by Brian Clegg
Later, Maxwell would also work on solo books on his electromagnetic theory (Treatise on Electricity and Magnetism) and a title taking in his own contribution to thermodynamics and a far wider exploration of the subject in the Theory of Heat, which was where he exposed to a wider world the ‘finite being’ that Thomson had turned into Maxwell’s demon.
Glenlair life
The demon might never have existed. In 1865, enjoying his first summer at Glenlair that was more than just a vacation, Maxwell had been out riding. He was on an unfamiliar horse† and lost control sufficiently to end up riding fast under low trees. A branch hit him across the head – the injury was little more than a bad scrape, but it became seriously infected with the skin condition erysipelas. With no medical solution other than to keep the patient comfortable and wait for the fever to break, it was touch and go for a few weeks.
Although Maxwell would continue to work – science would never have left his mind for long – being at Glenlair was also an opportunity to have more of a family life. For several years the majority of the Maxwells’ time would be dedicated to the place that was always more home than anywhere else. It was a chance to improve the estate and make changes to the house, alterations that had been suggested ever since Maxwell’s father’s day, but which no one had found time to carry out.
It was also the Maxwells’ last chance to have children. Katherine was 41 when they moved to Glenlair full-time. While not impossibly old to have a first child, it was certainly unusually so at the time. There seems little doubt that Maxwell would have loved to have had children, both from the enthusiasm he displayed in playing with the children of others and bearing in mind his sense of duty to the estate, which would have been seen at the time as a responsibility to have an heir. His contemporary biographers note:
[I]t is impossible not to recall the ready kindness with which, in later life, he would devote himself to the amusement of children. There is no trait by which he is more generally remembered by those with whom he had private intercourse.
As is often the case with spouses of the period, we know a lot less of Katherine’s nature, though, as we have seen, there were hints that she was less outgoing than Maxwell. We certainly don’t know whether she was interested in children at all, but for whatever reason, it was not to be.
Once Maxwell had recovered from his infection, his time at Glenlair also gave him the opportunity to revisit experiences and modify ideas of the past. Some scientists may be happy to publish a piece of work and then move on, but Maxwell’s character seems to have encouraged repeated consideration of a topic, sometimes just fine-tuning what had come before, and at other times – as he had with his electromagnetic model – totally reworking the way he approached the problem.
One opportunity to rework a feature of his younger days was one that many would consider more terrifying than appealing. I am sure I am not alone in having occasional nightmares where I find myself sitting a university exam – in maths or physics – and realise I haven’t revised any of it and don’t know any of the equations I will need. But when Cambridge University asked Maxwell to take a look at the Mathematics Tripos – the complex and by that time dated exam structure that had brought him to the attention of academia – Maxwell was happy to return to the fray. The exams were mired in the past, unchanged since well before Maxwell had taken them. He was asked to make the content and structure more current and relevant to the modern mathematician, a challenge he took on with enthusiasm.
Back to viscosity
Maxwell was also able to work more on his experiments on the viscosity of gases, with the help of Katherine. All the evidence was that he was correct in thinking that viscosity varied directly with temperature rather than with its square root. But the theory of the time did not support this. As an expert on mechanical models, he realised it was the model itself that was at fault. His first step was to move away from the notion of mean free path (see page 110), which had been useful for getting an idea of the size of molecules, and instead to employ a model where gas molecules exerted force on each other.
This was a significant change to the thinking. To make things simple in his earlier work on kinetic theory, the molecules in the gas had been treated like colliding billiard balls‡ which head towards each other at full speed until they come into contact, bounce apart, and immediately head away from each other at full speed.§ With his electromagnetic experience, Maxwell tried instead representing the interaction between the molecules as that of two electrically charged particles repelling each other. Here, the acceleration effects will start earlier as the repulsion acts well before the molecules are in contact and grows rapidly as they come closer with the inverse square of the distance between them.
At the same time, Maxwell threw in another concept, relaxation time, which allows for the way that a system, after being disturbed, returns to a state of equilibrium. Think, for example, of dropping a blob of milk into a cup of tea without stirring it. Before adding the milk, the tea molecules will be in equilibrium, bouncing off each other and keeping the temperature throughout the tea roughly the same. Add the cold milk and there will be a concentration of coldness in one point. The system will have been disturbed. But over time, the milk molecules will disperse through the tea and the tea/milk system will settle down to a new, different equilibrium. The time for the system to undergo this process is the relaxation time.¶
With his modifications to the model of a gas producing its viscosity, Maxwell was now able to match observation and theory. This new version of the model gave a gas a viscosity that was proportional to temperature just as the Maxwells’ experiments had shown. It still produced the same velocity distribution for the molecules – his original paper was not wrong in this – but in this new formulation, Maxwell was able to drop one of the limitations of the original model which meant that it would work only if there was no relationship between molecules with particular velocities. He completed this work in 1866 and published it in 1867.
He would revisit his theory six years later, but this was more a matter of tweaking the approach and presentation to match the latest findings from other physicists. He would also present his theory in 1873 to the Chemical Society, where he had to pile in as much evidence as he could for the existence of molecules to justify his use of them. Although chemists made use of the concept, they largely considered molecules to be a useful accounting fiction, rather than actual physical bodies that could perform the complex statistical ballet Maxwell portrayed.
