Maxwell's Equations & the Electromagnetic Wave

Four laws that quietly imply a wave

James Clerk Maxwell did not set out to discover light. Between 1861 and 1865 he unified everything then known about electricity and magnetism into four coupled equations — Gauss's law for E, Gauss's law for B, Faraday's law of induction, and Ampere's law — and found that his own addition to Ampere's law, a term he called the displacement current, was needed to keep the equations mathematically consistent when charge is not conserved locally in a naive way. That single addition had a consequence nobody had asked for: a changing electric field, on its own, could now generate a magnetic field exactly the way a changing magnetic field generates an electric field, and the two effects could feed each other indefinitely, even in empty space with no charges or currents anywhere nearby.

From four equations to one wave equation

In a vacuum, with no charges or currents, Maxwell's equations reduce to a coupled pair that link the rate of change of E to the curl of B and vice versa. Taking the curl of one and substituting the other eliminates one field entirely, leaving a standard wave equation for the other:

∇²E = μ₀ε₀ ∂²E/∂t²         (and the identical equation for B)

Plane-wave solution travelling along z:
  E = E₀ cos(kz − ωt) x̂
  B = B₀ cos(kz − ωt) ŷ

wave speed  c = ω/k = 1/√(μ₀ε₀)  ≈  3.00 × 10⁸ m/s
amplitude ratio  |E| = c|B|  everywhere, at every instant

The speed that falls out, c = 1/√(μ₀ε₀), is built entirely from two constants that had already been measured independently in ordinary electrostatics and magnetostatics experiments — nothing about light was assumed anywhere in the derivation. When Maxwell evaluated the number and found it matched the experimentally measured speed of light to within the accuracy of the day, he wrote that the agreement "can scarcely be considered as a mere coincidence" and concluded that light itself must be an electromagnetic phenomenon.

Why E, B and the direction of travel are mutually perpendicular

The plane-wave solution above is not the only mathematically conceivable one — it is the one consistent with all four of Maxwell's equations simultaneously. Gauss's laws (zero divergence for E and B in vacuum) rule out any field component along the direction of travel, which is why an electromagnetic wave is purely transverse. Faraday's and Ampere's laws then lock the remaining two transverse directions together: a changing B along y forces a curling E along x, and that changing E in turn regenerates B along y, always 90 degrees apart from the direction the wave is heading. There is no configuration of E and B that satisfies all four equations and points E or B along the propagation axis.

Linear vs circular polarisation

The wave equation only fixes that E is transverse and oscillates in step with B — it says nothing about which transverse direction E has to point along, or whether that direction can change as the wave travels. In linear polarisation, E oscillates back and forth along a single fixed line, as in the simplest plane-wave solution above. Circular polarisation is produced by superposing two linearly polarised waves of equal amplitude along perpendicular axes, 90 degrees out of phase — the resulting E vector keeps constant magnitude but rotates steadily around the propagation axis, tracing a helix in space as the wave advances. Unequal amplitudes or a phase offset other than 90 degrees give the general case, elliptical polarisation, with linear and circular as its two special-case extremes.

Frequently asked questions

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