field force enersy and momentum

8/11/2019 Field Force Enersy and Momentum

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Field, Force, Energy and Momentum in Classical Electrodynamics

Masud Mansuripur

College of Optical Sciences

The University of Arizona, Tucson

James Clerk Maxwell13 June 1831 5 November 1879


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To Annegret, Kaveh, and Tobias


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Contents

Preface ...................................................................................................................................... i

Keywords................................................................................................................................... iii

Chapter 1: Scalar and Vector Fields1.1. Introduction......................................................................................................... 3

1.2. Space and time .................................................................................................... 3

1.3. Scalar and vector fields....................................................................................... 51.4. Gradient of a scalar field..................................................................................... 6

1.5. Integration of fields over time and/or space ....................................................... 7

1.6. Divergence of a vector field................................................................................ 91.7. Theorem of Gauss ............................................................................................... 10

1.8. Curl of a vector field........................................................................................... 10

1.9. Theorem of Stokes .............................................................................................. 111.10. Longitudinal and transverse vector plane-waves.............................................. 13

General References ............................................................................................ 14Problems............................................................................................................. 15

Chapter 2: Foundations of the Classical Maxwell-Lorentz Theory of Electrodynamics

2.1. Introduction......................................................................................................... 22

2.2. Definition: Permittivity oof free-space ............................................................. 23

2.3. Definition: Permeability oof free space ........................................................... 232.4. Speed of light cand impedance of free spaceZo ................................................ 23

2.5. Sources of electromagnetic fields ....................................................................... 23

2.6. Electric fieldEand magnetic fieldH.................................................................. 252.7. Electric displacementDand magnetic inductionB ............................................ 25

2.8. Rules of the game................................................................................................ 262.9. Rule 1: Maxwells first equation......................................................................... 262.10. Rule 2: Maxwells second equation.................................................................. 27

2.11. Continuity equation of charge and current........................................................ 30

2.12. Rule 3: Maxwells third equation ..................................................................... 32

2.13. Rule 4: Maxwells fourth equation ................................................................... 332.14. Macroscopic versus microscopic equations...................................................... 34

2.15. Bound charge and bound current associated with polarization

and magnetization ............................................................................................. 342.16. Magnetic bound charge and bound current....................................................... 35

2.17. Maxwells boundary conditions........................................................................ 36

2.18. Rule 5: Energy in electromagnetic systems...................................................... 382.19. Rule 6: Momentum density of the electromagnetic field.................................. 41

2.20. The Einstein-box gedanken experiment............................................................ 43

2.21. The thought experiment of Balazs .................................................................... 442.22. Rule 7: Angular momentum density of the electromagnetic field.................... 45

2.23. Rule 8: Force density exerted by electromagnetic fields on material media .... 46

2.24. Conservation of linear momentum ................................................................... 47

2.25. Rule 9: Torque density exerted by electromagnetic fields on material media.. 47


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2.26. Conservation of angular momentum................................................................. 48

General References ............................................................................................ 49

Problems............................................................................................................. 50

Chapter 3: Mathematical Preliminaries

3.1. Introduction......................................................................................................... 623.2. Elementary special functions .............................................................................. 623.3. The Fourier transform operator........................................................................... 66

3.4. The Fourier theorem ........................................................................................... 67

3.5. Fourier transformation in higher dimensions...................................................... 683.6. Bessel functions and their properties .................................................................. 71

General References ............................................................................................. 75

Problems.............................................................................................................. 76

Chapter 4: Solving Maxwells Equations

4.1. Introduction......................................................................................................... 82

4.2. Plane-wave solutions of Maxwells equations.................................................... 834.3. Electric field produced by a stationary point-charge (electrostatics).................. 86

4.4. Electric field of a line-charge (electrostatics) ..................................................... 87

4.5. Electric field of a uniformly-charged plate (electrostatics) ................................ 874.6. Magnetic field of a long, thin wire carrying a constant current

(magnetostatics) ................................................................................................. 88

4.7. Magnetic field of a hollow cylinder carrying a constant current

(magnetostatics) ................................................................................................. 894.8. Electric field produced by a point-dipole (electrostatics)................................... 90

4.9. Fields radiated by an oscillating point-dipole (electrodynamics) ....................... 91

4.10. Radiation by an oscillating current sheet (electrodynamics) ............................ 94

4.11. Radiation by an oscillating line-current (electrodynamics) .............................. 964.12. Radiation by a hollow cylinder carrying an oscillating current

(electrodynamics).............................................................................................. 98

General References ............................................................................................ 102

Problems............................................................................................................. 103

Chapter 5: Solving Maxwells Equations in Space-time: The Wave Equation

5.1. Introduction......................................................................................................... 113

5.2. Scalar potential ( , )t r as the solution of a 2nd

-order partial

differential equation ............................................................................................ 113

5.3. Vector potential ( , )tA r as the solution of a 2nd

-order partialdifferential equation ............................................................................................ 114

5.4. Meaning of the Laplacian operator acting on a vector field ............................... 115

5.5. Relating scalar and vector potentials to their sources in the

space-time domain ............................................................................................... 116

5.5.1. Example: Oscillating point-dipole ............................................................. 118

5.5.2. Example: Infinitely-long, thin, current-carrying wire radiating

cylindrical waves ...................................................................... 119


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5.5.3. Example: Infinite sheet of oscillating current radiating plane-waves........ 120


Problems.............................................................................................................. 125

Chapter 6: The Lorentz Oscillator Model

6.1. Introduction......................................................................................................... 1406.2. Mass-and-spring model of an atomic dipole....................................................... 140

6.3. Generalization to the case of multi-electron atoms and molecules..................... 1426.4. Drude model of the conduction electrons ........................................................... 142

6.4.1. Example..................................................................................................... 143

6.5. The Clausius-Mossotti relation........................................................................... 144

6.6. Dependence of the real and imaginary parts of C() on frequency ...................1456.7. Phase and group velocities.................................................................................. 147

6.7.1. Example 1.................................................................................................. 148

6.7.2. Example 2.................................................................................................. 149

6.7.3. Example 3.................................................................................................. 1506.7.4. Example 4.................................................................................................. 151

6.8. Step-response and Impulse-response .................................................................. 152

6.9. The Kramers-Kronig relations ............................................................................ 154


Problems.............................................................................................................. 157

Chapter 7: Plane Electromagnetic Waves in Isotropic, Homogeneous, Linear Media

7.1. Introduction......................................................................................................... 1637.2. Complex vector algebra of the electromagnetic field ......................................... 164

7.3. Plane electromagnetic waves and their properties .............................................. 166

7.4. Plane-waves in isotropic, homogeneous, linear media ....................................... 1677.5. Energy flux and the Poynting vector .................................................................. 168

7.6. Reflection and transmission of plane-waves at a flat interface between

adjacent media.................................................................................................... 169

7.6.1. Case of TM or p-polarized incident plane-wave at a flat interfacelocated atz= 0 ........................................................................................171

7.6.2. Case of TE or s-polarized incident plane-wave at a flat interface

located atz= 0 ........................................................................................172

7.7. Fresnel reflection and transmission coefficients in several casesof practical interest .............................................................................................. 172

7.7.1. Special Case 1: normal incidence .............................................................. 1737.7.2. Special Case 2: Brewsters angle............................................................... 173

7.7.3. Special Case 3: total internal reflection ..................................................... 174

7.8. Concluding remarks............................................................................................ 174


Problems.............................................................................................................. 176


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11.2. Mass-and-spring model of polarization exhibiting spatial dispersion ............. 247

11.3. Dispersion relations ......................................................................................... 24811.4. Case of s-polarized incident plane-wave ......................................................... 249

11.5. Case of p-polarized incident plane-wave ......................................................... 250

11.6. Mechanical energy density, energy loss rate, and a mechanical Poynting

vector................................................................................................................. 252General References ........................................................................................... 254

Chapter 12: The Reciprocity Theorem

12.1. Introduction....................................................................................................... 25512.2. Electromagnetic field radiated by an oscillating electric dipole....................... 257

12.3. Electromagnetic field radiated by an oscillating magnetic dipole .................... 25912.4. Reciprocity in a system containing electrically-polarizable media ................. 259

12.5. Reciprocity in systems containing both electric and magnetic media ............. 262

12.6. Reciprocity in the presence of spatial dispersion ............................................. 263

12.7. Comparison with standard proofs of reciprocity .............................................. 264

12.8. Summary and Concluding Remarks ................................................................ 267References......................................................................................................... 269

Problems ........................................................................................................... 270

Solutions to Selected Problems .............................................................................................. 271

Appendix A: Vector Identities................................................................................................. 319

Appendix B: Vector Operations in Cartesian, Cylindrical, and Spherical Coordinates.......... 320

Appendix C:Useful Integrals and Identities ........................................................................... 321

Index......................................................................................................................................... 323


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Preface

This book grew out of a graduate-level course in electrodynamics that I have taught at the University

of Arizonas College of Optical Sciences over the past six years. A typical student enrolled in the course

is a first year graduate student in Optical Sciences, Electrical Engineering, or Physics, who has had some

prior exposure to electromagnetic theory. The level of mathematics required for this subject is not

particularly advanced; students are expected to be familiar with calculus, vector algebra, complex

numbers, ordinary differential equations, and elementary aspects of the Fourier transform theory. Most of

the mathematical tools and techniques needed for developing the theory of electrodynamics are in fact

interwoven with the course material in the form of a section here, a chapter there, or a few problems at the

end of each chapter. The student is thus motivated to learn the required mathematics in the relevant

physical context whenever the need arises.

