Multipole Theory in Electromagnetism
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1 Multipole Theory in Electromagnetism Classical, quantum, and symmetry aspects, with applications R. E. RAAB О. L. DE LANGE School of Chemical and Physical Sciences, University of Natal, Pietermaritzburg, South Africa CLARENDON PRESS-OXFORD 2005
2 CONTENTS Classical multipole theory Multipole expansion for the potential of a finite static charge distribution Dependence of electric multipole moments on origin Permanent and induced multipole moments Force and torque in an external electrostatic field Potential energy of a charge distribution in an electrostatic field Multipole expansion for the vector potential of a finite distribution of steady current Dependence of magnetic multipole moments on origin Force and torque in an external magnetostatic field Potential energy of a current distribution in a magnetostatic field Multipole expansions for the dynamic scalar and vector potentials The far- and near-zone limits Macroscopic media Maxwell's macroscopic equations: multipole forms for D and H Discussion Primitive moments versus traceless moments A charge distribution Macroscopic media 28 References 29 Quantum theory of multipole moments and polarizabilities Semi-classical quantum mechanics Electrostatic perturbation Buckingham's derivation of electrostatic multipole moments Magnetostatic perturbation Time-dependent fields: standard gauge Time-dependent fields: the Barron-Gray gauge Polarizabilities for harmonic plane wave fields Absorption of radiation Additional static magnetic polarizabilities Symmetries 52 ix
3 X CONTENTS 2.11 Macroscopic multipole moment and polarizability densities Phenomenology of the wave-matter interaction 54 References 56 Space and time properties Coordinate transformations Vectors Cartesian tensors Time reversal The space and time nature of various tensors Symmetry and property tensors Origin dependence of polarizability tensors A pictorial determination of symmetry conditions Discussion 82 References 82 Linear constitutive relations from multipole theory Constitutive relations Origin independence Symmetries The "Post constraint" Comparison with direct multipole results Electric dipole order Electric quadrupole-magnetic dipole order Electric octopole-magnetic quadrupole order Discussion 95 References 98 Transmission and scattering effects: direct multipole results The wave equation Intrinsic Faraday rotation in a ferromagnetic crystal Natural optical activity Time-odd linear birefringence in magnetic cubics Optical properties in the Jones calculus Gyrotropic birefringence Linear birefringence in non-magnetic cubic crystals (Lorentz birefringence) Intrinsic Faraday rotation in magnetic cubics The Kerr effect in an ideal gas Forward scattering theory of the Kerr effect Birefringence induced in a gas by an electric field gradient: forward scattering theory Forward scattering by a molecule 128
4 CONTENTS XI Induced moments Forward scattering by a lamina The electrostatic field Radiated field for linearly polarized light Field-gradient-induced birefringence Comparison between theory and experiment Birefringence induced in a gas by an electric field gradient: wave theory Discussion 140 References Reflection effects: direct multipole results Reflection and the reflection matrix The principle of reciprocity Equations of continuity Matching conditions in multipole theory The reflection matrix for non-magnetic uniaxial and cubic crystals Solutions of the wave equation Reflection coefficients Tests of translational and time-reversal invariance Discussion 169 References Transformations of the response fields and the constitutive tensor Gauge transformations of the 4-vector potential "Gauge transformations" of response fields Faraday transformations Transformations of linear constitutive relations in multipole theory 174 References Applications of the gauge and Faraday transformations Electric dipole order Electric quadrupole-magnetic dipole order, non-magnetic medium Electric quadrupole-magnetic dipole order, magnetic medium Discussion 186 References Transmission and reflection effects: transformed multipole results The wave equation and transmission 191
5 Xll CONTENTS 9.2 Reflection from non-magnetic uniaxial and cubic crystals Explicit results for non-magnetic uniaxial crystals Explicit results for non-magnetic cubic crystals Tests of translational and time-reversal invariance Reflection from antiferromagnetic СГ2О3: first configuration Reflection from antiferromagnetic СГ2О3: second configuration Comparison with experiment for СГ2О Uniqueness of fields Summary 206 References 210 A Transformations involving J 211 В Magnetostatic field 213 С Magnetostatic force 214 D Magnetostatic torque 215 E Integral transformations 216 F Origin dependence of a polarizability tensor 218 G Invariance of transformed tensors 220 Glossary of symbols 221 Index 229
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