Faraday, Maxwell, and the Electromagnetic Field Page 20
An obstacle was embedded deep in the scientific thinking of the time. Immersed in Newton's clockwork universe, people had thought that all physical phenomena resulted from some kind of mechanical action (combined, where appropriate, with forces such as gravity, which acted instantaneously at a distance) and that all would be clear to us if, and only if, we discovered the true mechanisms. Despite all his warnings, people couldn't understand that Maxwell's model didn't purport to represent nature's actual mechanism, but that it was merely a temporary aid to thought, a means of arriving at the relevant mathematical relationships by using an analogy. His analogy happened to use spinning cells, but that was by the way; it was the mathematical relationships that were important. Maxwell had conceded that the model was “somewhat awkward,” and to many of his contemporaries it was nothing more than an ingenious but flawed attempt to portray the true mechanism, for which the search would continue.
Probably not even Maxwell recognized the full measure of his achievement. Using only the familiar tools and materials of Newtonian mechanics, he had succeeded in building a bridge to unimagined new regions of scientific knowledge. The bridge was a bizarre and ungainly construction, but it served the purpose, and the wonder was that it was built at all; nobody but Maxwell had seen a need for it. What was it about Maxwell that set him apart from his contemporaries? Two characteristics stand out.
The first may seem paradoxical: he was in one sense a truer follower of Newton than most of his predecessors and contemporaries. As we've seen, the first mathematical laws of electricity and magnetism had been modeled on the law of gravity. According to Newton, the gravitational force between two masses was proportional to their product divided by the square of the distance between them. Simply replace masses with charges or pole strengths, and you had the basic laws of electricity and magnetism. But with the work of Coulomb, Ampère, Poisson, and others had come an assumption that the forces resulted from instantaneous action at a distance between the masses, poles, or charges. Newton himself had been careful not to make any such assumption—indeed, he had, as we've seen, described action at a distance as “so great an absurdity, that I believe no man, who has in philosophical matters a competent way of thinking, can ever fall into it.”12 But this warning had been forgotten, and throughout the early and middle 1800s, the only prominent physicists to challenge action at a distance openly were Faraday and Maxwell.
The second characteristic that set Maxwell apart from his peers is epitomized by his prediction of displacement currents in empty space, and consequently of electromagnetic waves. No hint of either had come from experimental results; nor were they prompted by logic. However long is spent on the search for an explanation, one is forced back to a single word—genius.
Faraday was by this time sinking into senility and was not up to reading Maxwell's paper, but if he had been able to give an opinion, it probably wouldn't have been wholly favorable. He had liked Maxwell's first paper, in which lines of force were represented, in analogy, by the smooth flow of a fluid, but this one possessed some features he wouldn't have liked at all. To Faraday, lines of electric and magnetic force were fundamental, freestanding, self-sufficient, but in Maxwell's model they had been, in a sense, demoted—they were merely the effect of the motion of his tiny cells and even smaller particles. And for all Maxwell's warnings that his model wasn't intended to portray nature's actual mechanism, it did seem to hypothesize that nature in some way operated like an assembly of atoms—objects that Faraday thought fanciful; he had said of them “why assume the existence of that of which we are ignorant, which we cannot conceive, and for which there is no philosophical necessity?”13 Although Faraday had been the inspiration for Maxwell's model, one suspects that he would not have recognized in it much of his own view of physical reality. In particular, he would have objected to its reliance on a medium—in his vision, lines of force transmitted their own vibrations without the need for any medium. One aspect of reality, however, was beyond Faraday's reach—mathematical relationships—and the whole purpose of Maxwell's construction-kit model had been to discover these. The aim had been achieved, and what Maxwell sought now was a way of freeing the theory from any arbitrary physical hypotheses.
