Faraday, Maxwell, and the Electromagnetic Field

by Nancy Forbes

  I think that the results which each man arrives at in his attempts to harmonize his science with his Christianity ought not to be regarded as having any significance except to the man himself, and to him only for a time, and should not receive the stamp of a society. For it is the nature of science, especially those branches of science which are spreading into unknown regions, to be continually [changing].4

  Like Faraday and like Newton, Maxwell believed that God made the universe, that the laws of physics were God's laws, and that every discovery was a further revelation of God's great design. At the same time, as a devout Christian, he believed that the true nature of God was to be found in the Holy Bible, which he knew as well as any scholar of divinity. In his scientific work he treated all theories, including his own, as provisional until they had been backed by experimental results. How was this approach to be reconciled with a Christian faith that required absolute trust and belief in the absence of any material evidence? Much of the answer lay in how the scriptures were interpreted. It was no longer necessary to accept as literal truth the account in Genesis of God creating the world in seven days; that, and many other passages in the Bible, could be taken as metaphors. This process of interpretation wasn't easy, but for Maxwell it was necessary. He couldn't keep his science and religion in separate compartments—every crack between them had to be examined and repaired—but he found no formula to help with the task and thought himself no better qualified than anyone else to explain the connection between the spiritual and physical worlds.

  Katherine was by now in her early forties, and it was clear that she and James would have no children. We don't know the reason but can be sure that it wasn't from choice: Maxwell greatly enjoyed playing with the children on the estate and, remembering the delights of his own childhood, loved to entertain them with tricks and games. There was also the matter of the succession. Without an heir, Glenlair would pass to a cousin on the Clerk side of the family, to whom it would be no more than a pleasant country estate. But one of Maxwell's mottos was that there is no use moping over what might have been. Outwardly, at least, he put the disappointment to one side, and made the most of life at is was.

  The six years Maxwell spent at Glenlair were in no sense a time of retirement. He went to British Association meetings all over the country, sometimes acting as president of the Mathematics and Physics section, and he made annual visits to Cambridge, where, as examiner for the Mathematical Tripos, he did much to make the questions more interesting and more relevant to everyday experience. Meanwhile, his own work went on apace. The biggest project was his Treatise on Electricity and Magnetism. Here he set out not merely to present his own theory of electromagnetism but also to bring together everything that was known and to make this subject, still widely regarded as arcane, accessible to all scientists. It was a monumental feat—the book ran to almost a thousand pages and was eventually published in 1873.

  The Treatise was a constant background task, but there was much else to do. His work for the British Association's committee on electrical standards had not stopped with the report on units and the production of the world's first standard of electrical resistance. Another difficult experiment was urgently needed—the measurement of the ratio of the electromagnetic and electrostatic units of electric charge. Much depended on the result because the speed of light, according to Maxwell's theory of electromagnetism, was exactly equal to this ratio. As we've seen, the ratio had already been measured experimentally by Kohlrausch and Weber, and their result, by Maxwell's interpretation, gave a theoretical value for the speed of light very close to the actual speed that had been measured by direct experiment. With so much at stake, it was important to confirm Kohlrausch and Weber's result by carrying out another experiment to measure the ratio of the units, preferably using a different method. Maxwell took on the job, this time in partnership with Charles Hockin, of St. John's College, Cambridge, and they carried out the experiment in London in the spring of 1868. By balancing the attractive electrostatic force between two oppositely charged metal discs against the repulsive electromagnetic force between two current-carrying wire coils, they measured the value of the ratio of the units of charge (which was also the speed of Maxwell's waves) at 288,000 meters per second, 7 percent below Kohlrausch and Weber's value and 8 percent below Fizeau's direct measurement of the speed of light. At first this seemed a disappointing result, but, on reflection, the experiment was a success. Maxwell's theory of electromagnetism had been strengthened because two independent experimental results now gave predicted wave speeds that, with reasonable allowance for experimental error, corresponded to the measured speed of light. We now know that the both the Fizeau and the Kohlrausch and Weber results were too high and Maxwell's too low, with the true value lying roughly midway between them.

