Electromagnetics, Third Edition

1213 Pages·2018·16.76 MB·English

by  Cloud

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Electromagnetics
Third Edition
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Electromagnetics
Third Edition
Edward J. Rothwell
Michael J. Cloud
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CRC Press
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To Professor Kun-Mu Chen: Thank you for three decades of
guidance, support, encouragement, and inspiration.
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Contents
Preface xxi
Authors xxv
1 Introductory concepts 1
1.1 Notation, conventions, and symbology . . . . . . . . . . . . . . . . . . . . 1
1.2 The field concept of electromagnetics . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 Historical perspective . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.2 Formalization of field theory . . . . . . . . . . . . . . . . . . . . . 4
1.3 The sources of the electromagnetic field. . . . . . . . . . . . . . . . . . . . 5
1.3.1 Macroscopic electromagnetics. . . . . . . . . . . . . . . . . . . . . 6
1.3.1.1 Macroscopic effects as averagedmicroscopic effects . . . 7
1.3.1.2 The macroscopic volume charge density . . . . . . . . . 7
1.3.1.3 The macroscopic volume current density . . . . . . . . 10
1.3.2 Impressed vs. secondary sources . . . . . . . . . . . . . . . . . . . 11
1.3.3 Surface and line source densities . . . . . . . . . . . . . . . . . . . 12
1.3.4 Charge conservation . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3.4.1 The continuity equation . . . . . . . . . . . . . . . . . . 16
1.3.4.2 The continuity equation in fewer dimensions . . . . . . 18
1.3.5 Magnetic charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2 Maxwell’s theory of electromagnetism 23
2.1 The postulate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.1.1 The Maxwell–Minkowskiequations . . . . . . . . . . . . . . . . . 24
2.1.1.1 The interdependence of Maxwell’s equations . . . . . . 25
2.1.1.2 Field vector terminology . . . . . . . . . . . . . . . . . 26
2.1.1.3 Invariance of Maxwell’s equations . . . . . . . . . . . . 27
2.1.2 Connection to mechanics . . . . . . . . . . . . . . . . . . . . . . . 27
2.2 The well-posed nature of the postulate . . . . . . . . . . . . . . . . . . . . 28
2.2.1 Uniqueness of solutions to Maxwell’s equations . . . . . . . . . . . 29
2.2.2 Constitutive relations . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2.2.1 Constitutive relations for fields in free space . . . . . . 33
2.2.2.2 Constitutive relations in a linear isotropic material. . . 34
2.2.2.3 Constitutive relations for fields in perfect conductors . 36
2.2.2.4 Constitutive relations in a linear anisotropic material . 37
2.2.2.5 Constitutive relations for biisotropic materials . . . . . 38
2.2.2.6 Constitutive relations in nonlinear media . . . . . . . . 40
2.3 Maxwell’s equations in moving frames . . . . . . . . . . . . . . . . . . . . 41
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viii Contents
2.3.1 Field conversions under Galilean transformation . . . . . . . . . . 41
2.3.2 Field conversions under Lorentz transformation . . . . . . . . . . 45
2.3.2.1 Lorentz invariants . . . . . . . . . . . . . . . . . . . . . 49
2.3.2.2 Derivation of Maxwell’s equations from Coulomb’s law 51
2.3.2.3 Transformation of constitutive relations . . . . . . . . . 51
2.3.2.4 Constitutive relations in deforming or rotating media . 53
2.4 The Maxwell–Boffi equations . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.4.1 Equivalent polarization and magnetization sources . . . . . . . . . 55
2.4.2 Covariance of the Boffi form . . . . . . . . . . . . . . . . . . . . . 56
2.5 Large-scale form of Maxwell’s equations . . . . . . . . . . . . . . . . . . . 58
2.5.1 Surface moving with constant velocity . . . . . . . . . . . . . . . . 59
2.5.1.1 Kinematic form of the large-scale Maxwell equations . . 61
2.5.1.2 Alternative form of the large-scale Maxwell equations . 64
2.5.2 Moving, deforming surfaces . . . . . . . . . . . . . . . . . . . . . . 65
2.5.3 Large-scale form of the Boffi equations . . . . . . . . . . . . . . . 66
2.6 The nature of the four field quantities . . . . . . . . . . . . . . . . . . . . 67
2.7 Maxwell’s equations with magnetic sources . . . . . . . . . . . . . . . . . . 69
2.8 Boundary (jump) conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 71
2.8.1 Boundary conditions across a stationary, thin source layer. . . . . 71
2.8.2 Boundary conditions holding across a stationary layer of field
discontinuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
2.8.3 Boundary conditions at the surface of a perfect conductor . . . . . 77
2.8.4 Boundaryconditionsacrossastationarylayeroffielddiscontinuity
using equivalent sources . . . . . . . . . . . . . . . . . . . . . . . . 77
2.8.5 Boundary conditions across a moving layer of field discontinuity . 78
2.9 Fundamental theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
2.9.1 Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
2.9.2 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
