Introduction to Electromagnetic Waves with Maxwell's Equations

Discover an innovative and fresh approach to teaching classical electromagnetics at a foundational level

Introduction to Electromagnetic Waves with Maxwell’s Equations delivers an accessible and practical approach to teaching the wellknown topics all electromagnetics instructors must include in their syllabus. Based on the author’s decades of experience teaching the subject, the book is carefully tuned to be relevant to an audience of engineering students who have already been exposed to the basic curricula of linear algebra and multivariate calculus.

Forming the backbone of the book, Maxwell’s equations are developed step-by-step in consecutive chapters, while related electromagnetic phenomena are discussed simultaneously. The author presents accompanying mathematical tools alongside the material provided in the book to assist students with retention and comprehension. The book contains over 100 solved problems and examples with stepwise solutions offered alongside them. An accompanying website provides readers with additional problems and solutions.

Readers will also benefit from the inclusion of:

• A thorough introduction to preliminary concepts in the field, including scalar and vector fields, cartesian coordinate systems, basic
• vector operations, orthogonal coordinate systems, and electrostatics, magnetostatics, and electromagnetics
• An exploration of Gauss’ Law, including integral forms, differential forms, and boundary conditions
• A discussion of Ampere’s Law, including integral and differential forms and Stoke’s Theorem
• An examination of Faraday’s Law, including integral and differential forms and the Lorentz Force Law

Perfect for third-and fourth-year undergraduate students in electrical engineering, mechanical engineering, applied maths, physics, and computer science, Introduction to Electromagnetic Waves with Maxwell’s Equations will also earn a place in the libraries of graduate and postgraduate students in any STEM program with applications in electromagnetics.

Ozgur Ergul, PhD, is Professor at the Middle East Technical University in Ankara, Turkey. His research focus is on the development of fast and accurate algorithms for the solution of electromagnetics problems involving large and complicated structures, integral equations, iterative methods, parallel programming, and high-performance computing.

Preface 15

Mathematical Notation 23

List of Symbols 27

Special Functions 31

Frequently Used Identities 33

Tools to Understand Maxwell’s Equations 37

0 Preliminary 39

0.1 Scalar and Vector Fields 40

0.2 Cartesian Coordinate Systems 42

0.3 Basic Vector Operations 42

0.4 Orthogonal Coordinate Systems 43

0.4.1 Properties of a Cartesian Coordinate System 43

0.4.2 Cylindrical Coordinate System 44

0.4.3 Spherical Coordinate System 45

0.5 Electrostatics, Magnetostatics, and Electromagnetics 47

0.6 Time in Electromagnetics 49

0.7 Final Remarks 51

1 Gauss’ Law 53

1.1 Integral Form of Gauss’ Law 54

1.1.1 Differential Surface With Direction 55

1.1.2 Dot Product 56

1.1.3 Flux of Vector Fields 62

1.1.4 Meaning of Gauss’ Law and Its Application 66

1.1.5 Examples 67

1.2 Using the Integral Form of Gauss’ Law 69

1.2.1 Examples 71

1.3 Differential Form of Gauss’ Law 73

1.3.1 Electric Charge Density 73

1.3.2 Divergence of Vector Fields 75

1.3.3 Divergence Theorem and the Differential Form of Gauss’ Law 81

1.3.4 Examples 83

1.4 Using the Differential Form of Gauss’ Law 85

1.4.1 Examples 88

1.5 Boundary Conditions for Normal Electric Fields 89

1.6 Static Cases and Coulomb’s Law 92

1.6.1 Superposition Principle 93

1.6.2 Coulomb’s Law and Electric Force 99

1.6.3 Examples 101

1.7 Gauss’ Law and Dielectrics 106

1.7.1 Electric Dipole 112

1.7.2 Polarization 113

1.7.3 Equivalent Polarization Charges 115

1.7.4 Examples 120

1.8 Final Remarks 123

1.9 Exercises 124

1.10 Questions 127

2 Ampere’s Law 133

2.1 Integral Form of Ampere’s Law 134

2.1.1 Differential Length With Direction 135

2.1.2 Circulation of Vector Fields 137

2.1.3 Meaning of Ampere’s Law and Its Application 140

2.1.4 Examples 143

2.2 Using the Integral Form of Ampere’s Law 145

2.2.1 Examples 147

2.3 Differential Form of Ampere’s Law 151

2.3.1 Electric Current Density 152

2.3.2 Cross Product 154

2.3.3 Curl of Vector Fields 157

2.3.4 Stoke’s Theorem and the Differential Form of Ampere’s Law 164

2.3.5 Examples 165

2.4 Using the Differential Form of Ampere’s Law 169

2.4.1 Examples 172

2.5 Boundary Conditions for Tangential Magnetic Fields 173

2.6 Gauss’ Law and Ampere’s Law 176

2.7 Static Cases, Biot-Savart Law, and Ampere’s Force Law 179

2.7.1 Superposition Principle 180

2.7.2 Ampere’s Force Law and Magnetic Force 190

2.7.3 Examples 194

2.8 Ampere’s Law and Magnetic Materials 200

2.8.1 Magnetic Dipole 206

2.8.2 Magnetization 208

2.8.3 Equivalent Magnetization Currents 210

2.8.4 Examples 217

2.9 Final Remarks 218

2.10 Exercises 219

2.11 Questions 221

3 Faraday’s Law 225

3.1 Integral Form of Faraday’s Law 226

3.1.1 Meaning of Faraday’s Law and Its Application 227

3.1.2 Lorentz Force Law 229

3.2 Using the Integral Form of Faraday’s Law 231

3.2.1 Examples 236

3.3 Differential Form of Faraday’s Law 240

3.4 Boundary Conditions for Tangential Electric Fields 242

3.5 Combining Faraday’s Law with Gauss’ and Ampere’s Laws 244

3.6 Static Cases and Electric Scalar Potential 246

3.6.1 Gradient of Scalar Fields 248

3.6.2 Examples 252

3.6.3 Gradient Theorem 253

3.6.4 Gradient in Gauss’ Law, Ampere’s Law, and Faraday’s Law 254

3.6.5 Electric Potential Energy 257

3.6.5.1 Electric Potential Energy of Discrete Charge Distributions 261

3.6.5.2 Stored Electric Potential Energy by an Electric Dipole 263

3.6.5.3 Stored Electric Potential Energy in Charge Distributions 265

3.6.5.4 Electric Potential Energy and Electric Force 269

3.6.6 Examples 272

3.6.7 Poisson’s Equation and Laplace’s Equation 276

3.6.8 Examples 283

3.6.9 Finding Electric Scalar Potential From Electric Field Intensity 283

3.6.10 Examples 286

3.6.11 Electrostatic Boundary Value Problems 288

3.6.12 Examples 291

3.7 Final Remarks 294

3.8 Exercises 294

3.9 Questions 296

4 Gauss’ Law for Magnetic Fields 299

4.1 Integral and Differential Forms of Gauss’ law for Magnetic Fields 300

4.1.1 Meaning of Gauss’ law for Magnetic Fields 302

4.1.2 Examples 304

4.2 Boundary Conditions for Normal Magnetic Fields 306

4.2.1 Examples 307

4.3 Static Cases and Magnetic Vector Potential 308

4.3.1 Magnetic Vector Potential and Coulomb’s Gauge 309

4.3.2 Examples 318

4.3.3 Magnetic Potential Energy 321

4.3.3.1 Magnetic Potential Energy of Discrete Current Distributions 323

4.3.3.2 Stored Magnetic Potential Energy by a Magnetic Dipole 324

4.3.3.3 Stored Magnetic Potential Energy in Current Distributions 326

4.3.3.4 Magnetic Potential Energy and Magnetic Force 329

4.3.4 Examples 332

4.4 Combining All Maxwell’s Equations 334

4.4.1 Wave Equations 336

4.4.2 Wave Equations for Potentials 343

4.4.3 Time-Harmonic Sources and Helmholtz Equations 349

4.4.4 Examples 354

4.5 Final Remarks 359

4.6 Exercises 360

4.7 Questions 363

5 Basic Solutions of Maxwell’s Equations 365

5.1 Summary of Maxwell’s Equations, Wave Equations, and Helmholtz Equations 366

5.1.1 Examples 375

5.2 Electromagnetic Propagation and Radiation 377

5.2.1 Hertzian Dipole 382

5.2.2 Examples 385

5.3 Plane Waves 389

5.3.1 Examples 400

5.3.2 Polarization of Plane Waves 401

5.3.3 Examples 407

5.3.4 Power of Plane Waves 409

5.3.5 Reflection and Refraction of Plane Waves 412

5.3.6 General Case for Reflection and Refraction 416

5.3.6.1 Perpendicular Polarization 418

5.3.6.2 Parallel Polarization 421

5.3.7 Examples 423

5.3.8 Total Internal Reflection 427

5.3.9 Total Transmission 430

5.3.10 Examples 434

5.3.11 Reflection and Transmission for Two Parallel Interfaces 437

5.4 Final Remarks 440

5.5 Exercises 440

5.6 Questions 443

6 Analyses of Conducting Objects 447

6.1 Ohm’s Law 449

6.2 Joule’s Law 452

6.3 Relaxation Time 453

6.4 Boundary Conditions for Conducting Media 456

6.5 Analyses of Perfectly Conducting Objects 457

6.5.1 Electric Scalar Potential for PECs 458

6.5.2 Boundary Conditions for PECs 458

6.5.3 Basic Responses of PECs 460

6.5.4 Concerns in Geometric Representations of PECs 462

6.5.5 Electrostatics for PECs 464

6.5.6 Method of Images 466

6.5.7 Examples 470

6.6 Maxwell’s Equations in Conducting Media 474

6.6.1 Complex Permittivity 476

6.6.2 Power and Energy in Conducting Media 478

6.6.3 Plane Waves in Conducting Media 479

6.6.4 Power of Plane Waves in Conducting Media 483

6.6.5 Reflection from PECs 484

6.6.6 Examples 494

6.7 Capacitance 503

6.7.1 Capacitance and Electric Potential Energy 504

6.7.2 Parallel-Plate Capacitors 505

6.7.3 Spherical Capacitors 513

6.7.4 Cylindrical Capacitors 518

6.7.5 Examples 520

6.8 Resistance 528

6.8.1 Examples 535

6.9 Inductance 544

6.9.1 Examples 553

6.10 Final Remarks 559

6.11 Exercises 560

6.12 Questions 565

7 Transmission of Electromagnetic Waves 569

7.1 Antennas and Wireless Transmission 570

7.1.1 Basic Properties of Antennas 571

7.1.2 Antenna Design Parameters 582

7.1.3 Antenna Types 585

7.1.3.1 Antenna Arrays 588

7.1.4 Friis Transmission Equation 600

7.1.5 Examples 603

7.2 Waveguides 613

7.2.1 Transverse and Axial Fields 614

7.2.2 Rectangular Waveguides 617

7.2.2.1 Transverse Magnetic Modes 618

7.2.2.2 Transverse Electric Modes 620

7.2.2.3 Non-Existing Modes 623

7.2.2.4 Important Properties of Modes 624

7.2.3 Parallel-Plate Waveguides 628

7.2.4 Examples 630

7.3 Transmission Line Theory 635

7.3.1 Telegrapher’s Equations 637

7.3.1.1 Transmission Line With a Load 641

7.3.1.2 Special Cases 643

7.3.1.3 Common Cases 646

7.3.2 Voltage and Current Patterns 647

7.3.3 Examples 651

7.4 Concluding Remarks 658

7.5 Exercises 658

7.6 Questions 663

8 Concluding Chapter 669

8.1 Electromagnetic Spectrum 670

8.1.1 Radio Waves (3 Hz to 300 GHz) 671

8.1.2 Microwaves (300 MHz to 300 GHz) 672

8.1.3 Infrared Radiation (300 GHz to 400 THz) 673

8.1.4 Visible Range (400 THz to 800 THz) 674

8.1.5 Ultraviolet Radiation (800 THz to 30 PHz) 675

8.1.6 X-Rays (30 PHz to 30 EHz) 676

8.1.7 Gamma Rays (Above 30 EHz) 678

8.2 Brief History of Electromagnetism (Electricity, Magnetism, and a Little Optics) 679

8.3 Electromagnetism in Action 685

8.3.1 Snapshots From Nature 686

8.3.1.1 Blue Sky, Bright Sun, Red Sunset 686

8.3.1.2 Rainbow in Pocket 687

8.3.1.3 Green Leaf, Red Apple, Blue Sea 688

8.3.1.4 Electromagnetic Waves From Space 689

8.3.1.5 Magnetic Earth 690

8.3.2 Snapshots From Technology 691

8.3.2.1 Telegraph to Cellular Phones 691

8.3.2.2 Home: Where Electromagnetism Happens 693

8.3.2.3 Looking Inside Body 694

8.3.2.4 Seeing World with Sensors and Radars 696

8.3.2.5 Atoms Under Microscope 699

8.4 How to Solve Maxwell’s Equations 700

8.4.1 Full-Wave Methods 705

8.4.1.1 Differential-Equation Solvers 706

8.4.1.1.1 Finite-Difference Time-Domain Method (FDTD): 706

8.4.1.1.2 Finite Element Method (FEM): 707

8.4.1.2 Integral-Equation Solvers 708

8.4.1.2.1 Method of Moments (MoM): 709

8.4.1.2.2 Acceleration Algorithms: 709

8.4.1.2.3 FMM and MLFMA: 711

8.4.2 Asymptotic Techniques 711

8.4.2.0.1 Quasistatic Approximations: 712

8.4.2.0.2 Geometrical Optics: 713

8.4.2.0.3 Uniform Geometrical Theory of Diffraction: 713

8.4.2.0.4 Physical Optics: 714

Bibliography 717

Index 725

The New copy of this book will include any supplemental materials advertised. Please check the title of the book to determine if it should include any access cards, study guides, lab manuals, CDs, etc.

The Used, Rental and eBook copies of this book are not guaranteed to include any supplemental materials. Typically, only the book itself is included. This is true even if the title states it includes any access cards, study guides, lab manuals, CDs, etc.