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# Classical Electromagnetic Radiation, Third Edition

592## Overview

This newly corrected, highly acclaimed text offers intermediate-level juniors and first-year graduate students of physics a rigorous treatment of classical electromagnetics. The authors present a very accessible macroscopic view of classical electromagnetics that emphasizes integrating electromagnetic theory with physical optics. The survey follows the historical development of physics, culminating in the use of four-vector relativity to fully integrate electricity with magnetism.

Starting with a brief review of static electricity and magnetism, the treatment advances to examinations of multipole fields, the equations of Laplace and Poisson, dynamic electromagnetism, electromagnetic waves, reflection and refraction, and waveguides. Subsequent chapters explore retarded potentials and fields and radiation by charged particles; antennas; classical electron theory; interference and coherence; scalar diffraction theory and the Fraunhofer limit; Fresnel diffraction and the transition to geometrical optics; and relativistic electrodynamics. A basic knowledge of vector calculus and Fourier analysis is assumed, and several helpful appendices supplement the text. An extensive Solutions Manual is also available.

## Product Details

ISBN-13: | 9780486490601 |
---|---|

Publisher: | Dover Publications |

Publication date: | 12/19/2012 |

Series: | Dover Books on Physics |

Edition description: | Third Edition |

Pages: | 592 |

Product dimensions: | 9.00(w) x 6.10(h) x 1.20(d) |

## About the Author

Professor Emeritus Mark A. Heald was on the faculty of Swarthmore College's Department of Physics and Astronomy for 33 years.

Jerry Marion (1929–52) published the original edition of this volume in 1965 under the imprint of Academic Press.

## Read an Excerpt

#### Classical Electromagnetic Radiation

**By Mark A. Heald, Jerry B. Marion**

**Dover Publications, Inc.**

**Copyright © 2012 Mark A. Heald and Adelia Marion**

All rights reserved.

ISBN: 978-0-486-28342-5

All rights reserved.

ISBN: 978-0-486-28342-5

CHAPTER 1

**Fundamentals of Static Electromagnetism**

In this book we shall be concerned mainly with radiation phenomena associated with electromagnetic fields. We shall study the generation of electromagnetic waves, the propagation of these waves in space, and their interaction with matter of various forms. The fundamental equations that govern all of these processes are *Maxwell's equations*. These are a set of partial differential equations that describe the space and time behavior of the electromagnetic field vectors. By way of review, we shall first examine briefly the *static* and *steady-state* properties of the electromagnetic field. In **Chapters 2** and **3** we shall discuss two topics that are usually not considered at great length in introductory accounts of electromagnetism—*multipole analysis* and solutions of *Laplace's equation*—because these subjects are of importance in radiation phenomena. In **Chapter 4** we shall treat time-varying electromagnetic fields and arrive at the four partial differential equations—Maxwell's equations—which give a full description of the classical behavior of electromagnetic fields. The remainder of the book is concerned primarily with radiation problems.

To describe electromagnetism, we use four vector fields:

**E** [equivalent to] *electric field or electric intensity* (statvolt/cm)

**D** [equivalent to] *dielectric displacement field or simply displacement* (statvolt/cm)

**B** [equivalent to] *magnetic field or magnetic induction or magnetic flux density* (gauss)

**H** [equivalent to] *magnetic intensity or magnetic field* (oersted)

The fields **E** and **B** are the fundamental fields because they represent the microscopic space-time averages within matter. The fields **D** and **H** are auxiliary fields, which involve the electromagnetic properties of matter and prove to be useful in treating macroscopic problems. Surprisingly, the *verbal* names of these four fields are less well standardized in the literature than the *symbols***E, D, B,** and **H.**

* * *

**1.1 UNITS**

To discuss electromagnetic phenomena, it is necessary to adopt one of many possible systems of units. Most likely the reader is already familiar with the *Système International* (SI), the system most popular for practical or engineering problems. The SI units (ampere, volt, tesla, etc.) are the calibration of choice for laboratory instruments. But in the study of the interaction of electromagnetic fields with the fundamental constituents of matter (atoms, molecules, electrons, etc.), the *Gaussian* system of units is commonly preferred and is used in this book. The Gaussian system is an amalgam of two 19th-century systems: electric quantities are measured in *electrostatic units* (esu) and magnetic quantities are measured in *electromagnetic units* (emu).