The approach of this book to classical electrodynamics is rather unconventional. It begins with a

minimum set of postulates that are considered fundamental in the sense that they cannot be derived from

each other or from other laws of classical physics. The set of postulates, of course, must be self-

consistent, as well as consistent with the conservation laws and with special relativity. These postulates

are described in their most general form at the outset, with no apologies for their sudden appearance and

no attempt to motivate them, say, by tracing the historical path that led to their discovery. The laws of

nature are what they are; it may have taken man a long and tortuous path to their discovery, but once thelaws are known, one should simply accept them and try to understand their consequences.

In this context, an analogy with a board game such as chess is constructive. Before one sets out to

play the game, one must learn the configuration of the board, the identity of the pieces, and the governing

set of rules in their detailed and complete form. For most practical purposes, it is irrelevant how the game

has evolved over the years, how the rules may have changed, and whether or not there is any justification

for the rules. The important thing is to learn the rules and play the game. In the case of physics, of course,

the postulates are justified because their consequences agree with observations. This, however, is

something that one will appreciate later, as one begins to understand the subject and learns how to deduce

the logical consequences that flow from the basic principles. The task before the student, therefore, is to

master the nomenclature and learn the basic rules of electrodynamics, then try to deduce their

consequences.

In the presence of known sources of radiation (i.e., sources whose spatio-temporal distributions aregiven a priori) we will use the method of plane-wave decomposition and superposition to derive general

expressions for electromagnetic fields and potentials. This will enable us to examine several idealized

situations in Chapters 4 and 5, thereby gaining insight into the nature of electromagnetic fields and

radiation. In the course of this analysis, I find it useful to move back and forth between the space-time

domain, where the fields and their sources reside, and the Fourier domain, which is home to various

plane-waves whose superposition reproduces the fields and the sources. The mathematical methods used

in the two domains may differ, but the final results pertaining to physical observables of any given system

are invariably the same.Throughout the book I have striven to be brief yet precise. Whenever possible, I develop a general

formalism to tackle a given class of problems, then specialize the solution to examine specific problems

within that class. For example, in dealing with plane-wave propagation in isotropic, homogeneous, linear

media (Chapter 7), Maxwells equations are solved in a way that is applicable to transparent as well asabsorptive media, encompassing both propagating and evanescent waves while accommodating arbitrary

states of polarization (i.e., linear, circular, elliptical). Once the general solution is at hand, a few specific

examples show its application to problems such as propagation in transparent or absorptive media, total

internal reflection, incidence at Brewsters angle, etc. The student is thus equipped with the tools needed

to tackle problems within a broad class, without having to learn each specific case as an isolated instance.

The theory of electrodynamics is too broad, and its applications too diverse, to allow coverage in a

brief textbook such as this one. My goal, therefore, is not to be comprehensive, but rather to build a

foundation upon which one could base future learning and further investigations. Throughout the book,

i


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Field, Force, Energy and Momentum in Classical Electrodynamics

Masud Mansuripur

Keywords for Chapter 1:

Scalar Field, Vector Field, Divergence, Gradient, Curl, Gausss Theorem, Stokess Theorem,Longitudinal Field, Transverse Field.


Maxwells Equations, Electromagnetic Field, Displacement Field, Magnetic Induction, Free

Charge, Free Current, Bound Charge, Bound Current, Continuity Equation, Impedance of Free

Space, Electromagnetic Energy, Field Momentum, Field Angular Momentum.


Fourier Transform, Fourier Operator, Fourier Theorem, Diracs Delta-Function, Sifting Property

of Delta-Function, Bessel Functions.


Plane-wave Solutions, Maxwells Equations, Electromagnetic Radiation, Scalar Potential, Vector

Potential, Lorenz Gauge, Point Charge, Point Dipole, Line Current, Current Sheet, OscillatingElectric Dipole, Oscillating Magnetic Dipole, Oscillating Hollow Cylinder.


Solutions of Maxwells Equations, Vector Potential, Scalar Potential, Lorenz Gauge, LaplacianOperator, Oscillating Point Dipole, Cylindrical Wave, Current-Carrying Wire, Oscillating Sheet

of Current.


Lorentz Oscillator, Mass-and Spring Model, Single-electron Lorentz Model, Multi-electron

Lorentz Model, Drude Model, Conduction Electrons, Clausius-Mossotti Relation, Phase

Velocity, Group Velocity, Dispersion, Kramers-Kronig Relations.


Plane-wave, Plane Electromagnetic Waves, Linear Media, Isotropic Media, Homogeneous

Media, Energy Flux, Poynting Vector, Reflection Coefficient, Transmission Coefficient, FresnelCoefficients, p-polarization, s-polarization, Normal Incidence, Oblique Incidence, Brewsters

Angle, Total Internal Reflection.


Plane Electromagnetic Waves, Multilayer Stack, Reflection Coefficient, Transmission

Coefficient, Parallel-Plate Slab, Optical Cavity, Cavity Resonator, Perfectly Matched Layer,

Finite Difference Time Domain Method.

iii


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Solution of Maxwells Equations, Cylindrical Coordinates, Linear Media, Isotropic Media,

Homogeneous Media, Circular Symmetry, Cylindrical Symmetry, Bessel Functions, Plane-waveSuperposition, Hankel Functions, Guided Modes, Surface Plasmon Polariton, Energy Flux,

Poynting Vector.


Electromagnetic Momentum, Electromagnetic Angular Momentum, Force, Torque, Radiation

Pressure, Momentum of a Light Pulse, Transparent Slab, Semi-Transparent Slab, Brewsters

Angle Incidence, Spherical Glass Bead, Optical Vortex, Circular Polarization, MomentumConservation.


Plane-wave Propagation, Plane-wave Solutions of Maxwells Equations, Linear Media,Homogeneous Media, Isotropic Media, Spatial Dispersion, Mass-and-Spring Model, Dispersion

Relations, s-Polarized Incidence, p-Polarized Incidence, Mechanical Energy Density, MechanicalPoynting Vector.


Reciprocity, Classical Electrodynamics, Electromagnetic Field, Oscillating Electric Dipole,

Oscillating Magnetic Dipole, Electrically-Polarizable Media, Magnetic Media, Spatial

Dispersion, Standard Proofs of Reciprocity.

iv


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The tendency of modern physics is to resolve the whole material universe into waves,and nothing but waves. These waves are of two kinds: bottled-up waves, which wecall matter, and unbottled waves, which we call radiation or light. If annihilation ofmatter occurs, the process is merely that of unbottling imprisoned wave-energy andsetting it free to travel through space. These concepts reduce the whole universe to aworld of light, potential or existent, so that the whole story of its creation can be toldwith perfect accuracy and completeness in the six words: God said, Let there be light.

Sir James Jeans (1877-1946)

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Field, Force, Energy and Momentum in Classical Electrodynamics,2011, 3-21 3

CHAPTER 1

Scalar and Vector Fields

The aim of exact science is to reduce the problems of nature to the determination of quantities byoperations with numbers.

James Clerk Maxwell (1831-1879)

Abstract. The concepts of scalar and vector fields, which are central to the theory of

electrodynamics, are introduced. These fields are generally defined in 3-dimensional Euclidean

space as complex-valued functions of the space-time coordinates (x,y,z, t). Integration and

differentiation in time and space, leading to such operations as gradient, divergence, and curl,

and subsequently to theorems of Gauss and Stokes, are developed. The intuitive approach taken

here avoids mathematical formalism in favor of physical understanding. Throughout the

chapter, examples based on complex-valued scalar and vector plane-waves help to illustrate the

various mathematical operations. The end-of-chapter problems should help refresh the readers

memory of elementary mathematical tools needed in this as well as in subsequent chapters.

