STUDY NOTES & REFERENCE GUIDE
Coursesity ‐ Stuvia | Page 1 of 292
,by Lee Bong on 08/09/23. Re-use and distribution is strictly not permitted, except for Open Access articles. STUDY NOTES & REFERENCE GUIDE
Contents
Preface vii
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
Coursesity ‐ Stuvia | Page 2 of 292
, STUDY NOTES & REFERENCE GUIDE
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
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
Coursesity ‐ Stuvia | Page 3 of 292
, STUDY NOTES & REFERENCE GUIDE
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
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.
8.3
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
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
Coursesity ‐ Stuvia | Page 4 of 292
Coursesity ‐ Stuvia | Page 1 of 292
,by Lee Bong on 08/09/23. Re-use and distribution is strictly not permitted, except for Open Access articles. STUDY NOTES & REFERENCE GUIDE
Contents
Preface vii
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
Coursesity ‐ Stuvia | Page 2 of 292
, STUDY NOTES & REFERENCE GUIDE
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
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
Coursesity ‐ Stuvia | Page 3 of 292
, STUDY NOTES & REFERENCE GUIDE
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
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.
8.3
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
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
Coursesity ‐ Stuvia | Page 4 of 292
