Handbook on Fundamentals of Electromagnetic Theory

Table of Contents

Introduction.. 7

1. Electrostatics.. 8
1.1. Coulomb’s law.. 12
1.2. Gauss’s Law.. 21
1.3. Laplace’s Equation.. 29
1.4. Uniqueness Theorem.. 35
1.5. Capacitance.. 42
1.6. Energy Density.. 45

2. Magnetostatics.. 48
2.1. Biot Savartz Law.. 49
2.2. Ampere’s Law.. 54
2.3. Faraday’s Law.. 58
2.4. Self inductance.. 63

3. Maxwell Equations.. 68

4. Electromagnetic Waves.. 72
4.1. Wave equation.. 72
4.2. Plane waves.. 74
4.3. Propagation of a uniform plane wave.. 78
4.4. Polarisation.. 82
4.5. Poynting Vector.. 86

5. Solved Problems.. 91

6. Appendix.. 130

Bibliography.. 140

Preface

Electromagnetic Theory plays an important role in modernizing human life and encompasses wide areas such as: generation, transmission, and distribution of electrical power, digital systems, satellite communications, signal processing, robotics, mechatronics, computer, control, artificial intelligence, and networks.

A 4 year engineering curriculum normally contains various modules of electromagnetic field theory. However, some curricula do not have enough slots to accommodate the two modules. This book, is designed for 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 understand the electromagnetic field theory and its applications. Several worked out examples are included to enhance the understanding of electromagnetic field theories. Each chapter also includes several practice problems with answers given at the end of the book, which would facilitate students’ understanding.

Several textbooks on electromagnetic theories already exist in the market. However, the Handbook on Electromagnetic Field Theory for Engineering is written for students with the following key features.

• Easy and logical presentation of each concept

• 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

Dr. K.S. Kiran & Dr. Thangadurai N

Acknowledgements

First and foremost, we acknowledge the contributions of the many researchers in the area of Field Theory on which most of the material in this text are based. It would have been extremely difficult to write this book without the support and assistance of a number of organizations and individuals who worked or working in the same domain.

Our sincere thanks to Dr. Chenraj Roychand, President, Jain University, who has always encouraged, inspired researchers and his passion, commitments and belief in education as a means of encouraging social welfare and national development have been a source of inspiration.

We would like to thank Dr. C. G. Krishnadas Nair - Chancellor, Dr. N. Sundararajan - Vice Chancellor, Jain University for their inspiration both as an mentor and as a teacher. We would like to thank Dr. Sandeep Shastri, Pro Vice chancellor, Jain University for his constant encouragement and expert advice.

Our deep gratitude goes to the Dr. Hariprasad S. A, Director, School of Engineering and Technology, Jain University, for his continuous advice and constant encouragement.

We thank our scholars and students, who have supported me with various questions and comments, have definitely enhanced it. In particular, We thank all my colleagues of Electronics and Communication Engineering and Physics department who has supported me with this manuscript. Also we thank all my coauthors who have equally contributed for bringing a shape to this manuscript.

Finally, let us thank the Management of Jain University who extended the Morale and Technical support and for their continuous encouragement in the preparation of the manuscript.

Dr. K.S. Kiran & Dr. Thangadurai N

Introduction

Today’s cutting edge technology - ‘ELECTROMAGNETIC COMMUNICATION’ or ‘WIRELESS COMMUNICATION’ originated during 1865 with the coining of the term ELECTROMAGNETIC WAVES by James Clark Maxwell who is regarded as the founder of electrodynamics.

Maxwell quantified the existing laws of electromagnetism (Gauss law, Amperes law, Faradays law) and also theoretically corrected Amperes law for varying current situations. According to his modified Amperes law [formulas are not part of this preview], varying electric field induces changing magnetic field .That is, magnetic field is produced by conventional flow of current and also by the rate of change of electric displacement. This can be observed in the case of capacitor connected in an AC circuit. So this source of magnetic field [formulas are not part of this preview] behaving similar to electric current was given the name ‘DISPLACEMENT CURRENT DENSITY’. It was this concept of displacement current density which led to the concept of electromagnetic waves. Hertz made use of this concept to develop his antenna to radiate and receive electromagnetic signals. This culminated in the invention of RADIO by Marconi.

Today as most of us are aware, the transmission of information interms of varying electric and magnetic fields through electromagnetic waves is the most efficient and useful method of communication.

Now to understand this whole process and improvise this ,one must understand the basic laws of electricity like Coulombs law, Gauss law, Laplaces Equation - to understand the techniques of determining electric field at a point due to static charges and Biot- Savartz law, Ampers law to learn the method of determining magnetic field at a point due to steady currents. Later, through Maxwell equations we can acquire an understanding about Electromagnetic waves.

