[PDF] The Mathematical Theory of Maxwells Equations

View - Download The Mathematical Theory of Maxwells Equations

Previous PDF

Next PDF

The Mathematical Theory of Maxwell's Equations

Andreas Kirsch and Frank Hettlich

Department of Mathematics

Karlsruhe Institute of Technology (KIT)

Karlsruhe, Germany

c

May 24, 2014

2

Preface

This book arose from lectures on Maxwell's equations given by the authors between 2007 and

2013. Graduate students from pure and applied mathematics, physics { including geophysics

{ and engineering attended these courses. We observed that the expectations of these groups of students were quite dierent: In geophysics expansions of the electromagnetic elds into spherical (vector-) harmonics inside and outside of balls are of particular interest. Graduate students from numerical analysis wanted to learn about the variational treatments of interior boundary value problems including an introduction to Sobolev spaces. A classical approach in scattering theory { which can be considered as a boundary value problem in the unbounded exterior of a domain { uses boundary integral equation methods which are particularly helpful for deriving properties of the far eld behaviour of the solution. This approach is, for polygonal domains or, more generally, Lipschitz domains, also of increasing relevance from the numerical point of view because the dimension of the region to be discretized is reduced by one. In our courses we wanted to satisfy all of these wishes and designed an introduction to Maxwell's equations which coveres all of these concepts { but restricted ourselves almost completely (except Section 4.3) to the time-harmonic case or, in other words, to the frequency domain, and to a number of model problems. The Helmholtz equation is closely related to the Maxwell system (for time-harmonic elds). As we will see, solutions of the scalar Helmholtz equation are used to generate solutions of the Maxwell system (Hertz potentials), and every component of the electric and magnetic eld satises an equation of Helmholtz type. Therefore, and also for didactical reasons, we will consider in each of our approaches rst the simpler scalar Helmholtz equation before we turn to the technically more complicated Maxwell system. In this way one clearly sees the analogies and dierences between the models. In Chapter 1 we begin by formulating the Maxwell system in dierential and integral form. We derive special cases as theEmode and theHmode and, in particular, the time harmonic case. Boundary conditions and radiation conditions complement the models. In Chapter 2 we study the particular case where the domainDis a ball. In this case we can expand the elds inside and outside ofDinto spherical wave functiuons. First we study the scalar stationary case; that is, the Laplace equation. We introduce the expansion into scalar spherical harmonics as the analogon to the Fourier expansion on circles inR2. This leads directly to series expansions of solutions of the Laplace equation in spherical coordinates. The extension to the Helmholtz equation requires the introduction of (spherical) Bessel- and Hankel functiuons. We derive the most important properties of these special functions in detail. After these preparations for the scalar Helmholtz equation we extend the analysis 3 4 to the expansion of solutions of Maxwell's equations with respect to vector wave functions. None of the results of this chapter are new, of course, but we have not been able to nd such a presentation of both, the scalar and the vectorial case, in the literature. We emphasize that this chapter is completely self-contained and does not refer to any other chapter { except for the proof that the series solution of the exterior problem satises the radiation condition. Chapter 3 deals with a particular scattering problem. The scattering object is of arbitrary shape but, in this chapter, with suciently smooth boundary@D. We present the classical boundary integral equation method and follow very closely the fundamental monographs [5, 6] by David Colton and Rainer Kress. In contrast to their approach we restrict ourselves to the case of smooth boundary data (as it is the usual case of the scattering by incident waves) which allows us to study the setting completely in Holder spaces and avoids the notions of \parallel surfaces" and weak forms of the normal derivatives. For the scalar problem we restrict ourselves to the Neumann boundary condition because our main goal is the treatment for Maxwell' equations. Here we believe { as also done in [6] { that the canonical spaces on the boundary are Holder spaces where also the surface divergence is Holder continuous. In order to prove the necessary properties of the scalar and vector potentials a careful investigation of the dierential geometric properties of the surface@Dis needed. Parts of the technical details are moved to the Appendix 6. We emphasize that this chapter is self countained and does not need any results from other chapters (except from the appendix). As an alternative approach for studying boundary value problems for the Helmholtz equation or the Maxwell system we will study the weak or variational solution concept in Chapter 4. We restrict ourselves to the interior boundary value problem with a general source term and the homogeneous boundary condition of an ideal conductor. This makes it possible to work (almost) solely in the Sobolev spacesH10(D) andH0(curl;D) of functions with vanishing boundary traces or tangential boundary traces, respectively. In Section 4.1 we derive the ba- sic properties of these special Sobolev spaces. The characteristic feature is that no regularity of the boundary and no trace theorems are needed. Probably the biggest dierence between the scalar case of the Helmholtz equation and the vectorial case of Maxwell's equations is the fact thatH0(curl;D) is not compactly imbedded inL2(D;C3) in contrast to the space H