To read the physics literature, one must be bilingual in these two systems. Each provides distinctive insights into the physics. The Gaussian system retains popularity because factors of the speed of light c appear explicitly and appropriately. SI's constants [member of]0 and μ0, while hiding the factors of *c*, have the perverse virtue of forcing the user to face up to the distinction between the fundamental and auxiliary fields (e.g., **B** vs. **H**). SI is the system of legal metrology. It regards the *ampere* as a fourth fundamental unit along with the meter, kilogram, and second (MKSA). This convention makes dimensional analysis easier than with the Gaussian units, in which the electromagnetic dimensions are fractional powers of the centimeter, gram, and second (CGS—see Problem 1-1). The systems also differ in the choice of where to place factors of 4π, a controversial esthetic called *rationalization*.

A conversion table between Gaussian and SI units is given in **Appendix D. Appendix E** summarizes the fundamental electromagnetic equations in both systems. The remainder of this chapter not only provides a review of the fundamental principles of steady-state electromagnetism but also accustoms the reader to the use of Gaussian units.

**1.2 THE LAWS OF COULOMB AND GAUSS**

The first experimental fact we wish to invoke is that the force between two point charges at rest is directed along the line connecting the charges, and the magnitude of the force is directly proportional to the magnitude of each charge and inversely proportional to the square of the distance between the charges. This is Coulomb's force law and in Gaussian units assumes the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)

for the force exerted on q1 by q2. (The Gaussian unit of charge is the *statcoulomb* or *esu.*) The quantity *r* is the distance between the charges, and **e**r is the unit vector in the direction from q2 to q1. If the charges carry the same sign, the force is repulsive; if the signs are opposite, the force is attractive. The force on a test charge *q* defines the electric field vector according to

F = qE (1.2)

Thus *coulomb's field law* due to a source charge *q'* is

E = [q'/r2]er (1.3)

An important property of the electric field (indeed, of the *electromagnetic* field) is that it is *linear*. That is, the principle of superposition applies, and the field due to a number of charges is just the vector sum of the individual fields. Were it not for this property, the analysis of electromagnetic phenomena would be exceedingly difficult.

The notation required to express superposition is a little cumbersome. If there is more than one source charge producing the **E** field, then we must deal with two *overlaid* coordinate systems: one to express the location of the charges and one to express the location of the point where the field is being evaluated. As shown in **Fig. 1-1**, we will let the primed radius vector **r'** locate

the *source point* and the unprimed radius vector **r** locate the *field point*. Thus the vector distance from a particular source charge to a field point is (r - r'), and we will denote the unit vector in this direction by er-r'. This notation is such a nuisance however that we will shorthand the magnitude and unit vector by *r* and er when the meaning is clear from context, as in Eq. (1.3). But, to be explicit, the general form of Coulomb's (field) law for an arbitrary array of point charges is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.4)

where the index *i* designates the *i*th charge at location r'i.

The integration of the normal component of the electric field **E** over a surface area [increment of S] is called the *flux* of **E** through this surface,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.5)

where *d***a = n***da* is the directed element of area. Now, if a charge *q* is enclosed by a *Gaussian surface S* (of arbitrary shape), the flux of **E** through this *closed* surface turns out to be equal to 4π times the enclosed charge. By superposition, if qencl is the *total net charge* enclosed within the surface, irrespective of its spatial distribution within the volume, we have the ITL∫ITL expression of *Gauss' law* for the electric field:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.6)

where the direction of *d***a** is that of the *outward* normal to the surface *S* (note that any charge *outside* the surface does not count!). The elementary formal proof is supplied in Problem 1-3. The factor of 4π comes from an integration over solid angle. Applications are given in Problems 1-4 through 1-8.

A vector field, such as **E**, is often pictorialized by drawing "lines of force" or *field-lines*, which are continuous curves everywhere parallel to the local direction of the field. Consider a set of field-lines that form the walls of a thin tubular region of space. Gauss' law shows that this construction is properly called a *flux tube*, and that the tube necessarily begins on positive charge and ends on negative charge. That is, a field-line is simply the limiting form of a flux tube of negligible cross section.

We may convert the integral relation of Eq. (1.6) into a *differential* relation as follows. The closed surface *S* defines the enclosed volume *V*, to which we can apply the divergence theorem, Eq. (A.53),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.7)

And the enclosed charge *q* can be expressed as the volume integral of the *charge-density* function ρ(**r**) [charge-per-unit-volume],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.8)

With these substitutions, Eq. (1.6) becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.9)

But this relation must be valid for *any* volume *V*, and so the integrands themselves must be equal at all points. The result is the *differential* expression of Gauss' law:

[div E = 4πρ] (*total* charge) (1.10)

This equation states that the divergence of **E** is zero except at locations occupied by charge, reaffirming that electrostatic field-lines always originate and terminate on electric charges. In Eq. (1.10)ρ is the *total* charge density; a different bookkeeping can be used in the presence of a dielectric medium [see Eq. (1.29)].