1.1. Introduction. This chapter introduces the concepts of scalar and vector fields in flat space-time, using Lorentzian coordinate systems. The fields are generally complex-valued, which is

convenient for algebraic manipulations. The physical fields, of course, are always real-valued

and, therefore, will be represented by the real parts of the complex entities that are used here todescribe the field strengths and their variations throughout space and time. There are various

ways to integrate and also to differentiate the fields in space-time, e.g., time integration and time

differentiation; spatial differentiation in the form of gradient, divergence, and curl operations;

and spatial line-, surface-, and volume-integrations. We will describe these operations in somedetail, prove the theorems of Gauss and Stokes, which pertain to vector fields and their spatial

derivatives and integrals, and provide examples in each case using a particularly useful field, the

plane-wave.

1.2. Space and time. Electromagnetic phenomena take place in space and time. An event inspace-time occurs at a point r in space at time t. We shall assume that all events occur in flat

space-time (i.e., in the absence of gravity and gravitational fields), and that all events are

observed by an inertial observer, namely, one whose motion is unaccelerated (relative to distant

stars). In addition, the observer uses a Lorentzian reference frame for all his observations. Thusevery event will be specified in an orthonormal coordinate system such as (r, t) = (x,y,z, t)

Cartesian or (,,z, t) cylindrical or (,,,t) spherical; see Fig.1.One may imagine that each point r in space has its own ideal (e.g., atomic) clock, which

runs at a fixed rate, and that the clocks at all locationsrare coordinated and synchronized by the

inertial observer in whose reference frame all observations are made. Think of a clock located at

r as a compact, tightly-wound spiral curve, with a pointer moving inexorablyalong the spiral in the same (forward) direction. Each point rhas its own spiral

clock. When an event takes place at r, the pointer will be somewhere along thespiral curve; the location of the pointer, as measured by the length of the spiral

from its starting point at t= 0, is the time associated with the event.

The reason we are using a tightly-wound spiral curve to represent the time axis, is that we

have run out of easily imaginable dimensions. In a 2-dimensional (2D) space, where events areconfined, for example, to a planar surface embedded within a 3D Euclidean space, one does not

Masud Mansuripur

All rights reserved 2011 Bentham Science Publishers


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Scalar and Vector Fields Field, Force, Energy and Momentum in Classical Electrodynamics 5

1.3. Scalar and vector fields. When a number, generally complex-valued, is associated with

each point or event (r, t) in space-time, we will have a complex functionf(r,t), generally referredto as ascalar field. The temperature T, specified at each point rin a room and at each instant of

time tis a good example of a real-valued scalar field T(r, t).

Similarly, when a vector V is associated with each point (r,t) in space-time, we have a

vector field V(r,t). In 3D space, vectors are generally specified by their 3 components alongspecific directions. Thus, in a Cartesian coordinate system we have, at each point (r,t), three

numbers (Vx,Vy,Vz) that specify the vector field. Similarly, in a cylindrical coordinate system, the

vector field is identified by (V,V,Vz), while in spherical coordinates the components of the field

are (V,V,V). Note that there are no good reasons to restrict the components of vectors to beingreal-valued numbers. Thus, in general, a vector field assigns to each point (r,t) a complex-valued

vector, i.e., a vector whose three components in Euclidean space are complex-valued. One way

to visualize a complex-valued vector field is to imagine a pair of ordinary, real-valued vectors

(V,V) attached to each point in space, while the magnitudes and directions of both Vand V

at each and every locationrchange arbitrarily with time. The complex-valued vector field is then

described by V(r,t) = V(r,t) + iV(r, t), where, in Cartesian coordinates,

Vx(r, t) = Vx (r, t) + iVx(r, t), (1a)

Vy(r, t) = Vy (r, t) + iVy(r, t), (1b)

Vz(r, t) = Vz(r,t) + iVz(r,t). (1c)

Similar expressions may be written for the components of a complex vector field in othercoordinate systems as well.

Of course, what distinguishes a vector field (real or complex) from a mere collection of

three scalar fields is vector algebra, namely, the rules of addition, subtraction, dot-multiplication,and cross-multiplication of vectors. For concreteness, we shall describe these rules in Cartesian

coordinates only, although the same applies in any orthonormal coordinate system as well. For

addition and subtraction of vector fields we haveV1(r, t) V2(r, t) = [Vx1(r, t) Vx2(r,t)]x

^+ [Vy1(r,t)Vy2(r, t)]y

^+ [Vz1(r,t)Vz2(r,t)]z

^. (2)

Note that the corresponding components of the two fields are simply added together or

subtracted from each other. The addition and subtraction rules for complex numbers being wellknown, the above rule for addition and subtraction of vector fields clearly applies to real as well

as complex vector fields.

For dot- and cross-multiplication of two vector fields such as V1(r, t) and V2(r, t), one must

express each field as the sum of its three components, Vxx^

+Vyy^

+Vzz^, then proceed to multiply

all the components of V1into all the components of V2using the following rules:

Dot: x^ x

^=y

^ y

^=z

^ z

^= 1; x

^ y

^=y

^ x

^=x

^z

^=z

^ x

^=y

^ z

^=z

^ y

^= 0. (3)

Cross: x^ x

^=y

^ y

^=z

^ z

^= 0; x

^ y

^= y

^ x

^=zz

^; y

^ z

^= z

^ y

^=x

^; z

^x

^= x

^ z

^=y

^. (4)

Once again, the multiplication rule for complex numbers being well known, it makes nodifference whether the components of the vector fields being multiplied together are real- or

complex-valued.

A good example of a real-valued vector field is the wind velocity within a wind tunnel,described as a function of time tfor each pointrin 3D space. To each pointris thus assigned a


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22 Field, Force, Energy and Momentum in Classical Electrodynamics,2011, 22-61

CHAPTER 2

Foundations of the Classical Maxwell-Lorentz Theory of Electrodynamics

"From a long view of the history of mankind, seen from, say, ten thousand years from now, there canbe little doubt that the most significant event of the 19th century will be judged as Maxwell's

discovery of the laws of electrodynamics. The American Civil War will pale into provincialinsignificance in comparison with this important scientific event of the same decade."

Richard P. Feynman (1918-1988)

Then came H. A. Lorentz's decisive simplification of the theory. He based his investigations withunfaltering consistency upon the following hypotheses: The seat of the electromagnetic field is theempty space. In it there are only one electric and one magnetic field vector. This field is generated byatomistic electric charges upon which the field in turn exerts ponderomotive forces. The onlyconnection between the electromagnetic field and ponderable matter arises from the fact thatelementary electric charges are rigidly attached to atomistic particles of matter. For the latterNewton's law of motion holds. Upon this simplified foundation Lorentz based a complete theory ofall electromagnetic phenomena known at the time, including those of the electrodynamics of movingbodies. It is a work of such consistency, lucidity, and beauty as has only rarely been attained in anempirical science.

Albert Einstein (1879-1955)

Abstract. The sources of electromagnetic fields are electric charge, electric current, polarization

and magnetization. The relationships among the fields and their sources, all of which

represented by functions of space and time, are described by Maxwells macroscopic equations.

The fields carry energy, whose rate-of-flow at each point in space at any instant of time is given

by the Poynting vector. At any location where one or more fields and one or more sources

reside simultaneously, there could occur an exchange of energy between the fields and the

sources. The time-rates of such exchanges are uniquely specified by the Poynting theorem,

which is a direct consequence of Maxwells macroscopic equations in conjunction with the

definition of the Poynting vector. Electromagnetic fields also carry momentum and angularmomentum, whose densities at all points in space-time are simple functions of the local

Poynting vector. A generalized version of the Lorentz law of force dictates the time-rate of

exchange of momentum between the fields and the sources in regions of space-time where they

overlap. There also exists a simple expression for the torque exerted by the fields on the

sources, which defines the time-rate of exchange of angular momentum between them. This

chapter is devoted to a precise and detailed description of the relations among the fields and

their sources, as well as their interactions involving electromagnetic force, torque, energy,

momentum, and angular momentum.

2.1. Introduction. Throughout this chapter we shall treat the classical theory of electrodynamics

as a game of chess. The board on which the game unfolds is the three-dimensional Euclidean

space; in other words, we assume a flat space-time, in which no gravitational deformations ofspace-time geometry are allowed. We choose an inertial (i.e., unaccelerated) observer, and use a

Lorentzian reference frame to assign coordinates to each and every point in space-time.Locations on our cosmic chessboard are thus uniquely identified (within the given Lorentzian

frame of reference) by their four-dimensional space-time coordinates (r,t). We shall identify the

pieces that reside within the 3D space (i.e., the chessboard) and move around through time.Also specified will be the rules of the game, according to which the pieces interact with each

Masud Mansuripur



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Foundations of the Classical Maxwell-Lorentz Field, Force, Energy and Momentum in Classical Electrodynamics 23

other and evolve in space and time. Throughout the chapter, the system of units will be MKSA

(meter, kilogram, second, ampere).

2.2. Definition: Permittivity o of free-space. The permittivity of free-space (i.e., vacuum,

empty space) is o= 8.8541012

farad/meter. In what follows we will clarify the meaning of

permittivity as well as the relationship between farad and the basic units of the MKSA system.

2.3. Definition: Permeability oof free space. The permeability of free space is o= 4107

henry/meter. In what follows we will clarify the meaning of permeability as well as the

relationship between henry and the basic units of the MKSA system.

2.4. Speed of lightcand impedance of free space Zo: The speed of light in vacuum, c, can bederived from Maxwells equations. The exact relation between c and the permittivity and

permeability of free space will be seen to be c =1/oo. Given that, in principle, cis preciselymeasurable (c =2.9979245810

8m/s), and that the exact value of ois 410

7henry/meter,

it is common practice to express the precise value of oas 1/(oc2). Equating the units on both

sides of the equation yields: faradhenry=sec

2

.Also, as will be seen later, the impedance of free space is defined as Zo=o/o377ohm.

(The symbol for ohm, the unit of electrical resistance, is .) Equating the units on the two sides

of the equation yields: henry/farad= 2. Combining this with the previously obtained relation

between farad and henry, we find: farad = sec/and henry = sec.

The special combinations of oand oin the expressions for candZoappear quite naturallyin electrodynamics equations. Therefore, whenever possible, we will use c and Zo to simplify

expressions that contain various combinations of oand o.

2.5. Sources of electromagnetic fields: There exist four material sources in electromagnetic

(EM) systems: free(r,t), Jfree(r,t), P(r, t), and M(r,t). These are continuous and differentiable

functions of spacerand time t. The density of free charge,free, is a scalar; its MKSA units arecoulomb/m

3. (Since currentIis the time-rate of flow of charge Q, namely,I = dQ/dt, the units of

charge are: coulomb = ampere sec.) The other three sources, namely, density of free currentJfree

(units = ampere/m2), polarization P (units = coulomb/m

2), and magnetization M (units =

henry ampere/m2= weber/m

2), are vector functions of r and t. Loosely speaking, these four

sources furnish some of the pieces of the aforementioned chess set: Charge, Current,Polarization, and Magnetization reside in 3D space (and change in time) in ways that are roughly

similar to the way in which pawns, bishops and knights occupy positions on the chessboard and

move around. The analogy is not perfect, of course, and one should not push it too far. Forinstance, the various sources of EM fields can overlap in the same region of space; their

magnitudes at each point in space can vary continuously with time; they produce the EM fields,

but also are influenced by these fields; the sources can convert from one form to another, forexample, current can contribute to charge density and vice versa; etc.

Digression: The current density Jcould arise from the motion of . So, for example, J(r, t) =(r, t)V(r, t), where V(r,t) is the velocity of charge at the point (r, t) in space-time. (Check theconsistency of the units of Jwith those of the product V.) However, there could also exist Jwithout a net . For example, when, in a typical copper wire, the positive charges (copper ions)


18/57

24Field, Force, Energy and Momentum in Classical Electrodynamics Masud Mansuripur

are stationary, while the negative charges (conduction electrons) move with some velocity V

along the length of the wire (under the influence of an electric field), assuming equal densities

for the two types of charge, we have a situation in which the net charge density is zero, yetthere exists a non-zero current densityJ.

Charge and current densities are intimately related via the continuity equation

Jfree(r,t) + free(r, t)/t = 0. Thus any net current flowing in or out of a given volume mustchange the total charge content of that volume.

Fig. 1. (a) An electric dipole consists of a pair of equal and opposite charges, q, separated by a

small distance d. The dipole moment is defined asp =qd, withdalways pointing from the negative

to the positive charge. (b) The total strength of the dipole momentspn(t) within a small volume V

surrounding a given pointr, when normalized by Vyields the polarizationP(r, t) of the material

medium at locationrand time t.

P(r, t) is the density of atomic electric dipole moments. Each atomic dipole is characterized

by its equal and opposite charges q, and the small separation d between these charges. Bydefinition, the direction ofdis from negative to positive charge. The dipole moment is defined as

p=qd(units of p= coulomb meter). To determine the polarization P(r,t), take a small volume

Vcentered on r, then find the vector sum of all the individual dipole moments pnwithin that

volume. Normalizing the total dipole moment within V by the volume V yields the localpolarizationP(r,t); units ofP= coulomb/m

2.

Fig. 2. (a, b) Magnetic dipoles are produced by spinning elementary particles, by the orbital

motion of charged particles, and also by various combinations of spin and orbital magneticmoments within atoms and molecules. In the case of a small, flat loop of area A carrying an

electric currentI, the magnitude and direction of the dipole moment are given bym=oIAz^, with

z being the surface normal in the direction determined by the right-hand rule in conjunction with

the sense of circulation of the currentI. (c) The total strength of the dipole momentsmn(t) within a

small volume Vsurrounding a given point r, when normalized by Vyields the magnetizationM(r, t) of the material medium at locationrand time t.

+p =qd

q

-q

Small volume V

(centered atr)

P(r,t) = (1/V)pn(t)

p1

p2

p3pn

d

Electric dipole moment

(a) (b)

m =oIAz

Small volume V

(centered atr)

M(r,t) = (1/V)mn(t)

m1

m2

m3mn

Orbital magneticmoment

Spin magneticmoment

m

(a) (b) (c)


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CHAPTER 3

Mathematical Preliminaries

" the enormous usefulness of mathematics in the natural sciences is something bordering on themysterious and that there is no rational explanation for it."

Eugene Wigner (1902-1995), in "The Unreasonable Effectiveness of Mathematics in the Natural Sciences."1

Abstract. In preparation for a Fourier analysis of Maxwells equations in the following chapter,

we describe here the mathematics of Fourier transformation, exploring certain properties of the

forward and reverse Fourier operators. Several special functions are also discussed notable

among them, Diracs delta-function and various Bessel functions which appear frequently in

Fourier analysis and elsewhere. Simple charge- and current-density distributions serve as

exemplary electromagnetic systems that can be readily transformed into the Fourier domain.

3.1. Introduction. This chapter provides a brief overview of the mathematical tools and

techniques needed for the analysis of Maxwells equations by means of Fourier transformation.

We begin by introducing a few elementary functions that are specially useful in the context ofFourier transform theory, explore their properties, and proceed to rely on them when describing

the properties of the Fourier operator. Another class of special functions, which appear

frequently in Fourier transform theory and elsewhere, are Bessel functions discussed at the endof the chapter.

The forward and reverse Fourier integrals are initially defined and analyzed for complex-

valued functions of a single real-valued variable; these will be referred to as one-dimensional(1D) Fourier transforms. Subsequently, we generalize the concept to higher-dimensional spaces,

where complex functions of two or more real variables are transformed back and forth between amulti-dimensional space and its corresponding Fourier domain. With regard to the solution of

Maxwells equations, which generally reside in Lorentzian space-time, the functions of interest

are usually four-dimensional (4D), as they depend on the space-time coordinates (r, t) = (x,y,z, t).The corresponding Fourier domain in this case will also be a 4D space whose coordinates, often

referred to as the spatio-temporal frequencies, are denoted by (k,) = (kx,ky,kz,).

3.2. Elementary special functions. Certain special functions play an important role in the theoryof Fourier transforms. In this section we provide a brief review of the various properties of these

functions, and demonstrate their usefulness in problems involving Fourier transforms.

a) The unit-step function Step(x) is equal to 0.0 when x < 0 and equal to 1.0 when x > 0. The

value of the function at x = 0 could be assigned arbitrarily, as it does not affect the properties of

the function. For the sake of completeness, however, we define Step(x) = whenx = 0. A plot ofthis function appears in Fig. 1(a).

b) The unit rectangular pulse function Rect(x) is equal to 1.0 when |x| < , and equal to 0.0 when

|x| > . The value of the function atx = could be assigned arbitrarily, as it does not affect theproperties of the function. For the sake of completeness, however, we define Rect(x) = when

x = . A plot of this function appears in Fig. 1(b). Note that the area under Rect(x) is unity.

Masud Mansuripur



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Mathematical Preliminaries Field, Force, Energy and Momentum in Classical Electrodynamics 63

c) The unit triangular pulse function Tri(x) is equal to 1 |x| when |x| < 1, and equal to 0.0 when|x| > 1, as shown in Fig. 1(c). Like the rectangular pulse, the area under the triangular pulse is

equal to 1.0.

d) The sinc function sinc(x) is defined as sin(x)/(x) over the entire x-axis. A plot of thisfunction appears in Fig. 1(d). Note that the value of the function at x = 0 is 1.0. Also, the area

under the function can be shown to be unity, that is,

sinc(x)dx =1.0.

e) The Dirac delta-function (x) does not have a simple definition, and cannot be easilyvisualized in a unique and unambiguous way. A good way to describe it would be as a very tall

and very narrow function ofx, centered atx = 0, symmetric around this central point, and with an

area equal to 1.0, that is, (x)dx =1.0. Thus 1Rect(x/), where is a small, real-valued,

positive constant would be the simplest representation of (x); see Fig.2(a). Similarly,1Tri(x/) approaches a delta-function in the limit when 0; see Fig.2(b). For sufficientlysmall , there is essentially no difference between 1Rect(x/) and 1Tri(x/), except for thelatter function being continuous and readily differentiable which would be useful if one were

interested in the first derivative (x) of (x).

Fig. 1. Plots of several elementary functions. (a) Unit-step function. (b) Unit rectangular pulse. (c)

Unit triangular pulse. (d) The sinc function.

Another embodiment of Diracs delta-function is 1sinc(x/) in the limit when 0. Thefact that sinc(x) has an infinite number of oscillations is of no consequence, so long as the chosen

value of is small enough to cram a large number of these oscillations into a small

neighborhood of the origin (x = 0), thus ensuring that the area under the function in thatneighborhood is as close to unity as is desired.

The various properties of (x) can be easily understood with the aid of the above definitionsand analogies. For example, the fact that (x) =(x) is a direct consequence of the requirementof symmetry aroundx = 0. Or, (2x) =(x) is clearly true given that, for sufficiently small ,the function 1Rect(2x/), is even, tall, narrow, and has an area equal to , as can be seen inFig.2(c). By the same token, (x)=(1/ || )(x) for any real-valued and (x +) =(1/||)[x+(/)]. The latter delta-function is centered atx =/, having an area equal to 1/|| .

x

Step(x)

x

Rect(x)

1.0

(a) (b)

(c) Tri(x)

(d) sinc(x)

x1.0

1.0

1.0

1.0

1.01.0

2.0

3.0

x


21/57


Figure 2(d) shows that a unit-step function, when smoothed-out over the interval (, )and differentiated with respect tox, yields the delta-function (x) in the limit when 0.

The most important property of (x), which is always true irrespective of the functionalform used to visualize the delta-function, is its sifting property. If an arbitrary function f(x)

happens to be continuous atx = 0, then the sifting property of (x) is stated as follows:

f(x)(x)dx =f(0). (1a)

More generally, whenf(x) is discontinuous atx = 0, we will have

f(x)(x)dx = [f(0+) +f(0)]. (1b)

Fig. 2. Visualizing Diracs delta-function (a) as a tall, narrow rectangular pulse, and (b) as a tall,

narrow triangular pulse. (c) Representing (2x) as 1Rect(2x/) for sufficiently small . (d) Aunit-step function whose transition from 0 to 1 has been softened and extended over the small

interval (, ) yields the rectangular-pulse approximation to a delta-function whendifferentiated with respect tox.

The sifting property can be explained by super-imposing the graph of f(x) on any one of the

visualizations of (x), then multiplying the two functions together and integrating over any

region of the x-axis that contains a small neighborhood of the origin (x = 0). It should also beobvious that the sifting property remains valid when (x) is shifted along the x-axis by anarbitrary amount. For instance, if (x) is shifted to x =x0, at which point f(x) happens to becontinuous, we will have

f(x)(xx0)dx =f(x0). (1c)

x

1Rect(x/)

1/

(a)

(d) Step(x)

/2

1.0

x

/2x

1Tri(x/)

1/

(b)

x

1Rect(2x/)

1/

(c)

/4 /4

/2/2x

d

1/

/2 /2

Step(x)dx


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CHAPTER 4

Solving Maxwells Equations

Lorentz proclaimed the very radical thesis which had never before been asserted with suchdefiniteness: The ether is at rest in absolute space. In principle this identifies the ether with absolutespace. Absolute space is no vacuum, but something with definite properties whose state is describedwith the help of two directed quantities, the electrical field and the magnetic field , and, as suchis called the ether.

Max Born (1882-1970)

I cannot but regard the ether, which can be the seat of an electromagnetic field with its energy andits vibrations, as endowed with a certain degree of substantiality, however different it may be fromall ordinary matter.

Hendrik Antoon Lorentz (1853-1928)

Abstract. We solve Maxwells macroscopic equations under the assumption that the sources of

the electromagnetic fields are fully specified throughout space and time. Charge, current,

polarization, and magnetization are thus assumed to have predetermined distributions asfunctions of the space-time coordinates (r, t). In this type of analysis, any action by the fields on

the sources will be irrelevant, in the same way that the action on the sources by any other force

be it mechanical, chemical, nuclear, or gravitational need not be taken into consideration. It

is true, of course, that one or more of the above forces could be responsible for the presumed

behavior of the sources. However, insofar as the fields are concerned, since the spatio-temporal

profiles of the sources are already specified, knowledge of the forces would not provide any

additional information. In this chapter, we use Fourier transformation to express each source as

a superposition of plane-waves. Maxwells equations then associate each plane-wave with other

plane-waves representing the electromagnetic fields. Inverse Fourier transformation then

enables us to express the electric and magnetic fields as functions of the space-time coordinates.

4.1. Introduction. Electromagnetic (EM) fields originate from the sources free(r, t), Jfree(r, t),P(r, t), and M(r, t). When these sources are fully specified throughout space and time, theresulting EM fields can be uniquely and unambiguously derived from Maxwells macroscopic

equations.

A simple yet powerful method of solving Maxwells equations under such circumstancesinvolves shuttling back and forth between the space-time domain (r, t) and the Fourier domain

(k, ). Since the sources are specified everywhere in space for all time, they can be Fouriertransformed and, therefore, expressed as superpositions of plane-waves. Through Maxwells

equations, these plane-waves may be related to other plane-waves that describe the scalar and

vector potentials (r, t),A(r, t), and also the EM fieldsE(r, t),D(r, t),H(r, t), andB(r, t). Oncethe amplitudes of the plane-waves representing the fields and/or the potentials for all values of

(k, ) are determined, the linearity of Maxwells equations allows one to compute the exactdistributions of the fields and/or the potentials by superposing these plane-waves via inverse

Fourier transformation.

In this chapter we employ the mathematical tools and techniques developed in Chapter 3 tocalculate the EM field distributions for several systems of general as well as practical interest.

The Fourier domain relations that connect the fields and the potentials to their sources are

derived in the next section. The utility of these relations are subsequently demonstrated throughseveral examples.

Masud Mansuripur



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Solving Maxwells Equations Field, Force, Energy and Momentum in Classical Electrodynamics 83

4.2. Plane-wave solutions of Maxwells equations. The scalar and vector potential functions in

the form of plane-waves in vacuum are given by

o( , ) exp[i( )],t t = r k r (1a)

o( , ) exp[i( )].t t= A r A k r (1b)

In general,kandAoare complex vectors, ois a complex scalar, and is a complex-valued

constant.One can always associate the curl of the vector potential with the B-field, for the simple

reason that the divergence of the curl of any vector field is identically zero, which, in the present

case, means that Maxwells 4th

equation, ( , ) 0,t =B r is automatically satisfied. We thus write

o( , ) ( , ) i exp[i( )].t t t= = B r A r k A k r (2)

Clearly, the complex B-field amplitude is o oi ,= B k A and the divergence of ( , )tB r

vanishes everywhere because 2 o oi ( ) ( ) 0. = =k k A k k A It might be helpful to think

momentarily ofkandAoas real-valued vectors, to realize thatBois related only to the transversecomponent o A of Ao, namely, the projection of Ao in the plane perpendicular to the k-vector.

The longitudinal component oA ofAois in no way constrained by our association of the B-field

with the vector potential in accordance with Eq.(2); therefore, oA remains free and available for

later adjustments. As will be seen below, this degree of freedom associated with the longitudinal

component ofAois intimately related to the so-called freedom to choose the gauge.Having eliminated Maxwells 4thequation, we now turn to his 3rdequation and rewrite it as

follows:

( , ) ( , ) / [ ( , )] /t t t t t = = E r B r A r ( , ) ( , ) / 0.[ ]t t t + =E r A r (3)

The above equation informs us that the vector field / t +E A is curl-free. It must,therefore, be equal to the gradient of some scalar field, because the gradient of anyscalar field is

curl-free as well. For instance, the gradient of (r,t) given by Eq.(1a) is oi exp[i( )],t k k r

whose curl 2 oi exp[i( )]t k k k r is identically zero due to the fact that k k = 0. By

convention, the scalar potential of electrodynamics is defined to be the field whose gradient is

( / ).t +E A We thus have

( , ) ( , ) ( , )/ .t t t t = E r r A r (4)

For the plane-wave solutions of Maxwells equations, the above identity yields

o o o( , ) exp[i( )] ( i i ) exp[i( )].t t t = = + E r E k r k A k r (5)

Equations (2) and (5) thus relate the electromagnetic E- and B-fields to the scalar and vector

potentials of Eq.(1) in such a way as to automatically satisfy Maxwells 3rd

and 4th

equations.Instead of having to find the six components ofEoandBo, one only needs to determine the scalar

entity oand the three components of Ao. It should also be borne in mind that the longitudinal

component oA ofAois as yet unconstrained.


24/57


Next, we turn to Maxwells 1stequation, assume that the contributions of free charge density

and polarization are combined in the total bound charge density total( , ),t r and that the total

charge density present in the entire space-time is given by total o( , ) exp[i( )].t t = r k r With

the aid of Eq.(5) we obtain

o total o o o o o o o( , ) ( , ) i ( ) .t t = = =E r r k E k k k A (6)

Equation (6) relates the scalar- and vector-potential amplitudes o and Ao to the charge-

density amplitude o. Taking advantage of the fact that o ok A = k A is as yet unconstrained, we

simplify Eq.(6) by setting k Ao equal to o/c2; the reason for this choice becomes clear

shortly. The above choice of the longitudinal component of the vector potential, generally

referred to as working in the Lorenz gauge, is equivalent to forcing A(r, t) and (r, t) into thefollowing relationship:

2( , ) (1/ ) ( , )/ 0.t c t t + =A r r (7)

With the Lorenz gauge thus set, Eq.(6) becomes an equation for o in terms of o, which

may readily be solved as follows:

oo 2 2

o

( , ) .( , )( / )[ ]k c

=

kk (8)

In the above equation, the fact that oand odepend on the plane-waves k-vector as well as

its frequency is explicitly emphasized. We also have written k k as k2to highlight its scalar

nature. In general, of course,k =k+ ikis a complex vector, resulting in k2=(k

2 k

2)+2ik k

being a complex scalar. Equation (8) clearly indicates that, in the Lorenz gauge, the scalar

potential is intimately related to the total charge-density distribution total ( , ),t r but not in any

explicit way to the current density distribution total ( , ),tJ r which is the only other source of

electromagnetic fields and radiation.

The remaining Maxwell equation, o total o o( , ) ( , ) ( , )/ ,t t t t = +B r J r E r may also be

written in terms of the scalar and vector potentials. Writing total o( , ) exp [i( )],t t= J r J k r and

substituting for ( , )tB r from Eq.(2) and for ( , )tE r from Eq.(5), we arrive at

2

o o o o o o oi ( ) i ( i i ) = +k k A J k A

2

o o o o o o o o o o( ) ( ) +k k A k A k = J k A

2 2 2o o o o o( / ) ( / ) .[ ]k c c +A = k A k J (9)

The vector identity ( ) ( ) ( ) = A B C A C B A B Chas been used in the above derivation.Using the fact that k Ao= o/c

2 (Lorenz gauge), Eq.(9) is simplified, yielding the relation

between the vector potential and the current density as follows:

o oo 2 2

( , ) .( , )( / )k c

=

J kA k (10)


25/57


CHAPTER 5

Solving Maxwells Equations in Space-time: The Wave Equation

A real field is then a set of numbers we specify in such a way that what happens at a pointdepends only on the numbers at that point. We do not need to know any more about whats going on

at other places. It is in this sense that we will discuss whether the vector potential is a real field.

Richard P. Feynman (1964), arguing in favor of according vector potential the

status of physical reality on the basis of the Aharonov-Bohm effect.1

For the rest of my life I will reflect on what light is.

Albert Einstein (1917), as quoted in S. Perkowitz,Empire of Light.

Abstract. The problem addressed in the present chapter is the same problem as discussed in the

preceding one, namely, the determination of fields for given distributions of charge, current,

polarization and magnetization. Here, however, we will derive expressions for the scalar and

vector potentials as functions of space and time coordinates for the given source distributions,

which are also specified in space-time. Once the potentials are obtained, the electromagnetic

fields will be calculated by straightforward differentiation. The integrals will look very different

from those encountered in Chapter 4, but the final results will be exactly the same.

5.1. Introduction. The first and third of Maxwells equations, namely, free( , )t =D r and

( , ) ( , )/ ,t t t = E r B r become equations that relate the E-field to its sources, free( , )t r and

bound ( , ) ( , ),t t = r P r whenever the B-field happens to be time-independent, that is,

( , )/ 0.t t =B r The first and third equations thus form a complete basis for electrostatics in the

absence of time-varying magnetic fields. Similarly, the second and fourth equations,

free( , ) ( , ) ( , )/ ,t t t t = +H r J r D r and ( , ) 0,t =B r form a complete basis for magnetostatics

whenever the displacement field happens to be time-independent, i.e., ( , )/ 0.t t =D r In the

absence of time-variation, therefore, the four equations split into two independent sets, each

containing only two equations.

Such splitting of the equations becomes impossible, however, when the fields are functions

of time, as the sources of theE- andB-fields become intertwined. Nevertheless, by working withpotentials rather than with the fields, and by staying in the Lorenz gauge, it is possible to arrive

at two 2nd

-order partial differential equations, one for the scalar potential ( , ),t r the other for

the vector potential ( , ),tA r with the scalar potential depending only on total charge distribution

total( , ),t r and the vector potential being a function only of the total current distribution

total ( , ).tJ r These equations, generally referred to as wave equations, will be derived and analyzed

in the following sections.

5.2. Scalar potential ( , )t r as the solution of a 2nd

-order partial differential equation. From

Maxwells first equation, free( , ) ,t =D r we have

o free total( , ) ( , ) ( , ) ( , )t t t t = =E r r P r r (1)

Using the relation between the E-field and the potentials, namely, / ,t = E A we may

rewrite Eq.(1) as follows:

Masud Mansuripur



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Solving Maxwells Equations in Space-time Field, Force, Energy and Momentum in Classical Electrodynamics 115

The last equation is the wave equation for the vector potential, relating ( , )tA r to total( , ).tJ r

Applying the Fourier transform operation to both sides of Eq.(6) yields

totalo

2 2

( , )( , ) .

( / )k c

=

kJA k (7)

Once again, one can easily go back and forth between Eq.(6) in the space-time domain andits counterpart in the Fourier domain, Eq.(7), as the need arises.

5.4. Meaning of the Laplacian operator acting on a vector field. If we expand the well-

defined operation ( , )[ ]t A r in Cartesian coordinates, we find it to be equal to another

well-defined operation, namely, 2( ) , A A where 2 2 2 2 ( ) ( ) ( ) .x y zA A A= + +A x y z

Since Ax, Ay, and Azare scalar functions of r and t, the meaning of the above terms should be

perfectly clear. Thus the wave equation for ( , ),tA r Eq.(6), is in fact a compact formula that

contains three equations, one for each Cartesian component of ( , ).tA r The problem with this

definition of the Laplacian operator is that it is valid only in Cartesian coordinates; one cannot

obtain 2 ( , )tA r in cylindrical or spherical coordinates, for example, by applying the scalar

Laplacian operator to the components ( , , )zA A A or ( , , )A A A of the vector field, then

combining the results to create a vector Laplacian field. Needless to say, one can always use the

operational definition, 2 ( , ) ( ) ( ),t =A r A A to find the Laplacian of a vector field inany coordinate system, but the algebra quickly becomes tedious.

