In the mid-nineteenth century electricity and magnetism looked like two phenomena that touched but stayed apart: Ørsted had shown that a current in a wire deflects a compass, and Faraday that a moving magnet induces a current in a wire. Faraday described these interactions intuitively, as invisible "lines of force" filling space; but he had no mathematics for them.

The Scottish physicist James Clerk Maxwell (1831–1879) turned that intuition into equations. After a series of papers, his 1865 "A Dynamical Theory of the Electromagnetic Field" united electricity and magnetism in a single concept of the "field". While arranging the equations, he was forced, for consistency, to add a new term to Ampère's law — the "displacement current". That addition produced an unexpected consequence: the field equations described a wave propagating through empty space.

Maxwell computed the speed of this wave from two measurable electric and magnetic constants in the equations. The number that came out was almost exactly the measured speed of light. The meaning was staggering: light itself was an electromagnetic wave — visible light, heat, and other as-yet-undiscovered waves were different frequencies of the same phenomenon. "We can scarcely avoid the inference," he wrote, "that light consists in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena."

Maxwell died at 48 without seeing his prediction confirmed. In 1887 Heinrich Hertz produced and measured these invisible waves in the laboratory; within a few decades radio, television, radar, and wireless communication were built upon them. The equations left another legacy: the speed of light emerging as independent of the observer would lead Einstein to special relativity in 1905. Einstein kept a portrait of Maxwell, not Newton, on his study wall.