1. Electrostatics

Cartesian coordinate system

Cylindrical coordinates

Divergence

Gradient

Curl

Spherical coordinates

Cartesian to spherical

Spherical to Cartesian

Divergence

Gradient

Curl

Vector joining two points

Equation for a straight line joining two points

Mid point on a line joining two points

Vector identities

Dot Product

Dot product of two vectors gives a scalar.

Dot product of A and B represents component of A in the direction of B.

Cross Product

Cross product of two vectors gives a scalar.

[Please note that formulas and images are not part of this preview.]

1.1. Coulomb’s law

The force of attraction or repulsion between two charged bodies is directly proportional to the product of their charges and inversely proportional to the square of the separating distance.

Vectorial form

Electrical field Intensity

It is the force experienced by a unit positive charge.

In the above diagram, the electric field intensity at q₂ due to q₁ is given by

Electric field is a vector. It can be represented as the gradient of potential along the direction of decreasing potential.

Electric intensity at a point due to many charges

By the law of superposition of electric fields the resultant electric field at a point due to various point charges is the vectorial sum of the individual electric fields.

Electric intensity due to a line charge distribution

Note: Electric field due to a line charge is perpendicular to the line of charges.

Assignment: The antenna of an FM transistor is held vertical to receive horizontally polarized signal. Explain.

Electric field due to a dipole

Electric intensity due to a volume charge distribution

Electric potential

It is the work done in bringing a unit positive charge from infinity to a point in the electric field.

Equipotential surface

It is a surface with uniform potential. The field lines are perpendicular to such a surface everywhere. No work is required to move a charge on an equipotential surface.

Ex

1. The surface of a charged conductor.

2. A sphere around a point charge.

Note

1. Infinity is said to be at zero potential.

Potential due to a point charge

Consider a charge ‘Q’ at ‘A’. The work done to bring a unit positive charge from infinity to point ‘A’ is given by W = V.

Potential due to a dipole

A dipole possesses two equal and opposite charges separated by a small distance.

Superposition of electric potential

Potential at a point due to various charges is equal to the algebraic sum of the individual potentials.

Relation between electric field intensity and electric potential

Electric potential due to a line charge

Note

Quadrupole : Two pairs of equal and opposite charges with each charge located at the vertices of a parallelogram in the order of alternating sign.

Linear quadrupole : It is an arrangement of two pairs of equal and opposite charges occupying positions on a straight line such that two charges of one kind lie together and on either side of which are distributed the other two charges.

Electric field at a point on the axis of a charged ring:

1.2. Gauss’s Law

It is a technique to evaluate electric intensity at a point due to a symmetric charge distribution. Gauss quantified Faradays postulates of electric flux.

Electric flux: Lines of force normal to the surface.

Lines of force emanate from positive charge and converge on negative charge.

Faradays Law of electric flux (not to be confused with the law of electromagnetic induction):

The number of lines of force emanating from a charge is directly proportional to the quantity of charge.

Electric flux density (D): [Electric displacement]

It represents the lines of force crossing unit area normally.

Statement of Gauss law

The total electric flux over a closed surface is equal to the charge enclosed by the surface.

Divergence

It represents the magnitude of a physical quantity emerging or converging at a point.

For example tip of a fountain head is a source of divergence. Electric fields are said to be divergent in nature.

Differential or point form of gauss Law

Gauss divergence theorem

Statement: The volume integral of the divergence of a vector function ‘F’ over a volume

‘V’ is equal to the surface integral of the normal component of the vector function ‘F’

over the surface enclosing the volume V.

Applications of Gauss law

To find an expression for field at a point due to an Infinite line charge:

Consider an infinite line charge of linear density along AB .The field intensity at a point P at a distance r is determined using gauss theorem as follows.

Note: A Gaussian surface would be symmetrical such that the electric flux density D is constant every where along the surface. Gauss law is not applicable if there is no symmetry.

To find an expression for field at a point due to a plane sheet of charge:

The electric field at a point P at a distance r from the charged plane is to be calculated.

Imagine a hollow right circular cylinder of radius X extending to a length r on either side of the plane normal to it .The flux is normal to the surface and emerges through the end faces each of area A.

Note:

Think why?

It is interesting to note that in the above expression, the electric field due to an infinitely large plane is independent of the distance and every where normal to the plane where as coulombs law demands the reduction in field as inverse square law.

To find Field at a point to a spherical shell of charge:

Case 1: At a point outside the charged spherical shell

Case 2: At a point on the spherical shell

Case 3: At a point inside the shell

Field due to a uniformly charged solid sphere

Case 1: Field at a point outside the sphere

Case 2 : Field at a point on the sphere

Case 3: Field at a point inside the sphere

Poisson and Laplace’s equation

Laplace equation is a method to determine the potential due to a charge distributed in space by knowing the potential along the boundary.

It is based on the principle that if the potential along the boundary is known, then the potential inside the boundary is unique. Laplace could be applied to calculate potential across a pn junction.

Derivation: we know that electric field is the gradient of potential,

[formulas and images are not part of this preview]

This is the Poissons equation.

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