Electromagnetism - Tamer Becherrawy - ebook

Electromagnetism ebook

Tamer Becherrawy

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This book deals with electromagnetic theory and its applicationsat the level of a senior-level undergraduate course for science andengineering. The basic concepts and mathematical analysis areclearly developed and the important applications are analyzed. Eachchapter contains numerous problems ranging in difficulty fromsimple applications to challenging. The answers for the problemsare given at the end of the book. Some chapters which open doors tomore advanced topics, such as wave theory, special relativity,emission of radiation by charges and antennas, are included. The material of this book allows flexibility in the choice of thetopics covered. Knowledge of basic calculus (vectors, differentialequations and integration) and general physics is assumed. Therequired mathematical techniques are gradually introduced. After adetailed revision of time-independent phenomena in electrostaticsand magnetism in vacuum, the electric and magnetic properties ofmatter are discussed. Induction, Maxwell equations andelectromagnetic waves, their reflection, refraction, interferenceand diffraction are also studied in some detail. Four additionaltopics are introduced: guided waves, relativistic electrodynamics,particles in an electromagnetic field and emission of radiation. Auseful appendix on mathematics, units and physical constants isincluded. Contents 1. Prologue. 2. Electrostatics in Vacuum. 3. Conductors and Currents. 4. Dielectrics. 5. Special Techniques and Approximation Methods. 6. Magnetic Field in Vacuum. 7. Magnetism in Matter. 8. Induction. 9. Maxwell's Equations. 10. Electromagnetic Waves. 11. Reflection, Interference, Diffraction and Diffusion. 12. Guided Waves. 13. Special Relativity and Electrodynamics. 14. Motion of Charged Particles in an Electromagnetic Field. 15. Emission of Radiation.

