Maxwell’s first scientific paper appeared when he was just 14, which suggests that he was a terrifying mathematical prodigy. In fact, Maxwell was a very clever boy but by no means exclusively scientific. Indeed, a poem of his was published in the Edinburgh Courant six months before his first scientific paper. He wrote the latter after meeting the decorative artist D R Hay, who was searching for a way to draw ovals. The 14-year-old Maxwell generalized the definition of an ellipse and succeeded in producing true ovals identical to those studied in the 17th century be René Descartes. Maxwell’s father showed the method to James David Forbes, an experimental physicist at Edinburgh University, who realized that it was correct. Forbes then presented the paper on Maxwell’s behalf at a meeting of the Royal Society of Edinburgh – a remarkable achievement for someone so young.

Student days

Maxwell began his studies at Edinburgh University in 1847 at the age of 16. He moved to Cambridge in 1850 to take the mathematical Tripos, which lasted for three years and a term. This unusually long undergraduate career, which resulted from the different ages at which students in England and Scotland then went to university, proved entirely beneficial for Maxwell. At Edinburgh he gained a broad education centred on philosophy, while Cambridge gave him an excellent training in applied mathematics and the most gruelling examination system the wit of man has devised. At both, he encountered first-class minds.

The first grand unification

On 5 January 1865, while at King’s, Maxwell ended a letter to his cousin Charles Cay about his latest scientific work with the casual remark, “I have also a paper afloat containing an electromagnetic theory of light, which, till I am convinced to the contrary, I hold to be great guns.” The judgment was correct. More than a new theory, this was a new kind of theory that entailed completely new views of scientific explanation, unifying as it did three different realms of physics – electricity, magnetism and light. This unification of nature’s basic forces is a goal that physicists are still working on today.

Before Maxwell there had been huge progress in optics and electromagnetism but troubling questions remained in both fields. The wave theory of light, originated by Thomas Young and Augustin Fresnel, was in one sense a marvellous success, leading to a flood of new discoveries. But in another way it was a worrying failure. At least 11 alternative theories existed, each of which tried to explain Fresnel’s and other formulae in terms of an underlying ether, but, as Stokes proved devastatingly in 1862, every one of them was flawed. Part of the miracle of Maxwell’s theory was that it almost magically swept the troubles with those theories away.

A different issue hampered electromagnetism, which had been discovered by the Danish physicist Hans Christian Oersted in 1820. Oersted had found that a compass needle brought near a current-carrying wire pointed at right angles to the direction of the current, which involved a twisting motion that could not be explained by any other force. Two explanations emerged. Ampère sought to reinterpret the twisting as an attraction of a more complex kind, while Faraday, who had shown that magnetism, the electric current and the resultant force on a body act perpendicularly to each other, took Oersted’s finding as an irreducible new fact.

Faraday saw the “lines of force”, which are revealed by sprinkling iron filings on a sheet of paper held over a magnet, not only as geometrical lines but also, more daringly, as physical lines rather like stretched elastic bands with an extra sideways repulsion. For him, these physical stresses could be used to explain magnetic force. Maxwell developed both aspects of Faraday’s thinking, devising in his second paper in 1861 an “ether” full of tiny “molecular vortices” aligned with the lines of force. Like tiny spinning Earths, Maxwell reasoned, each vortex shrinks axially and expands sideways, giving just the stress patterns that Faraday had hypothesized (see image “mechanical model”). To explain how the vortices rotate, Maxwell envisioned smaller “gearwheel particles” meshing with the vortices.

While emphasizing that this idea, especially the gearwheel particles, was speculative and not a real physical model, he nevertheless saw it as a useful way to understand electromagnetism. In a wire, the particles are free to flow and form an electric current. In space, they serve as counter-rotating idle wheels between vortices to make successive ones turn in the same direction. This machinery gave the right result; Maxwell had “explained” magnetic force in Faraday-like terms.

