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emlectures - UFIFT-HEP-10 Classical Electrodynamics Charles...

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UFIFT-HEP-10- Classical Electrodynamics Charles B. Thorn 1 Institute for Fundamental Theory Department of Physics, University of Florida, Gainesville FL 32611 Abstract 1 E-mail address: [email protected]
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Contents 1 Introduction 4 1.1 The Field Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Maxwell’s equations: Field Equations of Motion . . . . . . . . . . . . . . . . . 4 1.3 Heaviside-Lorentz (HL) Units . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Physical meaning of Maxwell’s equations . . . . . . . . . . . . . . . . . . . . 5 1.5 Charge conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.6 Potentials and Gauge Invariance . . . . . . . . . . . . . . . . . . . . . . . . . 7 2 Electrostatics 8 2.1 Point charge and the Dirac delta function . . . . . . . . . . . . . . . . . . . 8 2.2 Interfaces between different materials . . . . . . . . . . . . . . . . . . . . . . 9 2.3 Uniqueness of electrostatic solutions, Green’s theorem . . . . . . . . . . . . . 10 2.4 Green functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.5 Electrostatic energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.6 Capacitance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3 Electrostatic Boundary-Value problems 13 3.1 Method of Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2 Method of Separation of Variables . . . . . . . . . . . . . . . . . . . . . . . . 15 3.3 Angle Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.4 Problems with Azimuthal Symmetry . . . . . . . . . . . . . . . . . . . . . . 21 3.5 Green function between two concentric spheres . . . . . . . . . . . . . . . . . 23 3.6 Conductors with a Conical Singularity . . . . . . . . . . . . . . . . . . . . . 25 3.7 Cylindrical Coordinates and Bessel functions . . . . . . . . . . . . . . . . . . 26 3.8 Boundary-value problems in cylindrical coordinates . . . . . . . . . . . . . . 31 3.9 Green functions in cylindrical coordinates . . . . . . . . . . . . . . . . . . . . 32 3.10 A little more wisdom about Green functions . . . . . . . . . . . . . . . . . . 33 3.11 Electrostatics in 2 Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4 The Multipole expansion and Dielectric Materials 39 4.1 Electric Multipoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.2 Electrostatics in Dielectric Materials . . . . . . . . . . . . . . . . . . . . . . 42 4.3 Energy and Forces on Dielectrics . . . . . . . . . . . . . . . . . . . . . . . . 43 4.4 Boundary value problems with dielectrics . . . . . . . . . . . . . . . . . . . . 45 4.5 Models for χ e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 5 Magnetostatics 49 5.1 Circular Current Loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 5.2 Magnetic Multipoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 5.3 Magnetic Fields in Magnetic Materials . . . . . . . . . . . . . . . . . . . . . 52 5.4 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 1 c circlecopyrt 2010 by Charles Thorn
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5.5 Examples of Magnetic Boundary value Problems . . . . . . . . . . . . . . . . 53 5.6 Energy and Magnetic Materials . . . . . . . . . . . . . . . . . . . . . . . . . 56 5.7 Models of χ m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 5.8 Faraday’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 5.9 Inductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 5.10 Conductivity and the Quasi-static approximation . . . . . . . . . . . . . . . 63 6 Maxwell’s Equations 65 6.1 Ampere-Maxwell Equation in electromagnetic materials . . . . . . . . . . . . 65 6.2 Solving Maxwell’s equations with Green Functions . . . . . . . . . . . . . . . 66 6.3 Energy and Momentum and Their Conservation . . . . . . . . . . . . . . . . 68 6.4 Fields with Harmonic Time Dependence . . . . . . . . . . . . . . . . . . . . 69 6.5 The Dirac Monopole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 6.6 Symmetries of Maxwell Equations . . . . . . . . . . . . . . . . . . . . . . . . 72 7 Electromagnetic Plane Waves 76 7.1 Reflection and Refraction at a Plane Interface . . . . . . . . . . . . . . . . . 77 7.2 Brewster’s Angle and Internal Reflection . . . . . . . . . . . . . . . . . . . . 79 7.3 Action Principle for Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . 80 8 Lorentz Invariance and Special Relativity 81 8.1 Space-time symmetries of the wave equation . . . . . . . . . . . . . . . . . . 81 8.2 Einstein’s Insights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 8.3 Kinematics aspects of Lorentz transformations . . . . . . . . . . . . . . . . . 83 8.4 Space-time Tensors and their Transformation Laws . . . . . . . . . . . . . . 85 8.5 Lorentz covariance of Maxwell’s equations . . . . . . . . . . . . . . . . . . . 87 8.6 Action Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 8.7 Some particle motions in electromagnetic fields . . . . . . . . . . . . . . . . . 92 8.8 Electrodynamics of a Scalar Field . . . . . . . . . . . . . . . . . . . . . . . . 95 8.9 Lorentz Invariant Superconductivity: The Higgs Mechanism . . . . . . . . . 98 9 Propagation of Plane waves in Materials 102 9.1 Oscillator model for frequency dependence of a dielectric . . . . . . . . . . . 102 9.2 Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 9.3 Plasmas and the Ionosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 9.4 Group Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 9.5 Causality and Dispersion Relations . . . . . . . . . . . . . . . . . . . . . . . 107 9.6 Causal Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 10 Waveguides and Cavities 111 10.1 The approximation of perfect conductors . . . . . . . . . . . . . . . . . . . . 111 10.2 Waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 10.3 Rectangular Waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 2 c
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