A Brief Guide to the Great Equations

by Robert Crease

  Dissatisfaction, indeed, is a powerful driving force in science, and it can arise in many ways. The science of impossibility gives rise to a special and rare case of dissatisfaction. This kind of science often collides with our hopes and dreams – of limitless energy, of superluminal travel, of a crisp ontology where things are pinned to specific places at all times. Human beings seem hardwired to have such hopes, and hardwired to balk at the science that dashes them. Small wonder that the science of impossibility makes them dissatisfied. But science benefits in the end.

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  ‘The Most Significant Event of the Nineteenth Century’:

  MAXWELL’S EQUATIONS

  DESCRIPTION: A complete characterization of electromagnetism that among other things describes how changing magnetic fields produce electric fields; asserts that there are no magnetic monopoles; describes how electric currents and changing electric fields produce magnetic fields; and describes how electric fields are produced.

  DISCOVERER: James Clerk Maxwell

  DATE: 1860s; reformulated by Oliver Heaviside in 1884

  From a long view of the history of mankind – seen from, say, ten thousand years from now – there can be little doubt that the most significant event of the 19th century will be judged as Maxwell’s discovery of the laws of electrodynamics. The American Civil War will pale into provincial insignificance in comparison with this important scientific event of the same decade.

  – Richard Feynman, The Feynman Lectures on Physics

  Feynman is surely joking again, right? The American Civil War was one of the fiercest conflicts in history. It cost over 600,000 lives, destroyed $5 billion in property, liberated 4 million enslaved people, ended slavery in the U.S., and inflicted economic, political, and social wounds that have never healed. How could this terrible event whose effects are still felt today possibly be overshadowed by some equations, written by a modest Scotsman, who was trying to puzzle out how to describe a few odd effects of little or no practical value?

  This time, Feynman was not joking. Maxwell’s equations described a new kind of phenomenon – the electromagnetic field – that was unanticipated by Newtonian mechanics. These equations characterized this new phenomenon completely. They also predicted something novel: the existence of electromagnetic waves that could travel through space. And the understanding of electromagnetism that grew out of these equations helped transform it from a curiosity into a structural foundation of the modern era, embodied in electronic equipment and in any device based on electromagnetic waves, including radio, radar, television, microwave, and wireless communication. In the process, these equations affected human beings – how they live and interact with each other, themselves, and the world – far more profoundly than any war ever did, or could.

  Maxwell

  James Clerk Maxwell was born in Edinburgh in 1831, and raised by his parents on a family estate in Glenlair in the Galloway region of southwest Scotland, where he was taught by a private tutor. At the age of ten he was sent back to the city to a school called the Edinburgh Academy for more formal training. There his urbane peers dubbed him ‘Dafty’ for his country attire, strange accent, lack of flair, and childlike questions. But these questions – often a version of ‘What’s the go o’ that?’ – evidently stemmed from curiosity rather than stupidity. The youth’s intellect was further cultivated by William Thomson, the son of a friend of the family who was seven years older than Maxwell, scientifically inclined, and by 1846 already a professor at Glasgow, studying electricity. Maxwell began taking classes at Edinburgh University at the age of sixteen, and Cambridge University 4 years later, winding up at Trinity. After graduating from Cambridge in 1854, the 22-year-old wrote to Thomson that he was interested in studying electricity but was an ‘electrical freshman.’1 But Maxwell was a quick study, and soon brought himself up to speed.

  James Clerk Maxwell (1831–1879)

  The field of what was then sometimes called ‘electrical studies’ was in bits and pieces, contributed by a variety of people. The Danish physicist Hans Christian Ørsted (1777–1851) had shown, in 1820, that an electrical current generates magnetism around it. Shortly after, French physicist André-Marie Ampère (1775–1836) wrote an equation, now known as Ampère’s law, to characterize this phenomenon mathematically: the total magnetic force around a loop of wire is equivalent to the total current through it. In the 1840s, Maxwell’s mentor Thomson (1824–1907) had noticed similarities between the flow of electricity and heat, and wrote equations for electricity to exploit the analogy.

  The most extensive investigations had been carried out by British scientist Michael Faraday (1791–1867), who had conducted a long series of experiments before writing Experimental Researches in Electricity in 1844. Among other things, Faraday discovered induction – that a moving magnet creates current in a wire, and a changing current creates a current in another wire – and the ‘Faraday effect’ – when polarized light passes through glass in the presence of magnetism its plane of polarization is rotated, meaning that magnetism can affect light.

