A Brief Guide to the Great Equations

by Robert Crease

  He published these two revolutionary new features in the model as Part III of his paper in 1862.

  Keeping the Baby

  Two years later, Maxwell took a third key step in his paper entitled ‘A Dynamical Theory of the Electromagnetic Field’, written late in 1864 and published early in 1865. In it, he cites the earlier mechanical analogy only to abandon it, aiming to present all the results – including the displacement current and the idea that light is an electromagnetic wave – in the form of a set of freestanding equations. ‘Thus, then, we are led to the conception of a complicated mechanism capable of a vast variety of motion, but at the same time so connected that the motion of one part depends, according to definite relations, on the motion of other parts, these motions being communicated by forces arising from the relative displacement of the connected parts, in virtue of their elasticity. Such a mechanism must be subject to the general laws of Dynamics, and we ought to be able to work out all the consequences of its motion, provided we know the form of the relation between the motions of the parts.’9 Maxwell continued, a few paragraphs later, ‘In order to bring these results within the power of symbolical calculation, I then express them in the form of the General Equations of the Electromagnetic Field.’ He then lists twenty equations in eight general categories.10

  This brought to a close one of the most remarkable uses of analogy in science. His achievement is itself often expressed in terms of a famous analogy – ‘Maxwell threw out the bathwater and kept the baby’ – except that the bathwater begat the baby.

  The Treatise

  In 1873, Maxwell published A Treatise on Electricity and Magnetism, his complete presentation of the branch of science that he had developed by his remarkable analogy, and the form in which practically everyone for at least a decade would have to learn it. About a thousand pages long, it was rather difficult and even annoying to digest, for Maxwell made no effort to condense or simplify the work for the reader, aiming to be comprehensive rather than economical. For instance, in the key chapter, entitled ‘General Equations of the Electromagnetic Field’, Maxwell summarizes his work in twelve steps, labeled A to L, each involving an equation or group of equations. ‘These may be regarded as the principal relations among the quantities we have been considering’, he writes. Some could be combined, ‘but our object is not to obtain compactness in the mathematical formulae.’ Furthermore, these equations were based on concepts that were extremely difficult to use for those interested in practical applications, most notably A, the vector potential, and ψ, the scalar potential.

  Maxwell’s Treatise has also puzzled historians, because in it – and elsewhere – he is silent about how to make and find electromagnetic waves. The idea of electromagnetic waves was the single most thrilling and unexpected feature of Maxwell’s entire life’s work. His silence over how to make and find such waves seems as perverse as an astronomer whose studies predict the existence of a new planet, yet who does not think to go find a telescope to point at it, or tell someone to go do it. Maxwell’s silence is strange enough to demand explanation. Some historians say it is that he was less interested in electromagnetic waves than in light and the ether, others that he did not conceive of any way to produce and detect them, still others that he simply had no time to think on the subject. None of these explanations is really convincing, though it is true that Maxwell’s workload had dramatically increased by the time of the Treatise. In 1871, he was given charge of supervising the founding of the new Cavendish Laboratory in Cambridge, England, and in 1874, he was handed the task of editing the papers of the laboratory’s namesake, Henry Cavendish. Maxwell also became the scientific co-editor of the ninth edition of the Encyclopaedia Britannica. These projects left him little time for research.

  Maxwell did, however, retain his interest in seeing if the ‘great ocean of ether’, as he called it, could somehow be detected. It is invisible and we know little about it. We do not even know, he wrote in his Encyclopaedia Britannica entry on ‘Ether’, if dense bodies like the earth pass through this ocean the way fish pass through water, dragging some small portion of it with them; or the ether might pass though them ‘as the water of the sea passes through the meshes of a net when it is towed along by a boat.’ As he wrote beautifully and somewhat anxiously:

  There are no landmarks in space; one portion of space is like every other portion, so that we cannot tell where we are. We are, as it were, on an unruffled sea, without stars, compass, soundings, wind, or tide, and we cannot tell in what direction we are going. We have no log which we can cast out to take a dead reckoning by; we may compute our rate of motion with respect to the neighboring bodies, but we do not know how these bodies may be moving in space.11

