 Additional Exercises for

Modern Problems in

Classical Electrodynamics

If you have additional exercises that would be appropriate for Modern Problems in Classical Electrodynamics, and would be willing to share them, e-mail them to me at the address below. I will be more than pleased to post them on this web site and give you credit for submitting them. Solutions would be useful, too! (I won't post the solutions.)

A large collection of interesting exercises in classical electrodynamics has been posted on the web by Kirk MacDonald, along with links to still more sites.

0. Prologue

0.1. Introduction

0.2. Electrostatics

0.2.1. Charges

0.2.2. Forces and Electric Fields

0.3. Magnetostatics

0.3.1. Currents

0.3.2. Forces and Fields

0.3.3. Vector Potential

0.4. Electrodynamics

0.4.1. Conservation of Charge

0.4.2. Faraday's Law

0.4.3. Energy in the Magnetic Field

0.5. The Maxwell Equations and Electromagnetic Waves

0.5.1. The Maxwell-Ampere Law

0.5.2. Electromagnetic Waves

0.5.3. Potentials and Gauges

0.6. Conservation Laws

0.6.1. Poynting's Theorem

0.6.2. Conservation of Momentum

1. Relativistic Kinematics

1.1. The Principles of Special Relativity

1.1.1. Historical Overview

1.1.2. Einstein's Postulates

1.1.3. Intervals

1.1.4. Proper Time

1.2. The Lorentz Transformation

1.2.1. Rotation in 4-Space

1.2.2. Time Dilation and Length Contraction

1.2.3. Velocity Transformation

1.3. 4-Vectors and 4-Tensors

1.3.1. Cartesian Tensors

1.3.2. Relativistic Metric and Lorentz Transformation

1.3.3. 4-Vector Calculus

1.4. Electromagnetic Fields

1.4.1. The 4-Tensor Electromagnetic Field

1.4.2. Transformation of Electromagnetic Fields

2. Relativistic Mechanics and Field Theory

2.1. Relativistic Free Particle

2.1.1. Hamilton's Principle and the Calculus of Variations

2.1.2. Langrangian for a Free Particle

2.1.3. Energy and Momentum

2.1.4. de Broglie Waves

2.1.5. Rotational Invariance and Angular Momentum

2.2. Charged Particle in a Vector Potential

2.2.1. Langrangian Mechanics

2.2.2. Canonical Momentum

2.2.3. Canonical Equations of Motion

2.3. The Maxwell Equations

2.3.1. Equations of Motion of a Vector Field

2.3.2. Proca Mass Term

2.4. Invariance and Conservation Laws

2.4.1. Gauge Transformations

2.4.2. Symmetric Stress Tensor for the Electromagnetic Field

3. Time-Independent Electromagnetic Fields

3.1. Electrostatics

3.1.1. Coulomb's Law

3.1.2. Energy in Electrostatic Fields

3.1.3. Multipole Moments

3.2. Boundary-Value Problems with Conductors

3.2.1. Boundary Conditions and Uniqueness Theorems

3.2.2. Energy and Capacitance

3.2.3. Method of Images

3.2.4. Separation of Variables

3.2.5. Spheroidal Coordinates

3.2.6. Spherical Harmonics

3.2.7. Variational Methods

3.2.8. Numerical Methods

3.2.9. Green Functions

3.3. Magnetostatics

3.3.1. Biot-Savart Law

3.3.2. Forces and Energy

3.3.3. Multipole Moments

3.3.4. Magnetic Scalar Potential

4. Electromagnetic Waves

4.1. Plane Waves

4.1.1. Electric and Magnetic Fields in Plane Waves

4.1.2. Charged Particle in a Plane Wave

4.2. Canonical Equations of an Electromagnetic Field

4.2.1. Fourier Decomposition of the Field

4.2.2. Spontaneous Emission by a Harmonic Oscillator

4.2.3. Canonical Equations of the Electromagnetic Field

4.2.4. Blackbody Radiation and the Einstein Coefficients

4.3. Waves in Plasmas

4.3.1. Transverse Electromagnetic Waves

4.3.2. Longitudinal Electrostatic Waves

5. Fourier Techniques and Virtual Quanta

5.1. Fourier Transformation

5.1.1. Fourier's Theorem

5.1.2. Asymptotic Behavior of Fourier Transforms

5.1.3. Delta-Functions

5.1.4. Autocorrelation Functions and the Wiener-Khintchine Theorem

5.1.5. Pulse Compression