10(D) for the scalar problem. This makes it necessary to introduce the Helmholtz decompo-

sition. The only proof which is beyond the scope of this elementary chapter is the proof that the subspace ofH0(curl;D) consisting of divergence-free vector elds is compactly imbedded inL2(D;C3). For this part some regularity of the boundary (e.g. Lipschitz regularity) is needed. Since the proof of this fact requires more advanced properties of Sobolev spaces it it transponed to Chapter 5. We note that also this chapter is self contained except of the beforementioned compactness property. The nal Chapter 5 presents the boundary integral equation method for Lipschitz domains. The investigation requires more advanced properties of Sobolev spaces than those presented in Section 4.1. In particular, Sobolev spaces on the boundary@Dhave to be introduced and the correponding trace operators. Perhaps dierent from most of the traditional approaches we rst consider the case of the cube (;)3R3and introduce Sobolev spaces of periodic functions by the proper decay of the Fourier coecients. The proofs of imbedding and trace theorems are quite elementary. Then we use, as it is quite common, the partion of unity and 5 local maps to dene the Sobolev spaces on the boundary and transfer the trace theorem to general Lipschitz domains. We dene the scalar and vector potentials in Section 5.2 analogously to the classical case as in Chapter 3 but have to interpret the boundary integrals as certain dual forms. The boundary operators are then dened as traces of these potentials. In this way we follow the classical approach as closely as possible. Our approach is similar but a bit more explicit than in [16], see also [10]. Once the properties of the potentials and corresponding boundary operators are known the introduction and investigation of the boundary integral equations is almost classical. For example, for Lipschitz domains the Dirichlet boundary value problem for the scalar Helmholtz equation is solved by a (properly modied) single layer ansatz. This is preferable to a double layer ansatz because the corresponding double layer boundary operator fails to be compact (in contrast to the case of smooth boundaries). Also, the single layer boundary operator satises a Garding's inequality; that is, can be decomposed into a coercive and a compart part. Analogously. the Neumann boundary value problem and the electromagnetic case are treated. In our presentation we try to show the close connection between the scalar and the vector cases. Starting perhaps with the pioneering work of Costabel [7] many important contributions to the study of boundary integral operators in Sobolev spaces for Lipschitz boundaries have been published. It is impossible for the authors to give an overview on this subject but instead refer to the monograph [10] and the survey article [4] from which we have learned a lot. As mentioned above, our approach to introduce the Sobolev spaces, however, is dierent from those in, e.g. [1, 2, 3]. In the Appendix 6 we collect results from vector calculus and dierential geometry, in par- ticular various forms of Green's theorem and the surface gradient and surface divergence for (smooth) functions on (smooth) surfaces. We want to emphasize that it was not our intention to present a comprehensive work on Maxwell's equations, not even for the time harmonic case or any of the beforementioned sub- areas. As said before this book arose from { and is intended to be { material for designing graduate courses on Maxwell's equations. The students should have some knowledge on vec- tor analysis (curves, surfaces, divergence theorem) and functional analysis (normed spaces, Hilbert spaces, linear and bounded operators, dual space). The union of the topics covered in this monograph is certainly far too much for a single course. But it is very well possible to choose parts of it because the chapters are all independent of each other. For example, in the summer term 2012 (8 credit points; that is in our place, 16 weeks with 4 hours per week plus exercises) one of the authors (A.K.) covered Sections 2.1{2.6 of Chapter 2 (only interior cases), Chapter 3 without all of the proofs of the dierential geometric properties of the surface and all of the jump properties of the potentials, and Chapter 4 without Section 4.3. Perhaps these notes can also be useful for designing courses on Special Functions (spherical harmonics, Bessel functions) or on Sobolev spaces. One of the authors wants to dedicate this book to his father, Arnold Kirsch (1922{2013), who taught him aboutsimplication of problems without falsication(as a concept of teaching mathematics in high schools). In Chapter 4 of this monograph we have picked up this concept by presenting the ideas for a special case only rather than trying to treat the most 6 general cases. Nevertheless, we admit that other parts of the monograph (in particular of Chapters 3 and 5) are technically rather involved.

