Electromagnetic field theories for engineering

Electromagnetic Field Theories for Engineering

Md. Abdus Salam

Electromagnetic Field

Theories for Engineering

2123

Md. Abdus Salam

Department of Electrical and Electronic Engineering

Institute Technology Brunei

Darussalam

Brunei Darussalam

ISBN 978-981-4585-65-1 ISBN 978-981-4585-66-8 (eBook)

DOI 10.1007/978-981-4585-66-8

Springer Singapore Heidelberg New York Dordrecht London

Library of Congress Control Number: 2014932423

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Springer is part of Springer Science+Business Media (www.springer.com)

To all my teachers, colleagues, and students

who have encouraged and helped me to

develop professionally over the years. Also,

to my wife Asma Ara Bagum, my son Syeed

Hasan, and my daughters, Yusra binti Salam

and Sundus Salam for their love, patience,

and support.

Preface

Electrical engineering plays an important role in modernizing human life and

encompasses wide areas such as: generation, transmission, and distribution of elec￾trical power, digital systems, satellite communications, signal processing, robotics,

mechatronics, computer, control, artificial intelligence, and networks.

A 4 year electrical and electronic engineering curriculum normally contains two

modules of electromagnetic field theories during the first 2 years. However, some

curricula do not have enough slots to accommodate the two modules. This book,

Electromagnetic Field Theories, is designed for electrical and electronic engineer￾ing undergraduate students to provide fundamental knowledge of electromagnetic

fields and waves in a structured manner. A comprehensive fundamental knowledge

of electric and magnetic fields is required to understand the working principles of

generators, motors, and transformers. This knowledge is also necessary to analyze

transmission lines, substations, insulator flashover mechanism, transient phenomena,

etc.

This book is written in a simple way so that the students will find it easy to under￾stand the electromagnetic field theory and its application in electrical engineering.

Several worked out examples are included to enhance the understanding of electro￾magnetic field theories. Each chapter also includes several practice problems with

answers given at the end of the book, which would facilitate students’understanding.

The basic parameters in electromagnetic fields are discussed in Chap. 1, while

vector calculus and orthogonal coordinate systems are explained in Chap. 2. In

Chap. 3, the basics of electrostatics, Coulomb’s law, electric field intensity, Gauss’

law, Ohm’s law, and energy have been discussed. Poisson’s and Laplace’s equations,

uniqueness theorem, and their analysis on geometric shapes have been introduced

in Chap. 4. The current and its density, resistance, capacitance, continuity equation,

etc., have been discussed in Chap. 5. Chapter 6 explains Lorentz’s force, magnetic

flux density, Biot–Savart law, Ampere’s circuital law, vector magnetic potential, air

gap, and series and parallel magnetic circuit. Faraday’s law, conduction current,

displacement current, Maxwell’s equation, and basics of transformer, have been

discussed in Chap. 7. Chapter 8 deals with transmission line equations, velocity

of wave propagation, wavelengths, lossless propagation, distortionless transmission

line, power, and Smith chart. Plane waves and its analysis are included in Chap. 9,

and basics of antenna have been discussed in Chap. 10.

vii

viii Preface

Features

Several textbooks on electromagnetic theories already exist in the market. However,

the book on Electromagnetic Field Theories for Engineering is written for electrical

and electronic engineering students with the following key features.

• Easy and logical presentation of each article

• Interpretation of each theory with proper mathematical expressions

• Emphasis on engineering mathematics to understand electromagnetic field

theories

• Detailed description of fundamental laws of electromagnetic field theories

• Step-by-step problem solving procedures

• Inclusion of solved examples and practice problems

• Large number of exercise problems at the end of each chapter

• Inclusion of answers to practice and exercise problems

Aids for Instructors

The solution manual will be provided to instructors who will adopt this as a textbook,

and they may obtain the solution manual by directly contacting the publishers.

Acknowledgments

The author has written this textbook based on his years of teaching experience. The

author would like to acknowledge with gratitude the following faculty members

for their inspiration, comments, and suggestions during the preparation of the first

edition of this book.