2.9.2.1 Duality of electric and magnetic point source fields . . 82
2.9.2.2 Duality in a source-free region . . . . . . . . . . . . . . 83
2.9.3 Reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
2.9.4 Similitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
2.9.5 Conservation theorems . . . . . . . . . . . . . . . . . . . . . . . . 87
2.9.5.1 The system concept in the physical sciences. . . . . . . 87
2.9.5.2 Conservation of momentum and energy in mechanical
systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
2.9.5.3 Conservation in the electromagnetic subsystem . . . . . 91
2.9.5.4 Interpretation of the energy and momentum conserva-
tion theorems . . . . . . . . . . . . . . . . . . . . . . . 94
2.9.5.5 Boundary conditions on the Poynting vector . . . . . . 97
2.9.5.6 An alternative formulation of the conservation theorems 98
2.10 The wave nature of the electromagnetic field . . . . . . . . . . . . . . . . . 98
2.10.1 Electromagnetic waves . . . . . . . . . . . . . . . . . . . . . . . . 99
2.10.2 Wave equation for bianisotropic materials . . . . . . . . . . . . . . 102
2.10.3 Wave equation using equivalent sources . . . . . . . . . . . . . . . 105
2.10.4 Wave equation in a conducting medium . . . . . . . . . . . . . . . 105
2.10.4.1 Scalar wave equation for a conducting medium . . . . . 106
2.10.5 FieldsdeterminedbyMaxwell’sequationsvs.fieldsdeterminedby
the wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
2.10.6 Transient uniform plane waves in a conducting medium . . . . . . 107
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Contents ix
2.10.7 Propagationof cylindrical waves in a lossless medium . . . . . . . 114
2.10.8 Propagationof spherical waves in a lossless medium . . . . . . . . 117
2.10.9 Energy radiated by sources . . . . . . . . . . . . . . . . . . . . . . 120
2.10.10 Nonradiating sources . . . . . . . . . . . . . . . . . . . . . . . . . 122
2.11 Application: single chargedparticlemotioninstatic electricandmagnetic
fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
2.11.1 Fundamental equations of motion . . . . . . . . . . . . . . . . . . 124
2.11.2 Nonrelativistic particle motion in a uniform, static electric field. . 126
2.11.3 Nonrelativistic particle motion in a nonuniform, static electric
field; electron optics . . . . . . . . . . . . . . . . . . . . . . . . . . 130
2.11.4 Nonrelativistic particle motion in a uniform, static magnetic field 132
2.11.5 Nonrelativistic particle motion in uniform, static electric and
magnetic fields: E B drift . . . . . . . . . . . . . . . . . . . . . 138
×
2.11.6 Nonrelativistic particle motion in a nonuniform, static magnetic
field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
2.11.7 Relativistic particle motion in a uniform, static magnetic field . . 142
2.11.8 Relativistic particle motion in uniform, static electric and
magnetic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
2.12 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
3 The static and quasistatic electromagnetic fields 151
3.1 Statics and quasistatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
3.2 Static fields and steady currents . . . . . . . . . . . . . . . . . . . . . . . . 151
3.2.1 Decoupling of the electric and magnetic fields. . . . . . . . . . . . 152
3.2.2 Static field equilibrium and conductors . . . . . . . . . . . . . . . 153
3.2.3 Steady current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
3.3 Electrostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
3.3.1 Direct solutions to Gauss’s law . . . . . . . . . . . . . . . . . . . . 158
3.3.2 The electrostatic potential and work . . . . . . . . . . . . . . . . . 160
3.3.2.1 The electrostatic potential . . . . . . . . . . . . . . . . 160
3.3.3 Boundary conditions. . . . . . . . . . . . . . . . . . . . . . . . . . 163
3.3.3.1 Boundary conditions for the electrostatic field . . . . . 163
3.3.3.2 Boundary conditions for steady electric current . . . . . 163
3.3.4 Uniqueness of the electrostatic field . . . . . . . . . . . . . . . . . 165
3.3.5 Poisson’s and Laplace’s equations . . . . . . . . . . . . . . . . . . 166
3.3.5.1 Uniqueness of solution to Poisson’s equation . . . . . . 167
3.3.5.2 IntegralsolutiontoPoisson’sequation: thestaticGreen’s
function. . . . . . . . . . . . . . . . . . . . . . . . . . . 167
3.3.5.3 Useful derivative identities . . . . . . . . . . . . . . . . 169
3.3.5.4 The Green’s function for unbounded space . . . . . . . 171
3.3.5.5 Coulomb’s law . . . . . . . . . . . . . . . . . . . . . . . 171
3.3.5.6 Green’s function for unbounded space: two dimensions 172
3.3.5.7 Dirichlet and Neumann Green’s functions . . . . . . . . 173
3.3.5.8 Reciprocity of the static Green’s function . . . . . . . . 175
3.3.5.9 Image interpretation for solutions to Poisson’s equation 176
3.3.6 Force and energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
3.3.6.1 Maxwell’s stress tensor . . . . . . . . . . . . . . . . . . 181
3.3.6.2 Electrostatic stored energy . . . . . . . . . . . . . . . . 182
3.3.7 Multipole expansion . . . . . . . . . . . . . . . . . . . . . . . . . . 185
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