commentary How Maxwell’s equations came to light Basil Mahon The nineteenth-century Scottish physicist James Clerk Maxwell made groundbreaking contributions to many areas of science including thermodynamics and colour vision. However, he is best known for his equations that unified electricity, magnetism and light. he famous American physicist a string of infinitesimal current segments, Richard Feynman predicted that each of which could be treated as a point T“from a long view of the history of source. To work out the total magnetic force mankind — seen from, say, ten thousand exerted by the wire he simply summed the years from now — there can be little effects from all segments mathematically. doubt that the most significant event of To Faraday, the idea that Oersted’s the nineteenth century will be judged compass needle had been propelled by a set as Maxwell’s discovery of the laws of of straight-line attractions and repulsions electrodynamics.” This may well turn out between it and the wire seemed wrong. to be correct, but Feynman knew that What happened, he felt, was that the Maxwell’s discovery of the laws of the current-carrying wire induced a circular electrodynamics didn’t happen all at once1. force in the space around itself, and he If pressed to name a single occasion, he devised a brilliantly simple experiment to would probably have chosen Maxwell’s test the idea. Faraday fixed a bar magnet presentation of his paper ‘A Dynamical in a vertical position in the middle of a Theory of the Electromagnetic Field’ to small basin and poured mercury into the the Royal Society on 27 October 18642. basin until only the top end of the magnet Maxwell’s radical new theory, formally showed above the surface. Then he dangled published the following year, was expansive a wire into the pool. The wire and the and was only later condensed into the © GL ARCHIVE/ALAMY mercury formed part of an electric circuit four now-famous equations by one of his and, when he passed a current through it, followers. Nevertheless, it is right that these James Clerk Maxwell was one of the most the end of the wire that was in contact with are called Maxwell’s equations, and it is influential figures of nineteenth-century physics. the mercury moved in rapid circles around right that we should now celebrate their the magnet. He had made the world’s first 150th anniversary. electric motor. Before 1820, scientists had believed handicap may have worked to his advantage Ampère had shown how to make electricity and magnetism to be distinct by freeing his thinking. He asked questions magnetism from electricity — surely it phenomena. Then Hans Christian Oersted that had not occurred to others, devised should be possible to make electricity from reported a remarkable result: when he experiments that no one else had thought of, magnetism. For a decade, scientists tried placed a magnetized compass needle near and saw possibilities that others had missed. and failed. Then, in 1831, Faraday found a current-carrying wire, the needle moved, One of his contemporaries, André why the goal had been so elusive: to make aligning itself at right angles to the wire. Marie Ampère, had latched on to Oersted’s a current flow in a wire you had to change Astounded scientists everywhere began discovery with amazing speed and, the magnetic state of the space around the to investigate the connection between within a few months, had produced a full wire. All you needed to do was to move electricity and magnetism. Among them was mathematical theory. Any loop of current, a magnet in the neighbourhood of an Michael Faraday. he said, creates a magnetic force that acts electric circuit (or vice versa) and a current The self-taught son of a London through the loop. Ampère’s theory, like that would flow in the circuit. But exactly what blacksmith, 29 year-old Faraday was of Coulomb before him, was modelled on was this magnetic state in space? Faraday working at the Royal Institution under Newton’s law of gravitation. According to remembered the familiar pattern that iron Humphry Davy, and building a reputation as Coulomb, electric and magnetic forces acted filings make when scattered on paper over a resourceful and reliable analytical chemist. instantaneously in straight lines between a magnet and became convinced that a As soon as he could free himself from point charges and magnetic poles, and magnet was not just a piece of iron with other duties, he began to experiment with were inversely proportional to the square interesting properties, it was the centre of electric currents and magnets. Knowing no of the distance between them. To extend a system of curved magnetic lines of force mathematics, he was ostensibly less well- the theory to electric circuits, Ampère that spread out through space and had a equipped than his