Electromagnetic wave equation

The electromagnetic wave equation is what falls out of Maxwell's equations in source-free vacuum (ρ = 0, J = 0) when you take the curl of Faraday's law, substitute Ampère–Maxwell to eliminate ∂B/∂t, and use the vector identity ∇×(∇×E) = ∇(∇·E) − ∇²E together with ∇·E = 0. The result is ∇²E = μ₀ε₀ ∂²E/∂t², identically for B, which is the classical wave equation with propagation speed c = 1/√(μ₀ε₀) ≈ 2.998 × 10⁸ m/s. The same equation holds component-by-component for each Cartesian component of E and B.

This derivation is Maxwell's 1865 triumph. The constants μ₀ and ε₀ were known from laboratory measurements of electrostatics and magnetostatics. Plugging in the measured values gave Maxwell a predicted wave speed that matched Fizeau's 1849 optical measurement of the speed of light to within experimental error. The identification was immediate: light is an electromagnetic wave. Every subsequent electromagnetic wave phenomenon — radio, microwaves, X-rays, gamma rays — is a solution of this same equation at a different frequency. In media with refractive index n, the wave equation is modified to give propagation speed v = c/n; in conductors, a dissipative term ∝ σ ∂E/∂t is added, producing exponential decay with skin depth δ.