The Coulomb field, Eq. (1.3), is a "central force," and therefore the electrostatic field is *conservative*. That is, the line integral from one point to another is *independent of path* (see Problem 1-9). It follows that the line integral around any closed path must vanish:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.11)

Using Stokes' theorem, Eq. (A.54), this integral equation can be transformed into the equivalent differential statement of the conservative law,

[curl E = 0] (1.12)

The conservative property allows us to define a *scalar potential* Φ for the electric field, such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.13)

in which r0 is the location of an arbitrary reference point where Φ is defined to be zero, and the integral is carried out along *any path* from r0 to the "field point" r. The inverse operation gives the field in terms of the potential,

[E = - grad Φ] (1.14)

Gauss' law, Eq. (1.10), may now be expressed in terms of the potential as

div grad Φ = - 4 πρ

The scalar derivative operator div **grad** is more commonly written as [nabla]2, known as the *Laplacian operator*. Thus we have *Poisson's equation*,

[[nabla]2Φ = - 4 πρ] (1.15)

which expresses the physical content of Coulomb's law as a second-order differential equation for the scalar potential. In regions of space that contain no charge, Poisson's equation reduces to *Laplace's equation*,

[[nabla]2Φ = 0] (1.16)

Solutions of these important equations will be discussed in **Chapter 3**.

*(Continues...)*

Excerpted fromClassical Electromagnetic RadiationbyMark A. Heald, Jerry B. Marion. Copyright © 2012 Mark A. Heald and Adelia Marion. Excerpted by permission of Dover Publications, Inc..

All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.

Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.