  There is one trick we might play to detect it, he realized, thanks to the fact that a wave flowing through a medium moves at different speeds depending on the speed of the medium. Sound, for instance, always travels at the same speed – about 1,100 feet a second in air – due to the properties of the medium (air molecules) that propagate it. If a wind’s blowing, the sound still travels at the same rate in the air, but because the air carries the sound waves along with it, these will seem to be traveling faster or slower than usual from someone on the ground. If a wind’s blowing, sound waves thus travel at different rates in different directions.

  The same should be true of light. In moving around the sun, the earth might ‘drag’ some small amount of ether with it, but would have some changing velocity with respect to the ether; there would be an ether wind or ether drift. The speed of light would be different in different directions. The difference would be tiny – one part in a hundred million – from the velocity of light in ether at rest. Was this measurable?

  On Earth probably not. If experimenters shot beams of light back and forth in different directions, the hundred-millionth-part difference in travel time would be ‘quite insensible’, Maxwell wrote. ‘The only practicable method is to compare the values of the velocity of light deduced from the observation of the eclipses of Jupiter’s satellites when Jupiter is seen from the earth at nearly opposite points of the ecliptic.’ And so in March 1879, he contacted the director of the Nautical Almanac Office, in Cambridge, England, to ask if any research on this subject had been done. ‘I am not an astronomer’, he wrote with his usual modesty in making the inquiry, but ‘the only method, so far as I know’ of measuring the ether drift would be to make precise measurements of the apparent retardation of eclipses of the satellites of Jupiter.12

  By this time, Maxwell was showing symptoms of what turned out to be abdominal cancer. That November he died. The career of this prodigious, inventive force of nature – whose quiet pursuits transformed the world, Feynman claimed, more profoundly than the Civil War – died at the age of only forty-eight.

  Maxwell left unfinished business – exciting ideas suggested by his work that, for one reason or another, he had not pursued. One was the question of producing and detecting electromagnetic waves; another was measuring the ether drift; a third was revamping his series of equations in a concise way for practical use – which was becoming increasingly important with the expansion of telegraphs. All three of these were carried out in the decade following Maxwell’s death.

  Heinrich Hertz and the Discovery of Electromagnetic Waves

  Heinrich Hertz (1857–1894) was born and raised in Hamburg, and in 1878 began to study in Berlin under Hermann von Helmholtz, who was investigating Maxwell’s electrodynamics. Helmholtz tried to entice the bright 22-year-old to compete for a prize to be awarded for the person who solved an experimental problem, devised by Helmholtz himself, that would confirm a certain feature of Maxwell’s theory. The youngster declined, afraid the work would absorb several years and not result in a big enough effect to be decisive, and finished his doctoral dissertation instead. In 1885, Hertz moved to Karlsruhe, where he had access to a well-equipped laboratory that he put to use inventively. In 1886, the chance observation that an oscillating current caused sparks to jump a
cross small gaps in a nearby loop of wire set Hertz on a path that led to the publication, in the July 1888 issue of Annalen der Physik, of an article entitled ‘On Electromagnetic Waves in Air and Their Reflection.’ Hertz was able to measure the wavelength of these electromagnetic waves, and showed they had the properties of other kinds of waves – including the ability to reflect, refract, interfere, and be polarized, and had a finite speed – in stunning confirmation of Maxwell’s theory.

  Meanwhile, a physics professor at Liverpool in England named Oliver Lodge had noted that oscillating currents created waves in wires. In July 1888, Lodge completed a paper on his results and boarded a train to the Alps for a hiking holiday. En route, he pulled out his reading material – that month’s issue of the Annalen – to learn of Hertz’s work. Lodge was dismayed; he was planning to attend the annual meeting of the British Association for the Advancement of Science that September in Bath and had expected to be celebrated for his discovery, but now realized that Hertz’s work would overshadow his. Yet Lodge also found himself thrilled by the elegance of Hertz’s experiments, which were much more extensive than his own, for Hertz had detected electromagnetic waves not just in wires but also in air.

