Mathematical Theory of Electromagnetism Piero Bassanini University of Rome, “La Sapienza” Rome, Italy [email protected] Alan Elcrat Wichita State University, Wichita, Kansas [email protected] DOI: 10.1685/SELN09001 - Vol. 2 - 2009 Licensed under Attribution-Non-Commercial-No Derivative Works Contents Contents i Preface 1 1 Maxwell Equations For Bodies At Rest 3 1.1 Preliminaries ........................... 4 1.1.1 Mathematical Preliminaries ............... 4 1.1.2 Electric charge and electric field. Displacement vector. 8 1.1.3 Conduction current. ................... 9 1.1.4 Magnetic induction. .................... 13 1.1.5 Magnetic field intensity. ................. 15 1.2 Maxwell equations in integral form ............... 17 1.3 Maxwell equations. Constitutive relations. ........... 22 1.3.1 Maxwell field equations and constitutive relations. ... 22 1.3.2 Some consequences of the linear constitutive relations. 27 1.4 Matching conditions ....................... 33 1.5 Energy balance and uniqueness theorem ............ 41 1.5.1 Electromagnetic energy and Poynting vector. ...... 41 1.5.2 Uniqueness theorem. ................... 47 1.6 Stationary Maxwell equations .................. 49 i ii CONTENTS 1.6.1 Mathematical analysis. .................. 51 1.6.2 Electrostatics. ....................... 59 1.6.3 Magnetostatics. ...................... 62 1.7 Quasi-stationary fields in conductors .............. 63 1.8 Polarization and magnetization. ................. 66 Appendix ................................ 69 Exercises ................................. 70 2 Electrostatics and Magnetostatics 79 2.1 Electrostatic field in a dielectric ................. 81 2.1.1 Volume potentials and single layer potentials. ..... 83 2.1.2 Double layer potentials. ................. 85 2.1.3 Green’s identities. ..................... 89 2.2 Single conductor in a homogeneous dielectric .......... 92 2.2.1 Potential and charge problem. .............. 92 2.2.2 Influence problems: the Green function. ........ 97 2.3 The fundamental problems of Electrostatics .......... 101 2.3.1 Multiple conductors .................... 101 2.3.2 Condensers. ........................ 105 2.4 Kelvin’s and Earnshaw’s theorems. ............... 107 2.5 Magnetic field of a permanent magnet ............. 111 Exercises ................................. 116 3 Steady Currents In Conductors 129 3.1 Neumann vector fields. ...................... 130 3.2 Inhomogeneous Maxwell equations. ............... 136 3.2.1 Impressed electric field .................. 136 3.2.2 Potential jump model for toroidal conductors. ..... 138 CONTENTS iii 3.3 Single toroidal conductor ..................... 140 3.3.1 The magnetic field: Biot-Savart law. .......... 140 3.3.2 Ohm’s law and energy balance. ............. 145 3.3.3 The exterior electric field. ................ 149 3.4 The inductance matrix ...................... 152 3.5 Magnetic field due to an infinitely thin wire. .......... 153 3.6 Magnetic field of a solenoid ................... 161 3.7 Quasi-stationary fields: electric circuits. ............. 167 Exercises ................................. 173 4 Electromagnetic Waves 179 4.1 Maxwell’s equations and wave propagation ........... 180 4.2 Plane waves ............................ 182 4.2.1 Homogeneous and heterogeneous waves. ........ 183 4.2.2 Evanescent waves in dielectrics. ............. 189 4.2.3 Wavegroups. ........................ 190 4.3 Cauchy problem for the Maxwell equations in vacuo. ..... 192 4.3.1 Spherical means. ..................... 195 4.3.2 Loss of derivatives. Huygens’ principle. ......... 197 4.4 Radiation problem ........................ 200 4.4.1 Electrodynamic potentials and gauge transformation. 200 4.4.2 Retarded potentials. ................... 203 4.5 Telegraph equation ........................ 207 4.6 Weak solutions of the Cauchy problem ............. 211 4.6.1 Energy estimates. ..................... 212 4.6.2 Weak solution: existence and uniqueness. ........ 215 4.7 Characteristics and geometrical optics .............. 219 iv CONTENTS 4.7.1 Characteristics of the Maxwell equations. ........ 219 4.7.2 High frequency approximation. ............. 225 4.8 Reflection, refraction and Snell’s law. .............. 228 4.8.1 Snell’s law. ........................ 228 4.8.2 Total reflection and evanescent waves. .......... 231 4.8.3 Phase shift. ........................ 233 4.9 Interference in thin films. Reflection reduction ......... 235 4.10 Wave reflection from a system of layers ............. 242 4.10.1 Normal incidence. ..................... 244 4.10.2 Oblique incidence. .................... 250 Exercises ................................. 255 5 Electrodynamics Of Moving Bodies 263 5.1 Lorentz invariance of the Maxwell equations .......... 265 5.2 The Maxwell equations in four dimensions ........... 269 Exercises ................................. 273 6 Anisotropy, Dispersion and Nonlinearity 279 6.1 Hereditary constitutive relations ................. 281 6.1.1 Memory functions. .................... 281 6.1.2 Dispersion relations. ................... 287 6.2 Evaluation of the memory functions ............... 289 6.3 Approximation by local constitutive relations. ......... 295 6.4 Anisotropic dielectric media. Kerr effect ............ 298 6.5 Plane waves in uniaxial crystals. ................. 302 6.6 Anisotropic conductors. Hall effect. ............... 305 6.7 Second-harmonic generation in nonlinear optics. ........ 308 6.7.1 The nonlinear constitutive relations. .......... 308 CONTENTS v 6.7.2 The “laser problem”. ................... 312 6.7.3 Crystals of class 32-D3. .................. 315 6.7.4 Propagation along the x−axis. ............. 316 6.7.5 Propagation along the y−axis. .............. 326 6.7.6 Propagation along the optic axis. ............ 330 6.7.7 Crystals of class 6-D6. .................. 336 6.7.8 Propagation along the x−axis. ............. 337 6.7.9 Propagation along the optic axis. ............ 338 Exercises ................................. 362 References 371 Index 374 vi CONTENTS List of Figures 1.1 Current tube ........................... 11 1.2 Right-handed screw rule ..................... 14 1.3 Orientation for the surface S ................... 18 1.4 The Ampere circuital law for the Biot-Savart magnetic field . 21 1.5 Experimental confirmation of Lenz’s law ............ 28 1.6 Oersted’s experiment ....................... 31 1.7 The domain Ωh .......................... 35 1.8 Hysteresis loop .......................... 68 2.1 Gauss’ solid angle formula .................... 88 2.2 Isolated conducting sheath .................... 100 2.3 Bar magnet ............................ 112 0 3.1 Irreducible curves Γ = Γi, Γ = Γe for the torus ........ 132 3.2 Branch surface for an infinitely thin wire ............ 155 3.3 Lines of force of H and B for the solenoid and of B for the magnet165 3.4 Lines of force of H for the magnet ............... 166 3.5 RLC-circuits ........................... 167 3.6 Displacement current between the condenser plates ...... 168 3.7 Ideal transformer ......................... 173 vii viii LIST OF FIGURES 4.1 Model of a telegraphic line .................... 208 4.2 Geometry in Snell’s law ..................... 230 6.1 Behavior of Re ²(ω) and of Im ²(ω) ............... 293 Preface These lecture notes treat the mathematical theory of electromagnetism. They are written at a level appropriate for graduate students in mathematics. Very little is assumed about prior exposure to the physical theory, but some math- ematical sophistication is assumed at various points in the exposition. The subject is treated as a continuum theory with only brief mention of underly- ing molecular origins of phenomena. Also, thermodynamical considerations are not emphasized. The basic organization of these lectures was made by the first author and used in lectures given at Universita di Roma, La Sapienza. The purpose of them is to give a self contained treatment of electromagnetism for students of mathematics in order to lay the foundation for the many applications of these ideas in applied mathematics. In earlier times it might have been assumed that students of mathematics that had use for this material would take courses in physics or electrical engineering departments, but the crowded curriculum and prerequisite structure make this difficult today. In addition an exposition written from the point of view of applied mathematics is natural for students of mathematics. This applies in particular to the treatment of steady currents in chapter 3 and of nonlinear optics in chapter 6. For these reasons lecture notes of the kind presented here can play an important role for students of mathematics. 1 2 Chapter 1 Maxwell Equations For Bodies At Rest An exposition of the Maxwell theory of electromagnetism must perforce be- gin by stating the system of physical units chosen, as the Maxwell equations change (albeit slightly) according to this choice. The basic well-known ex- perimental fact underlying the theory is the existence of the electric charge, which is convenient to view as an autonomous physical quantity endowed with an independent physical dimension. As unit charge we might think in principle of taking the charge of the electron, an elementary particle carry- ing a negative electric charge −e. This electron charge, though, is so small that it turns out to be impractical in view of describing electromagnetic phe- nomena at a macroscopic level, which is the level of description targeted in Maxwell’s model. A convenient practical charge unit is the coulomb , chosen so that the