Faraday, Maxwell, and the Electromagnetic Field

by Nancy Forbes

  Having read Faraday, then Ampère, he turned indefatigably to the other authors and sent a progress report to Lewis Campbell:

  I am working away at Electricity again and have been working my way into the views of the heavy German writers. It takes a long time to reduce to order all the notions one gets from those men but I hope to see my way through the subject and arrive at something intelligible in the way of a theory.

  He did arrive at a theory—though the task took three stages spread over nine years—and it was based on Faraday's concept of lines of force in space. He had already seen that mathematical writers were quite wrong to dismiss lines of force as an idle speculation or flight of fancy—the concept had evolved over years of solid experimentation and painstaking thought. The immediate task became clear—to find a way of expressing Faraday's ideas in mathematical language. By doing so, he hoped to demonstrate their equivalence to other theories, to confound Faraday's critics, and, with luck, to establish a base on which to build a fuller theory.

  Thomson had opened the door a crack with his analogy between electric lines of force and heat flow. Maxwell now looked for a more general analogy, and he found one: the steady flow of an imaginary fluid through a porous medium. Amazingly, by this simple means, he was able to model all the known properties of static electric and magnetic fields, and to show that the relevant formulas could be derived equally well from assumption of action at a distance or from Faraday's lines of force.

  Maxwell's fondness and flair for analogies were evident from the essay “Are There Any Real Analogies in Nature?” which he had written a few years earlier for the Apostles, and stemmed from his study of philosophy at Edinburgh. He remembered especially how Hamilton had stressed Immanuel Kant's proposition that all human knowledge is of relations rather than of things. Other physicists, less versed in philosophy, thought his use of analogies idiosyncratic, and few really understood what he was getting at. Some took him literally and couldn't see what the mechanics of fluid flow had to do with electricity and magnetism. But they should have heeded his clear warning to readers not to take his model to represent any kind of physical reality. It offered, he was at pains to stress, no physical theory of electricity or magnetism. He was simply attempting to “shew how, by a strict application of the ideas and methods of Faraday, the connection of the very different order of phenomena which he has discovered may be clearly placed before the mathematical mind.”6 The fluid itself was merely an aid to thought—“a collection of imaginary properties”—its purpose was to enable one to find the appropriate mathematical relationships without being committed to any particular physical theory.7

  Maxwell's imaginary fluid was weightless, friction-free, and incompressible. This last property was the key to the analogy. It meant that the fluid had its own built-in inverse-square law: the speed of a particle of fluid flowing directly outward from a point source was inversely proportional to the square of its distance from the source. This, as he explained, was a just a matter of geometry. The amount of fluid that emerges per second from any sphere centered on the point source is the same, whatever the size of the sphere. So, as the sphere's surface area (4πr²) is proportional to the square of its radius, the fluid must move outward at a speed inversely proportional to the square of its distance from the source. If the source is replaced by a sink, the same applies in reverse—this time the velocity is inward. And Maxwell showed that if there were any number of sources and sinks of any shape in any configuration, the speed and direction of fluid flow at any point could, in principle, be calculated by summing mathematically all the flows from each point on each source and to each point on each sink.

  Electric and magnetic forces were known to follow a similar law—the force between two electric charges or two magnetic poles was inversely proportional to the square of their distance apart—so the basis of the analogy was set. The direction and speed of flow of fluid at any point represented the direction and strength of either the electric force or the magnetic force; the faster the flow, the stronger the force. It was a strange analogy—moving fluid representing static force—but it served Maxwell's purpose. And the beauty of it all was that the streamlines of fluid flow represented Faraday's electric or magnetic lines of force.

  Faraday had thought of the lines as discrete—he always talked of the number of lines—but Maxwell merged them into a continuous entity called flux. Electric or magnetic flux was the total amount of force that acted through any given cross section—one might think of it as akin, for example, to the amount of sunlight striking a particular patch of ground. In any small region in space, the flux had both a direction and a concentration, or density. A high density of flux corresponded to a high concentration of Faraday's discrete lines—the higher the flux density in that part of space, the stronger was the electric or magnetic force there. In Maxwell's analogy, the direction of his fluid flow in any part of space corresponded to direction of the electric or magnetic flux there, and the speed of flow corresponded to the flux density. To track the fluid's motion, Maxwell constructed imaginary tubes for it to flow along. The tubes behaved as though they had real walls because the lines of flow never crossed one another, and the whole system of tubes fitted together, leaving no gaps. The fluid traveled fast where the tubes were narrow and slower where they broadened out. Electric and magnetic flux were similarly contained in tubes; and, by analogy, forces were strong where the tubes were narrow and the flux dense, and weaker where the tubes were wide and the flux sparse.