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Table of Contents

Preface

List of Symbols

Chapter 1. Prologue

1.1. Scalars and vectors

1.2. Effect of rotations on scalars and vectors

1.3. Integrals involving vectors

1.4. Gradient and curl, conservative field and scalar potential

1.5. Divergence, conservative flux, and vector potential

1.6. Other properties of the vector differential operator

1.7. Invariance and physical laws

1.8. Electric charges in nature

1.9. Interactions in nature

1.10. Problems

Chapter 2. Electrostatics in Vacuum

2.1. Electric forces and field

2.2. Electric energy and potential

2.3. The two fundamental laws of electrostatics

2.4. Poisson’s equation and its solutions

2.5. Symmetries of the electric field and potential

2.6. Electric dipole

2.7. Electric field and potential of simple charge configurations

2.8. Some general properties of the electric field and potential

2.9. Electrostatic energy of a system of charges

2.10. Electrostatic binding energy of ionic crystals and atomic nuclei

2.11. Interaction-at-a-distance and local interaction*

2.12. Problems

Chapter 3. Conductors and Currents

3.1. Conductors in equilibrium

3.2. Conductors with cavities, electric shielding

3.3. Capacitors

3.4. Mutual electric influence of conductors

3.5. Electric forces between conductors

3.6. Currents and current densities

3.7. Classical model of conduction, Ohm’s law and the Joule effect

3.8. Resistance of conductors

3.9. Variation of resistivity with temperature, superconductivity

3.10. Band theory of conduction, semiconductors*

3.11. Electric circuits

3.12. Problems

Chapter 4. Dielectrics

4.1. Effects of dielectric on capacitors

4.2. Polarization of dielectrics

4.3. Microscopic interpretation of polarization

4.4. Polarization charges in dielectric

4.5. Potential and field of polarized dielectrics

4.6. Gauss’s law in the case of dielectrics, electric displacement

4.7. Electrostatic equations in dielectrics

4.8. field and potential of permanent dielectrics

4.9. Polarization of a dielectric in an external field

4.10. Energy and force in dielectrics

4.11. Action of an electric field on a polarized medium

4.12. Electric susceptibility and permittivity

4.13. Variation of polarization with temperature

4.14. Nonlinear dielectrics and non-isotropic dielectrics

4.15. Problems

Chapter 5. Special Techniques and Approximation Methods

5.1. Unicity of the solution

5.2. Method of images

5.3. Method of analytic functionns

5.4. Method of separation of variable

5.5. Laplace’s equation in Cartesian coordinates

5.6. Laplace’s equation in spherical coordinates

5.7. Laplace’s equation in cylindrical coordinates

5.8. Multipole expansion

5.9. Other methods

5.10. Problems

Chapter 6. Magnetic field in Vacuum

6.1. Force exerted by a magnetic field on a moving charge

6.2. Force exerted by a magnetic field on a current, Laplace’s force

6.3. Magnetic flux and vector potentialt

6.4. Magnetic field of particles and currents, Biot-Savart’s law

6.5. Magnetic momen

6.6. Symmetries of the magnetic field

6.7. Ampère’s law in the integral form

6.8. field and potential of some simple circuits

6.9. Equations of time-independent magnetism in vacuum, singularities of B

6.10. Magnetic energy of a circuit in a field B

6.11. Magnetic forces

6.12. Question of magnetic monopoles*

6.13. Problems

Chapter 7. Magnetism in Matter

7.1. Types of magnetism

7.2. Diamagnetism and paramagnetism

7.3. Magnetization current

7.4. Magnetic field and vector potential in the presence of magnetic matter

7.5. Ampère’s law in the integral form in the presence of magnetic matter

7.6. Equations of time-independent magnetism in the presence of matter

7.7. Discontinuities of the magnetic field

7. 8. Examples of calculation of the field of permanent magnets

7.9. Magnetization of a body in an external field

7.10. Magnetic susceptibility, nonlinear mediums and non-isotropic mediums

7.11. Action of a magnetic field on a magnetic body

7.12. Magnetic energy in matter

7.13. Variation of magnetization with temperature

7.14. Ferromagnetism

7.15. Magnetic circuits

7.16. Problems

Chapter 8. Induction

8.1. Induction due to the variation of the flux, Faraday’s and Lenz’s laws

8.2. Neumann’s induction

8.3. Lorentz induction

8.4. Lorentz induction and the Galilean transformation of fields

8.5. Mutual inductance and self-inductance

8.6. LR circuit

8.7. Magnetic energy

8.8. Magnetic forces acting on circuits

8.9. Some applications of induction

8.10. Problems

Chapter 9. Maxwell’s Equations

9.1. Fundamental laws of electromagnetism

9.2. Maxwell’s equations

9.3. Electromagnetic potentials and gauge transformation

9.4. Quasi-permanent approximation

9.5. Discontinuities on the interface of two mediums

9.6. Electromagnetic energy and Poynting vector

9.7. Electromagnetic pressure, Maxwell’s tensor

9.8. Problems

Chapter 10. Electromagnetic Waves

10.1. A short review on waves

10.2. Electromagnetic waves in infinite vacuum and dielectrics

10.3. Polarization of electromagnetic waves

10.4. Energy and intensity of plane electromagnetic waves

10.5. Momentum and angular momentum densities, radiation pressure

10.6. A simple model of dispersion

10.7. Electromagnetic waves in conductors

10.8. Electromagnetic waves in plasmas

10.9. Quantization of electromagnetic waves

10.10. Electromagnetic spectrum

10.11. Emission of electromagnetic radiations

10.12. Spontaneous and stimulated emissions

10.13. Problems

Chapter 11. Reflection, Interference, Diffraction and Diffusion

11.1. General laws of reflection and refraction

11.2. Reflection and refraction on the interface of two dielectrics

11.3. Total reflection

11.4. Reflection on a conductor

11.5. Reflection on a plasma

11.6. Interference of two electromagnetic waves

11.7. Superposition of several waves, conditions for observable interference

11.8. Huygens-Fresnel’s principle and diffraction by an aperture

11.9. Diffraction by an obstacle, Babinet’s theorem

11.10. Diffraction by several randomly distributed identical apertures

11.11. Diffraction grating

11.12. X-ray diffraction

11.13. Diffusion of waves*

11.14. Cross-section*

11.15. Problems

