Electromagnetic Fields

Professor Jean Van Bladel is an eminent researcher and educator in fundamental electromagnetic theory and its application in electrical engineering. Over a distinguished career, he has been the recipient of many awards and honors. A Fellow of the IEEE, he was awarded the Henrich Hertz Medal of the IEEE in 1995 and the Distinguished Achievement Award of the IEEE Antennas and Propagation Society in 1997. With the International Union of Radio Science (URSI), he was secretary general from 1979 to 1993 and was named Honorary President in 1999. He is currently Professor Emeritus at Ghent University in Belgium.

Preface xiii

1. Linear Analysis 1

1.1 Linear Spaces 2

1.2 Linear Transformations 5

1.3 The Inversion Problem 8

1.4 Green’s Functions 11

1.5 Reciprocity 14

1.6 Green’s Dyadics 17

1.7 Convergence of a Series 19

1.8 Eigenfunctions 20

1.9 Integral Operators 23

1.10 Eigenfunction Expansions 26

1.11 Discretization 30

1.12 Matrices 33

1.13 Solution of Matrix Equations: Stability 36

1.14 Finite Differences 38

1.15 Perturbations 43

2. Variational Techniques 51

2.1 Stationary functionals 52

2.2 A Suitable Functional for the String Problem 53

2.3 Functionals for the General l Transformation 55

2.4 Euler’s Equations of Some Important Functionals 58

2.5 Discretization of the Trial Functions 60

2.6 Simple Finite Elements for Planar Problems 62

2.7 More Finite Elements 65

2.8 Direct Numerical Solution of Matrix Problems 69

2.9 Iterative Numerical Solution of Matrix Problems 70

3. Electrostatic Fields in the Presence of Dielectrics 77

3.1 Volume Charges in Vacuum 77

3.2 Green’s Function for Infinite Space 80

3.3 Multipole Expansion 83

3.4 Potential Generated by a Single Layer of Charge 86

3.5 Potential Generated by a Double Layer of Charge 91

3.6 Potential Generated by a Linear Charge 94

3.7 Spherical Harmonics 98

3.8 Dielectric Materials 102

3.9 Cavity Fields 105

3.10 Dielectric Sphere in an External Field 108

3.11 Dielectric Spheroid in an Incident Field 111

3.12 Numerical Methods 115

4. Electrostatic Fields in the Presence of Conductors 125

4.1 Conductivity 125

4.2 Potential Outside a Charged Conductor 127

4.3 Capacitance Matrix 133

4.4 The Dirichlet Problem 134

4.5 The Neumann Problem 137

4.6 Numerical Solution of the Charge Density Problem 139

4.7 Conductor in an External Field 142

4.8 Conductors in the Presence of Dielectrics 146

4.9 Current Injection into a Conducting Volume 148

4.10 Contact Electrodes 153

4.11 Chains of Conductors 158

5. Special Geometries for the Electrostatic Field 167

5.1 Two-Dimensional Potentials in the Plane 167

5.2 Field Behavior at a Conducting Wedge 171

5.3 Field Behavior at a Dielectric Wedge 175

5.4 Separation of Variables in Two Dimensions 177

5.5 Two-Dimensional Integral Equations 181

5.6 Finite Methods in Two Dimensions 185

5.7 Infinite Computational Domains 188

5.8 More Two-Dimensional Techniques 192

5.9 Layered Media 196

5.10 Apertures 199

5.11 Axisymmetric Geometries 203

5.12 Conical Boundaries 207

6. Magnetostatic Fields 221

6.1 Magnetic Fields in Free Space: Vector Potential 221

6.2 Fields Generated by Linear Currents 224

6.3 Fields Generated by Surface Currents 227

6.4 Fields at Large Distances from the Sources 229

6.5 Scalar Potential in Vacuum 232

6.6 Magnetic Materials 234

6.7 Permanent Magnets 236

6.8 The Limit of Infinite Permeability 239

6.9 Two-Dimensional Fields in the Plane 244

6.10 Axisymmetric Geometries 249

6.11 Numerical Methods: Integral Equations 251

6.12 Numerical Methods: Finite Elements 253

6.13 Nonlinear Materials 258

6.14 Strong Magnetic Fields and Force-Free Currents 260

7. Radiation in Free Space 277

7.1 Maxwell’s Equations 277

7.2 The Wave Equation 280

7.3 Potentials 282

7.4 Sinusoidal Time Dependence: Polarization 286

7.5 Partially Polarized Fields 290

7.6 The Radiation Condition 293

7.7 Time-Harmonic Potentials 296

7.8 Radiation Patterns 300

7.9 Green’s Dyadics 303

7.10 Multipole Expansion 307

7.11 Spherical Harmonics 313

7.12 Equivalent Sources 320

7.13 Linear Wire Antennas 327

7.14 Curved Wire Antennas: Radiation 333

7.15 Transient Sources 337

8. Radiation in a Material Medium 357

8.1 Constitutive Equations 357

8.2 Plane Waves 370

8.3 Ray Methods 377

8.4 Beamlike Propagation 388

8.5 Green’s Dyadics 392

8.6 Reciprocity 397

8.7 Equivalent Circuit of an Antenna 402

8.8 Effective Antenna Area 409

