Lehner G. Electromagnetic Field Theory for Engineers and Physicists
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Electromagnetic Field Theory for Engineers
and Physicists
G
¨
unther Lehner
Electromagnetic Field
Theory for Engineers
and Physicists
Translated by Matt Horrer
123
Prof. Dr.rer.nat. G
¨
unther Lehner (em.)
Universit
¨
at Stuttgart
Fak.05 Informatik, Elektrotechnik
und Informationstechnik
Pfaffenwaldring 47
70569 Stuttgart
Translator
Matt Horrer
USA
mhorrer@ieee.org
ISBN 978-3-540-76305-5 e-ISBN 978-3-540-76306-2
DOI 10.1007/978-3-540-76306-2
Springer Heidelberg Dordrecht London New York
Library of Congress Control Number: 2009943738
c
Springer-Verlag Berlin Heidelberg 2010
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The Author’s Preface
This book deals with the fundamental principles of electrodynamics, i .e. the theory
of electromagnetic fields as given by Maxwell's equations. It is an outgrowth from
the lectures, which the author has been giving to the students of electrical
engineering at the University of Stuttgart, Germany, for approximately a quarter of
a century. For the textbook, the contents of the lectures have been supplemented by
a chapter on numerical methods for the solution of boundary and initial value
problems, which provides a rough first survey over the methods available only,
without going into details. Furthermore, there are several appendices devoted to
some more special topics, as among others to the problem of the possibility of an
extremely small but nonzero restmass of the photon, which would lead to Proca’s
equations, a modified version of Maxwell’s equations; to the important question of
eventually existing magnetic monopoles; to the deeper meaning of the
electromagnetic potentials in view of quantum mechanics and the Bohm-
Aharonov-effects. The last appendix covers a brief survey of special relativity,
because this, in principle, is an essential part of electrodynamics, which is
inevitably needed for its real understanding.
The treatment is based on Maxwell’s equations from the beginning. They are
described and explained in Chapter 1. The following chapters are devoted to
electrostatics; to the important mathematical tools of electromagnetic field theory
(method of separation of variables using cartesian coordinates, cylindrical
coordinates, and spherical coordinates; conformal mapping for plane problems); to
stationary current density fields; to magnetostatics; to quasi stationary time
dependent problems as field-diffusion, skin effect etc.; and finally electromagnetic
waves and dipole radiation. Everything in these chapters is derived from Maxwell's
equations, except the additionally necessary assumptions characterizing various
media, their conductivity, polarizability, and magnetizability.
The basic concepts of vector analysis are also developed from the beginning
together with Maxwell’s equations. The divergence (div) is defined as the small
volume limit of the surface integral (flux) of a vector field and the rotation (curl) as
small surface limits of three line integrals (circulations) of a vector field. These
definitions immediately clarify the plausible meaning of both of these operators of
vector analysis. The divergence being the volume density of sources or sinks, the
rotation being the three dimensional surface density of circulation. The integral
theorems of Gauss and Stokes are immediately plausible consequences of these
definitions also. This procedure provides an easy and well comprehensible access
to the realm of vector analysis. It also very clearly demonstrates the physical
meaning of Maxwell’s equations. Helmholtz’s theorem (presented in one of the
appendices) teaches us that each vector field is completely defined by its
divergence and its rotation. So it is obvious that we need four equations to describe
electric and magnetic fields, two for their sources and sinks and two for their
circulations. Thus, Maxwell’s equations, often considered to be almost
incomprehensible, are hoped to become really plausible.
vi The Author’s Preface
The different chapters contain a variety of analytical solutions of boundary
and initial value problems. It is quite often claimed that this is no longer of interest
and that such problems nowadays are usually treated numerically by computers.
The author cannot share this opinion. People trying to solve electrodynamical
boundary and initial value problems numerically, without having studied and
understood the theoretical background and not having seen examples of a variety
of fields, often if not mostly obtain faulty results. Having little or no feeling for the
matter, they may believe in the correctness of their results. It is not at all easy to test
if the solutions are really correct. The availability of many analytical solutions is a
very valuable and even indispensable tool for testing numerical programs. It is
always advisable to solve similar problems analytically when doing numerical
work and to test the numerical methods by comparing the results.
Initially conceived for students of engineering and physics, the textbook
turned out to be useful for professionally working engineers and physicists also.
That is why six editions of the original German version of the book have appeared
already. The author together with the translator hopes that the present English
translation will be as useful for its readers.
Stuttgart, 2009 Günther Lehner
Table of Contents
List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
1 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Charge and Coulomb’s Law . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Electric Field Strength E and Displacement Field D . . . . . . . . . . . . 4
1.4 Electric Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.5 Divergence of a Vector Field and Gauss’ Integral Theorem . . . . . . . . 8
1.6 Work and the Electric Field . . . . . . . . . . . . . . . . . . . . . . . . 12
1.7 The Rotation of a Vector Field and Stoke’s Integral Theorem . . . . . . . 14
1.8 Potential and Voltage . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.9 The Electric Current and Ampere’s Law . . . . . . . . . . . . . . . . . . 22
1.10 The Principle of Charge Conservation and Maxwell’s First Equation . . . 27
1.11 Faraday’s Law of Induction . . . . . . . . . . . . . . . . . . . . . . . . 30
1.12 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.13 System of Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2 Basics of Electrostatics . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.1 Fundamental Relations . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.2 Field Intensity and Potential for a given Charge Distribution . . . . . . . 41
2.3 Specific Charge Distributions . . . . . . . . . . . . . . . . . . . . . . . 44
2.3.1 One-dimensional, Planar Charge Distributions . . . . . . . . . . . . . . . 44
2.3.2 Spherically Symmetric Distributions . . . . . . . . . . . . . . . . . . . . . . . 45
2.3.3 Cylindrically Symmetric Distributions . . . . . . . . . . . . . . . . . . . . . . 48
2.4 The Field Generated by two Point Charges . . . . . . . . . . . . . . . . 51
2.5 The Ideal Dipole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
2.5.1 The Ideal Dipole and its Potential . . . . . . . . . . . . . . . . . . . . . . . . . 57
2.5.2 Volume Distribution of Dipoles . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
2.5.3 Surface Distributions of Dipoles (Dipole Layers) . . . . . . . . . . . . . . 63
2.5.4 Line Dipoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
2.6 Behavior of a Conductor in an Electric Field . . . . . . . . . . . . . . . . 72
2.6.1 Metallic Sphere in the Field of a Point Charge . . . . . . . . . . . . . . . 73
2.6.2 Metallic Sphere in a Uniform Electric Field . . . . . . . . . . . . . . . . . . 77
2.6.3 Metallic Cylinder in the Field of a Line Charge . . . . . . . . . . . . . . . 80
2.7 The Capacitor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
2.8 E and D inside Dielectrics . . . . . . . . . . . . . . . . . . . . . . . . . 84
2.9 The Capacitor with a Dielectric . . . . . . . . . . . . . . . . . . . . . . 88
2.10 Boundary Conditions for E and D and Refraction of Force Lines . . . . . 90
2.11 A Point Charge inside a Dielectric . . . . . . . . . . . . . . . . . . . . . 93
2.11.1 Uniform Dielectric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
2.11.2 Plane Boundaries between two Dielectrics . . . . . . . . . . . . . . . . . . 95
2.12 A Dielectric Sphere in a Uniform Electric Field . . . . . . . . . . . . . . 97
2.12.1 The Field of a Uniformly Polarized Sphere . . . . . . . . . . . . . . . . . . 97
2.12.2 An Externally Applied Uniform Field as the Cause of Polarization 101
2.12.3 Dielectric Sphere (
ε
i
) and Dielectric Space (ε
a
) . . . . . . . . . . . . . 101
2.12.4 Generalization: Ellipsoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
2.13 Polarization Current . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
2.14 Energy Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
2.14.1 Energy Principle in its General Form . . . . . . . . . . . . . . . . . . . . . . 108
2.14.2 Electrostatic Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
2.15 Forces in the Electric Field . . . . . . . . . . . . . . . . . . . . . . . 114
2.15.1 Force on the Plate of a Capacitor . . . . . . . . . . . . . . . . . . . . . . . . 114
2.15.2 Capacitor with two Dielectrics . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
3 Formal Methods of Electrostatics . . . . . . . . . . . . . . . . . . . . 117
3.1 Coordinate Transformations . . . . . . . . . . . . . . . . . . . . . . . 117
3.2 Vector Analysis for Curvilinear, Orthogonal Coordinate Systems . . . . 121
3.2.1 Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
3.2.2 Divergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
3.2.3 Laplace Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
3.2.4 Circulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