Maxwell addressed the electric force – the crux of his discussion – after submitting two papers on the magnetic force for publication. The key issue was where the energy resides. Previous theories had assumed that the energy was located at or on magnets or electrically charged bodies. In Maxwell’s theory, however, the magnetic energy was in the surrounding space, or “field”, as he called it. The energy was, in other words, the kinetic energy of the vortices.

Drawing on insights from William Thomson (the future Lord Kelvin), Maxwell proceeded to make his ether elastic, with the electric force being the result of the potential energy needed to distort the ether. Intrigued by the fact that an elastic ether ought to transmit waves, Maxwell decided to calculate the speed at which they would move in terms of electric and magnetic forces, doing the calculations while at Glenlair.

On returning to London, he looked up the ratio for magnetic to electric forces, which had been determined experimentally in 1858 by the German physicist Wilhelm Weber. Weber had measured the ratio because it played an important, but not well understood, part in his own theory of electromagnetism. A velocity appeared in his theory also, but with a different numerical value that had no obvious physical meaning. Maxwell plugged Weber’s force ratio into his equations and discovered to his utter astonishment that the velocity exactly equalled the speed of light, which was then known experimentally to an accuracy of 1%. With excitement manifest in italics, he wrote, “We can scarcely avoid the inference that light consists in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena.”

Having made this epoch-making discovery, Maxwell moved from his visionary model to hard fact. In a paper that has a good claim to be the foundation of dimensional analysis, in 1863 he proved that the ratio of the magnetic and electric forces indeed contains a velocity that equals the speed of light, c. The importance of this result to physics is hard to overstate. Before Maxwell, c was just one velocity among many. Now it was privileged, pointing the way forward to Einstein and relativity.

Maxwell’s vortex-ether began as an attempt at a mechanical explanation of Faraday’s magnetic stresses. Another person might have been tempted to improve and refine it. Maxwell saw that no such effort was necessary. He had by now assembled a series of equations relating electric and magnetic quantities; he could deduce wave propagation from them. Instead of explaining electromagnetism or light, he had connected these two apparently different classes of phenomena using equations that took two forms. The first, which appeared in his 1865 paper and again in his Treatise, consisted of eight groups of equations. The second, in 1868, contains the four equations that we now know as “Maxwell’s equations”. The differences are somewhat technical: the eight equations include the concept of a “vector potential” and the incorrectly named “Lorentz force law”. (Devotees of Ockham’s razor should notice a remark by Maxwell in his Treatise that “to eliminate a quantity which expresses a useful idea would be a loss rather than a gain in this stage of our inquiry”.)

Maxwell’s theory predicted many new phenomena, such as radiation pressure. But its most remarkable consequence – as Maxwell at once realized – was that it pointed to the existence of an electromagnetic spectrum. This “great storehouse of nature” might contain other radiation of higher and lower frequencies, a thought that was vindicated over the next 30 years with the discovery of radio waves, X-rays and gamma radiation. As for relativity, Maxwell introduced Hamilton’s word, in the way that physicists now understand it, in his small book Matter and Motion of 1877. Poincaré read the work; Einstein learned of it from Poincaré; and the rest is history.

Maxwell’s legacy

When Einstein visited Cambridge in the 1920s, someone remarked, “You have done great things but you stand on Newton’s shoulders.” His reply was, “No, I stand on Maxwell’s shoulders.”

He was correct, but much else in modern physics also rests on Maxwell. It was after all Maxwell who introduced the methods that underlie not only Maxwell–Boltzmann statistics but the quantum-mechanical Fermi–Dirac and Bose–Einstein statistics governing photons and electrons. It was even he, in two innocent-seeming discussions in the 1870s, who first emphasized what we now call the “butterfly effect” – the fact that tiny differences in initial conditions can produce huge final effects, the starting point of chaos theory. In a similar vein, Maxwell’s scientific contributions have had dramatic effects on the future course of physics, notably the quest to unify nature’s fundamental forces. Sadly Maxwell died of cancer on 5 November 1879 and never lived to see the applications of radio or the demystifying of equipartition. But the power of his scientific insights lives on.