Karlsruhe, March 2014 Andreas Kirsch

Frank Hettlich

Contents

Preface2

1 Introduction 9

1.1 Maxwell's Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.2 The Constitutive Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.3 Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.4 Boundary and Radiation Conditions . . . . . . . . . . . . . . . . . . . . . . 18

1.5 The Reference Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2 Expansion into Wave Functions 27

2.1 Separation in Spherical Coordinates . . . . . . . . . . . . . . . . . . . . . . . 27

2.2 Legendre Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.3 Expansion into Spherical Harmonics . . . . . . . . . . . . . . . . . . . . . . . 40

2.4 Laplace's Equation in the Interior and Exterior of a Ball . . . . . . . . . . . 50

2.5 Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

2.6 The Helmholtz Equation in the Interior and Exterior of a Ball . . . . . . . . 62

2.7 Expansion of Electromagnetic Waves . . . . . . . . . . . . . . . . . . . . . . 72

2.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

3 Scattering From a Perfect Conductor 91

3.1 A Scattering Problem for the Helmholtz Equation . . . . . . . . . . . . . . . 91

3.1.1 Representation Theorems . . . . . . . . . . . . . . . . . . . . . . . . 92

3.1.2 Volume and Surface Potentials . . . . . . . . . . . . . . . . . . . . . . 99

3.1.3 Boundary Integral Operators . . . . . . . . . . . . . . . . . . . . . . 112

3.1.4 Uniqueness and Existence . . . . . . . . . . . . . . . . . . . . . . . . 116

3.2 A Scattering Problem for the Maxwell System . . . . . . . . . . . . . . . . . 124

7

8CONTENTS

3.2.1 Representation Theorems . . . . . . . . . . . . . . . . . . . . . . . . 125

3.2.2 Vector Potentials and Boundary Integral Operators . . . . . . . . . . 135

3.2.3 Uniqueness and Existence . . . . . . . . . . . . . . . . . . . . . . . . 141

3.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

4 The Variational Approach to the Cavity Problem 147

4.1 Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

4.1.1 Basic Properties of Sobolev Spaces of Scalar Functions . . . . . . . . 148

4.1.2 Basic Properties of Sobolev Spaces of Vector Valued Functions . . . . 157

4.1.3 The Helmholtz Decomposition . . . . . . . . . . . . . . . . . . . . . . 160

4.2 The Cavity Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

4.2.1 The Variational Formulation and Existence . . . . . . . . . . . . . . . 163

4.2.2 Uniqueness and Unique Continuation . . . . . . . . . . . . . . . . . . 173

4.3 The Time{Dependent Cavity Problem . . . . . . . . . . . . . . . . . . . . . 183

4.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

5 Boundary Integral Equation Methods for Lipschitz Domains 201

5.1 Advanced Properties of Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . 201

5.1.1 Sobolev Spaces of Scalar Functions . . . . . . . . . . . . . . . . . . . 203

5.1.2 Sobolev Spaces of Vector-Valued Functions . . . . . . . . . . . . . . . 216

5.1.3 The Case of a Ball Revisited . . . . . . . . . . . . . . . . . . . . . . . 235

5.2 Surface Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

5.3 Boundary Integral Equation Methods . . . . . . . . . . . . . . . . . . . . . . 263

5.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

6 Appendix 277

6.1 Table of Dierential Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 277

6.2 Results from Linear Functional Analysis . . . . . . . . . . . . . . . . . . . . 279

6.3 Elementary Facts from Dierential Geometry . . . . . . . . . . . . . . . . . 281

6.4 Integral Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286

6.5 Surface Gradient and Surface Divergence . . . . . . . . . . . . . . . . . . . . 288

Bibliography 292

Index294

Chapter 1

Introduction

In this introductory chapter we will explain the physical model and derive the boundary value problems which we will investigate in this monograph. We begin by formulating Maxwell's equations in dierential form as our starting point. In this monograph we consider exclusively linearmedia; that is, the constitutive relations are linear. The restriction to special cases leads to the analogous equations in electrostatics or magnetostatics and, assuming periodic time dependence when going into the frequency domain, to time harmonic elds. In the presence of media the elds have to satisfy certain continuity and boundary conditions and, if the region is unbounded, a radiation condition at innity. We nish this chapter by introducing two model problems which we will treat in detail in Chapters 3 and 4.

