CHAPMAN C SANDRA COPYRIGHT ELECTRODYNAMICS CORE Core Electrodynamics Sandra C. Chapman June 29, 2010 Preface CHAPMAN C This monograph is based on a final year undergraduate course in Electro- dynamics that I introduced at Warwick as part of the new four year physics degree. When I gave this course my intention was to engage the students in the elegance of electrodynamics and special relativity, whilst giving them the tools to begin graduate study. Here, from the basis ofSANDRA experiment we first derive the Maxwell equations, and special relativity. Introducing the mathematical framework of generalized tensors, the laws of mechanics, Lorentz force and the Maxwell equations are then cast in manifestly covari- ant form. This provides the basis for graduate study in field theory, high energy astrophysics, general relativity and quantum electrodynamics. As the title suggests, this book is ”electrodynamics lite”. The journey through electrodynamics is kept as brief as possible, with minimal diversion into details, so that the elegance of the theory can be appreciated in a holistic way. It is written in an informalCOPYRIGHT style and has few prerequisites; the derivation of the Maxwell equations and their consequences is dealt with in the first chapter. Chapter 2 is devoted to conservation equations in tensor formulation, here Cartesian tensors are introduced. Special relativity and its consequences for electrodynamics is introduced in Chapter 3 and cast in four vector form, here we introduce generalized tensors. Finally in Chapter 4 Lorentz frame invariant electrodynamics is developed. Supplementary material and examples are provided by the two sets of problems. The first is revision of undergraduate electromagnetism, to ex- pand on the material in the first chapter. The second is more advanced corresponding to the remaining chapters, and its purpose is twofold: to expand on points that are important, but not essential, to derivation of manifestly covariant electrodynamics, and to provide examples of manip- ulation of cartesian and generalized tensors. As these problems introduce material not covered in the text they are accompanied by full worked solu- ELECTRODYNAMICStions. The philosophy here is to facilitate learning by problem solving, as well as by studying the text. Extensive appendices for vector relations, unit conversion and so forth CORE 1 2 are given with graduate study in mind. As SI units are used throughout here, conversions for units and equations are also given. There have been many who have contributed to the existence of this book. Thanks in particular go to Nick Watkins for valuable discussions and to George Rowlands for his insightful reading of the final draft. David Betts, the editor of this series, has also shown remarkable patience and tenacity as the many deadlines have come and gone. The completion of the book was also much assisted by a PPARC personal fellowship. Finally, my thanks go to the students themselves; their lively receptionCHAPMAN of the original course and their insightful questions were theC inspiration for this book. If the reader finds that this book provides a shortcut to experience the beauty of electrodynamics, without sacrificing the rigour needed for further study, then I have suceeded. Sandra Chapman, June, 1999 SANDRA COPYRIGHT ELECTRODYNAMICS CORE Contents CHAPMAN C 3.5 Four Vectors and Four Vector Calculus. 60 3.5.1 Some mechanics, Newton’s laws. 60 3.5.2 Some four vector calculus. 63 3.6 A Frame Invariant Electromagnetism . 64 3.6.1 Charge conservation. .SANDRA . 64 3.6.2 A manifestly covariant electromagnetism . 65 4 The Field Tensors 67 4.1 Invariant Form for E and B: TheEMFieldTensor. 67 4.2 Maxwell’s Equations in Invariant Form. 70 4.3 Conservation of Energy-Momentum. 73 4.4 LorentzForce........................... 74 4.4.1 Manifestly covariantelectrodynamics. 75 4.5 Transformation of the Fields .COPYRIGHT . 76 4.6 Field from a Moving Point Charge. 77 4.7 Retarded Potential. 81 3.5 Four Vectors and Four Vector Calculus. 60 3.5.1 Some mechanics, Newton’s laws. 60 3.5.2 Some four vector calculus. 63 3.6 A Frame Invariant Electromagnetism . 64 3.6.1 Charge conservation. 64 3.6.2 A manifestly covariant electromagnetism . 65 4 The Field Tensors 67 4.1 Invariant Form for E and B: TheEMFieldTensor. 67 4.2 Maxwell’s Equations in Invariant Form. 70 4.3 Conservation of Energy-Momentum. 73 4.4 LorentzForce........................... 74 ELECTRODYNAMICS4.4.1 Manifestly covariantelectrodynamics. 75 4.5 Transformation of the Fields . 76 4.6 Field from a Moving Point Charge. 77 CORE 3 4 CONTENTS 4.7 Retarded Potential. 81 1 A Brief Tour of Electromagnetism. 9 1.1 The Building Blocks. 