Dr. M. H. Rashid, Professor, University of West Florida, USA

Dr. Akhtar Kalam, Professor, Victoria University, Australia

Dr. M. Saifur Rahman, Professor, Virginia Tech and State University, USA

Dr. Jim Cathey, Professor, University of Kentucky, Lexington, USA

Dr. Mohammod Ali, Professor, University of South Carolina, USA

Dr. M. Bashir Uddin, Professor, Dhaka University of Engineering andTechnology,

Gazipur, Bangladesh

Dr. M. A. Rahman, Professor, Memorial University of Newfoundland, Canada

Dr. S. M. Islam, Professor, Curtin University of Technology, Perth, Australia

Dr. Hussein Ahmad, Professor, Universiti Teknologi Malaysia

Dr. M. M. A. Hashem, Professor, Khulna University of Engineering and

Technology, Bangladesh

Dr. Khaled Ellithy, Professor, Qatar University, Doha, Qatar

Dr. Saleh Al Alawi, Professor, Sultan Qaboos University, Oman

Preface ix

Dr. Md. Rafiqul Islam, Associate Professor, International Islamic University,

Malaysia

Dr. Mohammad A. Kashem, Associate Professor, University of Wollongong,

NSW, Australia

Dr. Quazi Delwar Hossain, Assistant Professor, Chittagong University of

Engineering and Technology, Bangladesh

Dr. Q. M. Rahman, Assistant Professor, Western University, London, Canada

The author would like to thank Dk. Nurul Saidatul Mirzuana and Umi Farina,

B. Eng. students for typing the solution manual of this book. The author would

also like to thank Dr. Loyola D’Silva, Production Manager, Springer Asia Pvt. Ltd.,

Singapore, and the production staff for their help in bringing the first edition of the

book to completion.

Brunei Darussalam Md. Abdus Salam PhD

January 2014

Contents

1 Basics of Electromagnetics ....................................... 1

1.1 Introduction ............................................... 1

1.2 Field Parameters and SI Units ................................ 1

1.2.1 Electric Flux Density and Field Intensity ............... 2

1.2.2 Magnetic Flux Density and Field Intensity ............. 3

1.2.3 Current Density.................................... 4

1.3 Exercise Problems .......................................... 4

Bibliography .................................................... 5

2 Vector Analysis and Coordinate Systems ........................... 7

2.1 Introduction ............................................... 7

2.2 Vectors and Scalars ......................................... 7

2.3 Vector Components ......................................... 8

2.4 Unit Vectors ............................................... 9

2.5 Vector Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.6 Vector Subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.7 Vectors Multiplication and Division . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.8 Dot Product of Two Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.9 Cross Product of Two Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.10 Orthogonal Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.10.1 Cartesian Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . 18

2.10.2 Circular Cylindrical Coordinate System . . . . . . . . . . . . . . . 22

2.10.3 Spherical Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . 28

2.11 Potential Gradient and Gradient of a Scalar Field . . . . . . . . . . . . . . . . 35

2.12 Divergence of a Vector Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.13 Curl of a Vector Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.14 Two Important Vector Identities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.15 Exercise Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

xi

xii Contents

3 Electrostatic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.2 Coulomb’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.3 Electric Field Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.4 Gauss’ Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.5 Electric Field of Continuous Charge Distribution . . . . . . . . . . . . . . . . 60

3.6 Electric Field Due to an Infinite Sheet Charge . . . . . . . . . . . . . . . . . . . 63

3.7 Electric Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.8 Derivation of Electric Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.9 Line Integral of Irrotational Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.10 Potential Due to a Point Charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.11 Electric Dipole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.12 Materials for Static Electric Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3.13 Dielectric Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

3.14 Dielectric Material Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

3.15 Dielectric Boundary Conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

3.16 Refraction of Electric Field at Dielectric Boundary. . . . . . . . . . . . . . . 83

3.17 Electrostatic Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

3.18 Exercise Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4 Poisson’s and Laplace’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.2 Derivation of Poisson’s and Laplace’s Equations. . . . . . . . . . . . . . . . . 91