  As well as drafting most of his great Treatise while at Glenlair, Maxwell wrote another book, The Theory of Heat, and seventeen papers on a great variety of topics, each of which broke new ground. Most of this work lies outside our story, but one example will serve to demonstrate, from a different aspect, the almost-magical power of imagination that had enabled Maxwell to predict displacement currents and electromagnetic waves in empty space. In The Theory of Heat Maxwell introduced what became, perhaps, his best known creation: Maxwell's Demon. The demon is an imaginary tiny creature who guards a hole in a partition between two gas-filled chambers. To start with, the gas in both chambers is at the same temperature. Temperature depends on the average speed of the gas molecules (strictly, on the average of the square of their speeds) but, according to Maxwell's own theory, at any given temperature some molecules will be going faster than average and some slower. At his hole, the demon operates a shutter, allowing only fast molecules to pass from the left chamber to the right one, and only slow ones to pass the other way. The gas in the right chamber gets hotter, and that in the left chamber colder—the demon is making heat pass from a cold substance to a hot one, thus defying the second law of thermodynamics. Maxwell was making the point that the law was not a physical one: it was statistical. To say that heat cannot flow from cold to hot was, as he put it, like saying that when you throw a glassful of water into the sea you can't get the same glassful out again. At the same time, he was posing a deep puzzle, and the demon lived up to its name, perplexing and intriguing generations of physicists. Among other things, it sparked the creation of information theory, which underpins our digital communications. Oddly, it was Maxwell's less frolicsome colleague William Thomson who named the demon; Maxwell wanted to call him a valve!

  Whatever his business of the moment, Maxwell was apt to “feel the electrical state coming on”—thoughts on electricity and magnetism were never far away. His thoughts, like Faraday's, were often visual and no doubt included mental images of electric and magnetic forces and fluxes looping through space and embracing one another. These forces and fluxes were represented by vectors, mathematical entities that had both magnitude and direction. They also had a kind of three-dimensional geometry, but it was a very different geometry from anything in the textbooks of the time. It could be represented by equations, but these took a form that seemed arcane to most physicists and Maxwell looked for a way to demystify the subject. Was it possible to describe the geometry of vectors in a way that helped people to visualize the relationships between the physical quantities? Indeed it was. From his mental pictures, Maxwell coined three expressions that eventually became universal currency—curl, divergence, and gradient, the last two usually being abbreviated to div and grad. Maxwell originally proposed “convergence” and “slope,” and, in his Treatise, replaced curl with the more formal “rotation,” but, in essence, all the terms have stood the test of time. Once grasped, these images brought everything to life, and one senses that Faraday, with his acute visual imagination, would have recognized concepts that had already, if hazily, formed in his own mind's eye. They were concepts essential to the electromagnetic field.

  Curl is the essence of the relationsh
ip between electricity and magnetism; it explains how the force of each connects with the flux of the other. At any given point in space, any vector, like magnetic flux or the velocity of wind in air, has a curl, which is itself a vector, though it may take the value of zero. Curl isn't easy to visualize, but it can be done. Think of water flowing in a river. The vector here is the speed and direction of flow, and, in general, it varies from point to point in the river. Now imagine a tiny paddle wheel somehow fixed at a point in the river but with its axis free to take up any angle. If (and only if) the water is flowing faster on one side of the paddle wheel than the other, the wheel will spin, and its axis will take up the position that makes it spin fastest. The curl of the water flow at out point is a vector whose magnitude is proportional to the rate of spin and whose direction is along the axis of spin, by convention in the direction a right-handed screw would move if it turned the same way as the paddle wheel. If the wheel doesn't spin, the curl of the water flow is zero. Curl is at the heart of two of the four equations in which Oliver Heaviside later summed up Maxwell's theory. At a point in empty space they say that the curl of the electric field force at a point is proportional to the rate at which the magnetic field force there is changing, and vice versa.

