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The Power and Beauty of Electromagnetic Fields
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Table of Contents

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 14.3 Additional pulse waveforms 217 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 25.4 Radiation and diffraction 462 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 About the Author 519 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

About the Author

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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