Faraday, Maxwell, and the Electromagnetic Field

by Nancy Forbes

  The daily newspapers, as is well-known, usually contain in the autumn time paragraphs and leaders upon marvellous subjects which at other times make way for more pressing matters. The sea serpent is one of these subjects.3

  Heaviside went on to make sport of a press report about a boy with a stick who could detect water deep underground and only then got down to the business of the paper, which was a masterly account of a strangely neglected topic—the use of the earth as a return conductor in telegraphy. Heaviside loved pricking dignity, and one of his favorite targets was the church: he mocked the pompous solemnity of archbishops and, in one article, mischievously claimed that Ohm's law was evidence of divine creation—God had arranged it to save electricians the trouble of the laborious extra calculations that would have been needed had it not been true.

  Cheered by such diversions, Heaviside worked his way, bit by bit, through Maxwell's Treatise. At first he was as perplexed as anyone by the great man's theory of electromagnetism, but eventually he managed not only to master the theory but also to re-express it in a form that was much easier to grasp—the form, in fact, in which the theory is generally expressed today. The achievement was later described by another Maxwellian, George Francis Fitzgerald:

  Maxwell, like every other pioneer who does not live to explore the country he opened out, had not had time to investigate the most direct means of access to country nor the most systematic way of exploring it. This has been reserved for Oliver Heaviside to do. Maxwell's Treatise is encumbered with the debris of his brilliant lines of assault, of his entrenched camps, of his battles. Oliver Heaviside has cleared these away, has opened up a direct route, has made a broad road, and has explored a considerable trace of country. The maze of symbols, electric and magnetic potential, vector potential, electric force, current, displacement, magnetic force and induction, have been practically reduced to two, electric and magnetic force.4

  If asked how he did it, he would probably have said “by hard work,” and indeed it was. But there were two key components. One was the creation of a language for describing how vectors—quantities that have both magnitude and direction—vary in space. It became accepted as the natural language for the field. Called simply “vector analysis,” it fits the part so well that it is hard for today's students to imagine a time when it didn't exist. Each vector in three-dimensional space is represented by a single letter. (Pictorially it could be represented by an arrow of given length and direction.) With the single letters comes an algebra that enables mathematical relationships to be set out in simple, or at least simple-looking, equations that are independent of any coordinate system. The idea came from Maxwell's use, in his Treatise, of a form of quaternion representation. As we've seen, quaternions were an elegant but fearsomely complicated creation of the Irish mathematician Sir William Rowan Hamilton. They had, in effect, a vector part and another part that was an ordinary number, or scalar. Heaviside experimented with quaternions but found them largely useless, so he separated the vector and scalar parts and worked out a new algebra for the vectors. He later discovered that, in America, Josiah Willard Gibbs had also found disappointment in quaternions and had independently devised exactly the same vector algebra. Heaviside was happy to share the credit—he was always generous with praise where he thought it was due—and, in any case, it was an honor to share credit with a man like Gibbs.

  The second component in Heaviside's simplification of Maxwell's theory was to concentrate on the field forces and push the quantities called potentials into the background. The forces, to him were “real,” but the potentials were “metaphysical”5 and he decided “to murder the whole lot.”6 This way, with a little rearrangement, he was able to reduce Maxwell's twenty equations, or eight in “quaternion” format, to four. Here they are, for a point in empty space where no currents or charges are present:

  div E = 0

  div H = 0

  curl E = −μ∂H/∂t

  curl H = ε∂E/∂t

  E and H are the electric and magnetic field forces—the mechanical forces that would be exerted on a unit electric charge or a unit magnetic pole placed at the point. ∂E/∂t and ∂H/∂t are their rates of change with time, and μ and ε are the fundamental constants of magnetism and electricity.7 Div, short for divergence, and curl are ways of describing how the vectors vary in a small region surrounding the point and, as we've seen, had already been identified and named by Maxwell. When charges or currents are present, the equations acquire extra symbols to represent charge density and current density, but they are still astonishingly simple.8 They have been called the Mona Lisa of science; it seems a marvel even to professional physicists that they give rise to all the seemingly complicated phenomena of electricity and magnetism. The first two equations imply that electric and magnetic forces obey an inverse-square law, and the third and fourth imply that disturbances in the field will spread out as electromagnetic waves with speed 1/√(με), which is the speed of light.9

  Heaviside had given us the four equations that were to become famous. It is right that they are known as Maxwell's equations but they are, in part, Heaviside's creation, too.10

  With his background in telegraphy, Heaviside was intrigued by the way that electromagnetic energy moved. By Maxwell's theory, energy was located in space—at any instant, each part of space contained a definite amount of energy. When a change occurred in the field, some parts of space would gain energy and others lose it, but energy couldn't be simultaneously destroyed at one point in space and created at another, because this would involve actions occurring at a distance—the very concept that Maxwell and Faraday had set out to banish. Energy had to flow, and Heaviside worked out how it flowed. The rate of energy flow was equal to the product of the electric and magnetic field forces, its direction was at right angles to both, and it was greatest when the electric and magnetic forces were at right angles to one another. He expressed the law of energy flow compactly in vector form as:

  W = E × H

  where W is the energy-flow vector, and E × H is the vector product11 of the electric and magnetic field forces.12

  This was a great result, but soon after it was published in the Electrician, Heaviside found that he had been scooped by John Henry Poynting, a former student at the Cavendish who had left Cambridge shortly after Maxwell died and was now professor of physics at Birmingham University—Poynting had published the same result a few months earlier in the Royal Society's journal. Heaviside's response to the disappointment shows two sides of his character: he always generously acknowledged Poynting's priority but never failed to mention his own part in the matter. The Poynting vector, now known to all students of electromagnetism, bears its own aide-memoire, as it points in the direction of flow.