1.1 Maxwell's Equations

Electromagnetic wave propagation is described by four particular equations, theMaxwell equations, which relate ve vector eldsE,D,H,B,Jand the scalar eld. In dierential form these read as follows @B@t + curlxE= 0 (Faraday's Law of Induction) @D@t curlxH=J(Ampere's Law) div xD=(Gauss' Electric Law) div xB= 0 (Gauss' Magnetic Law): The eldsEandDdenote theelectric eld(inV=m) andelectric displacement(inAs=m2) respectively, whileHandBdenote themagnetic eld(inA=m) andmagnetic ux density (inV s=m2=T=Tesla). Likewise,Janddenote thecurrent density(inA=m2) and charge density(inAs=m3) of the medium. Here and throughout we use therationalized MKS-system, i.e. the elds are given with respect to the units Volt (V), Ampere (A), meter (m), and second (s). All elds depend both 9

10CHAPTER 1. INTRODUCTION

on the space variablex2R3and on the time variablet2R. We note that the dierential operators are always taken with respect to the spacial variablexwithout indicating this. The denition of the dierential operators div and curl, e.g., in cartesian coordinates and basic identities are listed in Appendix 6.1. The actual equations that govern the behavior of the electromagnetic eld were rst com- pletely formulated by James Clark Maxwell (1831{1879) inTreatise on Electricity and Mag- netismin 1873. It was the ingeneous idea of Maxwell to modify Ampere's Law which was known up to that time in the form curlH=Jfor stationary currents. Furthermore, he collected the four equations as a consistent theory to describe electromagnetic phenomena. As a rst observation we note that in domains where the equations are satised one derives from the identity divcurlH= 0 the well known equation of continuity which combines the charge density and the current density. Conclusion 1.1Gauss' Electric Law and Ampere's Law imply theequation of continuity @@t = div@D@t = divcurlHJ=divJ: Historically, and more closely connected to the physical situation, the integral forms of Maxwell's equations should be the starting point. In order to derive these integral relations, we begin by lettingSbe a connected smooth surface with boundary@Sin the interior of a region

0inR3where electromagnetic waves propagate. In particular, we require that the

unit normal vector(x) forx2Sis continuous and directed always into \one side" ofS, which we call the positive side ofS. By(x) we denote a unit vector tangent to the boundary ofSatx2@S. This vector, lying in the tangent plane ofStogether with a second vector n(x) in the tangent plane atx2@Sand normal to@Sis oriented such thatf;n;;gform a mathematically positive system; that is,is directed counterclockwise when we look atS from the positive side, andn(x) is directed to the outside ofS. Furthermore, let 2R3be an open set with boundary@ and outer unit normal vector(x) atx2@ ThenAmpere's lawdescribing the eect of the external and the induced current on the magnetic eld is of the formZ @S

H d`=ddt

Z S

D ds+Z

S

J ds:(1.1)

It is named by Andre Marie Ampere (1775{1836).

Next,Faraday's law of induction(Michael Faraday, 1791{1867), which isZ @S

E d`=ddt

Z S

B ds;(1.2)

describes how a time-varying magnetic eld eects the electric eld. Finally, the equations includeGauss' Electric LawZ

D ds=Z

dx(1.3)

1.2. THE CONSTITUTIVE EQUATIONS11

descibing the sources of the electric displacement, andGauss' Magnetic Law Z

B ds= 0 (1.4)

which ensures that there are no magnetic currents. Both named after Carl Friedrich Gauss (1777{1855). In regions where the vector elds are smooth functions andand"are at least continuous we can apply the following integral identities due to Stokes and Gauss for surfacesSand solids lying completely inD. Z S curlF ds=Z @S

F d`(Stokes);(1.5)

