On December 8, 1864, 33-year-old Scottish physicist James Clerk Maxwell (1831–1879) read his crowning masterpiece to the Royal Society of London, published in Philosophical Transactions of the Royal Society (Vol. 155, 1865) under the title “A Dynamical Theory of the Electromagnetic Field”.

Maxwell presented his set of 20 Electromagnetic Field Equations, unifying electricity, magnetism, and optics into a single comprehensive physical theory.

The Displacement Current & Electromagnetic Waves

Maxwell introduced the crucial concept of Displacement Current ($\mathbf{J}_D = \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}$), modifying Ampère’s Law to satisfy electric charge conservation:

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

Derivation of the Electromagnetic Wave Equation

By taking the curl of Faraday’s Law ($\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$), Maxwell derived 3D wave equations for electric and magnetic fields in vacuum:

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

Speed of Light Equality

The propagation velocity $c$ of these transverse electromagnetic waves is given by:

$$c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} \approx 2.998 \times 10^8\,\text{m/s}$$

Maxwell famously concluded:

“The agreement of the results seems to show that light and magnetism are affections of the same substance, and that light is an electromagnetic wave propagated through the field according to electromagnetic laws.”

Foundation for Quantum Electrodynamics (QED) & Relativity

Maxwell’s equations represent one of the greatest unifications in scientific history, inspiring Albert Einstein’s Special Relativity (1905) and serving as the classical limit of Quantum Electrodynamics (QED).

Key Takeaways

  • Year: 1864
  • Key Figure: James Clerk Maxwell (Scottish Physicist)
  • Core Discovery: Published “A Dynamical Theory of the Electromagnetic Field”, introducing displacement current and deriving the electromagnetic wave equation.
  • Great Unification: Unified electricity, magnetism, and optics ($c = 1/\sqrt{\mu_0\epsilon_0}$).
  • Quantum Relevance: Forms the classical foundation for Quantum Electrodynamics (QED) and photon gauge field theory ($A_\mu$).