5.2. Method of Virtual Quanta

5.2.1. Fourier Decomposition of the Field of a Relativistic Charge

5.2.2. Bremsstrahlung

5.2.3. Excitation by a Fast Charged Particle

5.2.4. Transition Radiation

6. Macroscopic Materials

6.1. Polarization and Magnetization

6.1.1. The Macroscopic Form of the Maxwell Equations

6.1.2. The Constitutive Relations

6.1.3. Boundary Conditions

6.1.4. Magnetic Scalar Potential

6.1.5. Conservation of Energy, and Poynting's Theorem

6.2. Properties of Dielectric and Magnetic Materials

6.2.1. Dielectric Materials

6.2.2. Magnetic Materials

7. Linear, Dispersive Media

7.1. Linear Media

7.1.1. Waves in a Nondispersive Medium

7.1.2. Constitutive Relations in Dispersive Media

7.1.3. Kramers-Kronig Relations

7.1.4. Plane Waves in Dispersive Media

7.1.5. Phase Velocity and Group Velocity

7.1.6. Conservation of Energy in Dispersive Media

7.1.7. Lorentz-Drude Model

7.2. Reflection and Refraction at Surfaces

7.2.1. Boundary Conditions

7.2.2. Dielectric Reflection

7.2.3. Metallic Reflection

7.2.4. Surface Waves

7.3. Energy Loss by Fast Particles Traveling Through Matter

7.3.1. Ionization and Excitation

7.3.2. Relativistic Limit and the Density Effect

8. Nonlinear Optics

8.1. Nonlinear Susceptibility

8.1.1. Nonlinear Polarization

8.1.2. Anisotropic Materials

8.2. Multiphoton Processes

8.2.1. Coupled-Wave Equation

8.2.2. Second-Harmonic Generation

8.2.3. Sum-Frequency Generation

8.3. Nonlinear Index of Refraction

8.3.1. Third-Order Susceptibility

8.3.2. Wave Equation with a Nonlinear Index of Refraction

8.3.3. Phase-Conjugate Reflection

8.4. Raman Processes

8.4.1. Raman Scattering

8.4.2. Coherent Raman Amplification

9. Diffraction

9.1. Geometrical Optics

9.1.1. Eikonal Approximation

9.1.2. Rays in Geometrical Optics

9.1.3. Integral Theorems

9.2. Gaussian Optics and Laser Resonators

9.2.1. Paraxial Approximation

9.2.2. Laser Resonators and Mode Spacing

9.2.3. Transverse Modes and Resonator Stability

9.3. Diffraction

9.3.1. Scalar Diffraction Theory

9.3.2. Fraunhofer Diffraction (Far Field)

9.3.3. Fresnel Diffraction (Near Field)

10. Radiation by Relativistic Particles

10.1. Angular and Spectral Distribution of Radiation

10.1.1. Fourier Decomposition of the Fields

10.1.2. Retarded Fields and Lienard-Wiechert Fields

10.1.3. Multipole Radiation

10.1.4. Spectral Distribution of Radiation from a Point Charge

10.1.5. Angular Distribution of Radiation from a Point Charge

10.1.6. Total Power Radiated by a Point Charge

10.2. Bremsstrahlung and Transition Radiation

10.2.1. Bremsstrahlung

10.2.2. Transition Radiation

10.3. Thomson Scattering

10.3.1. Linear Thomson Scattering

10.3.2. Nonlinear Thomson Scattering

10.4. Synchrotron Radiation and Undulator Radiation

10.4.1. Synchrotron Radiation

10.4.2. Undulator Radiation

10.5. Coherent Emission from Multiple Particles

10.5.1. Coherence and Form Factor

10.5.2. Coherent Radiative Processes

10.6. Radiation from Relativistic Particles Traveling Through Matter

10.6.1. Angular Spectral Fluence

10.6.2. Cherenkov Radiation

11. Fundamental Particles in Classical Electrodynamics

11.1. Electromagnetic Mass and the Radiation Reaction

11.1.1. Difficulties in the Classical Theory

11.1.2. The 4/3 Problem and Poincaré Stresses

11.1.3. Point Particles and the Radiation Reaction

11.1.4. Extended Particles

11.2. Magnetic Monopoles

11.2.1. The Maxwell Equations

11.2.2. Magnetic Monopoles and Charge Quantization

11.3. Spin

11.3.1. Relativistic Equations of Motion

11.3.2. Thomas Precession and Spin-Orbit Coupling

For information on this web page, contact charles.a.brau@vanderbilt.edu.

Last updated on 6/19/06.