Chapter 12. Guided Waves

12.1. Transmission lines

12.2. Guided waves

12.3. Waveguides formed by two plane and parallel plates

12.4. Guided electromagnetic waves in a hollow conductor

12.5. Energy propagation in waveguides

12.6. Cavities

12.7. Applications of waveguides

12.8. Problems

Chapter 13. Special Relativity and Electrodynamics

13.1. Galilean relativity in mechanics

13.2. Galilean relativity and wave theory*

13.3. The 19th Century experiments on the velocity of light

13.4. Special theory of relativity

13.5. Four-dimensional formalism

13.6. Elements of relativistic mechanics

13.7. Special relativity and wave theory*

13.8. Elements of relativistic electrodynamics

13.9. Problems

Chapter 14. Motion of Charged Particles in an Electromagnetic field

14.1. Motion of a charged particle in an electric field

14.2. Bohr model for the hydrogen atom*

14.3. Rutherford’s scattering *

14.4. Motion of a charged particle in a magnetic field

14.5. Motion in crossed electric and magnetic fields

14.6. Magnetic moment in a magnetic field

14.7. Problems

Chapter 15. Emission of Radiation

15.1. Retarded potentials and fields

15.2. Dipole radiation

15.3. Electric dipole radiation

15.4. Magnetic dipole radiation

15.5. Antennas

15.6. Potentials and fields of a charged particle*

15.7. Case of a charged particle with constant velocity *

15.8. Radiated energy by a moving charge

15.9. Problems

Answers to Some Problems

Appendix A. Mathematical Review

Appendix B. Units in Physics

Appendix C. Some Physical Constants

Further Reading

Index

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First published 2012 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

ISTE Ltd 27-37 St George's Road London SW19 4EU UK

www.iste.co.uk

John Wiley & Sons, Inc. 111 River Street Hoboken, NJ 07030 USA

www.wiley.com

© ISTE Ltd 2012

The rights of Tamer Bécherrawy to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Cataloging-in-Publication Data

Bécherrawy, Tamer. Electromagnetism : Maxwell equations, wave propagation, and emission / Tamer Bécherrawy. p. cm. Includes bibliographical references and index. ISBN 978-1-84821-355-5 1. Electromagnetism. 2. Maxwell equations. 3. Electromagnetic waves. 4. Field emission. I. Title. QC670.B37 2012 537--dc23

2012009826

British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN: 978-1-84821-355-5

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Preface

The scientific study of electric and magnetic forces started as two distinct sciences during the second half of the 18th Century. The concepts of electric and magnetic fields were introduced as independent constructs to facilitate the calculation of forces. However, after the discovery by Oersted in 1819 that an electric current produces a magnetic field, and the discovery by Faraday in 1831 that a variable magnetic field induces currents, it became clear that electric and magnetic fields are related and that they are very important physical concepts. In 1873, Maxwell unified electricity and magnetism in a single theory, called electromagnetism, based on four fundamental equations. An important prediction of this theory was the existence of electromagnetic waves that propagate with the speed of light. This prediction was confirmed experimentally by Hertz in 1887.

Thanks to the discovery of induction, the large-scale production of electricity became possible, opening the door to a new technological era in the second half of the 19th Century. The discovery of electromagnetic waves and the development of electronics generated a real revolution in telecommunications in the 20th Century with considerable economical, social, cultural and political impact.

The electromagnetic field, which is an association of the electric and magnetic fields, is a real physical object with energy, momentum, and angular momentum, which may be static or propagating as waves exactly like sound, elastic waves, or even particles. This is the first example of field theories in modern physics. It was followed by the discovery of the gravitational field in the framework of General Relativity and quantum fields in the framework of Quantum Electrodynamics and Quantum Chromodynamics. On the other hand, Maxwell‘s theory solved the very long-standing problem of the nature of light; it is an electromagnetic wave of short wavelength. Thus, Maxwell‘s work unified electricity, magnetism and optics in a single theory. Electromagnetic theory is in such complete agreement with experiments that any theory in conflict with it should be modified or abandoned.

The formulation of the electromagnetic theory was a major event in the history of physics in its incessant search to explain the maximum of phenomena with the minimum of basic principles. Furthermore, electromagnetism is the prototype of the so-called gauge theories in modern physics. They include the unification of electromagnetic and weak interactions by Glashow, Salam and Weinberg around 1967, Quantum Chromodynamics around 1973 and the so-called Grand Unification Theories that try to unify all interactions in Nature.

The electromagnetic theory posed two challenging problems, which produced real revolutions in physics and even in philosophy at the beginning of the 20th Century. The first was the disagreement of the propagation of light with the Galilean transformation, which is one of the basic principles of Classical Mechanics. This was shown by several experiments (namely Michelson‘s historical experiment) and it is fundamental since Maxwell‘s equations, which are obeyed by light as electromagnetic waves, are not covariant in the Galilean transformation. This contradiction was solved by the Special Theory of Relativity that modified the Galilean transformation, and had far-reaching consequences. The second problem was the understanding of the black body radiation and the discrete emission spectrum of atoms, which contradict both Classical Mechanics and the electromagnetic theory. Its solution led to the formulation of Quantum Theory. At present, the interaction of electromagnetic radiations with matter remains a very important subject both in theoretical physics and in various domains of applied physics.

