Faraday, Maxwell, and the Electromagnetic Field

by Nancy Forbes

  What of Maxwell's classes? For all his strong and progressive ideas on teaching, he was, sadly, not very good at it himself. Yet the students liked him. They were allowed to borrow only two books at a time from the college library, but Maxwell took out more for them, something professors were allowed to do for friends, and, when challenged, he replied that the students were his friends. He prepared his lessons carefully and would start well but then be drawn into what Lewis Campbell called “the spirit of indirectness and paradox that, though he was aware of the dangers, would often take possession of him against his will.” He would throw in illustrations and metaphors that were intended to help but left most of the class bewildered. To add to his troubles, he made many algebraic errors on the blackboard that took time to find and correct. Many students, nevertheless, remembered him with affection. One reports:

  But much more notable [than the other professors] was Clerk Maxwell, a rare scholar and scientist as the world came to know afterwards; a noble-souled Christian gentleman with a refined delicacy of character that bound his class to him with a devotion which his remarkably meagre qualities as a teacher could not undo.9

  And, to some, he was truly inspiring. David Gill, who became director of the Royal Observatory at the Cape of Good Hope, recalled:

  After the lectures, Clerk Maxwell used to remain in the lecture room for hours, with three or four of us who desired to ask questions or discuss any points suggested by himself or ourselves, and would show us models of apparatus he had contrived and was experimenting with at the time, such as his precessional top, color box, etc. These were hours of the purest delight to me.10

  Gill had less fond memories of Katherine. The delightful afternoon sessions sometimes ended, he said, when Maxwell's “awful wife” appeared and called him home to an early dinner. Perhaps the early dinners were on the days when Maxwell gave his evening classes at the Mechanics’ Institution. When talking to the working men, he seemed to be able to avoid the “spirit of indirectness and paradox,” and his classes were remembered long after he left Aberdeen. One farmer recalled how his friend had been made to stand on a mat while the professor “pumpit him fu’ o’ electricity” so that his hair stood on end.11

  It seems paradoxical that such a fine writer, who had strong and sound views on the principles of education, should have such travails in the classroom. As Campbell observed, Maxwell found it hard to bridge the gulf between his vast erudition and the students’ modest compass. His quicksilver mind was always making connections, allusions, analogies, and comparisons that were quite beyond most students, but after all the years of free conversation with his father, who always understood his meaning from the slightest nuance, he found it hard to suppress them. The problem didn't arise on formal occasions, when he was obliged to slow down and present the words as he would when writing. We can be sure that in teaching, as in everything else, Maxwell did his best.

  Marischal College was not the only university in Aberdeen; there was also King's College—this at a time when there were only three other universities in the whole of Scotland. Should they not unite, to gain economies of scale? Some important people thought so, and a royal commission had been set up to make a judgment on the matter. There was talk of “union”—common management of otherwise little-changed functions—but the commission decided in favor of “fusion,” a complete merger that would halve the number of professors. The new, combined, University of Aberdeen needed only one professor of natural philosophy, and the man they chose was Maxwell's opposite number at King's. One reason for this extraordinary decision was that to lose Maxwell was the cheaper option, as he had not served long enough to qualify for a pension. Another was that his rival was well dug-in and a polished negotiator, who was known as “Crafty” Thomson. As for Maxwell's research, very few people at the time had any idea of its importance, and none of them lived in Aberdeen.

  Just at this time, Maxwell heard that his old mentor James Forbes had accepted the principalship at St. Andrews University, leaving the chair of natural philosophy at Edinburgh vacant. This would be a wonderful job and, naturally, Maxwell applied. But the post was equally attractive to his old friend P. G. Tait, who wanted to return to Scotland from Belfast, and he applied, too. Once again they were rival candidates, and this time Tait was preferred. The electors’ choice was not as strange as it may seem to us—Tait was a first-rate physicist and a fine lecturer with a commanding presence. He was also the first person Maxwell turned to for mathematical advice, when needed. Twice spurned in his own country, Maxwell looked elsewhere and saw that King's College, London, wanted a professor of natural philosophy. He applied for the post and was selected.

  There was plenty to do meanwhile. Besides preparing his great paper on kinetic theory for publication, he wrote another, on elastic spheres, and sent a report on his color vision experiments to the Royal Society of London—work for which he was soon rewarded with the Society's Rumford Medal. At home there was estate business to see to and, as laird of Glenlair, he was expected to play a leading part in local affairs. Following his father's example, Maxwell took on this role wholeheartedly—for example, helping to raise funds for the endowment of a new church in the nearby village of Corsock. During the summer, he went to a horse fair and bought a handsome bay pony for Katherine. Soon after returning he became severely ill with a high fever. It was smallpox, almost certainly contracted at the fair, and it almost killed him. Maxwell was in no doubt that Katherine's devoted nursing had saved his life. He was laid up for several weeks, but strength and vitality came back little by little, and he was able to break in Charlie, the new pony, riding side-saddle with a carpet taking the place of a lady's riding habit. In October 1860, after an incident-packed year, he and Katherine made the long journey south to London.

  King's College, in the Strand, had been founded in 1829 as an Anglican alternative to the new nonsectarian London University, now University College, which had itself been founded as a secular alternative to the strictly Church of England universities of Oxford and Cambridge. Its educational mission was to prepare young people for life and work in a rapidly changing world. Unlike the traditional fare provided by Cambridge and Aberdeen, its courses were much like those at today's universities. King's not only gave classes in modern subjects like chemistry, physics, botany, and economics but also ran purpose-built courses in law, medicine, and engineering.

  At the age of twenty-nine, Maxwell delivered his second inaugural lecture. Experience had confirmed the soundness of the theme he had introduced at Aberdeen—his job was to help people think for themselves—and he developed it further:

  In this class, I hope you will learn not merely results, or formulae applicable to cases that may possibly occur in our practice afterwards, but the principles on which those formulae depend, and without which the formulae are mere mental rubbish.

  I know the tendency of the human mind is to do anything rather than think. But mental labor is not thought, and those who have with labor acquired the habit of application often find it much easier to get up a formula than to master a principle.1

  He finished the lecture with a message that seemed to be addressed to himself as much as to the students and that turned out to be extraordinarily prophetic:

  Last of all we have the Electrical and Magnetic sciences, which treat of certain phenomena of attraction, heat light and chemical action, depending on conditions of matter, of which we have as yet only a partial and provisional knowledge. An immense mass of facts has been collected and these have been reduced to order, and expressed as a number of experimental laws, but the form in which these laws are ultimately to appear as deduced from general principles is as yet uncertain. The present generation has no right to complain of the great discoveries already made, as if they left no room for improvement. They have only given science a wider boundary, and we have not only to reduce to order the regions already conquered but to keep up operations on a continually increasing scale.

  Within four years, h
e was to turn rhetoric into fact by opening up vast new regions of scientific knowledge.

  The Maxwells rented a house in the newly developed district of Kensington, close to the big open space of Hyde Park and Kensington Gardens, which was a fine place for strolling and for riding. James had a vigorous four-mile walk to work on fine days, with the alternative of a horse-drawn bus ride. The walk took him first through the park and then along Piccadilly, passing within a few yards of the Royal Institution in Albemarle Street. Faraday was, by now, retired and living at Hampton Court, but he still called in regularly at the Institution and, though there is no record of it, we can be fairly certain that he and Maxwell met there from time to time for a chat. It was probably at one such occasion that Faraday asked Maxwell to give one of the Institution's now-famous Friday Evening Discourses. Maxwell naturally accepted and chose to talk about color vision.

  It was a perfect occasion to demonstrate the three-color principle—the eye's three sets of receptors channel their separate signals to the brain, which then combines them to manufacture the color that you “see.” But Maxwell's color top was far too small for people in the back seats to see clearly and his color box could only be used by one person at a time. Something else was needed—what about a color photograph? The techniques of black-and-white photography were by now well-known, and one of his new colleagues, Thomas Sutton, was an expert. They devised a simple scheme. Take three ordinary photographs of the same object, one through a green filter, one through a red filter and one through a blue filter; then project them through the same filters on to a screen, superimposing the three beams of light to form a single image. The experiment worked beautifully; the audience at the Royal Institution sat spellbound as the image of a tartan ribbon appeared on the screen in glorious color. Maxwell had produced the world's first color photograph.2

  The time had come to give voice to the thoughts on electricity and magnetism that had been forming in “the department of the mind conducted independently of consciousness.” In his first paper, six years earlier, he had taken the flow of a hypothetical weightless fluid as an analogy and shown that the known formulas for static electric and magnetic fields did not depend on the orthodox assumption that forces resulted from material bodies acting on one another at a distance; they could be derived equally well from Faraday's idea of lines of force in space. As we've seen, Maxwell was struck early on by the integrity and power of Faraday's writings, and the years of subconscious “decocting” of ideas had served to convince him more and more that Faraday was right—fields of force truly existed in space.

  In his inaugural address, Maxwell had, in effect, set himself a manifesto: to produce a theory that explained all the known experimental laws of electricity and magnetism by deduction from general principles. These laws were, briefly:

  Like electric charges repel one another and unlike ones attract, both with a force inversely proportional to the square of the distance separating them.

  Like magnetic poles repel one another and unlike ones attract, both with a force inversely proportional to the square of the distance separating them. But poles exist only in north/south pairs and all magnetism, even in permanent iron magnets, probably results from electric currents. (Law 3 implies that any loop of current acts as a magnet with a north pole on one side of the loop and a south pole on the other.)

  A current in a wire creates a circular magnetic field around the wire, its direction depending on that of the current.

  A changing magnetic field, or flux, that passes through a conducting circuit generates an electric current in the circuit, its direction depending on whether the flux is increasing or decreasing.3

  No satisfactory complete theory existed, though Wilhelm Weber had made a brave and ingenious attempt. His highly mathematical theory, based on action at a distance, required the force between electric charges to depend not only on their distance apart but also on their relative velocity and acceleration. Much as he respected Weber's work, Maxwell's intuition bridled at these assumptions and at the whole concept of action at a distance. He felt sure that the true theory lay instead on the road indicated by Faraday and that his best chance of finding it was by a suitable analogy.

  He sought a mechanical analogy that could represent changing fields as well as static ones. A tall order, but his thoughts eventually led to a promising idea. There were reasons to suppose that something rotational might be going on in a magnetic field. For one thing, this could help explain why magnetic force acted in a circle around an electric current. For another, Faraday had shown that when polarized light passed through a strong magnetic field, its plane of polarization of light was rotated. The epitome of rotation was a vortex in a fluid, and a vortex in a fluid had a natural tendency to contract along its spin axis and expand sideways. One could imagine all space filled with a fluid in which vortices could exist, and, in a region of space, a system of adjacent vortices that were all rotating the same way with their axes parallel. There would be a tension along their lengths and each would exert a sideways pressure against its neighbors. This property was exactly analogous to Faraday's magnetic lines of force, which exerted a tension along their lengths and a repulsive pressure against each other.

  As his thoughts developed, Maxwell came to replace the fluid vortices in his model with solid, tiny, close-packed spherical cells that could spin. As it spun, each cell would tend to expand at its equator and flatten at its poles, so the combined effect of many cells spinning with their axes aligned would be exactly the same as the vortices—longitudinal tension and sideways pressure, again corresponding to the properties of Faraday's magnetic lines of force. For simplicity, we'll make the change from vortices to cells now, a little earlier than Maxwell did.

  Two more components were needed: something to set the cells spinning and something to prevent the edges of neighboring cells from rubbing awkwardly against each other. Maxwell solved both problems in a single stroke. To stop neighboring cells from rubbing against each other, he put even smaller particles between them to act like ball bearings or like the “idle wheels” that an engineer places between two gears that need to rotate in the same direction. Then came the inspiration: suppose these tiny particles were particles of electricity. In the presence of an electromotive force, they would move along channels between the cells, constituting an electric current, and it was this movement that set the cells spinning.

  The contraction of the spinning cells along their aligned axes of spin represented magnetic lines of force; the faster the cells spun, the greater the contraction and the stronger the force. And the north-to-south direction of the forces, by Maxwell's convention, was the way a right-handed screw would move if it rotated the same way as the cells. The expansion of the spinning cells around their “equators” represented the sideways repulsion between, magnetic lines of force. Maxwell was on his way, but he still needed to accommodate another of Faraday's discoveries: different substances had different magnetic characteristics. Some, like iron and nickel, had a high magnetic inductive capacity (they conducted magnetic lines of force very well), while others, like wood, had a lower inductive capacity even than a vacuum (they were the diamagnetic materials). Maxwell solved the problem with his customary sureness of touch. His imaginary cells were everywhere, coexisting with any ordinary matter that occupied the same space, and he simply made the density of each cell proportional to the inductive capacity of whatever ordinary substance was present in the same space—the denser the cell, the more readily the substance conducted magnetic lines of force. By the same token, the higher the density of the cells, the greater would be the forces of longitudinal contraction and sideways expansion for a given rate of spin. In Maxwell's analogy, these forces represented the concentration, or density, of magnetic flux. But if the cells were everywhere, why were they not apparent, and how could they coexist with ordinary matter? Maxwell wasn't deterred by such awkward questions. The mass density of the cells could be very low indeed, so low that they offered no significan
t obstruction to ordinary matter and hence were undetectable by any known instrument. As long as they had some mass and rotated fast enough they would contract along their axes of spin and so generate the necessary forces. And, in any case, it was only a conceptual model, an aid to thought.

  Density wasn't the only property of the cells to vary in this way. In a part of space occupied by an insulator, the cells, or perhaps local groups of cells, would hang onto their particles of electricity; but in a good conductor, like a copper wire, the particles could move freely. This “stickiness” represented the electrical resistance of the material—an ideal conductor would have none, an ideal insulator would be perfectly sticky, and real-world materials filled the range in between. The tiny particles of electricity had rolling contact with the cells; there was no sliding. In a uniform, unchanging, linear magnetic field, the particles wouldn't move bodily; they would just rotate, along with the cells. But if a row of particles moved without rotating, thus forming an electric current, they would set the cells with which they had contact spinning—exactly the conditions needed to create a circular magnetic field around a current-carrying wire, law 3 above. If the particles rotated and moved, the circular field due to their movement would be superimposed on the linear one due to their rotation. In law 2, the magnetic forces had already been explained, and the inverse-square rule was intrinsic to the model—although the mechanism was more complicated than that in Maxwell's fluid model of his Cambridge days, this was, once again, essentially a matter of geometry.4