Classical Electrodynamics

A rigorous graduate-level treatment of classical electrodynamics—from Coulomb's law through Maxwell's equations, electromagnetic waves, and radiation theory—with full derivations, numerical simulations, and Fortran/Python examples.

Course Overview

Classical electrodynamics is one of the most beautiful and complete theories in physics. James Clerk Maxwell's unification of electricity, magnetism, and optics into four compact equations stands as one of the greatest intellectual achievements in the history of science. This course follows the graduate-level treatment in the tradition of Jackson and Griffiths, covering the full mathematical structure from first principles.

What You'll Learn

  • • Electrostatics: fields, potentials, energy
  • • Boundary value problems and multipole expansion
  • • Magnetostatics and vector potential
  • • Maxwell's equations in differential and integral form
  • • Conservation laws and the Poynting theorem
  • • Electromagnetic waves in vacuum and matter
  • • Reflection, refraction, waveguides, and resonators
  • • Radiation from accelerating charges and antennas
  • • Liénard–Wiechert potentials and Larmor formula
  • • Relativistic formulation of electrodynamics

Prerequisites

  • • Multivariable calculus and vector calculus
  • • Ordinary and partial differential equations
  • • Linear algebra
  • • Classical mechanics
  • • Basic electromagnetism (undergraduate level)
  • • Complex analysis (helpful)

References

  • • J. D. Jackson, Classical Electrodynamics (3rd ed.)
  • • D. J. Griffiths, Introduction to Electrodynamics (4th ed.)
  • • L. D. Landau & E. M. Lifshitz, Classical Theory of Fields
  • • A. Zangwill, Modern Electrodynamics

Maxwell's Equations — The Heart of the Course

Differential form (SI units):

$$\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0}$$$$\nabla \cdot \mathbf{B} = 0$$$$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$$$$\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0\epsilon_0 \frac{\partial \mathbf{E}}{\partial t}$$

Integral form:

$$\oint_S \mathbf{E} \cdot d\mathbf{A} = \frac{Q_{\rm enc}}{\epsilon_0}$$$$\oint_S \mathbf{B} \cdot d\mathbf{A} = 0$$$$\oint_C \mathbf{E} \cdot d\boldsymbol{\ell} = -\frac{d\Phi_B}{dt}$$$$\oint_C \mathbf{B} \cdot d\boldsymbol{\ell} = \mu_0 I_{\rm enc} + \mu_0\epsilon_0 \frac{d\Phi_E}{dt}$$

Course Structure

⚡

Part I: Electrostatics

Coulomb's law, Gauss's law, electric potential, energy, conductors, and dielectrics. Laplace and Poisson equations with boundary conditions.

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Part II: Magnetostatics

Biot-Savart law, Ampere's law, magnetic vector potential, multipole expansion, and magnetic materials.

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Part III: Electrodynamics

Faraday's law, Maxwell's displacement current, full Maxwell equations, conservation laws, and the Poynting theorem.

〰️

Part IV: EM Waves

Plane waves in vacuum and matter, reflection and transmission (Fresnel equations), dispersion, waveguides, and resonant cavities.

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Part V: Radiation

Retarded potentials, Liénard–Wiechert fields, radiation from accelerating charges, Larmor formula, and radiation reaction.

Key Results at a Glance

Coulomb's Law

$$\mathbf{F} = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r^2} \hat{r}$$

Force between two point charges separated by distance r

Lorentz Force

$$\mathbf{F} = q(\mathbf{E} + \mathbf{v} \times \mathbf{B})$$

Force on a charge q moving with velocity v

Poynting Vector

$$\mathbf{S} = \frac{1}{\mu_0}(\mathbf{E} \times \mathbf{B})$$

Energy flux density of the EM field (W/m²)

Speed of Light

$$c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} \approx 3 \times 10^8 \text{ m/s}$$

Emerges from Maxwell's equations — light is an EM wave

Larmor Radiation Formula

$$P = \frac{\mu_0 q^2 a^2}{6\pi c} = \frac{q^2 a^2}{6\pi \epsilon_0 c^3}$$

Power radiated by a non-relativistic accelerating charge

Wave Equation

$$\nabla^2 \mathbf{E} - \mu_0\epsilon_0 \frac{\partial^2 \mathbf{E}}{\partial t^2} = 0$$

Derived directly from Maxwell's equations in vacuum