9. Plane Boundaries 423

9.1 Plane Wave Incident on a Plane Boundary 423

9.2 Propagation Through a Layered Medium 442

9.3 The Sommerfeld Dipole Problem 448

9.4 Multilayered Structures 452

9.5 Periodic Structures 460

9.6 Field Penetration Through Apertures 478

9.7 Edge Diffraction 490

10. Resonators 509

10.1 Eigenvectors for an Enclosed Volume 509

10.2 Excitation of a Cavity 514

10.3 Determination of the Eigenvectors 517

10.4 Resonances 525

10.5 Open Resonators: Dielectric Resonances 529

10.6 Aperture Coupling 540

10.7 Green’s Dyadics 544

11. Scattering: Generalities 563

11.1 The Scattering Matrix 563

11.2 Cross Sections 568

11.3 Scattering by a Sphere 574

11.4 Resonant Scattering 582

11.5 The Singularity Expansion Method 586

11.6 Impedance Boundary Conditions 598

11.7 Thin Layers 601

11.8 Characteristic Modes 604

12. Scattering: Numerical Methods 617

12.1 The Electric Field Integral Equation 617

12.2 The Magnetic Field Integral Equation 624

12.3 The T-Matrix 629

12.4 Numerical Procedures 633

12.5 Integral Equations for Penetrable Bodies 639

12.6 Absorbing Boundary Conditions 646

12.7 Finite Elements 651

12.8 Finite Differences in the Time Domain 654

13. High- and Low-Frequency Fields 671

13.1 Physical Optics 671

13.2 Geometrical Optics 676

13.3 Geometric Theory of Diffraction 681

13.4 Edge Currents and Equivalent Currents 689

13.5 Hybrid Methods 692

13.6 Low-Frequency Fields: The Rayleigh Region 695

13.7 Non-Conducting Scatterers at Low Frequencies 696

13.8 Perfectly Conducting Scatterers at Low Frequencies 699

13.9 Good Conductors 707

13.10 Stevenson’s Method Applied to Good Conductors 711

13.11 Circuit Parameters 715

13.12 Transient Eddy Currents 719

14. Two-Dimensional Problems 733

14.1 E and H Waves 733

14.2 Scattering by Perfectly Conducting Cylinders 738

14.3 Scattering by Penetrable Circular Cylinders 743

14.4 Scattering by Elliptic Cylinders 746

14.5 Scattering by Wedges 749

14.6 Integral Equations for Perfectly Conducting Cylinders 751

14.7 Scattering by Penetrable Cylinders 759

14.8 Low-Frequency Scattering by Cylinders 764

14.9 Slots in a Planar Screen 770

14.10 More Slot Couplings 778

14.11 Termination of a Truncated Domain 786

14.12 Line Methods 792

16.2 Scattering by Bodies of Revolution: Integral Equations 908

16.3 Scattering by Bodies of Revolution: Finite Methods 912

16.4 Apertures in Axisymmetric Surfaces 915

16.5 The Conical Waveguide 918

16.6 Singularities at the Tip of a Cone 926

16.7 Radiation and Scattering from Cones 930

15. Cylindrical Waveguides 813

15.1 Field Expansions in a Closed Waveguide 814

15.2 Determination of the Eigenvectors 818

15.3 Propagation in a Closed Waveguide 822

15.4 Waveguide Losses 832

15.5 Waveguide Networks 837

15.6 Aperture Excitation and Coupling 844

15.7 Guided Waves in General Media 859

15.8 Orthogonality and Normalization 865

15.9 Dielectric Waveguides 873

15.10 Other Examples of Waveguides 882

16. Axisymmetric and Conical Boundaries 905

16.1 Field Expansions for Axisymmetric Geometries 905

17. Electrodynamics of Moving Bodies 943

17.1 Fields Generated by a Moving Charge 943

17.2 The Lorentz Transformation 946

17.3 Transformation of Fields and Currents 950

17.4 Radiation from Sources: the Doppler Effect 955

17.5 Constitutive Equations and Boundary Conditions 958

17.6 Material Bodies Moving Uniformly in Static Fields 960

17.7 Magnetic Levitation 962

17.8 Scatterers in Uniform Motion 966

17.9 Material Bodies in Nonuniform Motion 972

17.10 Rotating Bodies of Revolution 974

17.11 Motional Eddy Currents 979

17.12 Accelerated Frames of Reference 984

17.13 Rotating Comoving Frames 988

Appendix 1. Vector Analysis in Three Dimensions 1001

Appendix 9. Some Eigenfunctions and Eigenvectors 1105

Appendix 2. Vector Operators in Several Coordinate Systems 1011

Appendix 10. Miscellaneous Data 1111

Appendix 3. Vector Analysis on a Surface 1025

Appendix 4. Dyadic Analysis 1035

Appendix 5. Special Functions 1043

Appendix 6. Complex Integration 1063

Appendix 7. Transforms 1075

Appendix 8. Distributions 1089

Bibliography 1117

General Texts on Electromagnetic Theory 1117

Texts that Discuss Particular Areas of Electromagnetic Theory 1118

General Mathematical Background 1122

Mathematical Techniques Specifically Applied to Electromagnetic Theory 1123

Acronyms and Symbols 1127

Author Index 1133

Subject Index 1149