9 1.2 MaxwellIandII ........................ 11 1.2.1 Flux of a vector field . 13 1.2.2 Flux of E ........................ 15 1.2.3 Flux of B ........................ 18 1.2.4 Conservative and nonconservative fields. .CHAPMAN . 19 1.3 Maxwell III and IV . .C . 21 1.3.1 Faraday’s law and Galilean invariance. 21 1.3.2 Amp`ere’s law and conservation of charge. 25 1.4 Electromagnetic Waves. 26 1.5 Conservation Equations. 28 2 Field Energy and Momentum. 29 2.1 Tensors and Conservation Equations.SANDRA . 30 2.1.1 Momentum flux density tensor. 30 2.1.2 Momentum flux, gas pressure and fluid equations. 33 2.1.3 Cartesian tensors, some definitions. 34 2.2 Field Momentum and Maxwell Stress . 37 2.2.1 Energy conservation: Poynting’s theorem . 37 2.2.2 Momentum conservation: Maxwell stress . 39 2.3 Radiation Pressure. 41 COPYRIGHT 3 A Frame Invariant Electromagnetism 45 3.1 TheLorentzTransformation. 46 3.2 The Moving Charge and Wire Experiment. 50 3.3 Maxwell in Terms of Potentials. 55 3.4 Generalized Coordinates. 56 3.5 Four Vectors and Four Vector Calculus. 60 3.5.1 Some mechanics, Newton’s laws. 60 3.5.2 Some four vector calculus. 63 3.6 A Frame Invariant Electromagnetism . 64 3.6.1 Charge conservation. 64 3.6.2 A manifestly covariant electromagnetism . 65 4 The Field Tensors 67 4.1 Invariant Form for E and B: TheEMFieldTensor. 67 ELECTRODYNAMICS4.2 Maxwell’s Equations in Invariant Form. 70 4.3 Conservation of Energy-Momentum. 73 4.4 LorentzForce........................... 74 CORE CONTENTS 5 4.4.1 Manifestly covariantelectrodynamics. 75 4.5 Transformation of the Fields . 76 4.6 Field from a Moving Point Charge. 77 4.7 Retarded Potential. 81 A Revision Problems 91 A.1 Static Magnetic Fields. 91 A.2 Static Electric Fields. 93 CHAPMAN A.3 Conservation and Poynting’s Theorem. 93 A.4 The Wave Equation: Linearity and Dispersion. 94 C A.5 Free Space EM Waves I. 95 A.6 Free Space EM Waves II. 95 A.7 EM Waves in a Dielectric. 95 A.8 Dielectrics and Polarization. 96 A.9 EM Waves in a Conductor: Skin Depth. 96 A.10CavityResonator. .......................SANDRA 97 B Solutions to Revision Problems 99 B.1 Static Magnetic Fields. 99 B.2 Static Electric Fields. 101 B.3 Conservation and Poynting’s Theorem. 102 B.4 The Wave Equation: Linearity and Dispersion. 103 B.5 Free Space EM waves I. 104 B.6 Free Space EM Waves II. .COPYRIGHT . 105 B.7 EM Waves in a Dielectric. 105 B.8 Dielectrics and Polarization. 106 B.9 EM Waves in a Conductor: Skin Depth. 107 B.10 Cavity Resonator. 109 C Some Advanced Problems 111 C.1 Maxwell Stress Tensor. 111 C.2 Liouville and Vlasov Theorems: a Conservation Equation for PhaseSpace. .......................... 112 C.3 Newton’s Laws and the Wave Equation under Galilean Trans- formation............................. 112 C.4 Transformation of the Fields. 112 C.5 Metric for Flat Spacetime.1 . 113 ELECTRODYNAMICSC.6 Length of the EM field Tensor in Spacetime. 114 C.7 Alternative Form for the Maxwell Homogenous Equations. 114 C.8 Lorentz Transformation of the EM Field Tensor. 114 CORE 6 CONTENTS D Solution to Advanced Problems 117 D.1 Maxwell Stress Tensor. 117 D.2 Liouville and Vlasov Theorems: a Conservation Equation in PhaseSpace. .......................... 118 D.3 Newton’s Laws and the Wave Equation under Galilean Trans- formation............................. 119 D.4 Transformation of the Fields. 121 D.5 Metric for Flat Spacetime. 123 D.6 Length of the EM Field Tensor in Spacetime. .CHAPMAN . 126 D.7 Alternative Form for the Maxwell Homogenous Equations.C . 127 D.8 Lorentz Transformation of the EM Field Tensor. 127 E Vector identities 129 E.1 Differential Relations . 129 E.2 Integral Relations . 130 F Tensors SANDRA 133 F.1 CartesianTensors ....................... 133 F.2 Special Tensors . 135 F.3 Generalized Tensors . 135 F.3.1 General properties of spacetime . 136 F.3.2 Flat spacetime . 137 G Units and Dimensions 139 G.1 SI Nomenclature . 140 G.2 Metric PrefixesCOPYRIGHT . 141 H Dimensions and Units 143 H.1 Physical Quantities . 143 H.2 Equations . 146 I Physical Constants (SI) 147 ELECTRODYNAMICS CORE List of Figures CHAPMAN C 4.1 Charge q is at rest at the origin of the S! frame and is moving with velocity vˆx1 w.r.t. frame S. The origins of the S and S! frames coincide at t =0. .................. 78 4.2 Sketch of E and E at point P versus x! /γ = x vt. 79 1 2 1 1 − SANDRA 4.3 Sketch of the E! field around a charge at rest, and the E field around a charge moving at relative velocity v, in the x1,x2 plane. .............................. 80 4.4 A volume element is located at rest at the origin of the S! frame which is moving at +v in the x1 direction w.r.t. the S frame. In the S! frame the volume element contains charge ρ!dx1! dx2! dx3! .