This is an introductory textbook on electromagnetic fields suitable for undergraduate engineering courses in electrical engineering. It adopts a straightforward approach of presenting theoretical concepts in brief and using several worked-out examples of varied kinds to provide a further understanding of concepts. The material covered in the book broadly aims to develop formulations associated with the analysis of electrostatic, static magnetic and time-varying electromagnetic fields. A review of vector calculus, included as an appendix, provides the necessary mathematical background for understanding the treatment.

The salient features of the book are:

- clear and concise explanations
- a number of worked-out examples
- variety in exercises
- a coverage that matches the undergraduate curriculum
- inclusion of several problems drawn from previous university question papers

**, Principal, Vasireddy Venkitadri Institute of Technology, Nambur, Guntur, is a professor in the department of Electronics and Communications Engineering. He has more than 22 years of teaching experience. He obtained his MTech degree from JNTU Kakinada in 1990 and his PhD in 2009 from Osmania University, Hyderabad. He is a member of I(E) and ISTE, India. He has published research papers in journals of international repute and is currently engaged in research in the areas of radar signal and image processing.**

*Y Mallikarjuna Reddy**Preface*

**Electrostatics**

Introduction to Electrostatic Fields

Coulomb’s Law

Force in Terms of Rectangular Coordinates

Force Due to

Electric Field Intensity (

Charge Distributions

Electric Field Intensity Due to Charge Distributions

Electric Field Due to Infinite Line Charge

Electric Field Due to Finite Line Charge

Electric Field Strength Due to a Circular Ring of Charge

Electric Field Strength Due to an Infinite Sheet of Charge

Electric Field Strength Due to a Sheet of Circular Disc

Electric Field Strength at the Centre of a Half Circular Ring

Electric Flux and Flux Density

Electric Flux Density Due to Point Charge

Electric Flux Density Due to Charge Distributions

Gauss’ Law (Integral Form)

Gauss’ Law in Point Form (Maxwell’s First Law)

Divergence of Electric Flux Density

Divergence Theorem of Electric Flux Density

Electric Flux Density Due to Infinite Line Charge Using Gauss’ Law

Flux Density Due to an Infinite Sheet Charge Using Gauss’ Law

Flux Density for a Coaxial Cable

Flux Density for a Spherical Shell of Charge

Flux Density for a Uniformly Charged Sphere

Work Done in Moving a Point Charge in an Electrostatic Field

Electric Potential

Potential Due to Point Charge

Potential Due to N Point Charges

Potential Due to Charge Distribution

Potential Difference Due to an Infinite Line Charge

Potential Due to a Line Charge of Finite Length Potential Due to a Circular Ring

Potential Gradient

Relationship between E and V: Maxwell’s Second Equation

Potential Function (V)

Equipotential Surfaces

Coulomb’s Law

Force in Terms of Rectangular Coordinates

Force Due to

*N*Number of ChargesElectric Field Intensity (

*E*)Charge Distributions

Electric Field Intensity Due to Charge Distributions

Electric Field Due to Infinite Line Charge

Electric Field Due to Finite Line Charge

Electric Field Strength Due to a Circular Ring of Charge

Electric Field Strength Due to an Infinite Sheet of Charge

Electric Field Strength Due to a Sheet of Circular Disc

Electric Field Strength at the Centre of a Half Circular Ring

Electric Flux and Flux Density

Electric Flux Density Due to Point Charge

*Q*Electric Flux Density Due to Charge Distributions

Gauss’ Law (Integral Form)

Gauss’ Law in Point Form (Maxwell’s First Law)

Divergence of Electric Flux Density

Divergence Theorem of Electric Flux Density

Electric Flux Density Due to Infinite Line Charge Using Gauss’ Law

Flux Density Due to an Infinite Sheet Charge Using Gauss’ Law

Flux Density for a Coaxial Cable

Flux Density for a Spherical Shell of Charge

Flux Density for a Uniformly Charged Sphere

Work Done in Moving a Point Charge in an Electrostatic Field

Electric Potential

Potential Due to Point Charge

Potential Due to N Point Charges

Potential Due to Charge Distribution

Potential Difference Due to an Infinite Line Charge

Potential Due to a Line Charge of Finite Length Potential Due to a Circular Ring

Potential Gradient

Relationship between E and V: Maxwell’s Second Equation

Potential Function (V)

Equipotential Surfaces

*Additional Problems**Questions**Problems**Answers Multiple-Choice Questions**Answers***Conductors and Dipole**