  The Bath meeting was the first public presentation of Hertz’s discovery to the broader scientific community, and the circumstances were rather dramatic.13 The president of the Mathematics and Physics Section had fallen ill, and his last-minute replacement was Irish physicist George FitzGerald (1851–1901), who had been studying the possibility of producing electromagnetic waves for almost a decade, and who was therefore well-prepared to state the significance of Hertz’s work. So while this popular meeting featured a new wax phonograph by Thomas Edison, and a speech on ‘Social Democracy’ by George Bernard Shaw, FitzGerald all but stole the show with the news: electromagnetic force does not work through action at a distance, but by waves traveling through the ether. ‘The year 1888’, FitzGerald announced, ‘will be ever memorable as the year in which this great question has been experimentally decided by Hertz in Germany.’ Alerted by FitzGerald’s announcement, Time magazine called the news ‘epoch-making.’ Yet confirmation of Maxwell’s ideas also brought to the surface the deep and long-standing dissatisfaction with Maxwell’s impractical formulations: FitzGerald spoke of attempts by meeting participants to ‘murder ψ’ and at least revise the vector potential A, and the consensus of the gathering was that some conceptual homicide was necessary.

  The dramatic news of the creation and detection of electromagnetic waves – implied by Maxwell’s work but not discussed by him – also provided a classic illustration of the unexpected productivity of equations themselves. As Hertz once said of Maxwell’s equations, ‘One cannot escape the feeling that these mathematical formulae have an independent existence and an intelligence of their own, that they are wiser than we are, wiser even than their discoverers, that we get more out of them than was originally put into them.’14

  Albert Michelson and the Nonexistence of the Ether

  Maxwell’s letter about ether drift, sent to the director of the Nautical Almanac Office, was read to the Royal Society at the beginning of January 1880, 2 months after Maxwell’s death, and then published in Nature. One fascinated reader was American physicist Albert A. Michelson (1852–1931). A graduate of the U.S. Naval Academy in Annapolis, Maryland, who remained there to teach science, Michelson was entranced by the challenge of measuring the speed of light, making attempts in 1878 and 1879, playing hooky from the academy’s traditional July 4 celebration to pursue the work. The 1879 measurement, in which he shot a beam of light down a 2000-foot path and back, had an unprecedented precision, earning the 27-year-old a reputation among U.S. scientists and front-page mention in the August 29 edition of The New York Times. Michelson’s fame, however, did not impress the academy enough to release him from a scheduled sea-duty – but he managed to pull strings and secure a leave of absence, allowing him to travel to Europe at the beginning of 1880 to study physics in Helmholtz’s lab. After reading Maxwell’s posthumous letter in Nature in January 1880, Michelson invented a device, called an interferential refractometer, that used mirrors to split a beam of light by refracting (bending) it, then sent the two beams along two paths at right angles to each other and back. When the two beams were made to interfere, the difference due to their travels through the ether in different directions would be on the order of a fraction of a wavelength – but this tiny difference would be ‘easily measurable’, he wrote to Nature.15 Explaining the planned experiment to his children, Michelson asked them to imagine a race between ‘two swimmers, one struggling upstream and back, while the other, covering the same distance, just crosses the river and returns.’ The point, he said, is that ‘The second swimmer will always win, if there is any current in the river.’16

  A first experiment in 1881 detected no drift, and seemed to have design flaws. Michelson quit active duty, moved to the Case School of Applied Science, in Cleveland, Ohio, and collaborated with Edward Morley (1838–1923), another experimenter, to enlarge and revise the apparatus. This experiment, too, detected no drift, despite an astounding sensitivity of a quarter part per billion. Michelson was baffled and disappointed by the null result, and he and Morley abandoned their plans for further measurements. But other scientists, including George FitzGerald, Dutch physicist Hendrik Lorentz, and French physicist Henri Poincaré, undertook desperate attempts to trust both the Michelson-Morley experiment and the existence of the ether, efforts that set the stage for Albert Einstein’s discovery of special relativity. In 1907, for his role in the magnificent experiment that made it possible – inspired by Maxwell’s letter – Michelson became the tenth person, and the first American citizen, to win the Nobel Prize in Physics.