A good way to understand the essence of the Laplacian operation on a vector field is to

analyze it in the Fourier domain, where we have

4( , ) (2 ) ( , ) exp[i( )]d d[ ] { }t t

= A r A k k r k

4 2(2 ) i ( , ) exp[i( )]d d[ ] t

= k k A k k r k 4 2(2 ) ( , ) exp[i( )]d d .k t

= A k k r k (8)

Note that ( )A

extracts the longitudinal component of ( , )A k before multiplying it with k2,

thereby throwing out the information contained in the transverse component, ( , ),A k of the

field. In contrast, the operation ( ) A

retains the entire vector field in the Fourier domain,

but it adds an undesirable element, as follows:

4 2( , ) (2 ) i ( , ) exp[i( )]d d .[ ] [ ]t t

= A r k k A k k r k (9)

Using the vector identity ( ) ( ) ( ) , = A B C A C B A B C we now write Eq.(9) as follows:

4 2( , ) (2 ) ( , ) ( , ) exp[i( )]d d .[ ][ ]t k t

= A r A k A k k r k (10)

The undesirable term in Eq.(10) turns out to be the same expression as appears on the right-hand-

side of Eq.(8). Subtracting this term then yields


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The Lorentz Oscillator Model Field, Force, Energy and Momentum in Classical Electrodynamics 141

along thex-axis byx(t), the force exerted on the charge will consist of the external force qE(t),

the restoring spring force x(t)x^, and the dissipative friction force V(t)x^, where the velocityof the particle along thex-axis is V(t)= dx(t)/dt. The total force must then be equal to mass times

acceleration according to Newtons law of motion, namely,

md2x(t)/dt

2= qE(t) x(t) dx(t)/dt. (1a)

Defining two new parameters, o=/mand =/m, we rewrite the above equation as follows:

d2x(t)/dt

2+dx(t)/dt+o

2x(t) = (q/m)E(t). (1b)

For reasons that will become clear later, ois usually referred to as the resonance frequency and as the damping coefficient. Using complex notation, we write E(t)=Re[Exoexp(it)] andx(t)=Re[xoexp(it)], where, in general, xo=|xo|exp(io) is a complex constant. We thenrewrite Eq.(1b) as

2xo ixo +o2xo = (q/m)Exo, (2a)

which yields

Denoting byp(t) the electric dipole moment of the oscillating mass-and-spring system, namely,

p(t)=q x(t)x^

=Re[qxoexp(it)x^]=Re[poexp(it)], (3)

we will have

Let there be N such dipoles in a unit volume of space at a given location r, and define the

polarization as P(r, t)=Re[Npoexp(it)]. Defining the dimensionless function C() such that

P(r,t)=Re[oC()Exoexp(

it)x^

], we will have

Using a new parameter p=Nq2/(om), called plasma frequency, we rewrite Eq.(5) as follows:

In the MKSA system of units, ohas units offarad/meterwhile the units of the E-field arevolt/meter, yielding for the polarization P(r,t) the units of faradvolt/m

2, which is the same as

coulomb /m2. Also, one may readily confirm that phas units ofs

1.

Generalizing the above result, we now state that the polarizationP(r, t) at a given locationrin an isotropic and linearly-polarizable medium excited by the monochromatic E-field

E(r,t)=Re[E(r)exp(it)] is obtained via the complex-valued proportionality constant oC(),namely,

P(r, t)=Re[oC()E(r)exp(it)]. (7)

The complex-valued function of frequency C() depends on three effective material parameters:the plasma frequency p, the resonance frequency o, and the damping coefficient .

2 o2+ ixo=

(q/m)Exo . (2b)

2 o2+ i

po = (q

2/m)Exo . (4)

o22 i

C() =( q

2/om)

. (5)

o22 i

C() =p

2

. (6)


30/57


6.3. Generalization to the case of multi-electron atoms and molecules. A typical material is

made up of atoms or molecules that have many electrons associated with their individual atomicnuclei. Each electron may therefore be represented by a mass-and-spring system similar to that

depicted in Fig.1. Electrons occupying different orbitals, of course, must have different

parameters p,o, and. The polarizability coefficient C() of a medium whose constituent

atoms/molecules each haveKdifferent electrons may thus be expressed as

Note in Eq.(8) that we have kept pindependent of k, while introducing the new parameter fk,called the k

thoscillator strength, in order to account for differences in the effective mass mkand

effective charge qk of the various oscillators, as well as possible differences in their number

densities Nk. The parameters (p,fk,ok,k) of the Lorentz oscillator model where k rangesover all the electrons in distinct orbitals associated with the constituent atoms/molecules of the

medium must be obtained either from quantum-mechanical calculations or throughexperimental measurements of the material properties.

6.4. Drude model of the conduction electrons. The Lorentz oscillator model may be used to

describe the conduction electrons in a metal or semiconducting medium, provided that the spring

constant or, equivalently, the resonance frequency okfor these electrons is set equal to zero.In other words, since the conduction electrons are notbound to any particular atom/molecule,

they should not be subject to the restoring force of a spring. However, when 0, theconduction electron may still be imagined to respond to an oscillating E-field by vibrating

around its equilibrium position (whatever that term may mean for a delocalized electron), subjectonly to the force exerted by the E-field and the dynamic friction force represented by the

damping coefficient . The proportionality constant (actually not a constant but a function of )between the E-field and the polarization P(r,t) of conduction electrons is now represented by

e(), rather than C(), and referred to as the electricsusceptibilityof the conduction electrons.Setting o = 0 in Eq.(6), we find

The above expression is known as theDrude modelof the conduction electrons, even though

it is only a special case of the Lorentz oscillator model. At high frequencies where >>, Eq.(9)may be further simplified by ignoring the term i in the denominator. We then havee() (p/)

2, which is known as plasma susceptibility. The electric permittivity of the

plasma is thus given by ()=1+e()1 (p/)2.

Note: In isotropic linear media, the displacement field D(r,) = oE(r,) +P(r,) may be expressed aso[1+e()]E(r,). The dimensionless proportionality coefficient relatingD(r,) toE(r,) is generally known asthe electric permittivity () of the medium, and related to the electric susceptibility via ()=1+e(). Since theactual proportionality coefficient relatingD(r,) toE(r,) is o(), it is not uncommon to see () referred to asthe relativepermittivity. It is also common practice to call () the dielectric constant of the material medium,although a better name would be dielectric function, since permittivity is not a constant but a function of .

Similarly, the magnetic induction B(r,) =oH(r,) +M(r,) may be written o[1+m()]H(r,). Thedimensionless proportionality coefficient relatingB(r,) to H(r,) is known as the magnetic permeability () of

o k22 ik

CK() =fk p

2

. (8)k=1

K

2+ ie() =

p2

. (9)


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Masud Mansuripur


CHAPTER 7

Plane Electromagnetic Waves in Isotropic, Homogeneous, Linear Media

What, then, is light according to the electromagnetic theory? It consists of alternate and oppositerapidly recurring transverse magnetic disturbances, accompanied with electric displacements, the

direction of the electric displacement being at the right angles to the magnetic disturbance, and bothat right angles to the direction of the ray.

James Clerk Maxwell (1831-1879)

The velocity of light is one of the most important of the fundamental constants of Nature. Itsmeasurement by Foucault and Fizeau gave as the result a speed greater in air than in water, thusdeciding in favor of the undulatory and against the corpuscular theory. Again, the comparison of theelectrostatic and the electromagnetic units gives as an experimental result a value remarkably close tothe velocity of light a result which justified Maxwell in concluding that light is the propagation ofan electromagnetic disturbance. Finally, the principle of relativity gives the velocity of light a stillgreater importance, since one of its fundamental postulates is the constancy of this velocity under allpossible conditions.