  The river analogy also gives us an idea of divergence. Unlike curl, divergence is not a vector but a scalar—a term that mathematicians use to describe a quantity that has a magnitude (which can be positive, negative, or zero) but no direction. The divergence of the water flow at our fixed point is a measure of the excess of water flowing out of a small region surrounding the point compared with that flowing in. Assuming water is incompressible (very nearly true), the two amounts will be equal and the divergence zero. Unless, of course, we inject water in at our point, in which case the divergence will then have a positive value, or if we suck it out, in which case the divergence will be negative. The other two equations in which Heaviside summarized Maxwell's theory both employ divergence. At a point in empty space they say that the divergence of the electric field force and the divergence of the magnetic field force are both zero.

  Gradient is a vector property of a scalar quantity. Imagine something that varies from place to place, like the height of land above sea level. Height is a scalar quantity, and its gradient at any given point is the slope of the land there along the direction of greatest incline (and is conventionally taken to point in the downhill direction). The gradient of an electric or magnetic potential is defined in similar fashion and manifests itself as the electric or magnetic field intensity, or force.

  Along with curl, divergence, and gradient came another way of making the underlying physics clearer. Maxwell heard from his friend P. G. Tait about some strange mathematical entities called quaternions that represented rotations in three-dimensional space. They were the brainchild of the Irish mathematical genius Sir William Rowan Hamilton, namesake but no relation of the Sir William Hamilton who had taught Maxwell philosophy at Edinburgh University. Though he actually has much stronger claims to fame, Hamilton believed that quaternions were his greatest creation, and that they held the key to understanding all rotational phenomena in the physical universe. He had died in 1865, but not before recruiting a staunch disciple. Tait was bowled over by quaternions and became their vigorous champion. Not many followed his lead—quaternions were fearsomely complicated, and most people wanted nothing to do with them. For Maxwell, though, they presented an opportunity. Up to now he had written his various vector relationships as triple equations—one for each of the three space dimensions—but now he found that by using a form of quaternion representation, he could express the same relationships in single equations. Moreover, Hamilton had already built the mathematics of curl, divergence, and gradient into the quaternion system, so everything fit together beautifully. But very few people understood quaternions, and some of those who did understand them hated them, so Maxwell decided to play safe by including in his Treatise both the standard and the quaternion forms of the equations. A consequence was that he ran out of letters, having come close to exhausting the Roman and Greek alphabets! He then took what seemed to be the only course open and resorted to heavy Gothic Roman letters for his quaternion equations, so giving them a strange Teutonic air.

  Thanks to quaternions, Maxwell could now present his theory of electromagnetism in eight equations rather than twenty, but it remained impenetrable to most physicists of the time. One can see why. Maxwell regarded the theory as work in progress and wanted to keep all the options for further advance open, even if this confused his contemporaries. In the Treatise, he still presented nearly all of his working in triple-equation format, with separate equations for each of the x, y, and z directions, and it was hard to see the forest for the trees. (To appreciate this difficulty, one need only look at figure 14.1, a diagram in which Maxwell used the notion of right-handed screws to help explain the x, y, and z components of the mechanical force on a current-carrying conductor in a magnetic field.)

  Fig. 14.1. Maxwell's diagram using the notion of right-handed screws to help explain the components of force on a conductor in a magnetic field. (Used with permission from Lee Bartrop.)

  Only the final equations appeared in the alternative quaternion format, as a kind of optional extra—one that most people preferred to do without. So things were to stay until six years after Maxwell's death, when Oliver Heaviside reduced the number of equations to four and replaced the quaternion representation with a much simpler kind of vector algebra. He thereby incurred the fury of Tait, who accused him of mutilating Hamilton's beautiful quaternions, but, as we'll see in a later chapter, Heaviside gave as good as he got—both of them were masters of literary invective and enjoyed a good scrap.