For now, he dispatched thoughts on electromagnetism to “the department of the mind conducted independently of consciousness” and gave his attention to an urgently needed experiment on another topic. He had shown in his first paper on the kinetic theory of gases that the viscosity of a gas should be independent of pressure, but the prediction had yet to be tested. It was a make-or-break test for the kinetic theory. If the prediction turned out to be wrong, the theory would be demolished; but if the experimental results showed it to be true, the theory would be greatly boosted. Nothing at all similar had been done before, so the laboratory facilities at King's didn't serve the purpose, and he decided to try the experiment at home. Maxwell is generally regarded as a cerebral genius, as indeed he was. But he also enjoyed practical work, and countless hours in his improvised laboratory-cum-workshop at Glenlair had honed his skill. The gas viscosity problem posed a formidable challenge to any experimenter, but it needed to be tackled, and Maxwell rolled up his sleeves.
In his Kensington attic, with Katherine as assistant, he carried out one of the most spectacular home experiments in the history of science. A tripod taller than a man held a torsional pendulum enclosed in a huge glass case connected by a tube to a pump that was used to raise or lower the pressure of the air that, by virtue of its viscosity, damped the swings of the pendulum. First the pressure seals failed, then the glass case imploded with a great bang, but Maxwell persevered and eventually got a reliable set of readings that emphatically verified his prediction that viscosity was independent of pressure—a milestone in the development of the kinetic theory of gases.
There were many other calls on his time, and some were in the cause of technological progress. In the same way that Faraday had answered calls to work on nationally important projects like optical glass and lighthouses, Maxwell came to the aid of the telegraph industry. One problem in particular needed attention because it bedeviled the greatest technical enterprise of the day—the laying of a properly working telegraph cable under the Atlantic Ocean. The first Atlantic cable, laid in 1858, had failed after a few weeks, and subsequent examination of recovered sections had shown it to be of poor quality. William Thomson led a drive to bring in proper quality control in the manufacture and supply of cables, and the most pressing need was for a physical standard of electrical resistance so that cables supplied by the manufacturers could be properly tested against specification.
William Thomson had suggested an ingenious experiment for the purpose, and Maxwell led a team from the British Association for the Advancement of Science to carry it out at King's College. His colleagues were fellow Scots—Fleeming Jenkin, who had also attended Edinburgh Academy, and Balfour Stewart. The idea was to spin a copper-wire coil rapidly in Earth's magnetic field and thereby generate a current in the coil that would have its own magnetic field. This field would vary as the coil rotated but would act predominantly either to the east or to the west, depending on which way the coil was spun. A magnetic needle, delicately suspended at the center of the coil, would swing back and forth but eventually settle at a fixed angle to Earth's field. The wonder of Thomson's design was that the angle of the needle's deflection depended only on the resistance of the coil, along with known factors like the dimensions of the coil and its speed of rotation. Using the appropriate formula, the angle gave an absolute measure of the resistance of the coil, which could then be used to calibrate a conveniently transportable “standard” model resistance, one that could be easily reproduced. The copies could then be taken anywhere and used to measure the resistance of lengths of cable, or anything else.
Putting the elegant design into practice wasn't easy. For every reading, the coil had to be hand-cranked for about nine minutes, keeping the speed constant, and Jenkin made a special governing device fo
r the purpose. Many runs had to be aborted when there was a mechanical fault or when iron ships passing on the nearby river Thames distorted Earth's magnetic field, but months of patient work was finally rewarded: the world had its first standard of electrical resistance. And it soon had telegraph communication across the Atlantic Ocean. With William Thomson on the board of directors, the Atlantic Telegraph Company laid a sound cable in 1866, and more soon followed.