Albert Abraham Michelson (1852-1931)

Abstract. Material media typically react to electromagnetic fields by becoming polarized or

magnetized, or by developing charge- and current-density distributions within their volumes or

on their surfaces. The response of a material medium to the fields could be complicated, as

would be the case, for instance, when the relation between induced polarization and the electric

field is non-local, non-linear, or history-dependent, or when the induced magnetization is an

anisotropic function of the local magnetic and/or electric fields. In many cases of practical

interest, however, the media are homogeneous, isotropic, and linear, with the electric dipoles

responding only to the localE-field (and magnetic dipoles responding only to the local H-field)

in accordance with the Lorentz oscillator model of the preceding chapter. Irrespective of the

manner in which the charge-carriers or the dipoles of the medium respond to the fields, there is

always an additional complication that the fields are not merely those imposed on the mediumfrom the outside. The motion of the charges and/or the oscillation of the dipoles in response to

the fields give rise to new electromagnetic fields, which must then be added to the external

fields before the induced charge, current, polarization, or magnetization can be computed. In

other words, the entire system of interacting fields and sources, whether originating outside or

induced within the media, must be treated self-consistently. This chapter provides a detailed

analysis of plane-wave propagation within the simplest kind of material media, namely, those

that are homogeneous and isotropic, whose induced electric dipoles are linear functions of the

localE-field, and whose induced magnetic dipoles are linear functions of the localH-field.

7.1. Introduction. A plane electromagnetic wave is specified by its temporal frequency , apropagation vector k, an electric-field amplitude Eo, and a magnetic-field amplitude Ho. In

general, electromagnetic fields are real-valued vector-fields throughout space and time, that is,theE- andH-fields,E(r,t) andH(r, t), are real-valued vector functions of the spatial and temporalcoordinates, r = (x,y,z) and t. It turns out, however, that complex-valued functions of (r,t) in

which k, Eo and Ho are allowed to be complex vectors may be used to mathematically

representtheE- andH-fields, so long as the real parts of these complex functions are recognizedas corresponding to the actual fields. In this way, the physical fields always end up being real-

valued, while their complex-valued representationssimplify mathematical operations. Section 2

provides a detailed description of the complex vector algebra used throughout the chapter.


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While a traditional, real-valued k-vector is used to represent a homogeneous plane-wave in

free space or within a transparent medium, the use of a complex k-vector enables one to discussinhomogeneous plane-waves such as evanescent waves in transparent media, exponentially

decaying (or attenuating) plane-waves in absorptive media, and also exponentially growing (or

amplifying) plane-waves within gain media. Moreover, the use of complex-valued field

amplitudesEoandHoenables one to consider arbitrary states of polarization (i.e., linear, circular,elliptical) within the same mathematical formalism. A general discussion of complex-valued

plane-waves and their properties is given in section 3.

Isotropic, homogeneous, linear media are characterized by the uniformity and isotropy of

their electromagnetic properties, and by the fact that their dielectric susceptibility oe() relates

the polarization P(r,t) to the local E-field E(r,t), while their magnetic susceptibility om()

relates the magnetizationM(r, t) to the local H-fieldH(r, t). Plane-wave solutions to Maxwells

equations in such media are discussed in section 4, where relationships among the frequency ,

the k-vector, the field amplitudes (Eo,Ho), and material parameters ()=1+e() and

()=1+m() are derived from Maxwells macroscopic equations.The rate of flow of electromagnetic energy (per unit area per unit time) in an optical or

electromagnetic field is given by the Poynting vector S(r,t) =E(r, t)H(r, t). A generalexpression for the time-averaged Poynting vector of a monochromatic plane-wave is derived in

section 5. This result is applicable to all sorts of plane-waves, whether propagating in a

transparent medium, exponentially decaying within an absorptive medium, exponentiallygrowing inside a gain medium, or evanescent. Using the results of section 5, one can calculate

the energy flux of any plane-wave residing in an isotropic, homogeneous, linear medium, andconfirm the conservation of energy under various circumstances.

When an electromagnetic wave arrives at the interface between two (physically distinct)

media, a part of the wave is reflected at the interface, while the remaining part enters from thefirst (incidence) medium into the second. Assuming the two media are isotropic, homogeneous,

and linear each specified by its dielectric permittivity () and magnetic permeability ()

and also assuming that the two media occupy adjacent, semi-infinite, half-spaces separated at aflat interface, one can readily calculate the reflection and transmission coefficients for arbitrary

plane-waves that arrive at the interface. These so-called Fresnel reflection and transmissioncoefficients are derived in section 6. The results of section 6 are completely general, and may be

used in conjunction with transparent or absorptive media. They encompass reflection and

transmission of arbitrarily-polarized plane-waves directed at the interface between two arbitrary

media at an arbitrary angle of incidence. The media under consideration may be metals, plasmas,or dielectrics with positive or negative refractive indices. The results of section 6 are applicable

to ordinary reflection and transmission, total internal reflection, and reflection from plasmas. We

discuss several cases of practical interest in section 7.

7.2. Complex vector algebra of the electromagnetic field. This chapter describes thedependence of plane electromagnetic waves on space and time coordinates using a powerful

complex notation. A complex number such as c=a+ib is specified by its real and imaginary

parts a and b, respectively. The sum and difference of two complex numbers are defined as

c1c2=(a1a2)+ i (b1b2). The product of two complex numbers is c1c2=(a1a2b1b2)+i(a1b2+a2b1). Division is the inverse of multiplication, in the sense that if c3=c1/c2, then c1=c2c3.

The complex-conjugate of c=a+ib is defined as c*=a ib. The product cc*= a2+b

2 is always

real and non-negative. It is often useful to write c1/c2as (c1c2*)/(c2c2*), which has a real-valued


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Plane Electromagnetic Waves in Isotropic Field, Force, Energy and Momentum in Classical Electrodynamics 165

denominator; complex division is thus reduced to complex multiplication of c1and c2*, followed

by normalization by the real-valued c2c2*.

In polar representation, a complex number is written as c = |c|exp(ic). The magnitude (or

modulus) |c| of the complex-number cis a non-negative real number; its phase cis in the range


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CHAPTER 8

Simple Applications Involving Plane Electromagnetic Waves

Science is built up with facts, as a house is with stones. But a collection of facts is no more ascience than a heap of stones is a house.

Jules Henri Poincar (1854-1912)

The mere formulation of a problem is often far more essential than its solution, which may be merelya matter of mathematical or experimental skills. To raise new questions, new possibilities, to regardold problems from a new angle requires creative imagination and marks real advances in science.

Albert Einstein (1879-1955)

Abstract. Plane-waves are the building blocks of arbitrary electromagnetic waves residing in

homogeneous, linear media. A basic understanding of plane-wave properties and their behavior

upon arrival at a flat interface between two such media is sufficient for the analysis of a number

of interesting electrodynamic problems. This chapter presents several exemplary problems that

are easy to set up and to describe, yet the understanding and appreciation of their full impact

requires subtle arguments involving the properties of plane-waves and those of the Fresnel

reflection and transmission coefficients. These examples reveal certain interesting as well as

useful features of optical and electromagnetic systems that are frequently encountered in

practical applications.

8.1. Introduction. This chapter contains a number of examples that require only an elementary

knowledge of the properties of plane-waves, yet the conclusions reached in each case are far

from trivial or obvious. The examples cover a range of problems of practical interest, includingthe reflection and transmission properties of multilayer stacks, characteristics of a Fabry-Perot-

like optical resonator, and the intricacies of the perfectly-matched boundary layer used in certain

numerical solutions of Maxwells equations.

8.2. Transmission through a multilayer stack. A multilayer stack consists of N layers ofisotropic, homogeneous, linear materials, each specified in terms of its complex-valued n()

and n() as well as a thickness dn. The stack is surrounded by vacuum on all sides, as shown in

Fig. 1. For a plane-wave normally incident on this multilayer, we show that the complex-valued

Fresnel transmission coefficient does not depend on which side of the stack faces the light

source. This result can also be extended to oblique incidence at an arbitrary incidence angle .Let us split off the first layer from the rest of the stack to create a small gap between the two

parts, as shown in Fig. 2. Note that the first layer by itself, surrounded by vacuum, is symmetric,

and, therefore,1=1 and 1=1. Let the amplitude of the plane-wave inside the gap and

traveling from left to right beA1. Denoting by1and2the reflection coefficients from the two

exposed surfaces within the gap, and by 1the transmission coefficient of the first layer, we have

A1=1E(i)

+12A1exp(i2). (1)

Here 2=4d/ois the round-trip phase acquired by the beam within the gap, which will vanishin the limit when d0. From Eq.(1) we find

A1=1E(i)

/[112 exp(i2)]. (2)

Denoting by 2 the transmission coefficient of the rest of the stack, we see that the totaltransmission coefficient is

Masu

field force enersy and momentum

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