Maxwell's Equations (1865): Light as an Electromagnetic Wave

In 1865 the Scottish physicist James Clerk Maxwell published “A Dynamical Theory of the Electromagnetic Field”, a paper that united electricity, magnetism and light into a single theory. Read to the Royal Society in London on 8 December 1864, it showed that electromagnetic disturbances travel as waves at the speed of light. Maxwell’s equations became one of the foundations of physics and of the modern electrical and wireless world.

Background: fields and forces

Early in the nineteenth century, electricity and magnetism were found to be linked. In 1820 Hans Christian Ørsted showed that an electric current deflects a compass needle, and in 1831 Michael Faraday discovered electromagnetic induction, the production of a current by a changing magnetic field.

Faraday, who had little mathematics, pictured space around magnets and currents as filled with “lines of force”. Most physicists preferred to think of forces acting directly at a distance. Maxwell, a gifted mathematician, set out to give Faraday’s intuitive picture a precise mathematical form.

Maxwell’s path to the theory

Maxwell was born in Edinburgh on 13 June 1831 and studied at Edinburgh and Cambridge. His first major paper on the subject, “On Faraday’s Lines of Force”, appeared in 1855–1856. While professor at King’s College London from 1860, he published “On Physical Lines of Force” (1861–1862), which used an elaborate mechanical model of spinning vortices and tiny particles to represent the field.

That model led to a striking result. Maxwell calculated that waves in such a medium would travel at a speed very close to the measured speed of light, and he concluded that light consisted of waves in the same medium responsible for electric and magnetic effects.

The 1865 paper

In “A Dynamical Theory of the Electromagnetic Field”, published in the Philosophical Transactions of the Royal Society in 1865, Maxwell set aside the mechanical model. He built the theory instead on general principles of dynamics and on experimentally established laws, expressed in twenty equations in twenty variables.

A key ingredient was the “displacement current”, a term Maxwell had introduced earlier, which allowed a changing electric field to act like a current and produce a magnetic field. With it, the equations showed that changing electric and magnetic fields could sustain each other and travel through empty space as a wave.

The predicted speed depended only on measured electrical and magnetic constants, and it came out at about 310,000 kilometres per second, close to the speed of light measured by Hippolyte Fizeau and Léon Foucault. Maxwell argued that light is an electromagnetic disturbance, and that other, invisible waves of the same kind should exist.

Maxwell’s approach was also a new way of doing physics. Instead of explaining forces by particles pushing and pulling at a distance, he described the field itself as a physical reality that stores energy and momentum and changes from point to point in space and time. Later physicists would build theories of gravity and of subatomic forces on the same idea.

Maxwell was a remarkably versatile scientist. In the same years he produced the first colour photograph, a tartan ribbon shown at the Royal Institution in 1861, and developed the kinetic theory of gases, explaining the behaviour of gases through the statistics of moving molecules.

Reception and confirmation

The theory was mathematically demanding and was not widely accepted at first. In 1865 Maxwell retired to Glenlair, his Scottish estate, where he wrote his great Treatise on Electricity and Magnetism (1873). In 1871 he became the first Cavendish Professor of Experimental Physics at Cambridge. He died there of abdominal cancer on 5 November 1879, aged 48.

In the 1880s Oliver Heaviside reformulated the theory in vector notation, producing the four compact equations now known as Maxwell’s equations. Between 1887 and 1888 Heinrich Hertz generated and detected electromagnetic radio waves in Karlsruhe, confirming Maxwell’s prediction.

Legacy

Maxwell’s equations underpin electrical power, radio, television, radar, mobile phones and much of modern technology. They also showed that light, radio waves, X-rays and other radiation are all parts of one electromagnetic spectrum.

The theory raised a deep question: the speed of light appears in the equations as a fixed constant. Resolving how that speed could be the same for all observers led Albert Einstein to special relativity in 1905. Einstein kept portraits of Faraday and Maxwell on his study wall, and physicists often rank Maxwell alongside Newton and Einstein.

Timeline

• 1831, 13 June: James Clerk Maxwell is born in Edinburgh.
• 1855–1856: Maxwell publishes “On Faraday’s Lines of Force”.
• 1861–1862: “On Physical Lines of Force” links light and electromagnetism.
• 1864, 8 December: “A Dynamical Theory of the Electromagnetic Field” is read to the Royal Society.
• 1865: The paper is published in the Philosophical Transactions.
• 1873: Maxwell publishes his Treatise on Electricity and Magnetism.
• 1879, 5 November: Maxwell dies in Cambridge.
• 1887–1888: Heinrich Hertz confirms electromagnetic waves.

Why it matters

Maxwell’s 1865 paper was the first complete field theory in physics. By showing that light is an electromagnetic wave and predicting waves beyond visible light, it opened the way to radio and modern communications and set the stage for twentieth-century physics.