# Electromagnetics, 1st edition

• Branislav M. Notaros

1st edition

## Overview

Electromagnetics is a thorough text that enables readers to readily grasp EM fundamentals, develop true problem-solving skills, and really understand and like the material. It is meant as an "ultimate resource" for undergraduate electromagnetics.

Preface xi

Chapter 1 Electrostatic Field in Free Space 1
1.1 Coulomb’s Law 2
1.2 Definition of the Electric Field Intensity Vector 7
1.3 Continuous Charge Distributions 8
1.4 On the Volume and Surface Integration 9
1.5 Electric Field Intensity Vector due to Given Charge Distributions 10
1.6 Definition of the Electric Scalar Potential 16
1.7 Electric Potential due to Given Charge Distributions 18
1.8 Voltage 21
1.9 Differential Relationship between the Field and Potential in Electrostatics 22
1.11 3-D and 2-D Electric Dipoles 26
1.12 Formulation and Proof of Gauss’ Law 28
1.13 Applications of Gauss’ Law 31
1.14 Differential Form of Gauss’ Law 35
1.15 Divergence 36
1.16 Conductors in the Electrostatic Field 39
1.17 Evaluation of the Electric Field and Potential due to Charged Conductors 43
1.18 Electrostatic Shielding 46
1.19 Charge Distribution on Metallic Bodies of Arbitrary Shapes 48
1.20 Method of Moments for Numerical Analysis of Charged Metallic Bodies 49
1.21 Image Theory 51

Chapter 2 Dielectrics, Capacitance, and Electric Energy 61
2.1 Polarization of Dielectrics 62
2.2 Polarization Vector 63
2.3 Bound Volume and Surface Charge Densities 64
2.4 Evaluation of the Electric Field and Potential due to Polarized Dielectrics 68
2.5 Generalized Gauss’ Law 70
2.6 Characterization of Dielectric Materials 71
2.7 Maxwell’s Equations for the Electrostatic Field 75
2.8 Electrostatic Field in Linear, Isotropic, and Homogeneous Media 75
2.9 Dielectric-Dielectric Boundary Conditions 79
2.10 Poisson’s and Laplace’s Equations 82
2.11 Finite-Difference Method for Numerical Solution of Laplace’s Equation 84
2.12 Definition of the Capacitance of a Capacitor 86
2.13 Analysis of Capacitors with Homogeneous Dielectrics 88
2.14 Analysis of Capacitors with Inhomogeneous Dielectrics 95
2.15 Energy of an Electrostatic System 102
2.16 Electric Energy Density 104
2.17 Dielectric Breakdown in Electrostatic Systems 108

Chapter 3 Steady Electric Currents 124
3.1 Current Density Vector and Current Intensity 125
3.2 Conductivity and Ohm’s Law in Local Form 128
3.3 Losses in Conductors and Joule’s Law in Local Form 132
3.4 Continuity Equation 133
3.5 Boundary Conditions for Steady Currents 137
3.6 Distribution of Charge in a Steady Current Field 138
3.7 Relaxation Time 139
3.8 Resistance, Ohm’s Law, and Joule’s Law 140
3.9 Duality between Conductance and Capacitance 146
3.10 External Electric Energy Volume Sources and Generators 149
3.11 Analysis of Capacitors with Imperfect Inhomogeneous Dielectrics 152
3.12 Analysis of Lossy Transmission Lines with Steady Currents 156
3.13 Grounding Electrodes 162

Chapter 4 Magnetostatic Field in Free Space 173
4.1 Magnetic Force and Magnetic Flux Density Vector 174
4.2 Biot-Savart Law 177
4.3 Magnetic Flux Density Vector due to Given Current Distributions 179
4.4 Formulation of Ampère’s Law 185
4.5 Applications of Ampère’s Law 187
4.6 Differential Form of Ampère’s Law 193
4.7 Curl 195
4.8 Law of Conservation of Magnetic Flux 198
4.9 Magnetic Vector Potential 201
4.10 Proof of Ampère’s Law 204
4.11 Magnetic Dipole 206
4.12 The Lorentz Force and Hall Effect 209
4.13 Evaluation of Magnetic Forces 211

