Maxwell's Equations

By the 1860s, electricity, magnetism, and light had been studied for centuries — and looked like separate phenomena. Coulomb, Ampère, Faraday, and Gauss had each contributed pieces. Then, between 1861 and 1865, James Clerk Maxwell — a Scottish physicist with a gift for synthesis — wrote down a system of equations that unified all three. The equations said something nobody had anticipated: light is an electromagnetic wave, propagating at a speed Maxwell could compute from purely electrical and magnetic measurements — the number agreed almost exactly with the measured speed of light. Maxwell's equations are the first great unification in physics.

The four Maxwell equations, in modern differential form, relate the electric field 𝐄 and magnetic field 𝐁 to charge density ρ and current density 𝐉: Gauss's law (∇ · 𝐄 = ρ/ε₀) — field lines emerge from positive charges, end on negative ones; Gauss's law for magnetism (∇ · 𝐁 = 0) — no magnetic monopoles; Faraday's law (∇ × 𝐄 = −∂𝐁/∂t) — changing 𝐁 induces circulating 𝐄; Ampère-Maxwell law (∇ × 𝐁 = μ₀(𝐉 + ε₀ ∂𝐄/∂t)) — magnetic fields curl around currents and around regions of changing 𝐄. Maxwell's correction — the displacement current, the second term in the fourth equation — closed the system; without it the equations violate charge conservation in the time-dependent case. In empty space (no charges, no currents), the equations admit wave solutions: 𝐄 and 𝐁 oscillate together, perpendicular to each other and to propagation, at speed c = 1/√(μ₀ε₀). Maxwell calculated this number from electrical measurements alone and got 3 × 10⁸ m/s — the measured speed of light. Light is an electromagnetic wave. The full spectrum — radio, microwave, infrared, visible, ultraviolet, X-rays, gamma rays — is one phenomenon at different wavelengths. Maxwell's equations are linear: electromagnetic waves superpose without interfering at the level of the equations, the fact underlying classical optics, interference patterns, holography, and Fourier-based signal analysis. They are also Lorentz-invariant — the same form in every inertial frame — which Einstein noticed in 1905, leading directly to special relativity.