  Oliver Heaviside and ‘Maxwell’s Equations’

  The standardization was largely due to Oliver Heaviside (1850– 1925), a self-taught electrical engineer, eccentric, and maverick (and discoverer of what was once called the Heaviside layer and now called the ionosphere), who is often called ‘the last amateur of science.’17 He left home at sixteen, never had a job in a university, and struggled in poverty, supported by relatives, friends, and a government pension. His one and only job was a 4-year stint as a telegraph operator, and he was avidly interested in the practical issue of improving the flow of energy down telegraph wires. He picked up much of contemporary mathematics on his own, using it in novel ways to improve the state of electromagnetic theory; he introduced imaginary numbers into electricity, for instance. When Heaviside came upon Maxwell’s Treatise, his reaction to it was somewhat the same as Maxwell’s own to the then-current state of electrical science: far too complex to be useful to practical folk, for far too many things have to be held in one’s head simultaneously. Maxwell’s formulation of his theory – founded in the vector potential A and the electrostatic potential ψ, a relic of the ‘action-at-a-distance’ perspective – was particularly ill-suited to the increasingly urgent concerns of telegraphy, which involved the flow of electromagnetic energy down specific pathways.

  The demands of this practical technology, indeed, did much to advance the science of electromagnetism in the 1880s.18 Many electromagnetic researchers at the time made clever physical models, involving wheels and connecting bands, to picture to themselves how electrical energy flows from place to place in Maxwell’s theory. Many were frustrated in particular by Maxwell’s use of the potentials A and ψ.

  In 1883, in a series of articles in a magazine called the Electrician, Heaviside began to examine how Maxwell’s work might be adapted for the practical context of studying the flow of electricity in telegraph wires and circuits. ‘[I]t was only by changing its form of presentation that I was able to see it clearly’, Heaviside wrote later.19 His amateur, self-taught condition served him well, for he was not inhibited by current mathematical lore nor impressed by prevailing physical perspectives. His outlook was practical; what was important to him was the energy at each point, and calculating how that energy
flowed down a path such as a wire. He was prone to expressing that outlook charmingly, in simple and direct terms, as in the following lead sentence from a paragraph in one of his scientific papers: ‘When energy goes from place to place, it traverses the intermediate space.’20 He then boldly reworked Maxwell’s sets of equations in terms of E and H to represent the electric and magnetic forces at each state, and currents D and B. The result was a sweeping condensation of Maxwell’s work into four equations. These four were pleasingly symmetrical – two electric, two magnetic, and the parallel evident. And they are so thoroughly revamped that they are sometimes called ‘Heaviside’s equations.’21 The equations for free space are the following:

  div εE = ρ curl H = kE + eĖ

  div μH = 0 −curl E = μ

  and in their more complicated form in the presence of electric charges

  div εE = ρ curl (H − h0 − h) = kE + εĖ + uρ

  div μH = σ −curl (E − e0 − e) = gH + μ + uσ

  Heaviside himself modestly referred to his four equations as ‘Maxwell Redressed’,22 though he did promote them, enthusiastically and polemically, as superior to Maxwell’s own equations and to other revisions thereof. Shortly after the 1888 Bath meeting, for instance, he published a brief note savagely attacking the continued use in propagation equations of the electric potential ψ and the vector potential A as ‘metaphysical’ (a term of opprobrium for scientists) and as ‘a mathematical fiction.’23 What we measure, after all, are the electric force E and the magnetic force H, not potentials. These give us real information about the state of the field; these are what propagate when current flows. Keeping ψ and A results in ‘an almost impenetrable fog of potentials’ and even inconsistencies, and Heaviside, recalling the Bath conference, advocated their ‘murder.’ Maxwell’s theory works just fine, he concluded, ‘provided that we regard E and H as the variables.’