FUNDAMENTALS OF ELECTROMAGNETICS

FUNDAMENTALSOF

ELECTROMAGNETICS

AAAAA.V.V.V.V.V. MAHA. MAHA. MAHA. MAHA. MAHATMETMETMETMETME

B.E.(Electronics and Telecommunication), M.Tech.(VLSI)

Assistant Professor, Deptt. of Electronics and CommunicationKavikulguru Institute of Technology and Science Ramtek,

Distt. Nagpur, (Maharashtra)

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FUNDAMENTALS OF ELECTROMAGNETICS

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CONTENTS

Preface ix

Acknowledgement xi

1. Vector Analysis 1–30

1.1 Scalars and Vectors 11.2 Vector Algebra 21.3 Unit Vector 41.4 Component of Vector 41.5 The Cartesian Co-ordinate System 51.6 Application of Unit Vectors 81.7 Circular Cylindrical Co-ordinate System 171.8 Spherical Co-ordinate System 20

2. Coulomb’s Law and Electric Field Intensity 31–61

2.1 Coulomb’s Law 312.2 Electric Field Intensity 332.3 Electric Field due to Continuous Charge Distribution 432.4 Electric Field Intensity due to Uniform Ring of Charge at a Point on its Axis 59

3. Electric Flux Density, Gauss’s Law and Divergence 62–107

3.1 Faraday’s Experiment 623.2 Gauss’s Law 663.3 Application of Gauss’s Law due to Symmetrical Charge Distribution 683.4 Application of Gauss’s Law to Differential Volume Element 79

3.5 The Vector Operator (del) 923.6 Divergence Theorem 93

4. Energy and Potential 108–146

4.1 Electric Potential 1084.2 Equipotential Surface 1294.3 Conservative Property of Elctrostatic Field 1294.4 Gradient 1304.5 Energy Density in Electrostatic Field 133

5. Conductors and Dielectric 147–177

5.1 Current and Current Density 1475.2 Polarisation in Dielectric 1525.3 Law of Conservation of Charges 1545.4 Continuity Equation 1555.5 Conduction Current Density 1565.6 Conductor Properties and Boundary Conditions 157

6. Poisson’s and Laplace Equation 178–204

6.1 Poisson’s and Laplace Equation 1786.2 Uniqueness Theorem 1796.3 Examples of the Solution of Laplace’s Equation 181

7. The Steady Magnetic Field 205–254

7.1 Biot-savart Law 205

7.2 H Due to Infinitely Long Current Filament 207

7.3 H Due to Finite Current Filament 2157.4 Ampere’s Circuital Law 2257.5 Curl 2287.6 Stroke’s Theorem 237

8. Maxwell’s Equations for Time Varying Field 255–263

8.1 Faraday’s Law of Electromagnetic Induction 2558.2 Maxwell’s Equation Derived from Faraday’s Law of Electromagnetic Induction 2568.3 Convection Current Density 2578.4 Maxwell’s Equation for Time Varying Field Derived from Ampere’s Law 2588.5 Maxwell’s Equation for Time Varying Field Derived from Gauss’s Law

(Electric Field) 2608.6 Maxwell’s Equation for Time Varying Field Derived from Gauss’s Law

for Magnetism 2608.7 Time Harmonic Field 261

9. Electromagnetic Wave Propagation 264–314

9.1 Properties of Electromagnetic Wave 2649.2 Wave Equation for Free Space/Perfect Dielectric 2659.3 Vector Helmholtz Equation (Wave Equation for Free Space/Perfect Dielectric

in Phasor form) 2689.4 Transverse Nature of Uniform Plane Wave 271

9.5 Relation Between E and H of Uniform Plane Wave 2729.6 Wave Equation for Lossy Dielectric/Conducting Media 2859.7 Wave Equation for Lossy Dielectric/Conducting Media in Phasor form 2879.8 Intrinsic Impedance 2939.9 Conductor and Dielectric 295

(vi)

9.10 The Poynting Vector and Power Flow 3109.11 Wave Polarisation 312

10. Waves at Boundary Between Two Media 315–337

10.1 Wave Incident Normally on Boundary Between Perfect Dielectrics 31510.2 Wave Incident Obliquely on Boundary Between Perfect Dielectric 31810.3 Use of Direction Cosines 33110.4 Wave Incident Normally on Perfect Conductor 33210.5 Wave Incident Obliquely on Perfect Conductor and Polarised Perpendicular

to the Plane of Incidence 33410.6 Wave Incident Obliquely on Perfect Conductor and Polarised Parallel to the

Plane of Incidence 335Index 338–339

(vii)

PREFACE

This book entitled “Fundamentals of Electromagnetics” presents the basic concepts ofelectromagnetic field. One of the main objectives of this book is to provide the students aclear and logical presentation of basic concepts and principles of electromagnetic fields. Thevarious concepts of the subject are explained through simple reader friendly language. Largenumbers of problems with their step by step solutions are provided to ease the understandingof difficult topics. The classroom experience and the difficulties of students while teaching thissubject have been taken into account.

The book is broadly classified into three categories— static electric field, steady magneticfield and time varying field. Study of static electric field is explained from chapter 1 tochapter 6. Chapter 7 concentrates on steady magnetic field while chapter 8, 9 and 10 focuson time varying field.

