STUDY NOTES & REFERENCE GUIDE
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, 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
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, 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
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
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, 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
8.3 The Dot Product for 4-Vector ..............................................................177
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
StudyNexa Stuvia | Page 4 of 279
StudyNexa Stuvia | Page 1 of 279
, 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
StudyNexa Stuvia | Page 2 of 279
, 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
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
StudyNexa Stuvia | Page 3 of 279
, 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
8.3 The Dot Product for 4-Vector ..............................................................177
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
StudyNexa Stuvia | Page 4 of 279