Electromagnetism plays an important part in almost all branches of physics: atomic physics, molecular physics, solid-state physics, astrophysics, atmospheric physics, etc., and it even intervenes in chemistry and biology. In fact, almost all properties of matter are fundamentally electromagnetic on both the macroscopic scale and the atomic and molecular microscopic scale. On the other hand, electromagnetic waves play a fundamental part in the transfer of energy and information. Thus, a good understanding of electromagnetism is essential in any scientific activity and in the training of future physicists and engineers.

The purpose in writing this book is to study electromagnetism at the upper undergraduate level following teaching experience of several years. The goal is to understand the concept of electromagnetic fields, to obtain Maxwell‘s equations and to analyze some of their consequences regarding the propagation and emission of radiation.

Writing a book on electromagnetism is not an easy task for two reasons: the first is that the subject is so well established and so many excellent books already exist that one can expect originality only in didactical details: selection of topics, clear presentation of the material, choice of exercises, etc. The second is that electromagnetism is very connected to other subjects, namely quantum theory, relativity, properties of matter, and it has countless applications. Thus, it is hard to set the limits of the text.

Some authors prefer to start with Maxwell‘s equations as basic equations and then study time-independent phenomena and time-dependent phenomena. This approach is similar to starting Classical Mechanics with Newton‘s principles or, at a higher level, starting with Hamilton‘s principle and Lagrange equations. I think that the traditional approach, starting with the time-independent phenomena, is more pedagogical because of the mathematical complexity of the fields as functions of space and time, and the complexity of Maxwell‘s equations as partial differential equations for vector quantities. Thus, this text may be divided into four parts:

− The first part of seven chapters studies the time-independent electric and magnetic phenomena. This study goes beyond introductory electricity and magnetism by the use of vector calculus, differential and partial differential equations, etc. In this part, the basic concepts of electric and magnetic fields, energy and symmetries are analyzed, as well as the properties of dielectrics and magnetic matter. Conduction in solids is introduced, but we do not develop circuit analysis. In Chapter 5, some useful mathematical techniques (Legendre polynomials, Bessel‘s functions and multipole expansion) are introduced.

− The second part studies the time-dependent phenomena. It includes a detailed study of induction with some of its applications in Chapter 8 and the formulation of Maxwell‘s equations in Chapter 9.

− The third part studies the propagation effects. It includes a detailed study of electromagnetic waves in Chapter 10 (including propagation in dielectrics, in conductors and in plasmas, the quantization of radiation and its emission),reflection, interference, diffraction and diffusion in Chapter 11, and guided waves in Chapter 12.

− The fourth part includes Chapter 13 on the Special Theory of Relativity (including its applications to mechanics and electrodynamics), the motion of charged particles in electromagnetic fields (both non-relativistic and relativistic) in Chapter 14, and the emission of electromagnetic waves by antennas and particles in Chapter 15. The chapter on the Special Theory of Relativity is necessary as an introduction to the subject and for a better understanding of the electromagnetic theory.

Electromagnetism if one of the first physics courses in which vector calculus and partial differential equations are extensively used. The electromagnetic theory in vacuum requires one electric field and one magnetic field, and the electromagnetic theory in matter requires two more fields. All of them are vector fields. They may be represented by their 12 components measured with respect to convenient Cartesian

axes. The four Maxwell‘s equations couple these components to the charge and current densities. It is unthinkable to handle these equations and analyze their consequences without the use of vector calculus. Only this analysis allows us to study electromagnetism independently of the used frame and to use curvilinear coordinates, which are very often more convenient to solve the equations. Thus, some knowledge of mathematical analysis (both real and complex) and vector calculus are assumed. The required mathematical techniques are introduced as the need arises. Appendix A summarizes the principal mathematical formulas, integrals and vector analysis.

I have tried to use clear notations by assigning similar symbols for the various physical quantities: a boldfaced symbol for a vector quantity, an italic symbol for a scalar quantity or a component of a vector quantity, an underlined symbol for a complex quantity, and script symbol for a curve, a surface, a volume and some special quantities. Physical quantities of the same type are referred to by symbols with different indexes: for instance, FE, FM , f(ex), etc., for the different types of force. The charge densities, per unit volume, per unit surface and per unit length are respectively qv, qs and qL. To avoid confusion with the components of the electric field E, the energy is designated by U (UK for the kinetic energy, UE for the electric energy, etc.). The frequency is represented by , instead of the usual Greek symbol ν, to avoid confusing it with the velocity v.