Chapter 1 introduces the basic concept of scalar vectors and co-ordinate system. Chapter 2is concerned with the Coulomb’s law and determination of electric field intensity due todifferent charge distribution. Chapter 3 discusses Faraday’s experiment to clear the conceptsof electric flux. It also covers the Gauss’s law and its application to symmetrical chargedistribution. It illustrates how it leads to the concept of divergence when it is applied todifferential volume element. Chapter 4 focuses on the work done in electric field and explainsthe concepts of energy and potential. Chapter 5 covers the basic concepts of current, currentdensity, properties of conductors and also the boundary conditions. Chapter 6 deals with thePoisson’s and Laplace’s equation. Chapter 7 is devoted to the study of fundamentals ofsteady magnetic field. It covers the concepts of Curl and Strokes theorem. Chapter 8concentrates on Maxwell’s equations for time varying field. Chapter 9 presents the propagationof electromagnetic wave in free space, dielectric and conducting media. In Chapter 10 thebehavior of electromagnetic wave on the boundary between two media is discussed.

I am sure that this book will satisfactorily meet the requirements of those for whom it ismeant. Readers are welcome to offer productive suggestions for the improvement of the book.

— Author

ACKNOWLEDGEMENT

I take great pleasure to present this book. The completion of this book can not be comprehendedwithout the earnest support provided by various people associated with it.

I would like to heartily thank the management of Vodithala Education Society, particularlyto Shri V. Srinivasa Rao, Hon. Secretary of the society for providing the infrastructure. I ampleased to acknowledge the motivation received from Dr. Bhaskar Patel, Principal, KavikulguruInstitute of Technology and Science, Ramtek. I express my deep sense of gratitude to Prof.S. M. Kharad, Head of Electronics and Communications Department for his suggestions. I alsoextend my sincere thanks to my colleagues especially to Prof. C.K. Thadani for his kindco-operation.

I would like to express earnest thanks to my husband Prof. V.P. Mahatme for his constantencouragement and suggestions for writing this book. Like a beacon he guided me throughall the odds. I am especially grateful to my parents Shri Prakashchandra Belokar, Sau. MandaBelokar, my father-in-law Shri Pandurangji Mahatme and children Kaustubh and Madhura fortheir cooperation and enormous patience.

It was my students whose difficulties motivate me to think about writing a book. Theyhave offered valuable criticism and relevant feedback. Special thanks to all those students. Itake this opportunity to express my sincere thanks to all of them who have helped medirectly or indirectly.

— Author

INTRODUCTION

The electromagnetic field is study of relationship between electricity and magnetism and leadsto four Maxwells equations which form the basics of electromagnetics. As the electromagneticwave propogates in space, it deals with the space and to study this three dimentional view,it is necessary to study the co-ordinate system. In this chapter, we shall study the conceptsof vector algebra, vector calculus and co-ordinate system.

1.1 SCALARS AND VECTORS

1.1.1 Scalar

A quantity that has only magnitude and algebraic sign is called scalar quantity. Thus ascalar quantity may be represented by a single positive or negative real number. e.g., physicalquantities like mass, length, time are scalar quantities. Also density, pressure, temperature arescalar quantities. A scalar is represented by letter such as m, t. The length of wire is givenas l = 2 meters. The temperature of furnace is given as t = 100°C.

1.1.2 Vector

A quantity that has magnitude and direction is called vector quantity. A vector is

represented by a letter with an arrow on the top such as A

or A. e.g., force, velocity,acceleration are vector quantities.

1

ChaptChaptChaptChaptChaptererererer 1VECTOR ANALYSIS

2 FUNDAMENTALS OF ELECTROMAGNETICS

We shall be mostly concerned with the fields, it may be scalar field or vector field. A fieldis defined as “If at every point of a region, there is corresponding value of some physicalquantity, the region is called field”. Mathematically we can represent a field by a function.This function describe the physical quantity at all points in the space w.r.t. an arbitrary origin.We may also find some physical effect associated with a field such as movement of compassneedle in the earth’s magnetic field.

Field has been classified as either scalar or vector field

1. Scalar field: If value of physical function at every point is a scalar quantity, then thefield is scalar field. e.g., temperature distribution in a room, density at any point inthe earth.

2. Vector field: If the value of physical function at every point is a vector quantity, thenthe field is vector field. e.g., gravitational and magnetic field of earth, wind velocityof atmosphere.

1.2 VECTOR ALGEBRA

1.2.1 Vector Addition

Let A and B are two vectors, then the addition of two vectors is performed byparallelogram law as shown in Fig. 1.1.

R = A B

Figure 1.1. Vector addition.

1.2.2 Vector Subtraction

To obtain the vector subtraction ( )A B or ( )A B , the sign and direction of secondvector is reversed and this vector is then added to first by the rule of vector addition asshown in Fig. 1.2.

Figure 1.2. Vector subtraction.

Fundamental Of Electromagnetics

Publisher : Laxmi Publications ISBN : 9789381159729 Author : A. V. Mahatme

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Fundamental Of Electromagnetics