Introduction

Conductors

Behaviour of Conductors in an Electric Field

Electric Dipole

Potential Due to Electric Dipole

Dipole Moment

Electric Field Due to Dipole

Torque on an Electric Dipole in an Electric Field

Torque on a Dipole Due to the Field of Another Dipole

Poisson’s and Laplace’s Equations

Uniqueness Theorem

Electric Field between Two Concentric Conducting Spheres Using Laplace’s Equation

Electric Field of a Coaxial Cable Using Laplace’s Equation

Electric Field Due to Semi Infinite Conducting Planes

Electric Field Due to Two Axial Conducting Cones

Conductors

Behaviour of Conductors in an Electric Field

Electric Dipole

Potential Due to Electric Dipole

Dipole Moment

Electric Field Due to Dipole

Torque on an Electric Dipole in an Electric Field

Torque on a Dipole Due to the Field of Another Dipole

Poisson’s and Laplace’s Equations

Uniqueness Theorem

Electric Field between Two Concentric Conducting Spheres Using Laplace’s Equation

Electric Field of a Coaxial Cable Using Laplace’s Equation

Electric Field Due to Semi Infinite Conducting Planes

Electric Field Due to Two Axial Conducting Cones

*Additional Problems**Questions**Problems**Answers**Multiple-Choice Questions**Answers***Dielectrics and Capacitance**

Dielectric Materials

Polarization

Electric Displacement Vector in Dielectrics

Boundary Conditions

Capacitance

Capacitance between Two Concentric Spheres

Capacitance of a Coaxial Cable

Capacitance of Two Parallel Wires (Single-Phase Transmission Line)

Energy Stored in an Electrostatic Field

Energy Stored in Terms of

Energy Stored in a Capacitor

Energy Stored in a Coaxial Cable

Electric Current and Current Density

Current Density

Conductors and Conductivity

Point Form of Ohm’s Law (Relationship between J and E

Relationship between J and rv

Continuity Equation

Relaxation time (Tr )

Resistance and Power

Polarization

Electric Displacement Vector in Dielectrics

Boundary Conditions

Capacitance

Capacitance between Two Concentric Spheres

Capacitance of a Coaxial Cable

Capacitance of Two Parallel Wires (Single-Phase Transmission Line)

Energy Stored in an Electrostatic Field

Energy Stored in Terms of

*E*and*D*Energy Stored in a Capacitor

Energy Stored in a Coaxial Cable

Electric Current and Current Density

Current Density

Conductors and Conductivity

Point Form of Ohm’s Law (Relationship between J and E

Relationship between J and rv

Continuity Equation

Relaxation time (Tr )

Resistance and Power

*Additional Problems**Questions**Problems**Answers**Multiple-Choice Questions**Answers***Magnetostatics**

Introduction

Density

Density

*Additional Problems**Questions**Problems**Answers**Multiple-Choice Questions*

*Answers*

**Ampere’s Circuital Law and Its Applications**

Introduction

Ampere’s Circuital Law or Ampere’s Work Law

Magnetic Field Intensity Due to a Solid Conductor

Magnetic Field Intensity Due to Coaxial Cable

Magnetic Field Intensity Due to an Infinite Sheet of Current

Magnetic Field Intensity at Any Point in between Two

Infinite Parallel Surface Current Sheets

Differential or Point Form of Ampere’s Circuital Law (Maxwell’s Third Equation)

Stokes’ Theorem

Point Form of Magnetic Flux Density

Magnetic Field Intensity Due to a Solenoid Using Ampere’s Circuital Law

Magnetic Field Intensity Due to a Toroid Using Ampere’s Circuital Law

Ampere’s Circuital Law or Ampere’s Work Law

Magnetic Field Intensity Due to a Solid Conductor

Magnetic Field Intensity Due to Coaxial Cable

Magnetic Field Intensity Due to an Infinite Sheet of Current

Magnetic Field Intensity at Any Point in between Two

Infinite Parallel Surface Current Sheets

Differential or Point Form of Ampere’s Circuital Law (Maxwell’s Third Equation)

Stokes’ Theorem

Point Form of Magnetic Flux Density

Magnetic Field Intensity Due to a Solenoid Using Ampere’s Circuital Law

Magnetic Field Intensity Due to a Toroid Using Ampere’s Circuital Law

*Additional Problems**Questions**Problems**Answers**Multiple-Choice Questions**Answers***Force in Magnetic Fields**

