Faraday, Maxwell, and the Electromagnetic Field

by Nancy Forbes

  The two systems of lines of force, represented by the medium's elasticity and momentum, were intimately linked by Maxwell's displacement current. Any change in the electric force was accompanied by elastic distortion in the medium. In the act of distortion there would be some movement in the medium, which implied momentum, and that represented magnetic force. So any change in electric force generated magnetic force. Moreover, the same thing happened in reverse: any change in magnetic force generated electric force. This two-way interaction was the final link in the connection between electricity and magnetism, and it was what gave rise to electromagnetic waves.

  Maxwell also took account of two fundamental results from experiment: Ampère's finding that any loop of electric current acts as a magnet, and Faraday's finding that an electromotive force is generated in a circuit whenever there is a change in the number of magnetic lines of force that pass through it.

  The medium connected with the world of ordinary matter at what Maxwell called “driving points,” where real mechanical forces were exerted and real mechanical work was done, as in an electric motor or a generator. Any conducting circuit, for example, could be a driving point, or a driven point, or both at the same time. Every circuit was, as it were, geared to the medium by the lines of magnetic force that passed through it. These were the field's electromagnetic momentum, and the number of them that that passed through a circuit, or linked with it, determined how that circuit was geared to the rest of the field. The gear ratio, as it were, depended on the circuit's size, shape, and position.

  Using his medium together with Lagrange's formulation of the laws of dynamics, Maxwell was able to calculate how every part of the field interacted with every other part.

  We can get a rough idea of the process by taking a variant of his bell-ringers analogy. Imagine a long row of people pedaling exercise bicycles. None of the bicycles has the usual flywheel, but they all drive chains that run through holes in a wall and connect to the same unseen machinery (and hence, through the machinery, to all the other chains). Each rider has a different feeling through his pedals: to some they feel heavy; to others they feel light. Each is experiencing a different portion of the machinery's inertia through a feeling of weight in the pedals, and each is feeling some effect, though perhaps a tiny one, from every other rider's pedaling, his pedals being driven partly by his own efforts and partly by everyone else's. If one rider were suddenly to pedal harder, the effect would be felt to some degree by every other rider, though only after a delay while the change is transmitted through the medium—a delay so short as to be imperceptible to the riders.

  Transferring the imagery back to Maxwell's theoretical reasoning, each rider's set of pedals becomes an electric circuit somewhere in space; the pedals’ rate of rotation represents both the amount of current flowing in the circuit and the number of magnetic lines of force passing through it; the invisible machinery is Maxwell's all-pervasive momentum-carrying medium; and the chain linkage is the magnetic coupling of the circuit to that medium. Our exercise-bicycle analogy is partial—it doesn't illustrate how electrical effects are transmitted by means of the medium's elasticity—but Maxwell brought everything together and showed how the electrical and magnetic effects combined. Amazingly, the medium's properties of linked inertia and elasticity were enough to enable him to write equations that determined both the state of the field at any instant at a point in space and the physical forces exerted on any conducting circuits or charged bodies.

  Maxwell's medium had electrical elasticity, and it also had the electromagnetic momentum that corresponded to magnetic lines of force. These properties were enough for him to calculate the speed at which disturbances were propagated through the medium, and he showed that this speed could be equivalently expressed as the ratio of the electromagnetic and electrostatic units of electric charge—the ratio that he had already identified, using his spinning cells model, with the speed of light. He had now shown, without the aid of any model, that the speed of light depends only on the elementary properties of electricity and magnetism and, moreover, that any electromagnetic wave, light included, consists of both an electric wave and a magnetic wave, always in phase, each vibrating at right angles to the direction of travel and at right angles to each other, as is illustrated in figure 12.2 in the previous chapter.

  By using the Lagrangian formulation to arrive at the relevant equations, Maxwell had not only dispensed with the need for a mechanical model, but he had gone further. In his “Dynamical Theory” paper were the seeds of a truly revolutionary idea: some of nature's workings in the physical world not only do not need a mechanical model, but they cannot be explained in a mechanical way. For example, a current-carrying circuit “held” energy. This energy was real; it could be used in an electric motor to do mechanical work, but where was the energy? Not in the wire, but in the field—distributed through the surrounding space. It was kinetic energy, yet there was no evidence of movement. The magnetic lines of force that passed through the circuit constituted electromagnetic momentum, and this played a similar role to the familiar mechanical momentum, which was a body's mass multiplied by its velocity. But in electromagnetism, the momentum was disembodied; it was distributed throughout space. One can see why nineteenth-century scientists found it so difficult to take in these radical ideas: they had all been trained to think more in terms of things like colliding billiard balls that could be touched and measured.

  Maxwell was doing nothing less than changing our concept of reality. He was the first to recognize that the foundations of the physical world are imperceptible to our senses. All we know about them—possibly all we can ever know—are their mathematical relationships to things we can feel and touch. We may never understand what they are; we have to be content to describe them in an abstract way, giving them symbols and writing them in equations. As Freeman Dyson has aptly observed, Maxwell was in this way setting a prototype for the great triumphs of twentieth-century physics. Just as no one can truly picture Maxwell's electromagnetic momentum, so no one can visualize an electron, even though it can be rigorously defined in mathematical terms.

