, @LECTSOLUTIONSTUVIA
by Lee Bong on 08/09/23. Re-use and distribution is strictly not permitted, except for Open Access articles.
Contents
Preface vii
Downloaded from www.worldscientific.com
1. Why Electromagnetism? 1
2. Vector Algebra 5
2.1 Addition and Subtraction............................................................... 5
2.2 Vector Components ........................................................................ 7
2.3 The Dot Product ........................................................................ 10
2.4 The Cross Product ..................................................................... 12
3. Introducing Electricity 15
3.1 Electric Charge............................................................................ 15
3.2 Coulomb’s Law ........................................................................... 17
3.3 The Electric Field ...................................................................... 19
3.4 Principle of Superposition............................................................ 21
3.5 Field Lines .................................................................................. 27
3.6 Electric Flux ............................................................................... 30
3.7 Gauss’ Law.................................................................................. 35
3.8 Electric Potential Energy ........................................................... 42
3.9 Electric Potential ........................................................................ 46
3.10 Conservative Fields ..................................................................... 50
3.11 Superposition of Potential ............................................................ 52
3.12 Potential Energy of a System of Charges................................... 53
3.13 Electric Field as the Gradient of Potential ............................... 58
xi
, xii Electromagnetism — Principles and Modern Applications
4. A First Look at Circuits 63
4.1 Capacitance................................................................................... 63
4.2 Dielectrics..................................................................................... 66
4.3 Electric Current .......................................................................... 70
4.4 Resistivity and Resistance........................................................... 73
4.5 Electromotive Force and Circuits ............................................... 75
by Lee Bong on 08/09/23. Re-use and distribution is strictly not permitted, except for Open Access articles.
4.6 Energy and Power in Circuits .................................................... 82
5. Introducing Magnetism 85
5.1 The Magnetic Field .................................................................... 85
5.2 Magnetic Field Lines.................................................................. 87
5.3 Magnetic Flux.............................................................................. 88
5.4 Motion in a Magnetic Field......................................................... 90
5.5 The Magnetic Field of a Moving Charge .................................. 92
Downloaded from www.worldscientific.com
5.6 The Biot–Savart Law................................................................. 93
5.7 Force on a Wire ........................................................................... 96
5.8 Torque on a Current Loop .......................................................... 98
5.9 Amp`ere’s Law........................................................................... 100
5.10 Electromagnetic Induction .......................................................... 107
5.11 Inductance ................................................................................... 110
5.12 Magnetic Energy......................................................................... 114
6. A Second Look at Circuits 119
6.1 The R-C Circuit...................................................................... 119
6.2 The R-L Circuit ..................................................................... 123
6.3 The L-C Circuit...................................................................... 125
6.4 The L-R-C Circuit ................................................................ 128
6.5 Alternating Current ....................................................................131
6.6 The AC L-R-C Circuit ........................................................ 135
7. Maxwell’s Equations 139
7.1 The Displacement Current ......................................................... 139
7.2 Maxwell’s Equations ................................................................... 142
7.3 The Divergence........................................................................... 144
7.4 The Curl.................................................................................... 148
7.5 Maxwell Equations in Differential Form .................................. 153
7.6 Electromagnetic Waves .............................................................. 156
, @LECTSOLUTIONSTUVIA
Contents xiii
7.7 The Poynting Vector .................................................................162
7.8 Polarisation .................................................................................. 163
8. Relativity and Maxwell’s Equations 169
8.1 Relativistic Kinematics.............................................................170
8.2 Four Vectors ............................................................................... 172
8.3 The Dot Product for 4-Vector ..................................................177
by Lee Bong on 08/09/23. Re-use and distribution is strictly not permitted, except for Open Access articles.
8.4 The Metric Tensor ....................................................................179
8.5 The Velocity 4-Vector ..............................................................181
8.6 Tensors and Relativistic Equations............................................ 183
8.7 The 4-Potential and Field Strength ......................................... 185
8.8 The 4-Vector Current Density ..................................................189
8.9 The Covariant Maxwell Equations.......................................... 191
9. Maxwell’s Equations from Symmetry 199
Downloaded from www.worldscientific.com
9.1 The Need for Quantum Field Theory........................................ 199
9.2 Matter and the Dirac Equation .................................................202
9.3 Global Gauge Invariance ..........................................................203
9.4 Local Gauge Invariance ............................................................204
9.5 Gauge Invariance and the Other Forces .................................... 209
10. The Double Copy: From Electromagnetism to Gravity 213
10.1 Scattering Amplitudes in QFT ..................................................214
10.2 Gravity and General Relativity .................................................. 219
10.3 The Double Copy for Scattering Amplitudes............................ 223
10.4 The Double Copy for Classical Solutions ................................ 225
Appendix A Line and Surface Integrals 231
A.1 Line Integrals ........................................................................... 231
A.2 Surface Integrals ......................................................................... 236
Appendix B The Dirac Equation 243
Appendix C Solutions to Exercises 249
Index 271
by Lee Bong on 08/09/23. Re-use and distribution is strictly not permitted, except for Open Access articles.
Contents
Preface vii
Downloaded from www.worldscientific.com
1. Why Electromagnetism? 1
2. Vector Algebra 5
2.1 Addition and Subtraction............................................................... 5
2.2 Vector Components ........................................................................ 7
2.3 The Dot Product ........................................................................ 10
2.4 The Cross Product ..................................................................... 12
3. Introducing Electricity 15
3.1 Electric Charge............................................................................ 15
3.2 Coulomb’s Law ........................................................................... 17
3.3 The Electric Field ...................................................................... 19
3.4 Principle of Superposition............................................................ 21
3.5 Field Lines .................................................................................. 27
3.6 Electric Flux ............................................................................... 30
3.7 Gauss’ Law.................................................................................. 35
3.8 Electric Potential Energy ........................................................... 42
3.9 Electric Potential ........................................................................ 46
3.10 Conservative Fields ..................................................................... 50
3.11 Superposition of Potential ............................................................ 52
3.12 Potential Energy of a System of Charges................................... 53
3.13 Electric Field as the Gradient of Potential ............................... 58
xi
, xii Electromagnetism — Principles and Modern Applications
4. A First Look at Circuits 63
4.1 Capacitance................................................................................... 63
4.2 Dielectrics..................................................................................... 66
4.3 Electric Current .......................................................................... 70
4.4 Resistivity and Resistance........................................................... 73
4.5 Electromotive Force and Circuits ............................................... 75
by Lee Bong on 08/09/23. Re-use and distribution is strictly not permitted, except for Open Access articles.
4.6 Energy and Power in Circuits .................................................... 82
5. Introducing Magnetism 85
5.1 The Magnetic Field .................................................................... 85
5.2 Magnetic Field Lines.................................................................. 87
5.3 Magnetic Flux.............................................................................. 88
5.4 Motion in a Magnetic Field......................................................... 90
5.5 The Magnetic Field of a Moving Charge .................................. 92
Downloaded from www.worldscientific.com
5.6 The Biot–Savart Law................................................................. 93
5.7 Force on a Wire ........................................................................... 96
5.8 Torque on a Current Loop .......................................................... 98
5.9 Amp`ere’s Law........................................................................... 100
5.10 Electromagnetic Induction .......................................................... 107
5.11 Inductance ................................................................................... 110
5.12 Magnetic Energy......................................................................... 114
6. A Second Look at Circuits 119
6.1 The R-C Circuit...................................................................... 119
6.2 The R-L Circuit ..................................................................... 123
6.3 The L-C Circuit...................................................................... 125
6.4 The L-R-C Circuit ................................................................ 128
6.5 Alternating Current ....................................................................131
6.6 The AC L-R-C Circuit ........................................................ 135
7. Maxwell’s Equations 139
7.1 The Displacement Current ......................................................... 139
7.2 Maxwell’s Equations ................................................................... 142
7.3 The Divergence........................................................................... 144
7.4 The Curl.................................................................................... 148
7.5 Maxwell Equations in Differential Form .................................. 153
7.6 Electromagnetic Waves .............................................................. 156
, @LECTSOLUTIONSTUVIA
Contents xiii
7.7 The Poynting Vector .................................................................162
7.8 Polarisation .................................................................................. 163
8. Relativity and Maxwell’s Equations 169
8.1 Relativistic Kinematics.............................................................170
8.2 Four Vectors ............................................................................... 172
8.3 The Dot Product for 4-Vector ..................................................177
by Lee Bong on 08/09/23. Re-use and distribution is strictly not permitted, except for Open Access articles.
8.4 The Metric Tensor ....................................................................179
8.5 The Velocity 4-Vector ..............................................................181
8.6 Tensors and Relativistic Equations............................................ 183
8.7 The 4-Potential and Field Strength ......................................... 185
8.8 The 4-Vector Current Density ..................................................189
8.9 The Covariant Maxwell Equations.......................................... 191
9. Maxwell’s Equations from Symmetry 199
Downloaded from www.worldscientific.com
9.1 The Need for Quantum Field Theory........................................ 199
9.2 Matter and the Dirac Equation .................................................202
9.3 Global Gauge Invariance ..........................................................203
9.4 Local Gauge Invariance ............................................................204
9.5 Gauge Invariance and the Other Forces .................................... 209
10. The Double Copy: From Electromagnetism to Gravity 213
10.1 Scattering Amplitudes in QFT ..................................................214
10.2 Gravity and General Relativity .................................................. 219
10.3 The Double Copy for Scattering Amplitudes............................ 223
10.4 The Double Copy for Classical Solutions ................................ 225
Appendix A Line and Surface Integrals 231
A.1 Line Integrals ........................................................................... 231
A.2 Surface Integrals ......................................................................... 236
Appendix B The Dirac Equation 243
Appendix C Solutions to Exercises 249
Index 271