  Poynting took the glory, but it was Heaviside who did the most to explore the consequences of the new discovery. What he found almost defies belief, even today. In an electrical circuit, no energy passed through the wires themselves—they merely acted as a guide for the flow of energy through the surrounding space. The only energy flow inside the wires was inward, and that was just the portion of energy that was dissipated as heat! What about the electric current—didn't that flow in the wires? Yes it did, but its energy was borne by the accompanying fields—the lines of magnetic force that encircled a current-bearing wire and those of electric force that spread out radially from it, like spokes. By the new formula, the direction of energy flow was at right angles to both these fields and so ran parallel to the wire. Very nearly so, anyway: the lines of energy flow near the wire converged ever so slightly and, when they hit the wire, turned sharply inward to be converted into heat.

  By 1888, Heaviside had been living much the same life for fourteen years, writing papers that nobody seemed to read and rarely traveling farther from home than his feet could take him. The old feeling of self-sufficiency—that his discoveries were all the “meat and drink” he needed—was wearing thin. He wanted his voice to be heard. Then he happened to read a report o
f a talk by Oliver Lodge, professor of physics at the University College in Liverpool, and he saw his name mentioned. Referring to electromagnetic waves, Lodge had said:

  I must take this opportunity to remark what a singular insight into the intricacies of the subject, and what a masterly grasp of a most difficult theory are to be found in the eccentric, and sometimes repellent, writings of Mr. Oliver Heaviside.13

  Repellent was, presumably, a warning that some readers might find Heaviside's gratuitous observations on sundry topics like archbishops to be in poor taste. Anyway, what was repellent when set alongside masterly?—this was the first public recognition that Heaviside had received in his life, and he was overjoyed. He straightaway wrote to Lodge to ask for a full text of his talk and soon found that he had another admirer, Lodge's friend George Francis Fitzgerald, who was professor of natural and experimental philosophy at Trinity College, Dublin. Like Heaviside, Lodge and Fitzgerald had been captivated by Maxwell's work and both had been trying, first in isolation and then with mutual support, to carry it on. Now Heaviside, the independent recluse, had gained true friendship on his own terms, and the three of them, united in a common cause, became firm friends and formed the core of the group that came to be called the Maxwellians. They were soon to be joined by a fourth person from an unexpected quarter.

  Lodge was a clay merchant's son from Staffordshire who hated the trade but endured it until he had a chance to escape when reaching his majority. As a teenager, he had heard John Tyndall speak, and from that moment knew what he wanted to do in life. He worked his way to University College, London, and gained a doctorate before being appointed to the professorship at Liverpool. Forceful and extroverted, Lodge pursued his science with doggedness and passion and had a penchant for mechanical models. We can get an idea of his style from a contemporary's opinion of his book Modern Views on Electricity:

  Here is a book intended to expound the modern theories of electricity and to expound a new theory. In it there are nothing but strings which move around pulleys, which roll around drums, which go through pearl beads, which carry weights; and tubes which pump water while others swell and contract; toothed wheels which are geared to one another and engage hooks. We thought we were entering the tranquil and neatly ordered abode of reason, but we find ourselves in a factory.14

  Although he saw the need for mathematics and could, with effort, follow the work of others, Lodge was at his best when experimenting. Fitzgerald, on the other hand, was a gifted mathematician. Born into one of Ireland's patrician Protestant families, he sailed through his degree classes at Trinity College, Dublin, and gained one of the prized fellowships. Everything seemed to come easily to him, and, perhaps for this reason, he lacked the stern mental discipline that comes from sustained toil. Heaviside said of him: “He had, undoubtedly, the quickest and most original brain of anybody.”15 Fitzgerald often failed to follow up his own ideas, claiming in his unpretentious way that he was too lazy, but he dispensed them freely and had an influence on late nineteenth-century physics that extended way beyond his own published work. He commanded tremendous respect, and it was chiefly through his influence that other British physicists began to pay serious attention to Maxwell's theory and to Heaviside's work. Fitzgerald and Lodge met in 1878 at a conference in Dublin and quickly struck up a friendship. Both had become fascinated by Maxwell's writings and each had been trying, in his own way, to take the work forward.