Chapter 5 Magnetostatic Field in Material Media 221
5.1 Magnetization Vector 222
5.2 Behavior and Classification of Magnetic Materials 223
5.3 Magnetization Volume and Surface Current Densities 227
5.4 Generalized Ampère’s Law 234
5.5 Permeability of Magnetic Materials 236
5.6 Maxwell’s Equations and Boundary Conditions for the Magnetostatic Field 239
5.7 Image Theory for the Magnetic Field 241
5.8 Magnetization Curves and Hysteresis 243
5.9 Magnetic Circuits — Basic Assumptions for the Analysis 247
5.10 Kirchhoff’sLaws for Magnetic Circuits 250
5.11 Maxwell’s Equations for the Time-Invariant Electromagnetic Field 258

Chapter 6 Slowly Time-Varying Electromagnetic Field 263
6.1 Induced Electric Field Intensity Vector 264
6.2 Slowly Time-Varying Electric and Magnetic Fields 269
6.3 Faraday’s Law of Electromagnetic Induction 271
6.4 Maxwell’s Equations for the Slowly Time-Varying Electromagnetic Field 276
6.5 Computation of Transformer Induction 277
6.6 Electromagnetic Induction due to Motion 283
6.7 Total Electromagnetic Induction 289
6.8 Eddy Currents 294

Chapter 7 Inductance and Magnetic Energy 311
7.1 Self-Inductance 312
7.2 Mutual Inductance 318
7.3 Analysis of Magnetically Coupled Circuits 324
7.4 Magnetic Energy of Current-Carrying Conductors 331
7.5 Magnetic Energy Density 334
7.6 Internal and External Inductance in Terms of Magnetic Energy 342

Chapter 8 Rapidly Time-Varying Electromagnetic Field 351
8.1 Displacement Current 352
8.2 Maxwell’s Equations for the Rapidly Time-Varying Electromagnetic Field 357
8.3 Electromagnetic Waves 361
8.4 Boundary Conditions for the Rapidly Time-Varying Electromagnetic Field 363
8.5 Different Forms of the Continuity Equation for Rapidly Time-Varying Currents 364
8.6 Time-Harmonic Electromagnetics 366
8.7 Complex Representatives of Time-Harmonic Field and Circuit Quantities 369
8.8 Maxwell’s Equations in Complex Domain 373
8.9 Lorenz Electromagnetic Potentials 376
8.10 Computation of High-Frequency Potentials and Fields in Complex Domain 381
8.11 Poynting’s Theorem 389
8.12 Complex Poynting Vector 397

Chapter 9 Uniform Plane Electromagnetic Waves 408
9.1 Wave Equations 409
9.2 Uniform-Plane-Wave Approximation 411
9.3 Time-Domain Analysis of Uniform Plane Waves 412
9.4 Time-Harmonic Uniform Plane Waves and Complex-Domain Analysis 416
9.5 The Electromagnetic Spectrum 425
9.6 Arbitrarily Directed Uniform TEM Waves 427
9.7 Theory of Time-Harmonic Waves in Lossy Media 429
9.8 Explicit Expressions for Basic Propagation Parameters 433
9.9 Wave Propagation in Good Dielectrics 436
9.10 Wave Propagation in Good Conductors 439
9.11 Skin Effect 441
9.12 Wave Propagation in Plasmas 447
9.13 Dispersion and Group Velocity 452
9.14 Polarization of Electromagnetic Waves 458

Chapter 10 Reflection and Transmission of Plane Waves 471
10.1 Normal Incidence on a Perfectly Conducting Plane 472
10.2 Normal Incidence on a Penetrable Planar Interface 483
10.3 Surface Resistance of Good Conductors 492
10.4 Perturbation Method for Evaluation of Small Losses 497
10.5 Oblique Incidence on a Perfect Conductor 499
10.6 Concept of a Rectangular Waveguide 505
10.7 Oblique Incidence on a Dielectric Boundary 507
10.8 Total Internal Reflection and Brewster Angle 513
10.9 Wave Propagation in Multilayer Media 520

