This change in the conception of reality is the most profound and the most fruitful that physics has experienced since the time of Newton.

— Albert Einstein, at Maxwell's centenary (1931)

Maxwell died at 48, little known to the general public. In his lifetime, hardly anyone outside a small group of British physicists accepted his field theory. Today anyone using a mobile phone depends on it dozens of times a second.

Waves that carry information

Maxwell’s equations predicted electromagnetic waves of any frequency. After Hertz, in 1887, came Marconi’s wireless telegraphy, radio, radar, television, satellites, mobile phones, wi-fi, GPS and the microwave oven. The engineers who design an antenna, a chip or an optical fibre are solving, in one form or another, Maxwell’s equations.

A speed that depends on nobody

The equations give the speed of light, $c=\frac1{\sqrt{\mu_0\varepsilon_0}}$, without saying relative to what. For Newton’s physics this was a problem: speeds always depend on who measures them. Physicists at the end of the nineteenth century assumed a motionless ether, but the Michelson–Morley experiment of 1887 found none. In 1905 Einstein settled the question in the most radical way: the speed of light is the same for everyone, and what changes is time and space. Relativity was born from taking Maxwell’s equations seriously.

The physics of probabilities

The 1860 distribution of speeds was the beginning of statistical mechanics. The Austrian Ludwig Boltzmann generalised it and showed that entropy, the quantity that always increases, measures the number of ways of arranging the molecules. Maxwell’s demon provoked a century of debate and ended up linking thermodynamics to information theory and computing.

The first paper on control theory

In 1868 Maxwell published a short paper, “On governors”, about the centrifugal regulators that controlled the speed of steam engines. He showed how to decide, by a calculation with the roots of a polynomial, whether the governor would stabilise the engine or make it oscillate more and more. It is considered the first mathematical work on control theory, the discipline behind autopilots, thermostats and robots.

Vectors and the language of physics

The argument with Tait over quaternions, the words curl and convergence, and the rewriting by Heaviside and Gibbs made vectors the language of physics and engineering. The dot product, the cross product, divergence and curl taught in calculus courses were born to write down Maxwell’s equations.

What remained

  • Electromagnetism. The four equations, the displacement current and light as an electromagnetic wave.
  • Statistical physics. The distribution of speeds, the viscosity of gases and Maxwell’s demon.
  • Optics and colour. The three-colour theory, Maxwell’s triangle and the first colour photograph.
  • Astronomy. The structure of Saturn’s rings.
  • Engineering. Control theory and structural analysis: the Maxwell–Cremona diagram is still used to calculate trusses.
  • The laboratory. The Cavendish Laboratory, where the electron, the neutron and the structure of DNA were discovered.

To remember

  • James Clerk Maxwell lived from 1831 to 1879: Edinburgh, Cambridge, Aberdeen, London and Cambridge again.
  • Three numbers are enough to describe a perceived colour; with three photographs, filtered in red, green and blue, Maxwell made the first colour photo.
  • Saturn’s rings cannot be solid: the inner parts must move faster than the outer ones, $v=\sqrt{GM/r}$.
  • The speeds of the molecules of a gas follow the Maxwell distribution; for air at 27 °C, the peak is near 420 m/s.
  • Four equations describe all electric and magnetic phenomena. The displacement current term guarantees the conservation of charge.
  • Out of the equations comes a wave with speed $\frac1{\sqrt{\mu_0\varepsilon_0}}\approx300{,}000$ km/s: light.

Sources: biography of Maxwell, MacTutor · Basil Mahon, The Man Who Changed Everything (Wiley, 2003) · James Clerk Maxwell, “On governors”, Proceedings of the Royal Society 16 (1868).

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