A characteristic of Maxwell's work, indeed his life, was that he seemed to take everything in his stride—he was never hurried. Somehow, he and Katherine managed to go riding in the park most afternoons and, of course, they went on accumulating data on color vision, asking all new houseguests to have a go. They had installed the latest big color box near the window in an upstairs room, and people across the road were alarmed at first to see them peering into what looked like a coffin. Maxwell also found time to keep up with the scientific journals and to pass any useful information on to his students. A case in point was William Rankine's analysis of forces in structures like steel-girder bridges, and here Maxwell brought in a dramatic improvement of his own. He introduced so-called reciprocal diagrams. Lines that converged to a point in the real structure became polygons in the new diagrams, and this made it easy to work out the forces graphically without the need for laborious arithmetical calculations—a boon for engineers. One can see how Professor Charles Coulson, one of Maxwell's successors at King's, was doing no more than expressing a general view when he said of Maxwell: “There is scarcely a single topic that he touched upon which he did not change beyond recognition.”14
After a time of inner reflection, as was his way, Maxwell once more turned his conscious thoughts to electromagnetism. The result was a definitive work that will always stand as one of the world's greatest scientific achievements. He called the paper “A Dynamical Theory of the Electromagnetic Field,” and, for once, Maxwell, the most modest of men, went so far as to toot his own horn. At the end of a long letter to his cousin Charles Hope Cay, he wrote:
I also have a paper afloat, containing an electromagnetic theory of light, which, till I am convinced to the contrary, I hold to be great guns.1
In the first part of the paper he introduced to the world to a concept that had been Faraday's alone but was now also his—the field:
The theory I propose may therefore be called a theory of the Electromagnetic Field because it has to do with the space in the neighbourhood of the electric or magnetic bodies and it may be called Dynamical Theory, because it assumes that in the space there is matter in motion by which the observed electromagnetic phenomena are produced.2
Even the most creative scientific theorists generally produce one great work on a subject and then move on. Maxwell was unique in the way he could return to a topic and raise it to new heights by taking a completely fresh approach. In his first paper on electricity and magnetism, he had used the analogy of an incompressible fluid to give mathematical expression to Faraday's concept of lines of force. In his second, he had built an entirely different imaginary model from spinning cells and idle wheels—a model that he admitted was “somewhat cumbersome”—but one that had yielded remarkable results. With it, he had not only accounted for all known electromagnetic effects but also had predicted two startling new ones: (1) displacement currents and (2) electromagnetic waves that traveled at the speed of light. Even the most enlightened of his contemporaries thought that the next step would be to refine this rather bizarre model, but, instead, Maxwell decided to put the model to one side and build the theory ab initio using only the principles of dynamics.
This was a fundamental shift in approach. He was no longer building imaginary models but rather trying to discern new scientific truth directly from the well-established mathematical relationships that were known as the laws of dynamics. These were the laws of motion that had been discovered by Newton, with one addition—the principle that energy was conserved in any closed system. The concept of energy in space was central to Maxwell's new approach in this paper, and he emphasized it by including the terms dynamical and field in the paper's title, distinguishing it from his previous paper “On Physical Lines of Force,” where the focus was on forces.
He opened the paper by again outlining the differences between his own methods and those of theorists who assumed that forces could act at a distance without the aid of an intervening medium. He granted that action at a distance might seem, at first sight, to be the most natural mode of explaining the phenomena of electricity and magnetism, and he expressed his heartfelt respect for the work of men such as Weber but made it clear that he was looking beyond what seemed natural at first sight into something deeper. Weber's theory, he said, posed “mechanical difficulties” that “prevent me from considering this theory as an ultimate one.”3 This, it seemed, was Maxwell's modest way of implying that the new theory he was about to present might be, in a colloquial sense at least, the “ultimate one.”
The mathematical laws of dynamics, everyone thought, belonged to material objects, and especially to machines with their levers, pulleys, gears, and springs. Maxwell's aim was to apply the same laws not to material bodies but to the space that both contained and surrounded them in electric or magnetic conditions—the field. He had already attempted this, with some success, in his spinning cells model, but only by constructing an imaginary machine that filled all space. The model had done its job by providing a kind of scaffolding that had given access to surprising new predictions—displacement currents and electromagnetic waves—but Maxwell now wanted to kick away the scaffolding and establish a theory that stood on its own, independent of any particular physical hypothesis.