more formally educated calculated the magnetic force exerted by physical presence there. And this didn’t contemporaries but, paradoxically, this a current-carrying wire by regarding it as happen just with iron magnets: similar 2 NATURE PHOTONICS | VOL 9 | JANUARY 2015 | www.nature.com/naturephotonics © 2014 Macmillan Publishers Limited. All rights reserved commentary magnetic lines of force spread out from way, he showed that all the known formulae every current-carrying electric circuit. for electric and magnetic forces in static Faraday went further. Experimenting conditions could be derived equally well with electrically charged objects, he from the conventional action-at-a-distance concluded that each was the source of theories or from Faraday’s lines of force. A electric lines of force, which also curved stupendous achievement but, at the time, through space. Unlike magnetic lines of Maxwell couldn’t think how to deal with force, which formed continuous loops (they changing lines of force. As was his way, he didn’t stop at the poles of an iron magnet got on with other work while ideas brewed but ran all the way through it), electric at the back of his mind. ones always ran from a positively charged Six years later he came up with a new object to a negatively charged one; so every model. He filled all space with imaginary positive charge was balanced by a negative tiny spherical cells that could rotate and charge somewhere else. He also observed were interspaced with even smaller particles that both magnetic and electric effects were that acted like ball-bearings. By giving the not instantaneous but took time to act; in his cells a small but finite mass and a degree of view, this was the time taken for the systems elasticity, Maxwell constructed a mechanical of lines of force to establish themselves. analogy for magnetic and electric lines Faraday’s thoughts were running along of force, and showed that any change in lines quite different from those of everyone one induced a change in the other. This © GL ARCHIVE/ALAMY else. The general scientific opinion was still extraordinary model yielded not only all that electric and magnetic forces resulted The experiments performed by the English the known formulae of electricity and from material bodies acting on one another scientist Michael Faraday (pictured) were magnetism, it predicted electromagnetic at a distance with the intervening space instrumental in leading Maxwell to his unified waves that travelled at a speed determined playing only a passive role. The Astronomer theory of electromagnetism. solely by the basic properties of electricity Royal, Sir George Biddell Airy, spoke for and magnetism. This speed turned out to many when he described Faraday’s lines be within 1.5 per cent of the experimentally of force as “vague and varying”. One can In an astonishing and short career (he measured speed of light. An astounding understand this view. Action-at-a-distance died aged 48) Maxwell made fundamental result, but the response of fellow-scientists gave exact formulae, whereas Faraday discoveries in every branch of physics that was muted. The goal in any branch of supplied none. While they respected he turned his hand to, but, like Faraday, his physics, they believed, was to identify Faraday as a superb experimenter, most greatest work was in the field of electricity nature’s true mechanism, and they regarded scientists thought him ill-equipped to and magnetism. Born into a distinguished Maxwell’s model as an ingenious but flawed theorize as he knew no mathematics. Scottish family, he went to a top school attempt to do this for electromagnetism Conscious of such views, Faraday was in Edinburgh before studying at both and light. Everyone expected that Maxwell’s circumspect when publishing his thoughts Edinburgh University and Cambridge next step would be to refine the model but, on lines of force. Only once, in 1846, did University. After taking his bachelor’s degree instead, he put the model on one side and set he venture into speculation. A colleague, at Cambridge, where he came second in the out to build the whole theory from scratch, Charles Wheatstone, was due to speak at the famous Mathematical Tripos exam, he began using only the laws of dynamics. Royal Institution about one of his inventions to read Faraday’s Experimental Researches The result, two years later, was the paper but took fright at the last minute. Faraday in Electricity3 and was at once struck by ‘A Dynamical Theory of the Electromagnetic decided to give the talk himself but ran out the great man’s openness in publishing his Field’. Here, the spinning cells were replaced of things to say on the advertised topic well failures as well as his successes, and his by an all-pervading medium that had inertia before the allotted hour was up.