## Table of Contents

Introduction to the Dover Edition xi

Preface xiii

Chapter 1 Fundamentals of Static Electromagnetism 1

1.1 Units 2

1.2 The Laws of Coulomb and Gauss 3

1.3 Dielectric Media 9

1.4 The Laws of Biot-Savart and Ampère 15

1.5 The Lorentz Force 21

1.6 Magnetic Materials 22

1.7 Summary of Equations for Static Fields 28

1.8 Boundary Conditions on the Field Vectors 30

1.9 Point Charges and the Delta-Function 34

Problems 36

Chapter 2 Multipole Fields 43

2.1 The Electric Dipole 43

2.2 Multipole Expansion of the Potential 47

2.3 The Dipole Potential 49

2.4 The Quadrupole Potential and the Quadrupole Moment 51

2.5 Further Remarks Concerning Electric Multipoles 56

2.6 Magnetic Multipoles 58

2.7 Magnetic Scalar Potential and Fictitious Poles 63

2.8 Magnetic Circuits and the Magnetic Ohm's Law 67

Problems 70

Chapter 3 The Equations of Laplace and Poisson 76

3.1 General Properties of Harmonic Functions 77

3.2 Laplace's Equation in Rectangular Coordinates 79

3.3 Laplace's Equation in Spherical Coordinates 85

3.4 Spherical Harmonics 98

3.5 Laplace's Equation in Cylindrical Coordinates 100

3.6 Numerical Evaluation of Laplace Solutions 112

3.7 Poisson's Equation-The Space-Charge-Limited Diode 116

Problems 119

Chapter 4 Dynamic Electromagnetism 126

4.1 Conservation of Charge and the Equation of Continuity 127

4.2 Electromagnetic Induction 130

4.3 Maxwell's Modification of Ampère's Law 132

4.4 Maxwell's Equations 135

4.5 Potential Functions of the Electromagnetic Field 139

4.6 Energy in the Electromagnetic Field 143

4.7 Electrostatic Energy and Coefficients of Potential 147

4.8 The Maxwell Stress Tensor 152

4.9 The Lagrange Function for a Charged Particle in an Electromagnetic Field 156

4.10 Electromagnetism and Relativity 159

Problems 160

Chapter 5 Electromagnetic Waves 167

5.1 Plane Electromagnetic Waves in Nonconducting Media 167

5.2 Polarization 174

5.3 Poynting's Vector for Complex Fields 177

5.4 Radiation Pressure 181

5.5 Plane Waves in Conducting Media 183

5.6 Current Distribution in Conductors-The Skin Effect 188

Problems 195

Chapter 6 Reflection and Refraction 199

6.1 Reflection and Transmission for Normal Incidence on a Dielectric Medium 199

6.2 Oblique Incidence-The Fresnel Equations 203

6.3 Total Internal Reflection 211

6.4 Reflection from a Metallic Surface 214

6.5 Refraction into a Conducting Medium 218

Problems 221

Chapter 7 Waveguides 224

7.1 Two-Conductor Transmission Lines 225

7.2 Propagation of Waves Between Conducting Planes 231

7.3 Waves in Hollow Conductors 235

7.4 TE and TM Waves 238

7.5 Rectangular Waveguides 240

7.6 Optical Fibers 245

Problems 252

Chapter 8 Retarded Potentials and Fields and Radiation by Charged Particles 256

8.1 Retarded Potentials 256

8.2 Retarded Fields 261

8.3 The Liénard-Wiechert Potentials 263

8.4 The Liénard-Wiechert Fields 268

8.5 Fields Produced by a Charged Particle in Uniform Motion 271

8.6 Radiation from an Accelerated Charged Particle at Low Velocities 274

8.7 Radiation from a Charged Particle with Collinear Velocity and Acceleration 276

8.8 Radiation from a Charged Particle Confined to a Circular Orbit 279

Problems 285

Chapter 9 Antennas 289

9.1 Radiation by Multipole Moments 289

9.2 Electric Dipole Radiation 292

9.3 Complete Fields of a Time-Dependent Electric Dipole 296

9.4 Linear Antennas 303

9.5 Antenna Directivity and Effective Area 311

9.6 Electric Quadrupole Radiation 315

9.7 Antenna Arrays 322

9.8 Magnetic Dipole Radiation 327

Problems 331

Chapter 10 Classical Electron Theory 335

10.1 Scattering of an Electromagnetic Wave by a Charged Particle 336

10.2 Dispersion in Gases 340

10.3 Dispersion in Dense Matter 350

10.4 Conductivity of Metals 354

10.5 Wave Propagation in a Plasma 356

10.6 The Zeeman Effect 362

10.7 Radiation Damping 367

Problems 370

Chapter 11 Interference and Coherence 378

11.1 Wiener's Experiment and the "Light Vector" 379

11.2 Coherent and Incoherent Intensities 381

11.3 "Almost Monochromatic" Radiation 388

11.4 Interference by Division of Wave Fronts 392

11.5 Interference by Division of Amplitudes 396

11.6 Coherence Time and Lengths 398

11.7 Visibility of Interference Fringes 402

11.8 Multiple Apertures-Diffraction Grating 407

11.9 Multiple Reflections-Fabry-Perot Interferometer 411

Problems 420

Chapter 12 Scalar Diffraction Theory and the Fraunhofer Limit 423

12.1 The Helmholtz-Kirchhoff Integral 425

12.2 The Kirchhoff Diffraction Theory 427

12.3 Babinet's Principle 431

12.4 Fresnel Zones 433

12.5 Fraunhofer Diffraction 439

12.6 Single Slit 444

12.7 Double and Multiple Slits 445

12.8 Rectangular Aperture 448

12.9 Circular Aperture 451

Problems 457

Chapter 13 Fresnel Diffraction and the Transition to Geometrical Optics 462

13.1 The Fresnel Approximation 462

13.2 The Transition Between Wave and Geometrical Optics 470

13.3 Gaussian Beams and Laser Resonators 475

Problems 485

Chapter 14 Relativistic Electrodynamics 486

14.1 Galilean Transformation 487

14.2 Lorentz Transformation 490

14.3 Velocity, Momentum, and Energy in Relativity 494

14.4 Four-Vectors in Electrodynamics 499

14.5 Electromagnetic Field Tensors 503

14.6 Transformation Properties of the Field Tensor 508

14.7 Electric Field of a Point Charge in Uniform Motion 510

14.8 Magnetic Field due to a Long Wire Carrying a Uniform Current 512

14.9 Radiation by an Accelerated Charge 513

14.10 Motion of a Charged Particle in an Electromagnetic Field-Lagrangian Formulation 516

14.11 Lagrangian Formulation of the Field Equations 520

14.12 Energy-Momentum Tensor of the Electromagnetic Field 522

Problems 528

Appendix A Vector and Tensor Analysis 531

A.1 Definition of a Vector 531

A.2 Vector Algebra 533

A.3 Vector Differential Operators 534

A.4 Differential Operations in Curvilinear Coordinates 536

A.5 Integral Theorems 539

A.6 Definition of a Tensor 540

A.7 Diagonalization of a Tensor 542

A.8 Tensor Operations 543

Appendix B Fourier Series and Integrals 544

B.1 Fourier Series 544

B.2 Fourier Integrals 545

Appendix C Fundamental Constants 547

Appendix D Conversion of Electric and Magnetic Units 548

Appendix E Equivalence of Electromagnetic Equations in the SI and Gaussian Systems 549

Bibliography 551

Index 557