A unit vector is often represented by e, while the unit vectors of the axes are ex, ey and ex. In order to write summations in a condensed form, the Cartesian coordinates x, y and z are sometimes designated by x1, x2 and x3 respectively, and the components of a vector V by V1 ≡ Vx, V2 ≡ Vy and V3 ≡ Vz. The partial derivatives of u(x, y, z, t) are represented by etc. We also use the notation and occasionally ù for

Some sections, indicated by an asterisk (*), have some difficulty and may be omitted without loss of continuity. At the end of each chapter, I have included numerous problems, which are ordered according to the sections of the chapter. The answers to most of the problems are given in a special addendum entitled Answers to Some Problems, which enables the student to check the results.

I hope that this text makes the subject more accessible for students, and that it is utilized as a good teaching tool for professors.

T. Bécherrawy

May 2012

List of Symbols

Chapter 1

Prologue

Most physical phenomena are fundamentally electromagnetic. This makes electromagnetism a basic theory in many branches of physics (solid state physics, electronics, atomic and molecular physics, relativity, atmospheric physics, etc.) also in some other sciences and most technologies.

Although physics is an experimental science, it uses mathematical language to formulate its theories and its laws and analyze their consequences. Electromagnetism is a typical theory that is impossible to formulate without extensive use of vector analysis, differential equations, complex analysis, etc. The use of mathematics can even lead to the prediction of new physical laws and new phenomena (the discovery of the electromagnetic waves by Maxwell is a typical example). However, only experiments can decide whether a particular solution or prediction and even the whole theory is acceptable. Until now, no experiment has contradicted electromagnetic theory, both on the macroscopic scale and the microscopic scale (nuclear, atomic or molecular).

Although permanent magnets and electrification by rubbing were known in antiquity, scientific observations of magnetism began around 1270 with the French army engineer Pierre de Marincourt. The observation of electric effects began much later with the French botanist C. Dufay around 1734. Contrary to the gravitational interaction between masses, the large majority of objects around us are globally neutral and, if they become charged, they discharge rapidly in the surrounding air. The scientific study of electricity started with Franklin (17061790), Priestley (17331804), Cavendish (17311810), Coulomb (17361806), Laplace (17491827), Ampre (17751836), Gauss (17771855), and Poisson (17811840) who formulated the laws of electricity and magnetism. Faraday (17911867) introduced the notions of influence and fields and discovered electric induction, which allowed the large-scale production of electricity. Electricity and magnetism were unified in a single theory by Maxwell in 1864. This long itinerary led to the present technological era with the considerable influence of electromagnetism and its consequences on our industrial, economical and cultural environment.

In this chapter, we introduce some basic mathematical methods and some general invariances and symmetries that we use in the formulation of any theory and especially electromagnetic theory.

1.1. Scalars and vectors

The basic elementary concepts in the formulation of physical theories are position and time. The position is specified by the coordinates with respect to a reference frame Oxyz, supported by a material body and represented by an origin O and three mutually orthogonal axes. Although these concepts seem to be simple, their analysis poses deep practical and philosophical questions even in classical mechanics. In modern physics, their analysis has been one of the corner-stones of the special theory of relativity (see Chapter 13), general relativity, and quantum theory.

Some physical quantities are determined by a single algebraic quantity with no characteristic orientation. Mass, time, temperature, and electric charge are examples of such quantities; these are scalar quantities. They may be strictly positive (mass, pressure, etc.), positive or negative (position along an axis, potential energy, electric charge, etc.), or even complex (wave function, impedance, etc.). Other physical quantities A are specified, each one by a positive magnitude A and an orientation; these are said to be vector quantities. Displacement, velocity, acceleration, force, electric field, magnetic field, etc., are examples of vector quantities. A more precise definition of a vector quantity is given in section 1.2.

A vector A is conveniently specified by its Cartesian components Ax, Ay and Az with respect to a frame Oxyz (Figure 1.1a). We may write , where ex, ey and ez are the unit vectors of the axes Ox, Oy and Oz; they are the basis vectors of the reference frame Oxyz. To simplify the writing of summations, we use the numbers 1, 2 and 3 instead of x, y and z to label the components and we write

[1.1]

The component A1, for instance, is the projection of A on the axis Ox. It is well known that the decomposition [] is unique.

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