  Maxwell had achieved the seemingly impossible—he had derived the theory of the electromagnetic field directly from the laws of dynamics. But, some may say, he did it only by postulating the existence of an all-pervading medium, or aether, a concept that Faraday refuted and that has since been discredited. The very idea of an aether seems preposterous to us today—how can a substance be so rarefied as to be imperceptible to the senses yet enough of an elastic solid, or in some versions an elastic fluid, to be able to transmit lateral vibrations at the speed of light? Though valid to a degree, such criticism largely misses the point.

  Maxwell's theory was founded on facts, on the laws of electricity and magnetism that had been established by Faraday and others in experiments, and on the laws of dynamics that had been similarly proved. From these he made predictions—displacement currents and electromagnetic waves—that were later found by experiment also to be facts. Whereas other nineteenth-century physicists like William Thomson, Oliver Lodge, and George Francis Fitzgerald firmly believed in a material aether and gave defined mechanisms to their own versions of it, Maxwell gave his medium only properties. Though Maxwell had no idea of it, these properties turned out to be a prelude to the revelation by Einstein, in his special theory of relativity, of the fundamental properties of space and time. Maxwell had left his spinning cells model well behind and taken the endeavor to a new level. The field, with its intricately linked quantities that varied in space and time and were represented by abstract symbols, was to become the foundation for the great discoveries of the twentieth century, including the current theory of particle physics that is known as the Standard Model.

  Maxwell wrote up his findings and published the paper in seven parts, including a twelve-page section on the electromagnetic theory of light.7 When Maxwell introduced it at a presentation to the Royal Society in October 1864, the audience was bewildered; t
hey simply didn't know what to make of it. A theory based on a bizarre model was bad enough, but one based on no model at all was beyond comprehension. One can sympathize with both Maxwell and his audience. It was a long and complex paper, difficult to summarize in a single talk and nearly impossible to assimilate quickly. Moreover, the mathematics was difficult. It described how the various quantities interacted with one another and how they varied in space and time. Most of the quantities, such as the electric and magnetic field intensities and flux densities, were represented by vectors that had both magnitude and a direction in three-dimensional space. Few people understood the mathematics of vectors, and what made things especially difficult for newcomers was that each vector equation came as a triple, one equation for each of the three dimensions. Maxwell's theory contained eight equations, but six of them were vector triples, so the total looked like twenty. One can see how it must have appeared impenetrable. The theory is usually presented today in the tight form of the four famous “Maxwell's equations,” but Maxwell himself never summarized the theory in quite that way. He preferred to keep a more expansive arrangement, remarking that his eight equations might readily be condensed but that “to eliminate a quantity which expresses a useful idea would be a loss rather than a gain at this stage of our enquiry.”8 As usual, he was right: he had established a bridgehead in completely strange territory and it was prudent to keep all options for further advancement open. We'll see in later chapters how Oliver Heaviside arrived at the four equations now used by almost everyone.

  There was, however, a further impediment, far more profound than a failure to follow the mathematics. William Thomson, a fine mathematician who had no difficulty absorbing the mathematics, expressed the view of many when he said that Maxwell had “lapsed into mysticism.”9 He and other fellows of the Royal Society were rooted in their Newtonian world, where every natural phenomenon had a mechanical explanation, and they failed utterly to see that Maxwell had opened the way to a new and different world. This was a historic moment. In his “Dynamical Theory” paper, Maxwell heralded one of the very rare events in science that the historian of science Thomas Kuhn has called a paradigm shift—a fundamental change in the set of shared beliefs and methods that guide scientists’ thought and work—but, like most such changes, this one didn't properly take hold until several decades later, when a new generation of scientists with young and open minds had succeeded the old guard. As we will see, this process had its conflicts and unexpected twists.

  The theory's construction had been an immense creative effort, sustained over a decade and inspired, from first to last, by the work of Michael Faraday. Thanks to Faraday's meticulous recording of his findings and thoughts in his Experimental Researches in Electricity, Maxwell had been able to see the world as Faraday did, and, by bringing together Faraday's vision with the power of Newtonian mathematics, to give us a new concept of physical reality, using the power of mathematics. But mathematics would not have been enough without Maxwell's own near-miraculous intuition; witness the displacement current, which gave the theory its wonderful completeness. The theory belongs to both Maxwell and Faraday.

