Maxwell's equations and electromagnetic waves

In the 1860s James Clerk Maxwell combined all known laws of electricity and magnetism into four equations and drew an astonishing conclusion from them: there must be electromagnetic waves that travel through empty space at the speed of light. So light itself is an electromagnetic wave. In the late 1880s Heinrich Hertz produced such waves in the laboratory. Radio, Wi-Fi, mobile networks and X-rays are all solutions of Maxwell's equations.

Flux and circulation

Maxwell's equations are written with two kinds of integral. The flux through a closed surface S, ∮ E · dA, is the net number of field lines “leaving” the surface (dA points along the outward normal). The circulation around a closed loop C, ∮ E · dl, measures how much the field “swirls” along the loop; for the E field it is the work done on a unit charge carried around the loop.

The four equations

With the Stokes and divergence (Gauss–Ostrogradsky) theorems the equations can be put in differential form: ∇ · E = ρ/ε₀, ∇ · B = 0, ∇ × E = −∂B/∂t, ∇ × B = μ₀j + μ₀ε₀ ∂E/∂t. The integral form is handy for symmetric problems, the differential form for waves and numerical computation.

Using symmetry: Gauss and Ampère

The displacement current

Picture a charging capacitor and a loop around the wire. If the surface bounded by the loop cuts the wire, Ampère's law gives μ₀I; if a surface bounded by the same loop passes between the plates, no charge flows through it and the result is zero — a contradiction! Maxwell showed that the electric flux between the plates is changing and that the term ε₀ · dΦ_E/dt equals exactly I. This “displacement current” is not a flow of charge, yet it produces a magnetic field just like a conduction current.

Electromagnetic waves

In empty space (ρ = 0, j = 0) consider a plane wave travelling along x: E = E_y(x, t), B = B_z(x, t). Faraday's law on a small rectangle in the xy-plane gives ∂E_y/∂x = −∂B_z/∂t; the Ampère–Maxwell law on a rectangle in the xz-plane gives −∂B_z/∂x = μ₀ε₀ ∂E_y/∂t. Differentiating the first with respect to x and the second with respect to t and eliminating B yields the wave equation.