  But Faraday’s work was regarded with suspicion by many electrical scientists. They looked on electricity with Newtonian eyes, as caused by a particle- or fluidlike substance that flowed along wires and collected in certain materials, and governed by a force that, like gravity, jumped instantaneously over space to effect action at a distance. To such scientists, what was important to understanding phenomena of electricity and magnetism was the mathematics. Faraday, on the other hand, was convinced that an ether filled all space, and that both electricity and magnetism were caused by strains in this ether and mechanically transmitted by it, probably at some finite speed. As a result, he was convinced that magnetism and electricity affected things like wires and conductors even when these were not moving, creating what he called an electrotonic state. The mathematics was not enough; you needed to understand the activity. As Maxwell wrote later, in contrasting Faraday’s views with those of others:

  Faraday, in his mind’s eye, saw lines of force traversing all space where the mathematicians saw centres of force attracting at a distance: Faraday saw a medium where they saw nothing but distance: Faraday sought the seat of the phenomena in real actions going on in the medium, they were satisfied that they had found it in a power of action at a distance impressed on the electric fluids.2

  But Faraday’s most serious failing, to his scientific contemporaries, was his lack of mathematical sophistication. Faraday, indeed, was even a little afraid of mathematics and preferred to communicate his ideas in images. He likened strains in the ether, for instance, to ‘lines of force.’ He was inspired in part by the fact that, when you sprinkle iron filings over a piece of paper close to a magnet, the filings order themselves up in neat patterns, each filing undergoing induction and turning itself into a little magnet in turn, lining up tip to toe with other filings in smooth curves that depart from one pole of a magnet and return to the other. Faraday came to treat these patterns as observable manifestations of a real something that traversed space. The properties of electricity and magnetism, he felt, derived from how these lines spread, squished, and curved, for which he had only a rudimentary mathematical description. But Faraday’s peers felt that, while Faraday’s work had much experimental flesh, it lacked mathematical bone.

  Maxwell would give mathematical bone to Faraday’s experimental flesh. In the process, his impact on electrical studies would be like that of Euler’s on mathematics; Maxwell would integrate many areas that seemed independent and even conflicting. His achievement would be so vast and thorough that he would turn what had seemed to be the most independent and thriving of these areas – optics – into the subdivision of a new territory, electromagnetism. But while Euler had reorganized mathematics by fully exploiting the potential of one area – analysis – Maxwell would effect his reorganization, and create this new territory, by a process of analogy. Maxwell’s was one of the most brilliant uses of analogy in the history of science, and it helped to brin
g about one of the most surprising and decisive transformations in civilization’s history.

  Maxwell’s mentor Thomson once said, ‘I never satisfy myself unless I can make a mechanical model of a thing. If I can make a mechanical model I can understand it. As long as I cannot make a mechanical model all the way through I cannot understand.’3 Maxwell, too, was attracted to the technique. Shortly after graduating from Trinity, he gave a talk to an undergraduate club on the subject – lighthearted in tone, cryptic in argumentative structure, but deeply insightful.4 Analogies are not about resemblances but about relations, he told the students. Scientists find them valuable because nature is not like a magazine, where you hardly expect that what you find on one page will throw light on the next – but more like a novel, where subjects introduced at the beginning are apt to keep reappearing, in more complex and subtle form, all the way through. Thus exploring the extent to which a strange new phenomenon is like another well-known one, making adjustments where needed, can be a fruitful way to get a grip on the former.

  First Step: Mathematical Force

  By the time he had given that talk, Maxwell had already begun to use the method to transform the theory of electricity and magnetism. His first step was a paper entitled ‘On Faraday’s Lines of Force’, which he read to the Cambridge Philosophical Society in December 1855, when he was 24 years old.5 The state of electrical science, Part I began, is a mess. We have experimental data for some parts but for others none. Certain pieces have not been mathematicized, while in the pieces that have, the formulas do not all fit together. Anyone studying electricity must mentally store up so much complex and inconsistent information that it is hard to think clearly enough to make a contribution. We must simplify and reduce all the information to grasp it better. I am no experimenter, Maxwell admits, but I shall use physical analogies to develop a mathematics more suited to electrical science. Bear in mind that these are only analogies. If we do, we can think more clearly, for we will neither be too distracted by the mathematics on the one hand, nor too stuck on the physical conceptions from which these are borrowed on the other.

  Maxwell then mentioned several suitable analogies. One was Faraday’s idea that the force exerted by electricity resembles geometrical lines that curve in space. Another was Thomson’s idea that electricity flows through space the way heat flows through a fluid: the centre of charge is analogous to the source of heat, the effect of electrical attraction or repulsion analogous to heat flow, potential difference analogous to temperature difference, and so forth. A third was the hydrodynamic analogy that an electric charge is like a pump that forces out a stream of an incompressible fluid like water, with speed of the pump like the intensity of the force of the charge, and so forth.

  Maxwell continued by assuming Faraday’s ‘vague and unmathematical’ idea that an electric field consists of lines of force that spread from one charge to another, and fill all space. Each point on these lines is associated with a direction and intensity. Now suppose, he said in effect, that electricity behaves the way that an incompressible fluid (such as water) does – that is, that lines of force were like tiny tubes carrying the fluid, with the motion resisted by a force proportional to the velocity – but suppose we also correct this picture for the context of electricity by saying that the fluid does not have any inertia. Then a similar mathematical framework for handling fluid flow, which had been developed by Thomson, could be applied to Faraday’s conclusions about lines of force. Maxwell uses this picture to put induction and many other of Faraday’s various physical ideas – along with Ampère’s law – into a set of six laws within a consistent mathematical framework. In Part II of the paper, Faraday handled Faraday’s notion of the electrotonic state by developing a variable for it that today is known as the magnetic vector potential (or A), couched in a mathematical structure involving differential equations (used to describe properties that change continually over time) that he had learned from Thomson’s work. The framework he developed doesn’t ‘account for anything’ and lacks ‘even the shadow of a true physical theory’, Maxwell admitted; it does not appear to say anything new. But it does provide the ‘mathematical foundation’ of Faraday’s researches, which would be a necessary condition for any eventual physical theory.6