Z divFdx=Z

F ds(Gauss);(1.6)

see Appendix 6.3. To derive the Maxwell's equations in dierential form we chooseFto be one of the eldsH,E,BorD. With these formulas we can eliminate the boundary integrals in (1.1){(1.4). We then use the fact that we can vary the surfaceSand the solid inD arbitrarily. By equating the integrands we are led to Maxwell's equations in dierential form as presented in the begining. With Maxwell's equations many electromagnetic phenomena became explainable. For in- stance they predicted the existence of electromagnetic waves as light or X-rays in vacuum. It took about 20 years after Maxwell's work when Heinrich Rudolf Hertz (1857{1894) could show experimentally the existence of electromagenetic waves, in Karlsruhe, Germany. For more details on the physical background of Maxwell's equations we refer to text books as

J.D. Jackson,Classical Electrodynamics[11].

1.2 The Constitutive Equations

In the general setting the equations are not yet complete. Obviously, there are more un- knowns than equations. TheConstitutive Equationscouple them:

D=D(E;H) andB=B(E;H):

The electric properties of the material, which give these relationships are complicated. In general, they not only depend on the molecular character but also on macroscopic quantities as density and temperature of the material. Also, there are time-dependent dependencies as, e.g., the hysteresis eect, i.e. the elds at timetdepend also on the past. As a rst approximation one starts with representations of the form

D=E+ 4PandB=H4M

12CHAPTER 1. INTRODUCTION

wherePdenotes the electric polarization vector andMthe magnetization of the material. These can be interpreted as mean values of microscopic eects in the material. Analogously, andJare macroscopic mean values of the free charge and current densities in the medium. If we ignore ferro-electric and ferro-magnetic media and if the elds are relativly small one can model the dependencies by linear equations of the form

D="EandB=H

with matrix-valued functions":R3!R33, thedielectric tensor, and:R3!R33, the permeability tensor. In this case we call a mediumlinear. The special case of anisotropic mediummeans that polarization and magnetization do not depend on the directions. Otherwise a medium is calledanisotropic. In the isotropic case dielectricity and permeability can be modeled as just real valued functions, and we have

D="EandB=H

with scalar functions";:R3!R. In the simplest case these functions"andare constant and we call such a mediumhomo- geneous. It is the case, e.g., in vacuum. We indicated already that alsoandJcan depend on the material and the elds. Therefore, we need a further relation. In conducting media the electric eld induces a current. In a linear approximation this is described byOhm's Law:

J=E+Je

whereJeis the external current density. For isotropic media the function:R3!Ris called theconductivity. If= 0 then the material is called adielectric. In vacuum we have = 0 and"="08:8541012AS=V m,=0= 4107V s=Am. In anisotropic media, also the functionis matrix valued.

1.3 Special Cases

Under specic physical assumptions the Maxwell system can be reduced to elliptic second order partial dierential equations. They serve often as simpler models for electromagnetic wave propagation. Also in this monograph we will always explain the approaches rst for the simpler scalar wave equation.

Vacuum

Vacuum is a homogeneous, dielectric medium with"="0,=0, and= 0, and no charge distributions and no external currents; that is,= 0 andJe= 0. The law of induction takes the form 0@H@t + curlE= 0:

1.3. SPECIAL CASES13

Assuming suciently smooth functions a dierentiation with respect to timetand an ap- plication of Ampere's Law yields

00@2H@t

2+ curlcurlH= 0:

The termc0= 1=p"

00has the dimension of a velocity and is called thespeed of light.

From the identity curlcurl =rdiv where the vector valued Laplace operator is taken componentwise it follows that the components ofHare solutions of the linearwave equation 1c 20@ 2H@t

2H= 0:

Analogously, one derives the same equation for the cartesian components of the electric eld: 1c 20@ 2E@t

2E= 0:

Therefore, a solution of the Maxwell system in vacuum can also be described by a divergencequotesdbs_dbs47.pdfusesText_47

[PDF] maxwell's equations integral form

[PDF] may day flight crash

[PDF] may et might

[PDF] maybelline little rock jobs

[PDF] mayday calls meaning

[PDF] mayday mayday mayday

[PDF] mayday origin

[PDF] Maylis de Kerangal: dans les rapides

[PDF] mazée

[PDF] mblock

[PDF] mblock mbot

[PDF] mbot technologie college

[PDF] mcdo dangereux pour santé

[PDF] mcdo dans les pays musulmans

[PDF] mcdo experience