Chapter 11 Field Analysis of Transmission Lines 533
11.1 TEM Waves in Lossless Transmission Lines with Homogeneous Dielectrics 534
11.2 Electrostatic and Magnetostatic Field Distributions in Transversal Planes 538
11.3 Currents and Charges of Line Conductors 539
11.4 Analysis of Two-Conductor Transmission Lines 540
11.5 Transmission Lines with Small Losses 547
11.6 Attenuation Coefficients for Line Conductors and Dielectric 550
11.7 High-Frequency Internal Inductance of Transmission Lines 556
11.8 Evaluation of Primary and Secondary Circuit Parameters of Transmission Lines 557
11.9 Transmission Lines with Inhomogeneous Dielectrics 563
11.10 Multilayer Printed Circuit Board 567

Chapter 12 Circuit Analysis of Transmission Lines 576
12.1 Telegrapher’s Equations and Their Solution in Complex Domain 577
12.2 Circuit Analysis of Lossless Transmission Lines 581
12.3 Circuit Analysis of Low-Loss Transmission Lines 581
12.4 Reflection Coefficient for Transmission Lines 583
12.5 Power Computations of Transmission Lines 589
12.6 Transmission-Line Impedance 592
12.7 Complete Solution for Line Voltage and Current 597
12.8 Short-Circuited, Open-Circuited, and Matched Transmission Lines 601
12.9 Transmission-Line Resonators 608
12.10 Quality Factor of Resonators with Small Losses 610
12.11 The Smith Chart — Construction and Basic Properties 614
12.12 Circuit Analysis of Transmission Lines Using the Smith Chart 618
12.13 Transient Analysis of Transmission Lines 628
12.14 Thévenin Equivalent Generator Pair and Reflection Coefficients for Line Transients 630
12.15 Step Response of Transmission Lines with Purely Resistive Terminations 634
12.16 Analysis of Transmission Lines with Pulse Excitations 640
12.17 Bounce Diagrams 646
12.18 Transient Response for Reactive or Nonlinear Terminations 649

Chapter 13 Waveguides and Cavity Resonators 662
13.1 Analysis of Rectangular Waveguides Based on Multiple Reflections of Plane Waves 663
13.2 Propagating and Evanescent Waves 666
13.3 Dominant Waveguide Mode 668
13.4 General TE Modal Analysis of Rectangular Waveguides 671
13.5 TM Modes in a Rectangular Waveguide 676
13.6 Cutoff Frequencies of Arbitrary Waveguide Modes 677
13.7 Wave Impedances of TE and TM Waves 680
13.8 Power Flow along a Waveguide 681
13.9 Waveguides with Small Losses 684
13.10 Waveguide Dispersion and Wave Velocities 688
13.11 Waveguide Couplers 692
13.12 Rectangular Cavity Resonators 696
13.13 Electromagnetic Energy Stored in a Cavity Resonator 700
13.14 Quality Factor of Rectangular Cavities with Small Losses 703

Chapter 14 Antennas and Wireless Communication Systems 713
14.1 Electromagnetic Potentials and Field Vectors of a Hertzian Dipole 715
14.2 Far Field and Near Field 720
14.3 Steps in Far-Field Evaluation of an Arbitrary Antenna 722
14.6 Antenna Directivity and Gain 740
14.7 Antenna Polarization 745
14.8 Wire Dipole Antennas 745
14.9 Image Theory for Antennas above a Perfectly Conducting Ground Plane 751
14.10 Monopole Antennas 754
14.11 Magnetic Dipole (Small Loop) Antenna 758
14.12 Theory of Receiving Antennas 760
14.13 Antenna Effective Aperture 766
14.14 Friis Transmission Formula for a Wireless Link 768
14.15 Antenna Arrays 772

APPENDICES
1 Quantities, Symbols, Units, and Constants 791
2 Mathematical Facts and Identities 796
3 Vector Algebra and Calculus Index 801
4 Answers to Selected Problems 802
Bibliography 806
Index 809

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