  When Maxwell sent the paper to Faraday, Faraday responded that at first he was ‘almost frightened’ by the application of ‘such mathematical force’ to the subject, but then delighted that the effort succeeded.7

  Second Step: The Grand Analogy

  Maxwell’s second step was a paper called ‘On Physical Lines of Force’, written in 1861–62, and it contains one of the greatest uses of analogy in the history of science. Maxwell begins by announcing his intention to ‘examine magnetic phenomena from a mechanical point of view’, and refers to an analogy Thomson had used to understand the Faraday effect: if a magnetic field can shift the plane of polarization of light, Thomson said, it is as if each point on a magnetic line of force were a tiny, spinning ‘molecular vortex’ that passed along some of its spin to any waves of light flowing by.

  Maxwell then further develops the image. Let’s say a magnetic field consists of such rotating ‘cells’, as he calls them, whose axes are along magnetic lines of force as if threaded on a string; the stronger the field, the more rapidly the cells spin. But Maxwell knows it is mechanically impossible to have cells on neighboring strings spin the same way – clockwise, let’s say – for those on one string would rub the wrong way against those in the next. Maxwell rescues the picture by assuming that the space in between is filled with something similar to what engineers call ‘idle wheels’ – smaller beads, in contact with the cells, that rotate counterclockwise, permitting the cells to rotate clockwise. These beads stay in place when the neighboring cells are rotating at the same speed, but changes in the speeds of the vortices cause the beads to move in a line, and they are passed from one cell to another. Thus, Maxwell decides, these beads act much like an electric current.

  The model displayed the effects of electromagnetism – the way a changing magnetic field generates an electric current, and an electric current generates a magnetic field – as produced by mechanical motions of a medium. Push-pulls in the ether could produce all the electrical and magnetic effects that Faraday and others had noted. It even produced a mechanical conception of Faraday’s electrotonic state, or what was happening when there was a magnetic field but no electrical current; the electrotonic state was like the impulse of the idle wheels when they turned without moving.

  Maxwell’s image

  Maxwell’s model

  Maxwell wrote up the idea in the spring of 1861, and it was published in installments between March and May. He then left for his usual summer vacation at Glenlair. He was under no illusion that he had created a picture, a representation, of electromagnetism. All he wanted to claim was that this strange model did whatever electrical and magnetic phenomena did, and thus that its mathematics would also work for them. His model was, Maxwell remarked, like an ‘orrery’, or model of the solar system you often see in natural history museums in which the planets are balls placed on rods that mechanically swing about a central ball, the sun. The value of assembling such a model – putting everything you know into it – is that when you finish, and can survey how it works as a whole, you can often see even more than you got from the pieces.

  During the vacation, Maxwell realized that he had left something significant out of the model. The cells, he knew, had to have at least a little springiness or elasticity, as do all solid bodies. But this springiness would cause certain effects in his model that he had not accounted for. When the cells pushed the beads but the beads could not move (in an insulating material, say), the cells’ elasticity would push the beads a little bit anyway, like rubber balls pushing against an immovable force, until the motion is counterbalanced by forces in the material. If the force were removed, the cells and beads would spring back. Maxwell called this a ‘displacement of the electricity’, whose amo
unt depends on the strength of the electromotive force and the nature of the body. He realized he had to incorporate this into his mathematics, which would also involve introducing a small corrective factor to Ampère’s law in the process.

  Still more revolutionary: anything elastic can transmit energy from one place to another in the form of waves. Maxwell had shown that the ether – the medium of electrical and magnetic phenomena – must be at least a little elastic. The medium could pass energy in the form of waves from one part to another via leapfrogging electric and magnetic effects operating at right angles to each other – from idle wheels to cells and back to idle wheels again, and on and on, forever. These waves would act the way light does, reflecting, refracting, interfering, and polarizing. Maxwell set out to find the rate that these transverse vibrations travel through the ether, assuming it were passed by purely mechanical forces. The result he calculated, based on the work of Rudolph Kohlrausch and Wilhelm Weber – two German physicists who had measured electrical constants a few years earlier – is 310,740 kilometers, or 193,088 miles, per second. But the velocity of light, as measured by Armand Fizeau a dozen years previously, is 314,858 kilometers, or 195,647 miles, a second, suggestively close. Thus Maxwell wrote, ‘The velocity of transverse undulations in our hypothetical medium, calculated from the electro-magnetic experiments of MM Kohlrausch and Weber, agree so exactly with the velocity of light calculated from the optical experiments of M. Fizeau, that we can scarcely avoid the inference that light consists in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena.’8