Electromagnetism with Spacetime Algebra John F. Lindner Physics Department The College of Wooster 13 December 2018 2 Contents 1 Advertisement 11 1.1 Action at a Distance . 11 1.2 Classical Field theory . 13 2 Geometric Algebra of Space 15 2.1 Geometric Product . 15 2.2 Normalized, Antisymmetric Basis . 15 2.3 Multivectors . 16 2.4 Pseudoscalar . 17 2.5 Pseudovectors . 17 2.6 Inner & Outer Product Decomposition . 18 2.7 Symmetries . 19 2.8 Parallel & Perpendicular . 20 2.9 Magnitudes . 22 2.10 Geometric Interpretations . 23 2.11 Metric . 24 2.12 Rotations . 26 3 Geometric Calculus of Space 29 3.1 Geometric Derivative . 29 3.2 Differentiating Scalar Fields . 30 3.3 Differentiating Vector Fields . 31 4 Geometric Algebra of Spacetime 35 4.1 Basis . 35 4.2 Multivectors . 36 4.3 Pseudoscalars . 37 4.4 Pseudovectors . 38 4.5 Spacetime Metric . 38 4.6 Important Spacetime Vectors . 39 4.7 Spacetime Rotations . 41 3 4 CONTENTS 5 The Foundation of Electromagnetism 45 5.1 Field . 45 5.2 Source . 47 5.3 The Maxwell Field Equation . 48 5.4 The Lorentz Force Equation . 50 6 Interpreting Maxwell's Equations 53 6.1 Divergence & Flux . 53 6.2 Curl & Circulation . 55 6.3 Integral Form . 56 6.4 Interpretations . 58 6.5 Maxwell's Form . 59 6.6 Magnetic Monopole Form . 60 6.7 Matter Fields . 60 7 Electrostatics 63 7.1 Coulomb's Law . 63 7.2 Boundary Conditions . 66 7.3 Point Charges and the Dirac Delta . 67 7.4 Electric Potential . 68 7.5 Work & Energy . 72 7.6 Poisson's Equation . 75 7.7 Laplace's Equation . 76 7.8 Separation of Variables . 79 8 Magnetostatics 85 8.1 Biot-Savart's Law . 85 8.2 Boundary Conditions . 88 8.3 Magnetic Potential . 89 8.4 Poisson's Equation . 93 8.5 Statics Comparison . 94 9 Electrodynamics 97 9.1 Faraday's Law . 97 9.2 Multiply Connected Regions . 100 9.3 Amp`ere-Maxwell's Law . 103 9.4 Dynamic Potentials . 104 9.5 Gauge Transformations . 105 9.6 Coulomb Gauge . 106 9.7 Lorentz Gauge . 107 9.8 Field Momentum . 107 9.9 Energy & Momentum Density . 108 CONTENTS 5 10 Electromagnetic Radiation 113 10.1 Electromagnetic Waves . 113 10.2 Retarded Potentials . 115 10.3 Li´enard-Weichert Potentials . 118 10.4 Constant Velocity . 120 10.5 Arbitrary Motion . 122 10.6 Arbitrary Force . 127 10.7 Feynman's Formula . 128 10.8 Larmor's Formula . 129 Appendices A Coordinate Systems 131 A-1 Curvilinear Coordinates . 131 A-2 Polar Spherical Coordinates . 132 A-3 Cylindrical Coordinates . 134 B Product Rules 137 C Electromagnetic Units 139 6 CONTENTS List of Tables 2.1 Basis for geometric algebra of 3-dimensional space. 16 2.2 Special cases of geometric product of two vectors. 21 4.1 Basis for geometric algebra of 4-dimensional spacetime. 36 4.2 Pseudoscalar comparison. 37 4.3 Important spacetime vectors . 39 7.1 Summary of similar but different electrostatic formulas. 75 8.1 Comparison of electrostatics and magnetostatics . 94 A-1 Electromagnetic Units . 140 A-2 Geometric Units Dimensions . 141 7 8 LIST OF TABLES List of Figures 1.1 Force between two charges in arbitrary motion . 12 2.1 Bivector perpendicular to its dual vector . 20 2.2 Projection and rejection . 21 2.3 Magnitudes of the inner and outer products . 22 2.4 Visualization of the lowest grade multivectors. 23 2.5 Visualization of bivectors as oriented areas . 23 2.6 Skew frame . 25 2.7 Vector Rotation . 27 4.1 Spacetime Rotation . 42 5.1 Electromagnetic field, the Faraday bivector . 46 5.2 Electromagnetic source . 47 6.1 Flux through an infinitesimal cube . 54 6.2 Vector fields with simple divergences . 54 6.3 Circulation around an infinitesimal square . 55 6.4 Vector fields with simple circulations . 56 6.5 Interior fluxes cancel in pairs . 57 6.6 Interior flows cancel in pairs . 57 6.7 Electric and magnetic charges . 59 6.8 Matter Fields . 61 6.9 Cube and triangle . 62 7.1 Charged shell . 64 7.2 Electric field discontinuity . 66 7.3 Electric potential definition . 69 7.4 Charged shell . 70 7.5 Charged assembly . 74 7.6 Harmonic averaging . 77 7.7 Relaxation algorithm . 78 7.8 Separation of variables example . 80 8.1 Biot-Savart law . 86 9 10 LIST OF FIGURES 8.2 Long straight current . 87 8.3 Magnetic field discontinuity . 88 8.4 Magnetic potential . 92 8.5 Magnetic potential for constant field . 93 9.1 Linear electric generator . 98 9.2 Ring of charge . 99 9.3 Time-varying solenoid . 100 9.4 Multiply connected region . 101 9.5 Source and sink . 103 9.6 Solenoid and charges . ..