As he saw it, something in space must be storing electromagnetic energy and transmitting its forces, and there was every reason to suppose that this medium, whatever its form, should obey the laws of dynamics, just as mechanical systems did. But how could he get a mathematical grip on this medium? The means came through his friends Thomson and Tait, who were in the early stages of collaborating on their great Treatise on Natural Philosophy, the first ever proper textbook on physics.4 Part of their preparation was to study the writings of the great French mathematicians, whose work had been largely overlooked in Britain. Prominent among these was Italian-born Joseph Louis Lagrange, who had developed a formalized way of analyzing the motion of whole mechanical systems. Every system, no matter how large or complex, had a fixed number of independent modes of movement, and Lagrange had derived differential equations that showed how each of these was related to the whole system's kinetic and potential energy. The equations could be neatly lined up, like soldiers on parade, and solved to determine the system's motion for any set of starting conditions.5
A remarkable feature of Lagrange's method was that it treated the system as a “black box.” Knowledge of the inputs and the system's general characteristics was enough to be able calculate the outputs; you didn't need to know the details of the internal mechanism. Characteristically, Maxwell found a compelling analogy to illustrate the point:
In an ordinary belfry, each bell has one rope which comes down through a hole in the floor to the bellringer's room. But suppose that each rope, instead of acting on one bell, contributes to the motion of many pieces of machinery, and that the motion of each piece is determined not by the motion of one rope alone, but by that of several, and suppose, further, that all this machinery is silent and utterly unknown to the men at the ropes, who can only see as far as the holes above them.6
Nature's detailed mechanism could remain hidden, like the machinery in the belfry. As long as it obeyed the laws of dynamics, he should be able to derive the laws of the electromagnetic field without the need for any kind of model.
Nature's hidden mechanism was embodied in the field—the seat of energy in space—and, in Maxwell's Lagrangian formulation, the field became a coherent, connected system. It was, however, a system quite unlike anything seen or thought of before. The field was not a phantom: it held real energy
that could be made to do mechanical work, and it exerted mechanical forces of electric and magnetic attraction and repulsion. Yet, for the most part, its components had an abstract quality. They were quantities that obeyed equations when given mathematical symbols but whose physical existence lay beyond anything we can detect with our senses.
Maxwell distinguished between two kinds of energy held by the field: electric energy was potential energy, like that in a coiled spring; and magnetic energy was kinetic, or “actual” energy, like that in a flywheel. To accommodate this energy, he assumed that all space, whether empty or occupied by material bodies, was packed with a medium that was capable of being set in motion and of transmitting that motion from one part of the field to every other part. To hold potential energy, the medium had a kind of electrical elasticity; and to hold kinetic energy, it possessed inertia and so acquired what he called “electromagnetic momentum” whenever it moved.
When acted on by an electric force, the elastic medium underwent distortion—Maxwell called it displacement—thus storing potential energy and exerting a spring-back force. The lines along which the distortion, or displacement, occurred were electric lines of force. (When Maxwell wrote in this paper of the number of lines of electric or magnetic force, he meant the number of unit lines of force, each being one unit of electric or magnetic flux.) The greater the distortion in any part of space, the greater was the density of the electric lines of force there, and the restoring force manifested itself in the world of ordinary matter as the tangible force of attraction or repulsion between electrically charged bodies. Any change in displacement constituted a brief electric current—the displacement current that he had described in his earlier paper. And the medium's momentum represented magnetic lines of force; the greater the momentum in any part of space, the greater the density of the lines there. Two more properties were needed. Where the medium occupied the same space as an ordinary substance—everywhere, that is, except in a vacuum—its elasticity and inertia were modified according to the ability of that substance to conduct, respectively, electric and magnetic lines of force.