Introduction

Force and Torque on a Moving Charge

Force on a Differential Current Element

Ampere’s Force Law: Force Between Two Current Elements

Force between Two Straight Infinitely Long Parallel Conductors

Magnetic Torque Due to Rectangular Loop in a Magnetic Field

Magnetic Dipole and Dipole Moment

Boundary Conditions for Magnetic Field

Force and Torque on a Moving Charge

Force on a Differential Current Element

Ampere’s Force Law: Force Between Two Current Elements

Force between Two Straight Infinitely Long Parallel Conductors

Magnetic Torque Due to Rectangular Loop in a Magnetic Field

Magnetic Dipole and Dipole Moment

Boundary Conditions for Magnetic Field

*Additional Problems**Questions**Problems**Answers**Multiple-Choice Questions**Answers***Magnetic Potential and Inductance**

Introduction

Scalar Magnetic Potential ( Vm )

Magnetic Potential at the Centre of a Square Loop

Vector Magnetic Potential ( A)

The Vector Magnetic Potential for Line Current Element

Poisson’s Equation for Vector Magnetic Potential

Properties of Vector Magnetic Potential

Vector Magnetic Potential in the Field Due to Infinite Length Conductor

Vector Magnetic Potential Due to a Straight Line of Finite Length

Inductor and Inductance

Inductance of a Solenoid

Inductance of a Toroid

Inductance of a Coaxial Cable

Inductance of a Two-Wire Transmission Line

Mutual Inductance

Neumann’s Formula for Mutual Inductance

Mutual Inductance between Two Solenoids

Magnetic Energy

Energy Density Stored in the Magnetic Field

Energy Stored Due to Mutual Inductance

Magnetic Circuits

Magnetic Materials

Characteristics of Magnetic Materials

Scalar Magnetic Potential ( Vm )

Magnetic Potential at the Centre of a Square Loop

Vector Magnetic Potential ( A)

The Vector Magnetic Potential for Line Current Element

Poisson’s Equation for Vector Magnetic Potential

Properties of Vector Magnetic Potential

Vector Magnetic Potential in the Field Due to Infinite Length Conductor

Vector Magnetic Potential Due to a Straight Line of Finite Length

Inductor and Inductance

Inductance of a Solenoid

Inductance of a Toroid

Inductance of a Coaxial Cable

Inductance of a Two-Wire Transmission Line

Mutual Inductance

Neumann’s Formula for Mutual Inductance

Mutual Inductance between Two Solenoids

Magnetic Energy

Energy Density Stored in the Magnetic Field

Energy Stored Due to Mutual Inductance

Magnetic Circuits

Magnetic Materials

Characteristics of Magnetic Materials

*Additional Problems**Questions**Problems**Answers**Multiple-Choice Questions**Answers***Time-Varying Fields**

Introduction

Faraday’s Law

Induced EMF in an AC Generator

Induced EMF in a Coil

Faraday’s Disc Generator

Equation of Continuity for Time-Varying Fields

Modified Ampere’s Circuital Law for Time-Varying Fields

Displacement Current

Ratio between Conduction Current Density and Displacement Current Density

Differences between Conduction, Convection and Displacement Currents

Differences between Displacement Current Density and Conduction Current Density

Maxwell’s Equations for Static Fields

Maxwell’s Equation for Sinusoidal (Harmonic) Time-Varying Fields

Boundary Conditions

Poynting Theorem

Faraday’s Law

Induced EMF in an AC Generator

Induced EMF in a Coil

Faraday’s Disc Generator

Equation of Continuity for Time-Varying Fields

Modified Ampere’s Circuital Law for Time-Varying Fields

Displacement Current

Ratio between Conduction Current Density and Displacement Current Density

Differences between Conduction, Convection and Displacement Currents

Differences between Displacement Current Density and Conduction Current Density

Maxwell’s Equations for Static Fields

Maxwell’s Equation for Sinusoidal (Harmonic) Time-Varying Fields

Boundary Conditions

Poynting Theorem

*Additional Problems**Questions**Problems**Answers**Multiple-Choice Questions**Answers**Appendix A: Review of Vector Algebra*

*Definitions*

*Distance Vector*

*Vector Addition*

*Vector Multiplication*

*Dot Product*

*Cross Product*

*Scalar Triple Product*

*Vector Triple Product*

*The Cartesian or Rectangular Coordinate System*

*Circular Cylindrical Coordinate System*

*Spherical Coordinate System*

*Differential Elements*

*Transformation of Vectors*

*Gradient, Divergence and Curl of a Vector*

*Vector Identities*

*Del Operators*

*Appendix B: Symbols of Quantities*

Index

Index