4.3 Uniqueness Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

4.4 Solutions of Laplace’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.4.1 One-Dimension Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.4.2 Two-Dimension Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

4.5 Solution of Laplace’s Equation in Cylindrical Coordinates . . . . . . . . 104

4.6 Solutions of Poisson’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

4.7 Numerical Solution of Laplace’s Equation . . . . . . . . . . . . . . . . . . . . . . 106

4.8 Exercise Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

5 Electric Currents. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

5.2 Current and Current Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

5.3 Conductivity and Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

5.4 Power and Joule’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

5.5 Continuity Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

5.6 Current Density Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 125

5.7 Capacitance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

5.8 Parallel Plate Capacitor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

5.9 Determination of Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

5.10 Coaxial Capacitor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

Contents xiii

5.11 Spherical Capacitor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

5.12 Parallel Plate Capacitor with Two Dielectric Slabs . . . . . . . . . . . . . . . 136

5.13 Exercise Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

6 Static Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

6.2 Magnetic Flux Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

6.3 Biot–Savart’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

6.4 Magnetic Field of a Long Straight Conductor . . . . . . . . . . . . . . . . . . . 144

6.5 Ampere’s Circuital Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

6.6 Ampere’s Circuital Law in a Long Straight Conductor . . . . . . . . . . . . 149

6.7 Infinite Sheet of Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

6.8 Curl of a Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

6.9 Scalar and Vector Magnetic Potential . . . . . . . . . . . . . . . . . . . . . . . . . . 159

6.10 Magnetization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

6.11 Magnetic Field Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 164

6.12 Magnetic Field of Two Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

6.13 Magnetic Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

6.14 Series Magnetic Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

6.15 Parallel Magnetic Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

6.16 Magnetic Circuit with Air Gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

6.17 Hysteresis Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

6.18 Inductance and Mutual Inductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

6.19 Exercise Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

7 Time-Varying Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

7.2 Faraday’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

7.3 Motional Voltage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

7.4 Maxwell’s Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

7.5 Conduction and Displacement Currents . . . . . . . . . . . . . . . . . . . . . . . . 190

7.6 Maxwell’s Equation from Ampere’s Law . . . . . . . . . . . . . . . . . . . . . . . 192

7.7 Transformer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

7.8 Time-Varying Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

7.9 Field of a Series Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

7.10 Time-Harmonic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

7.11 Exercise Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

8 Transmission Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

8.2 Transmission Line Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

8.3 Phasor Form Solution of Transmission Line Equation . . . . . . . . . . . . 212

xiv Contents

8.4 Lossless Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

8.5 Low-Loss Transmission Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

8.6 Distortionless Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

8.7 Determination of Attenuation Constant. . . . . . . . . . . . . . . . . . . . . . . . . 223

8.8 A Finite Transmission Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

8.9 Input Impedance for Lossless Transmission Line . . . . . . . . . . . . . . . . 229

8.10 Power of Lossless Transmission Line . . . . . . . . . . . . . . . . . . . . . . . . . . 234

8.11 Basics of Smith Chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

8.12 Exercise Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

9 Uniform Plane Waves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

9.2 Time-Domain Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

9.3 Wave Equation in Time-Harmonic Fields . . . . . . . . . . . . . . . . . . . . . . . 245

9.4 Solution of a Wave Equation in the Frequency Domain . . . . . . . . . . . 246

9.5 Solution of a Wave Equation in the Time Domain . . . . . . . . . . . . . . . . 251

9.6 Wave Propagation in Lossy Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . 254

9.7 Wave Propagation in Good Conductors . . . . . . . . . . . . . . . . . . . . . . . . 257

9.8 Power Flow and Poynting Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260

9.9 Incident and Reflected Waves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

9.10 Uniform Wave Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

9.11 Exercise Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273

10 Basics of Antennas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

10.2 Working Principles of Antennas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

10.3 Potential Functions for Antennas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276

10.4 Hertzian Dipole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

10.5 Antenna Gain and Directivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

10.6 Long Dipole Antennas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289

10.7 Friis Transmission Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

10.8 Exercise Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295

Appendix A: Mathematical Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297

A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297

A.2 Basic Trigonometric Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297

A.3 Trigonometric Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298

A.4 Derivative and Integral Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

A.5 Exponential and Logarithmic Formulae . . . . . . . . . . . . . . . . . . . . . . . . 300

A.6 Integral Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300

Contents xv

Appendix B: Answers to Practice and Exercise Problems. . . . . . . . . . . . . . . 301

Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303

Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304

Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305

Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306

Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

Chapter 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308

Chapter 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311

Chapter 1

Basics of Electromagnetics

1.1 Introduction

The word electromagnetic field is the combination of electric and magnetic fields.

An electromagnetic field is sometimes known as EM field, and it is generated when

charged particles are at rest or in motion. There are two types of charges in elementary

physics, namely the positive charge and the negative charge. An electric field depends

mainly on these charges. The rate of change of charge generates current, which

produces a magnetic field. A field is a special distribution of a parameter, which may

or may not be a function of time. Time varying electric and magnetic fields are joined

together to form an EM field. The time-dependent EM field produces a wave that

radiates from the source.

The arcs or sparks are produced when the surface potential gradient (electric

field) of a conductor exceeds the dielectric strength of the surrounding air. These

arcs transmit energy to a certain distance. This kind of phenomena leads scientists or

engineers to work on communication systems. Principles of EM fields are applied in

designing microwaves, antennae, electric machines, communication systems, and

bioelectromagnetic and remote sensing systems. Some tools are required to study

EM fields. These include imagination, vector algebra, coordinate systems and

transformation. In this chapter, different parameters of EM fields will be discussed.

1.2 Field Parameters and SI Units

Electric charges and currents produce electric and magnetic fields. It is very impor￾tant to define all the field parameters and their standard units to get fundamental

knowledge of electromagnetism. The notation and units of EM field parameters

are mentioned in Table 1.1. The charge density is defined as the fixed amount of

charge per unit volume. The charge density is categorized as volume, surface and

line charge densities. For surface, the charge q is identified by an element whose

area is s. Similarly, for line, the charge q is identified by an element whose length

is l. The volume, surface and line charge densities can be expressed as

Md. A. Salam, Electromagnetic Field Theories for Engineering, 1

DOI 10.1007/978-981-4585-66-8_1, © Springer Science+Business Media Singapore 2014

2 1 Basics of Electromagnetics

Table 1.1 Symbols and units

of field parameters Name of field parameter Notation/Symbol Unit

Electric field intensity E V/m

Electric flux density or

electric displacement

D C/m

Magnetic field intensity H A/m

Magnetic flux density B Wb/m2 or

Tesla

Charge density ρ C/m3

Current density J A/m2

ρv = lim

v→0

q

v

(C/m3

) (1.1)

ρs = lim

s→0

q

s (C/m2

) (1.2)

ρl = lim

l→0

q

l (C/m) (1.3)

The time rate of change of charge is known as current. The current is symbolised by

I and its unit of measure is C/s or A.

In electromagnetic modelling, four fundamental units are required. These are

length, mass, time and current. The SI (International System of Units) unit is often

known as MKSA system, which is derived from the four basic units as mentioned in

Table 1.2.

1.2.1 Electric Flux Density and Field Intensity

There is a direct relationship between the electric flux density and the electric field

intensity. The relationship between the electric flux density and the electric field

intensity is represented as

D = εE, (1.4)

where ε is the proportionality constant and it is known as permittivity of the medium.

The permittivity of any medium is defined as

ε = ε0εr, (1.5)

where

ε0 is the permittivity of the free space and its value is 8.854 × 10−12 F/m and

εr is the relative permittivity of the medium.

Substituting Eq. (1.5) into Eq. (1.4) yields

D = ε0εrE. (1.6)