  The amount of fluid that flowed per second past any cross section of a tube was the same wherever the cross section was positioned. This rate of fluid flow corresponded to the amount of flux that acted through any cross section of the tube, and this quantity was also the same no matter where the cross section was placed. Maxwell defined a unit tube of flow as one that passed a single unit of volume of fluid per second, and, by analogy, a unit tube of flux as one that held a single unit of flux. A unit tube of flow was one that passed one milliliter of fluid per second, and the corresponding unit tube of flux held one unit of flux across any cross section throughout its length. Any quantity of flux could now be described as the relevant number of unit tubes—the calibration could be as fine as need be by making the unit of flux suitably small. Mathematical physicists could now interpret Faraday's “vague and varying” lines of force (to use Sir George Airy's description) as Maxwell's mathematically impeccable unit tubes of flux.8 Perhaps to emphasize the point, Maxwell used the term “unit line of force” as an alternative term for a “unit tube of flux.”

  What made Maxwell's fluid move was pressure difference. Along each tube, fluid flowed from a relatively high-pressure source to a lower-pressure sink, the pressure falling as the fluid passed along the tube. In the electrical analogy, a source was a positively charged body at a relatively high electrical potential and a sink was a negatively charged body at a lower potential. Pressure difference in the fluid model represented the difference in electrical potential, which we now call voltage, and the rate of fluid flow along one of Maxwell's tubes represented electric flux. In any small region, the rate of fluid flow was proportional to the pressure gradient (the fall in pressure per unit distance) there, and, similarly, the concentration, or density, of electric flux9 was proportional to the potential gradient (the fall in potential per unit of distance). Maxwell called this potential gradient the intensity, or simply the force, of the electric field.

  In Maxwell's fluid model of the static electric field, substances like metals, in which electric currents could flow freely, took no part except that their surfaces could act as sources or sinks. Electric lines of force occurred in insulators—substances in which currents did not flow. As Faraday had found, these substances varied in their ability to conduct electric lines of force—each had its own specific inductive capacity. For example, glass conducted electric lines of force more readily than wood. In his model, Maxwell accommodated this property simply by endowing each substance with the appro
priate amount of resistance to fluid flow—the lower the resistance, the smaller the pressure gradient necessary to produce a given speed of flow. By analogy, the greater the inductive capacity of the substance, the lower the potential gradient necessary to produce a given density of flux. A simple equation summed it up: The electric flux density at any point was equal to the electrical potential gradient there multiplied by the electrical inductive capacity of the substance. The analogy for static electric fields was complete.

  Maxwell had done something remarkable. By using fluid pressure as an analogy for electrical potential, he had connected Faraday's concept of lines of electric force, regarded by most mathematical physicists as “vague and varying,” with the abstract and precise concept of potential, which had come from mathematical astronomy. Pierre Simon Laplace had used the concept of gravitational potentials so successfully in his Mécanique céleste that it was natural for others to apply the same technique to electricity. Now Maxwell had given the mathematicians a direct route to Faraday's ideas, should they choose to take it.

  The corresponding analogy for magnetic fields was a little more complicated. Maxwell began by considering a special case—the field surrounding a permanent iron magnet, such as the familiar bar magnet. This could be modeled in exactly the same way as a static electric field: a source and a sink became the north and south poles at the ends of the magnet; the pressure gradient at a point became the intensity, or force, of the magnetic field there; the resistance of the medium, or rather its reciprocal, became the magnetic inductive capacity; and the speed and direction of flow became the magnetic flux density. The fluid flow represented Faraday's magnetic lines of force, alias tubes of flux, and its pattern was exactly that revealed by iron filings sprinkled on a piece of paper held over a magnet. So far, so good, but this model fell short of what was needed—it did not explain why Oersted's compass needle had twitched when placed near an electric current.

  The strange force that Oersted had discovered, acting at right angles to the current, was different from anything else encountered in nature. To tackle the problem, Maxwell drew inspiration from Faraday's longtime correspondent, Ampère. As we've seen, Ampère had found that a small loop of current acted like a magnet. Maxwell went further by showing that the magnetic effect of a large current loop, or circuit, was exactly the same as that of a magnetic shell. This shell was an imaginary surface bounded by the circuit, and the whole thing acted as a weird kind of magnet; the whole of one side of the surface was its north pole and the other side was its south pole, the strength of the poles being proportional to the current. As Maxwell explained, the shell worked because it could be thought of as many small loops of current, each acting as a magnet, put together in a mesh. In the mesh, all the internal currents canceled out because every part of every internal loop was shared with a neighboring loop and so carried equal and opposite currents that annulled one another. So the combined effect of all the small loops was exactly the same as that of the single large loop of current that ran around the edge of the mesh.

  Fig. 10.1. Equivalence of a magnetic shell and a current loop. (Used with permission from Lee Bartrop.)

  The magnetic effect of any current-carrying electric circuit could, therefore, be simulated by an imaginary magnetic shell, which would cause lines of force to circle from the north face of the shell around the circuit to the south face, forming the same pattern that iron filings take when sprinkled on paper over a magnet. They would “embrace” the circuit, to use Maxwell's word. Having completed a circuit, each line of force (or tube of flux) would join onto itself, forming a continuous loop.

