficulty and by so doing
the author hopes that the book can be used at several stages of a
degree or diploma course, as well as the Council of Engineering
Institutions examinations.
The author wishes to express her gratitude to the Council, for
permission to use questions from their examination papers. The
answers given here are the entire responsibility of the author.
The author would also like to express her appreciation to her
colleague, Mr D. H. Trevena, for making many constructive comments
and for improving the presentation of the material.
R. V. BUCKLEY
1 ELECTRON BALLISTICS
This chapter is concerned with the dynamics of particle motion in a
vacuum, the electron being visualised as a particle with a negative
electric charge -e (1.60 x 10-19 coulombs). It is associated with a
mass m which increases as the velocity of the electron increases
towards that of the velocity of light (3 x 108 m/s).
From the theory of relativity it can be shown that
m =
where m is the particle mass at a velocity v and c is the velocity of
light.
However, it is usual to assume that the mass is constant,
9.11 x 10-31 kg for all velocities up to one-third of the velocity of
light.
Note also a ~arameter often found in this work:
elm = 1.76 x 10 1 C/kg.
ACCELERATION OF AN ELECTRON
If the particle is free to move, it will be subject to Newton's laws
of motion. Assume the electron is placed in an electrostatic field
of ~~rength E volts/m. Then the force exerted on it is
= -
l- eE N
while the force required to produce an acceleration f is given by
F = mf N
therefore
f = - ~ E m/s2
m
If the potential at the point under consideration is V volts then
from electrostatic theory it is known that
av
E = - ax vol ts/m
1
where x is the displacement of the electron from some reference point.
Therefore
ENERGY RELATIONSHIPS
Figure 1.1
Consider figure 1.1 which shows two arbitrary points P and Q in an
electrostatic field. Clearly some form of energy will be needed to
move an electron from P to Q, where the potential difference is
(V2 - VI) volts. Therefore
acquired potential energy = -e(V2 - VI) J or coulomb volts
When (V2 - Vi) is positive, the negative sign is an indication of a
transfer of energy; alternatively it must be a gain in kinetic energy,
Let the velocities at P and Q be vI and v2 then
gain in kinetic energy = ~mv22
therefore
v2 2 - VI 2 = I2iei (V2 - VI)
In many problems the electron must start with zero velocity so
that vI = 0, v2 = v, and from a point of zero potential VI = 0,
V2 = V. Thus
This equation shows that the electron's speed at any point in the
field depends only on the potential V at that point.
Note that so far the equations do not involve any co-ordinate
system, except the above arbitrary distance x.
2
Uniform Electrostatic Field
Consider an electron entering the field at point 0 in figure 1.2 with
a velocity v making an angle e with the vertical. It is now necessary
to use a co-ordinate system to indicate which planes the electron will
move through.
y
Ymax
x
Figure 1.2
E = E = 0 • E - Y... volts/m
x z 'y d
Consider a point within the field, such as point A with co-ordinates
x, y. The vertical and horizontal components of velocity are v cos e
and v sin e so that from Newton's laws of motion (initial components)
x = v sin e x t
y v cos e x t + 21 f t2
Y
Now
f - ~ E
Y m Y
eliminate t between the equations for x and y hence
y
This is the equation of a parabola and the problem is often to find
the condition for the electron to strike the upper plate or to
continue on its trajectory and re-strike the lower plate.
Uniform Magnetic Field
A charged particle moving in a magnetic field experiences a force only
if it moves at an angle to the field and the force is perpendicular
to both the direction of motion and to the direction of the applied
field, as shown in figure 1.3.
3
Force = Bev sin n
~~ectron
motion
/
Equivalent current
Figure 1.3
Consider an electron injected into a magnetic field with velocity
v m/s as shown in figure 1.4. The field of strength B tesla is into
the plane of the paper as shown.
Field B tesla
Figure 1.4
Now the magnetic force due to the field, the charge on the electron
and its velocity is
F = Bev
It will be held on a circular path of radius R by a centifugal force
equal to mv2/R, therefore
Bev = IIDRV2 when a = 90 0
R = mv
eB
period of rotation T = 2uR = ~ s
v eB
Note that the period depends on the field strength only.
Combined Fields
The plane magnetron is developed as a result of the movement of an
electron in perpendicular electric and magnetic fields, when the path
of the electron starting from rest at the plane cathode will follow a
cycloidal path.
4,