  Early in 1869, Maxwell heard that James Forbes, the beloved mentor who had fired his interest in color vision and much else, had died. Maxwell felt the loss deeply, but Forbes's death meant that the post of principal at St. Andrews University was vacant, and a number of Maxwell's friends and associates urged him to stand for the job, among them Lewis Campbell, who was now professor of Greek at the university. Maxwell was at first reluctant, commenting that “my proper line is in working, not governing, still less in reigning and letting others govern.”5 However, he did believe passionately in good education, and his supporters were so enthusiastic that he began to feel that maybe he could do some good in the position. Eventually, he was persuaded to let his name go forward.

  It was a politically charged appointment, but Maxwell knew nothing of politics and wrote, rather pathetically, to a London acquaintance:

  I have paid so little attention to the political sympathies of scientific men that I do not know which of the scientific men I am acquainted with have the ear of the Government. If you can inform me, it would be of service to me.6

  Given such candid naivety, it is not surprising that Maxwell didn't get the job. Perhaps this was just as well—one thinks of a goldfish about to enter a tank of piranhas—but with Maxwell one can't be sure; for all we know, he might have transcended the political squabbles. The job went to the university's professor of humanity, John Campbell Shairp, an eminent scholar of poetry and a supporter of the newly forged Liberal Party, which had just come into power.

  After he lost his job at Aberdeen, Maxwell had been rejected for the chair of natural philosophy at Edinburgh University, but, shortly afterward, he was accepted for a similar post at King's College, London. Now, ten years later, the sequence was repeated, and once again Scotland's loss turned out to be England's gain. Shortly after St. Andrews had turned him away, Cambridge University asked Maxwell to accept an important new professorship in experimental physics. The university's chancellor, the duke of Devonshire, had offered a large sum toward the setting up of a new department and the building of a grand new laboratory. The task of getting the laboratory designed and built would fall to the first professor. Ideally, the university authorities wanted a first-rank scientist who had experience in running a leading research and education es
tablishment. William Thomson was the obvious choice, but he didn't want to leave Glasgow, where he had painstakingly built up a fine research center, starting in a converted wine cellar. Second on Cambridge's list was Hermann Helmholtz, but he had just accepted a top post in Berlin and wanted to stay there. Maxwell was the third choice, but a popular one among the younger fellows. Their spokesman was J. W. Strutt, who later, as Lord Rayleigh, succeeded to the same post, and he put their case fervently to Maxwell, imploring him to come.

  Once again, Maxwell was initially reluctant. He loved his home and had settled at Glenlair into a good life that combined laird's duties with scientific work in a pleasant and fruitful way. But he had affection for Cambridge, too, and saw what a rare opportunity the new project offered for the university and, indeed, the country. Was he really the man for the job, though? He wasn't sure. At length he accepted, on condition that he could leave after a year if he wished. This didn't imply any lack of commitment. As in all things, he was determined to do his best, but, being acutely aware of his lack of experience in directing a big and complex operation, he wanted to be free to withdraw if he found that he wasn't able to run the show well. In March 1871, twenty-one years after entering Cambridge University as a student, he was appointed its first professor of experimental physics. Once again, he and Katherine moved south.

  The first task at Cambridge was to draw up detailed specifications for the new laboratory building and its equipment. Everything had to be at the forefront of scientific progress, yet it was important to avoid mistakes that would be expensive or impossible to put right later, so Maxwell visited the best university laboratories in the country—the newly built Clarendon Laboratory at Oxford and William Thomson's laboratory at Glasgow—to learn all that he could from their experiences. There needed to be abundant light, plenty of space for bulky apparatus, a battery room, and a tower tall enough to provide water pressure to drive a powerful vacuum pump. To enable delicate experiments to be done on magnetism, one room needed to have table surfaces that were insulated from ordinary vibrations in the building, so Maxwell specified stone-topped tables that were four feet square, to be supported directly from the building's foundations by brick piers that came up through the floor without touching it. The job of producing a design to meet these and a host of other requirements was assigned to the young local architect William Fawcett, who had studied at Jesus College.