  Maxwell had put the “great guns” on parade, but it would be some time before they sounded. No one, probably not even Maxwell himself, recognized the full significance of his achievement, but he was by now a prominent figure in the scientific world, recognized and respected for his work on color vision and the kinetic theory of gases, and for his sterling efforts on electrical standards for the British Association for the Advancement of Science. His reputation brought much-appreciated kudos to King's College, and they lightened his burden by appointing a lecturer to assist with college duties. As to his own performance in the classroom, there is little reliable evidence, but the likelihood is that some of the problems that had plagued him at Aberdeen remained. Perhaps he had improved a little, and perhaps, as in Aberdeen, there were a few students like David Gill, who “could catch a few of the sparks that flashed as he thought aloud” and found him “supreme as an inspiration.”10

  Even with the extra help, Maxwell was finding it difficult to fit everything in. New ideas on kinetic theory and the theory of heat pressed on his mind, and he needed time to work them out. The work on electromagnetism, too, was far from finished. He wanted to write a substantial book, one that would bring much-needed order to the subject, help newcomers, and establish a solid base for his own further thinking. Another wish was to give more time to the estate and other local affairs at home, and to put in hand the grand enlargement to the Glenlair house that his father had designed and planned. Maxwell had thrived on the variety of work during the five years in London—college duties, home experiments, research on electromagnetism, and work for the British Association. He had also enjoyed the easy contact with fellow scientists; it was a joy simply being able to walk to meetings at the Royal Society and the Royal Institution. But he was still a country boy at heart and decided to resign his chair so that he and Katherine could take up a settled life in Galloway. The decision was made easier by knowing that he could safely hand the professorship to W. Grylls Adams, his very able assistant. Adams was younger brother to John Couch Adams, who, by discovering Neptune, had inspired the eponymous Adams Prize that Maxwell had won in 1857. The younger brother went on to have an illustrious career himself, becoming Astronomer Royal at Sydney. Maxwell agreed to help by returning to London in the winter and giving his usual course of evening lectures to working men.

  In the spring of 1965, James and Katherine carefully packed the big color box once more for travel and left their Kensington house for Glenlair.

  When Maxwell asked Katherine to marry him, he had couched the request in a poem. It ran:

  Will you come along with me,

  In the fresh spring tide,

  My comforter to be

  Through the world so wide

  Will you come and learn the ways

  A student spends his days

  On the bonny bonny braes

  Of our ain burnside.1

  This may not have been his finest piece of verse, but it wasn't meant for public scrutiny. It was an invitation to Katherine to share the home that meant so much to him: the ground that his parents had worked so hard to turn from stony scrub to pleasant and productive farmland; the hills and woods he had roamed as a boy with a walking staff his father had given him, alert to every movement of every creature that lived there. The “bonny bonny braes of our ain burnside” were those of the River Urr, which rippled brightly through the estate of Glenlair.

  Carrying on in his father's tradition, Maxwell had steadily improved the estate. From the start, John Clerk Maxwell had intended to extend the modest house by building on a taller and grander section, but funds didn't match the ambition. Maxwell had spent many hours with his father discussing possibilities and making drawings. Now there was a chance of bringing the scheme to fruition. He went over the plans, changing and trimming where necessary to keep within what could be afforded, and he arranged for builders to start work in the following spring. Life was going well, but one day out riding Maxwell scraped his head on a tree. The cut seemed trivial but became infected, and he became dangerously ill. Once again, Katherine's devoted nursing saved his life. He was laid up for a month but then began to regain strength, and it wasn't long before they were riding again.

  Maxwell remained shy with strangers, yet he left a strong impression on the memory of all he met; witness this recollection from somebody who met him in 1866:

  A man of middle height, with frame strongly knit, and a certain spring and elasticity in his gait; dressed for comfortable ease rather than elegance; a face expressive at once of sagacity and good humour, but overlaid with a deep shade of thoughtfulness; features boldly but pleasingly marked; eyes dark and glowing; hair and beard perfectly black, and forming a strong contrast to the pallor of his complexion…. He might have been taken, by a careless observer, for a country gentleman, or rather, to be more accurate, for a north country la
ird. A keener eye would have seen, however, that the man must be a student of some sort, and one of more than ordinary intelligence.2

  And first impressions were amply borne out, as the same observer reports on further acquaintance:

  He had a strong sense of humour, and a keen relish for witty or jocose repartee, but rarely betrayed enjoyment by outright laughter. The outward sign and conspicuous manifestation of his enjoyment was a peculiar and brightness of the eyes. There was, indeed, nothing explosive in his mental composition, and as his mirth was never boisterous, so neither was he fretful or irascible. Of a serene and placid temper, genial and temperate in his enjoyments, and infinitely patient when others would have been vexed or annoyed, he at all times opposed a solid calm to the vicissitudes of nature.

  One suspects that Maxwell would have been acutely embarrassed to hear any such analysis of his personal qualities. Serenity may have hidden internal struggles—he said he was as capable of wickedness as any man—but he rarely let inner thoughts escape. Introspection, he firmly believed, should never be performed in public. In an essay for the Apostles called Is Autobiography Possible? he had written:

  When a man once begins to make a theory of himself, he generally succeeds in making himself into a theory.

  …The stomach pump of the confessional ought only to be used in cases of manifest poisoning. More gentle remedies are better for the constitution in ordinary cases.3

  His religion, too, was an intensely personal matter. Though now a trustee of his local parish and an elder in the Church of Scotland, he was not bound by any particular doctrine. Over the years, he refused several invitations to join the Victoria Institute, a body that aimed to establish common ground between science and religion. On the final occasion, in 1875, he gave his reasons: