###### Relativistic Electrodynamics

### Relativistic Electrodynamics

###### Notes and problems from 2011 PHY450H1S.

# About the Book

These are my personal lecture notes for the Spring 2011, University of Toronto, Relativistic Electrodynamics course (PHY450H1S). This class was taught by Prof. Erich Poppitz, with Simon Freedman handling tutorials.

#### Table of Contents

- Copyright
- Document Version
- Dedication
- Preface
- Contents
- List of Figures
- COURSE NOTES
- 1 Principle of relativity
- 1.1 Distance as a clock
- 1.2 The principle of relativity
- 1.3 Enter electromagnetism
- 1.4 Einstein's relativity principle
- 2 Spacetime
- 2.1 Intervals for light like behaviour
- 2.2 Invariance of infinitesimal intervals
- 2.3 Geometry of spacetime: lightlike, spacelike, timelike intervals
- 2.4 Relativity principle in mathematical formulation
- 2.5 Geometry of spacetime
- 2.6 Proper time
- 2.7 More spacetime geometry
- 2.8 Finite interval invariance
- 2.9 Deriving the Lorentz transformation
- 2.10 More on proper time
- 2.11 Length contraction
- 2.12 Superluminal speed and causality
- 2.13 Problems
- 3 Four vectors and tensors
- 3.1 Introducing four vectors
- 3.2 The Special Orthogonal group (for Euclidean space)
- 3.3 The Special Orthogonal group (for spacetime)
- 3.4 Lower index notation
- 3.5 Problems
- 4 Particle action and relativistic dynamics
- 4.1 Dynamics
- 4.2 The relativity principle
- 4.3 Relativistic action
- 4.4 Next time
- 4.5 Finishing previous arguments on action and proper velocity
- 4.6 Symmetries of spacetime translation invariance
- 4.7 Time translation invariance
- 4.8 Some properties of the four momentum
- 4.9 Where are we?
- 4.10 Interactions
- 4.11 More on the action
- 4.12 antisymmetric matrices
- 4.13 Gauge transformations
- 4.14 What is the significance to the gauge invariance of the action?
- 4.15 Four vector Lorentz force
- 4.16 Chewing on the four vector form of the Lorentz force equation
- 4.17 Transformation of rank two tensors in matrix and index form
- 4.18 Where we are
- 4.19 Generalizing the action to multiple particles
- 4.20 Problems
- 5 Action for the field
- 5.1 Action for the field
- 5.2 Current density distribution
- 5.3 Review. Our action
- 5.4 The field action variation
- 5.5 Computing the variation
- 5.6 Unpacking these
- 5.7 Speed of light
- 5.8 Trying to understand ``c''
- 5.9 Claim: EM waves propagate with speed c and are transverse
- 5.10 What happens with a Massive vector field?
- 5.11 Review of wave equation results obtained
- 5.12 Review of Fourier methods
- 5.13 Review. Solution to the wave equation
- 5.14 Moving to physically relevant results
- 5.15 EM waves carrying energy and momentum
- 5.16 Energy and momentum of EM waves
- 5.17 Review. Energy density and Poynting vector
- 5.18 How about electromagnetic waves?
- 5.19 Problems
- 6 Lienard-Wiechert potentials
- 6.1 Solving Maxwell's equation
- 6.2 Solving the forced wave equation
- 6.3 Elaborating on the wave equation Green's function
- 6.4 Fields from the Lienard-Wiechert potentials
- 6.5 Check. Particle at rest
- 6.6 Check. Particle moving with constant velocity
- 6.7 Back to extracting physics from the Lienard-Wiechert field equations
- 6.8 Multipole expansion of the fields
- 6.9 Putting the pieces together. Potentials at a distance
- 6.10 Where we left off
- 6.11 Direct computation of the magnetic radiation field
- 6.12 An aside: A tidier form for the electric dipole field
- 6.13 Calculating the energy flux
- 6.14 Calculating the power
- 6.15 Types of radiation
- 6.16 Problems
- 7 Energy Momentum Tensor
- 7.1 Energy momentum conservation
- 7.2 Total derivative of the Lagrangian density
- 7.3 Unpacking the tensor
- 7.4 Recap
- 7.5 Spatial components of Tkm
- 7.6 On the geometry
- 7.7 Problems
- 8 Radiation reaction
- 8.1 A closed system of charged particles
- 8.2 Start simple
- 8.3 What is next?
- 8.4 Recap
- 8.5 Moving on to the next order in v over c
- 8.6 A gauge transformation to simplify things
- 8.7 Recap
- 8.8 Incorporating radiation effects as a friction term
- 8.9 Radiation reaction force
- 8.10 Limits of classical electrodynamics
- APPENDIXES
- A Professor Poppitz's handouts
- B Some tensor and geometric algebra comparisons in a spacetime context
- B.1 Motivation
- B.2 Notation and use of Geometric Algebra herein
- B.3 Transformation of the coordinates
- B.4 Lorentz transformation of the metric tensors
- B.5 The inverse Lorentz transformation
- B.6 Duality in tensor form
- B.7 Stokes Theorem
- C Frequency four vector
- D Non-inertial (local) observers
- D.1 Basis construction
- D.2 Split of energy and momentum (VERY ROUGH NOTES)
- D.3 Frequency of light from a distant star (AGAIN VERY ROUGH NOTES)
- E 3D GPS geometries
- F Playing with complex notation for relativistic applications in a plane
- F.1 Motivation
- F.2 Our invariant
- F.3 Change of basis
- G Waveguides: confined EM waves
- G.1 Motivation
- G.2 Back to the tutorial notes
- G.3 Separation into components
- G.4 Solving the momentum space wave equations
- G.5 Final remarks
- H Three dimensional divergence theorem with generally parametrized volume element
- H.1 A generally parametrized parallelepiped volume element
- H.2 On the geometry of the surfaces
- H.3 Expansion of the Jacobian determinant
- H.4 A look back, and looking forward
- I EM fields from magnetic dipole current
- I.1 Review
- I.2 Magnetic dipole
- J Yukawa potential note
- K Proof of the d'Alembertian Green's function
- K.1 An aside. Proving the Laplacian Green's function
- K.2 Returning to the d'Alembertian Green's function
- L Mathematica notebooks
- Bibliography

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