3.3 Some Important Coordinate Systems . . . . . . . . . . . . . . . . . . 124
3.3.1 Cartesian Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
3.3.2 Cylindrical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
3.3.3 Spherical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
3.4 Some Properties of Poisson’s and Laplace’s Equations (Potential
Theory) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
3.4.1 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
3.4.2 Green’s Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
3.4.3 Proof of Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
3.4.4 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
3.4.5 Dirac’s Delta Function (
δ-Function) . . . . . . . . . . . . . . . . . . . . . . . 132
3.4.6 Point Charge and
δ-Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
3.4.7 Potential in a Bounded Region . . . . . . . . . . . . . . . . . . . . . . . . . . 136
3.5 Separation of Laplace’s Equation in Cartesian Coordinates . . . . . . 139
3.5.1 Separation of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
3.5.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
3.5.2.1 Dirichlet Boundary Value Problem without Charges Inside . . . . 142
3.5.2.2 Dirichlet Boundary Value Problem with Charges Inside the
Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
3.5.2.3 Point Charge in Infinite Space . . . . . . . . . . . . . . . . . . . . . . . . . . 153
3.5.2.4 Appendix to Section 3.5: Fourier Series and Fourier Integrals . . 155
v iii Table of Contents
3.6 Complete Orthogonal Systems of Functions . . . . . . . . . . . . . . 159
3.7 Separation of Variables of Laplace’s Equation in Cylindrical
Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
3.7.1 Separation of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
3.7.2 Some Properties of Cylindrical Functions . . . . . . . . . . . . . . . . . . 167
3.7.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
3.7.3.1 A Cylinder with Surface Charges . . . . . . . . . . . . . . . . . . . . . . . . 171
3.7.3.2 Point Charge on the Axis of a Dielectric Cylinder . . . . . . . . . . . . 175
3.7.3.3 Dirichlet’s Boundary Value Problem and the Fourier-Bessel
Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
3.7.3.4 Rotationally Symmetric Surface Charges in the Plane z = 0
and the Hankel Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . 181
3.7.3.5 Charge Distributions that are not Rotationally Symmetric . . . . . 183
3.8 Separation of Variables for Laplace’s Equation in Spherical
Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
3.8.1 Separation of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
3.8.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
3.8.2.1 Dielectric Sphere in a Uniform Electric Field . . . . . . . . . . . . . . . . 193
3.8.2.2 A Sphere Carrying an arbitrary Surface Charge . . . . . . . . . . . . . 195
3.8.2.3 Dirichlet’s Problem for a Sphere . . . . . . . . . . . . . . . . . . . . . . . . . 199
3.9 Multi-Conductor Systems . . . . . . . . . . . . . . . . . . . . . . . . 200
3.10 Plane Electrostatic Problems and the Flux Function . . . . . . . . . . 206
3.11 Analytic Functions and Conformal Mappings . . . . . . . . . . . . . . 210
3.12 The Complex Potential . . . . . . . . . . . . . . . . . . . . . . . . . 217
4 The Stationary Current Density Field . . . . . . . . . . . . . . . . . . 232
4.1 The Basic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
4.2 Relaxation Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
4.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 237
4.4 The formal analogy between D and g . . . . . . . . . . . . . . . . . . 242
4.5 Some Current Density Fields . . . . . . . . . . . . . . . . . . . . . . 243
4.5.1 Point-like Current Source in Space . . . . . . . . . . . . . . . . . . . . . . . 243
4.5.2 Line Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
4.5.3 Mixed Boundary Value Problem . . . . . . . . . . . . . . . . . . . . . . . . . 247
5 Basics of Magnetostatics . . . . . . . . . . . . . . . . . . . . . . . . 256
5.1 Basic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
5.2 Some Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . 265
5.2.1 The Field of a Straight, Concentrated Current . . . . . . . . . . . . . . 265
5.2.2 Field of a Rotationally Symmetric Current Distribution in
Cylindrical Conductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
5.2.3 The Field of a simple Coil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
5.2.4 The Field of a Circular Current and a Magnetic Dipole . . . . . . . . 275
5.2.5 The Field of an Arbitrary Current Loop . . . . . . . . . . . . . . . . . . . . 282
5.2.6 The Field of a Conducting Loop in its Plane . . . . . . . . . . . . . . . . 284
Table of Contents ix