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# The Power and Beauty of Electromagnetic Fields. IEEE Press Series on Electromagnetic Wave Theory

• Published: December 2011
• Region: Global
• 678 Pages
• John Wiley and Sons Ltd

This revolutionary textbook takes a unique approach to electromagnetic theory, comparing both conventional and modern theories. It explores both the Maxwell-Poynting representation as well as the Alternate representation, which the author demonstrates is generally simpler and more suitable for analyzing modern electromagnetic environments. Throughout the text, students and researchers have the opportunity to examine both of these theories and discover how each one can be applied to solve problems.

The text is divided into four parts:

Part I: Basic Electromagnetic Theory includes Maxwell's equations, quasistatics, power and energy, stress and momentum, and electromagnetic wave theorems and principles

Part II: Four-Dimensional Electromagnetism includes four-dimensional vectors and tensors and energy-momentum tensors

Part III: Electromagnetic Examples includes statics and quasistatics, accelerating charges, plane waves, transmission lines, waveguides, antennas and diffraction, and ferrites

Part IV: Backmatter includes a summary, appendices, and references

Designed

Preface xxi

Acknowledgments xxvii

List of Figures xxix

PART I BASIC ELECTROMAGNETIC THEORY

1 Maxwell’s Equations 5

1.1 Mathematical notation 5

1.2 Free-space fields and forces 6

1.3 Vector and scalar potentials 10

1.4 Inhomogeneous wave equations for E and H 12

1.5 Static fields 12

1.6 Integration of the inhomogeneous wave equation 15

1.7 Polarizable, magnetizable, and conducting media 18

1.8 Boundary conditions 24

1.9 The complex Maxwell Equations 26

2 Quasistatic Approximations 29

2.1 Quasistatic expansions of a standing wave 30

2.2 Electroquasistatic (EQS) fields 31

2.3 Magnetoquasistatic (MQS) fields 33

2.4 Conduction problems 35

2.5 Laplacian approximations 37

3 Electromagnetic Power, Energy, Stress, and Momentum 39

3.1 Introduction 39

3.2 The Maxwell–Poynting representation 41

3.3 Quasistatic power and energy 43

3.4 Alternative representations 45

3.5 Differences between representations 54

4 Electromagnetic Waves in Free-Space 61

4.1 Homogeneous waves 61

4.2 One-dimensional waves 62

4.3 Harmonic uniform plane waves 63

4.4 Waves of high symmetry 64

4.5 Inhomogeneous scalar wave equations 66

5 Electromagnetic Waves in Linear Materials 67

5.1 Introduction 67

5.2 Electrically conducting media 67

5.3 Linear dielectric and magnetic media 70

6 Electromagnetic Theorems and Principles 77

6.1 Introduction 77

6.2 Complex power and energy theorems 78

6.3 Complex stress theorems 84

6.4 Complex momentum theorems 86

6.5 Duality 88

6.6 Uniqueness theorems 94

6.7 The equivalence principle 96

6.8 The induction theorem 97

6.9 Babinet’s Principle 98

6.10 The reciprocity theorem 100

PART II FOUR-DIMENSIONAL ELECTROMAGNETISM

7 Four-Dimensional Vectors and Tensors 105

7.1 Space–time coordinates 105

7.2 Four-vector electric-current density 106

7.3 Four-vector potential (Lorenz gauge) 106

7.4 Four-Laplacian (wave equation) 107

7.5 Maxwell’s Equations and field tensors 107

7.6 The four-dimensional curl operator 109

7.7 Four-dimensional “statics” 110

7.8 Four-dimensional force density 112

7.9 Six-vectors and dual field tensors 113

7.10 Four-vector electric and magnetic fields 113

7.11 The field tensors and Maxwell’s Equations revisited 115

7.12 Linear conductors revisited 116

8 Energy-Momentum Tensors 119

8.1 Introduction 119

8.2 Maxwell–Poynting energy-momentum tensor 121

8.3 Alternate energy-momentum tensors 121

8.4 Boundary conditions and gauge considerations 125

8.5 Electromagnetic beauty revisited 126

9 Dielectric and Magnetic Materials 129

9.1 Introduction 129

9.2 Maxwell’s Equations with polarization and magnetization 130

9.3 Amperian energy-momentum tensors 131

10 Amperian, Minkowski, and Chu Formulations 141

10.1 Introduction 141

10.2 Maxwell’s Equations in the Amperian formulation 141

10.3 Maxwell’s Equations in the Minkowski formulation 142

10.4 Maxwell’s Equations in the Chu formulation 143

10.5 Energy-momentum tensors and four-force densities 145

10.6 Discussion of force densities 148

10.7 The principle of virtual power 150

PART III ELECTROMAGNETIC EXAMPLES

11 Static and Quasistatic Fields 157

11.1 Spherical charge distribution 157

11.2 Electric field in a rectangular slot 158

11.3 Current in a cylindrical conductor 160

11.4 Sphere with uniform conductivity 163

11.5 Quasistatic analysis of a physical resistor 170

11.6 Magnetic diffusion 179

12 Uniformly Moving Electric Charges 183

12.1 Point charge 183

12.2 Surface charges separating at constant velocity 185

12.3 Expanding cylindrical surface charge 190

12.4 Expanding spherical surface charge 192

13 Accelerating Charges 195

13.1 Hertzian electric dipole 195

13.2 Hertzian magnetic dipole 200

13.3 Radiation from an accelerated then decelerated charge 202

14 Uniform Surface Current 207

14.1 Pulse excitations 207

14.2 Resistive-sheet detector 214

15 Uniform Line Currents 223

15.1 Axial current step (integral laws) 223

15.2 Axial current step (differential laws) 237

15.3 Superposition of axial line currents 240

15.4 Axial current with multiple pulses 246

15.5 Fields of a sinusoidal axial current 251

16 Plane Waves 255

16.1 Uniform TEM plane waves 255

16.2 Doppler-shifted TEM plane waves 257

16.3 Nonuniform plane waves 258

16.4 Skin-depth-limited current in a conductor 261

17 Waves Incident at a Material Interface 263

17.1 Reflected and transmitted plane waves 263

17.2 TE polarization 264

17.3 TM polarization 267

17.4 Elliptically polarized incident waves 269

18 TEM Transmission Lines 271

18.1 General time-dependent solutions 271

18.2 Parallel-plate TEM line in the sinusoidal steady state 274

18.3 TEM tapered-plate “horn” transformer 280

18.4 TEM line with parallel plates of high conductivity 282

18.5 Parallel-plate TEM line loaded with linear material 289

19 Rectangular Waveguide Modes 293

19.1 Introduction 293

19.2 Periodic potentials and fields 294

19.3 Waveguide dispersion 295

19.4 TEnm modes 296

19.5 TMnm modes 298

19.6 Null Alternate-power and Alternate-energy distributions 299

19.7 Uniqueness resolved 300

20 Circular Waveguide Modes 305

20.1 Introduction 305

20.2 TMnm modes 307

20.3 TEnm modes 310

20.4 Null Alternate power and energy distributions 323

20.5 Alternate energy momentum and photons 323

21 Dielectric Waveguides 335

21.1 Introduction 335

21.2 Symmetric TE modes 336

21.3 Antisymmetric TE modes 336

21.4 Dispersion relations 337

22 Antennas and Diffraction 341

22.1 Introduction 341

22.2 Half-wave dipoles 342

22.3 Self-complementary planar antennas 345

22.4 Traveling-wave wire antennas 345

22.5 The theory of simple arrays 349

22.6 Diffraction by a rectangular slit 356

22.7 Diffraction by a large circular aperture 360

22.8 Diffraction by a small circular aperture 369

22.9 Diffraction by the complementary screen 371

22.10 Paraxial wave equation 372

23 Waves and Resonances in Ferrites 377

23.1 Introduction 377

23.2 Ferrites 378

23.3 Large-signal equations 380

23.4 Linearized (small-signal) equations 381

23.5 Uniform precession in a small ellipsoid 383

23.6 Plane wave solutions 384

23.7 Small-signal power and energy 388

23.8 Small-signal stress and momentum 391

23.9 Quasiparticle interpretation (magnons) 393

24 Equivalent Circuits 395

24.1 Receiving circuit of a dipole 395

24.2 TEM transmission lines 398

24.3 Lossless tapered lines 406

24.4 Transients on transmission lines 408

24.5 Plane waves (oblique incidence) 411

24.6 Waveguides 413

24.7 The scattering matrix 418

24.8 Directional couplers 421

24.9 Resonators 421

25 Practice Problems 435

25.1 Statics 435

25.2 Quasistatics 448

25.3 Plane waves 458

25.5 Transmission lines 472

25.6 Waveguides 481

25.7 Junctions and couplers 485

25.8 Resonators 490

25.9 Ferrites 491

25.10 Four-dimensional electromagnetics 496

PART IV BACKMATTER

Summary 505

Electromagnetic Luminaries 511

Appendix A 521

A.1 Theory of Special Relativity 521

A.2 Transformations between fixed and moving coordinates 530

Appendix B 537

B.1 The unit step and uk (t ) functions 537

B.2 Three-dimensional vector identities and theorems 538

B.3 Four-dimensional vector and tensor identities 543

B.4 Four-space identities 544

Appendix C 547

C.1 Stationary spatially symmetric sources 547

C.2 Multipole expansions of static fields 550

C.3 Averaging property of Laplace’s Equation 553

C.4 Solutions of Laplace’s Equation 554

C.5 Laplace’s Equation in N dimensions 558

C.6 Ellipsoids in uniform fields 559

Appendix D 563

D.1 Alternate power, energy, stress, and momentum 563

D.2 Minkowski representations 568

D.3 Stress-momentum representations of torque 571

Appendix E 577

E.1 Fields of specified charges and currents 577

E.2 Fields of a moving point charge 578

E.3 Method of images 583

E.4 Characteristic impedances of TEM transmission lines 586

Appendix F 593

F.1 Bessel functions 593

F.2 Chebyshev polynomials 598

F.3 Hermite polynomials 600

Appendix G 601

G.1 Macsyma and Maxima 601

G.2 Macsyma program descriptions 602

G.3 Macsyma notebooks 605

G.4 Text of Macsyma/Maxima batch program 608

Appendix H 619

H.1 Animated fields of surface currents 619

H.2 Animated fields of a cylindrical volume current, Jz (t ) = Jou-1(t ) 620

H.3 Animated fields of a cylindrical surface current, Kz (t ) = Kou-1(t ) 621

H.4 Animated fields of line-current transients 622

H.5 Animated field of a radiating Hertzian dipole 623

H.6 Animated beauty-power fluxes of cylindrical waveguide modes 623

H.7 Macsyma animations and graphics 624

References 627

Index 631

Frederic R. Morgenthaler, PhD, joined the faculty of the Massachusetts Institute of Technology in 1960, becoming a Full Professor in 1968. He retired from MIT in 1996 and is currently Professor Emeritus of Electrical Engineering. Dr. Morgenthaler has served as a consultant to the U.S. government as well as private industry. A Fellow of the IEEE and the holder of approximately one dozen patents, Dr. Morgenthaler has authored over 100 scientific publications and papers.

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