  In Maxwell's fluid analogy, the magnetic shell became a kind of pump that drove the fluid from its “north” face around the surrounding space to its “south” face, each streamline of flow joining onto itself. But the magnetic shell didn't have a fixed shape. In fact, it could take on any shape; the only constraint was that it was bounded by the current-carrying circuit. You could construct the shell in a way that would include any point on any of the streamlines of fluid flow. This meant that the pumping action took place all the way around every line of flow; it caused the fluid pressure to fall continuously all the way around, and this was what made the fluid move.

  Here, the fluid analogy ran into a difficulty: you couldn't have continuously falling pressure all the way around a loop that joined onto itself. Maxwell didn't attempt any mechanical explanation, but he did explain what happened to the analogue of fluid pressure, the magnetic potential. It did, indeed, fall continuously as one went around the loop and was lower when one arrived back at the starting place than it had been before starting the journey. If one went round the loop the other way, against the direction of the analogous fluid flow, the potential was higher when one arrived back at the starting point than when one started, and if one repeated the process, the potential became higher and higher. So the potential at any point didn't have a single value; it depended on the number of times one had circled through the current loop. The difference in magnetic potential encountered in making one circuit around the current loop was the equal to the amount of mechanical work needed to move a unit magnetic pole around the loop. It represented the convertibility of electromagnetic energy into mechanical energy, and vice versa, that Faraday had discovered with his electric motor and generator.

  Bypassing the limitation in his analogy with fluid pressure, Maxwell used the fluid model in the same way that he had done in the electrical case to produce a simple formula: the magnetic flux density at any point was equal to the magnetic force there multiplied by the medium's magnetic inductive capacity. None of his findings from the fluid model were really new—they could all be derived from the assumption of action at a distance. But he had presented them in a wholly new light and shown that the known formulas of static electricity and magnetism could be explained equally well by Faraday's lines of force as by action at a distance.

  Maxwell extended the fluid analogy to model the flow of steady currents flowing through a resisting medium, but that was as far as it could take him. He had dealt with static electric and magnetic fields and steady currents, but the daunting task of working out the interactions of changing fields and currents lay ahead. It must have looked like a sheer cliff face, but there was more he could do even now. Faraday had conjectured that, even when things were static, a wire in a magnetic field was under a kind of strain that he called the electrotonic state. To Maxwell, this seemed a sound working hypothesis, and he signed off part 1 of his paper “On Faraday's Lines of Force” by signaling his intention to investigate the matter in part 2.

  I propose, in the following investigation to use symbols freely, and to take for granted the ordinary mathematical operations. By a careful study of the laws of elastic solids and of the motion of viscous fluids, I hope to discover a method of forming a mechanical conception of this electrotonic state adapted to general reasoning.

  In part 1 he had stuck to his original plan to represent Faraday's ideas using only word descriptions and simple equations. Now he needed some power tools and looked into what had already been done in the mathematics of vectors, quantities that had both magnitude and direction. The principal authors here were the great German mathematician Carl Friedrich Gauss, the Englishman George Green, and the Scot William Thomson. Making good use of their work, Maxwell derived a set of equations connecting electric and magnetic fields. They represented everything that was known at the time, though, as Maxwell was to discover later, a vital part of the linkage between electricity and magnetism was still missing. He also found a mathematical expression for the electrotonic state but was not yet able to assign it a physical role. All this was expressed in very complicated-looking mathematics.

  Though surpassed by his later writings, Maxwell's “On Faraday's Lines of Force”10 is, surely, one of the finest examples of creative thought in the history of science. In his book James Clerk Maxwell: Physicist and Natural Philosopher, Francis Everitt shrewdly characterizes Faraday as a
cumulative thinker, Thomson as an inspirational thinker, and Maxwell as an architectural thinker. Maxwell had not only found a way to express Faraday's ideas in mathematical language but also built a foundation for still-greater work yet to come.

  For now, he had done all he could. Maxwell presented his paper to the Cambridge Philosophical Society in two parts spanning the 1855 Christmas holiday and consigned his thoughts on electricity and magnetism to what he called “the department of the mind conducted independently of consciousness.”11 He believed strongly in the power of subconscious thought to generate insights and, as he often did, expressed the idea in a poem.

  There are powers and thoughts within us, that we know not till they rise

  Through the stream of conscious action, from where self in secret lies.

  But when will and sense are silent, by the thoughts that come and go

  We may trace the rocks and eddies in the hidden depths below.12

  Indeed, it would be six years before his next paper on electromagnetism appeared, and this set a pattern for the way he worked on his two great themes of electromagnetism and kinetic theory—he would write a paper during a few months of intense thought, then consign the topic back to the subconscious for several years (during which he wrote brilliant papers on other topics) before starting the next one. As we will see, much happened in the next six years, and what he produced then was a quite different set of analogies that represented changing fields as well as static ones and, in doing so, revealed what had been the missing part of the linkage between electricity and magnetism. They didn't know it, but his audience at the Cambridge Philosophical Society had witnessed the first of three stages in which Maxwell, inspired and guided by Faraday's ideas, created a theory of electromagnetism that would come to change our lives and to lay the foundations for twentieth-century developments in physics.