  Now they could share ideas, and Lodge set himself the goal of producing and detecting electromagnetic waves. According to Maxwell, they would be produced whenever an electric current changed; the problem lay in detecting them. But light waves, Lodge reasoned, were easy to detect, so why shouldn't he tackle the problem from the other end and try to produce light by electromagnetic means? He tried various methods—for example, passing a current across the contact point between a rapidly spinning carbon disc and another piece of carbon held against it—but failed utterly; he didn't get anywhere near the required frequency. Meanwhile, Fitzgerald had worked out theoretically that the amount of energy radiated by a pulsating electrical circuit was proportional to the fourth power of the frequency. This meant that at low frequencies, say several hundred cycles per second, the radiated energy would be weak, but that at frequencies of several million cycles per second, it should be strong enough to be detectable. And the wavelength, the distance from peak to peak of the wave, would then be a few meters, short enough to be measured in a laboratory. Moreover, Fitzgerald believed, the means of achieving such high frequencies already existed—all one had to do was to discharge a Leyden jar through a suitable circuit. One still had to detect the waves, but Fitzgerald had two good ideas here. One was to reflect a wave back toward its source and so form a standing wave (a wave that doesn't travel but just vibrates up and down in one place) which would be much easier to detect. The other was to use a detector circuit that was tuned to the frequency of the wave. These two elements did indeed prove crucial, but a third was lacking. As Fitzgerald put it, “the great difficulty is something to feel these rapidly alternating currents with.”16 No known instrument seemed to be up to the job, but, as we will see, somebody did find a simple and effective way to “feel” the currents. And it had been available all along.

  One might have expected practical-minded Lodge to take up the challenge, but for a while he was too busy with lectures and other work to carry out serious laboratory experiments. Then he accepted an invitation from the Society of Arts to give a series of talks on lightning protection, and, in preparation, he tried a few quick experiments with discharging Leyden jars—the sparking from the discharge, he thought, would simulate lightning. When Lodge discharged his jar, sparks appeared between the ends of wires connected to it. This was expected, but he found that the sparking could be made weaker or stronger by varying the lengths of the wires. This was interesting, but only when prompted by a junior colleague did Lodge realize that he had stumbled on evidence of Maxwell's electromagnetic waves. Waves streaming from the discharge through the space alongside the wires had been reflected from the wire ends, and what he had detected were the resulting standing waves—the stationary waves that vibrate in one place and occur whenever a traveling wave combines with its own reflection. He had discovered electromagnetic waves along wires. In his second talk to the Society of Arts in February 1888, Lodge presented a brief sketch of his new evidence supporting Maxwell's theory. He knew that more rigorous experiments would be needed to get the result fully recognized, but there was plenty of time to do this before the big September meeting of the British Association for the Advancement of Science in Bath.

  Preparations done, Lodge happily set off for a summer holiday walking in the Alps. For the journey he took some journals he hadn't had time to read. As the train pulled out of Liverpool, he turned to the July issue of the German journal Annalen der Physik und Chemie and was astonished to find that Dr. Heinrich Hertz of the Technische Hochschule at Karlsruhe had already produced and detected electromagnetic waves not only along wires but in free space. Moreover, he had measured the speed of the waves and had shown that they could be reflected, refracted, and polarized, just like light. Lodge was devastated—his own efforts seemed puny in comparison—but his disappointment was soon overtaken by admiration of Hertz's work and genuine pleasure at the results. His own findings would now play only a small part at the British Association's September meeting in Bath, but there was a much bigger story to tell. Hertz's results had provided clear proof of Maxwell's theory of the electromagnetic field and had finally laid action at a distance to rest. Still, many of the delegates at Bath would be asking: who is Heinrich Hertz?

  Hertz grew up in a comfortable home in Hamburg. His father, a barrister, was from a long line of Jewish merchants but had converted to Christianity, and his mother came from generations of Lutheran preachers in the south of Germany. With this eclectic background, the boy developed wide interests from an early age, and shone at everything: languages, classics, mathematics, and s
ports. Faced with a choice of “klassiker” or “techniker” classes, he managed to combine the two by changing schools several times and, at one stage, studying at home. He enjoyed practical work and, on leaving school, studied engineering at Dresden and Munich before realizing that his true vocation lay in mathematical and experimental physics. There was then only one place to go, and, at twenty-one, he moved to Berlin University, where he became the star pupil and then the assistant of Hermann von Helmholtz—the country's most celebrated scientist.17

  Helmholtz had a vast range of interests, but the topic that held most of his attention at this time was electromagnetism, and he was one of the few top-ranking physicists to take Maxwell's theory seriously. In Helmholtz's view, three theories were more or less equal contenders: those of his compatriots Wilhelm Weber and Franz Ernst Neumann, which were both based on action at a distance, and Maxwell's. It was important, he thought, to establish by experiment which was correct. Prompted by Helmholtz, Hertz tried an experiment to detect displacement currents, but he found nothing. The currents, if any, were too weak to register on the most sensitive instruments available. Still, the work honed his experimental skill, so when luck gave him the slenderest opportunity a few years later, he was ready to exploit it. Meanwhile, he needed to gain teaching experience and so for two years worked as an unpaid lecturer at Kiel University. In his free time, he turned to the theoretical aspects of Maxwell's theory. Amazingly, he arrived at the same equations as Heaviside, though in the old triple-equation format. When the two later came to know one another, Hertz graciously acknowledged Heaviside's priority and told him he believed: