In 1864, Scottish physicist James Clerk Maxwell (1831–1879) completed the formal mathematical formulation of Maxwell’s Equations, synthesizing electricity, magnetism, and light into four foundational field equations.

Maxwell’s equations proved that accelerating electric charges emit self-propagating transverse electromagnetic waves traveling at speed $c = 1/\sqrt{\mu_0 \epsilon_0}$, establishing the physical basis for radio, optics, radar, and quantum electrodynamics.

Differential Form of Maxwell’s Equations

In modern vector calculus notation (formalized by Oliver Heaviside in 1884):

1. Gauss’s Law for Electricity

$$\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0}$$

2. Gauss’s Law for Magnetism

$$\nabla \cdot \mathbf{B} = 0$$

3. Faraday’s Law of Induction

$$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$$

4. Ampère-Maxwell Law

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

Prediction of Electromagnetic Wave Speed ($c$)

In vacuum ($\rho = 0, \mathbf{J} = 0$), operating with $\nabla \times (\nabla \times \mathbf{E})$ yields the 3D wave equation:

$$\nabla^2 \mathbf{E} - \mu_0 \epsilon_0 \frac{\partial^2 \mathbf{E}}{\partial t^2} = 0 \implies v = \frac{1}{\sqrt{\mu_0 \epsilon_0}} \equiv c \approx 2.998 \times 10^8\,\text{m/s}$$

Key Takeaways

  • Year: 1864
  • Key Figure: James Clerk Maxwell (Scottish Physicist)
  • Core Discovery: Formulated Maxwell’s Equations, unifying electricity, magnetism, and optics.
  • Light Speed Prediction: Proved electromagnetic waves propagate at speed $c = 1/\sqrt{\mu_0 \epsilon_0}$.
  • Quantum Relevance: Classical gauge field theory foundation for Quantum Electrodynamics (QED).