Inevitably, given the gaps in knowledge at the time, there were limitations to the approach that Maxwell took. While his distribution has held up, there were aspects that didn’t fit with experimental measurements that were becoming increasingly accurate. This deviation primarily derived from the assumptions Maxwell (and Clausius) made about the ways that molecules could move. Maxwell had dealt with movements in the three dimensions of space and three potential axes of rotation, but he was thinking of the molecules as spheres. We now know that anything more than a single atom has a more sophisticated structure and is capable of rotating in different ways and of vibrating along the bonds between the atoms that make it up. It is this difference from Maxwell’s picture that accounts for the shortfall in his calculations – but given the information he had at the time, it was still a remarkable achievement.
Maxwell made significant progress, particularly in his written output, in his years at Glenlair, but it ought to be stressed that his time wasn’t all given over to work. As we have already seen, he spent a considerable amount of effort on improving the estate, and in 1867 he and Katherine took a tour of Italy – still as fashionable a destination for the British well-to-do as it had been in the time of Byron and Shelley some fifty years before. Their trip also got them out of the house while building work was under way, transforming Glenlair into a dwelling that was a little more like a manor house and less like a farmhouse. Being the Maxwells, though, their trip to the Continent was not the classic Grand Tour coveri
ng several countries and as many cultural locations as possible with the minimum effort put in by the travellers. The Maxwells made an attempt to learn Italian and seem to have immersed themselves more in the local culture than would have been expected at the time.
Over the years, Maxwell had been able to build up an effective laboratory at Glenlair – certainly sufficient to repeat his experiments on viscosity of gases – but there were some pieces of technology that were beyond the reach of his personal funding. Three years after the retreat to Scotland, he found himself back in London to try to refine his calculation for the speed of electromagnetic waves.
The wine merchant’s batteries
As we saw in Chapter 5, back in 1861, on his summer vacation at Glenlair, Maxwell had calculated the expected speed of electromagnetic waves in a vacuum to be around 310,700 kilometres per second, basing his calculation on the equivalent of the two factors used for conventional waves: the elasticity of the medium and its density. The equivalent properties for electromagnetism are known as the magnetic permeability and the electric permittivity of space, and the values of these properties were relatively poorly known at the time, so the result had a significant uncertainty.
With Charles Hockin, an electrical engineer based in Cambridge, Maxwell devised an experiment that would pin down these factors to far greater accuracy than ever before. The experiment balanced out the attraction from the electrical charges on two metal plates with the repulsion between two electromagnets with like poles facing each other. The bigger the effect, the more accurate the measurement could be – which meant hunting down an extremely high-voltage source.
Rather surprisingly, the owner of the most powerful batteries in the country turned out not to be a physics laboratory or an electrical power company, but a wine merchant based in Clapham, London. John Gassiot had spent his fortune on constructing an extravagant private laboratory. It was Gassiot who provided Maxwell and Hockin with a vast battery of cells, 2,600 in all, which between them put out around 3,000 volts.
The experiment proved highly successful, though the batteries went flat with unnerving rapidity, meaning that Maxwell and Hockin had to take measurements furiously quickly before the charge ran out. These more accurate values for magnetic permeability and the electric permittivity resulted in a calculated speed for electromagnetic waves of 288,000 kilometres per second (kps).
In the original calculation, Maxwell had come up with 310,700 kps to Fizeau’s 314,850 kps – so it might seem that his new calculation put him further off the mark. But in the intervening period, Maxwell had learned of an updated value for the speed of light from another French experimenter, Léon Foucault.|| Foucault had used an improved version of Fizeau’s equipment to get a more accurate speed of 298,000 kilometres per second** – and so Maxwell’s waves now seemed even more certain to be light.
This was immediately a useful support to his theory, but in other cases the significance of the work that Maxwell did would not become clear until much later. As we will see below, even a spin-off from Maxwell’s work could hold promise of great things – simply because he retained his child-like curiosity, seeing what others would regard as everyday phenomena and realising that here was something special and worth investigating.
Meet the governor
As we have seen, when Maxwell had been working on the electrical resistance standard at King’s College (page 183), his colleague Henry Fleeming Jenkin had designed a governor to keep a coil rotating at constant speed. Governors would be familiar to anyone who had seen a static steam engine. The early days of steam were riddled with explosions and runaway catastrophes when poorly understood technology was used outside its operating parameters. Back in the 1780s, James Watt had come up with the centrifugal governor. This elegant device has a pair†† of weights on hinged bars, which fly outwards as the vertical spindle they are mounted on rotates. The faster the spindle goes, the further out the balls move. The hinged bars are linked to a valve, closing it off if the balls fly too far out from the centre. This means that a steam engine with the governor in place never runs too fast – it automatically shuts itself off over a certain speed.
Thinking more about Jenkin’s governor, Maxwell began to explore the different ways in which this form of feedback mechanism, which automatically regulates a process, could be deployed (another, more modern version of a feedback-based governor present in almost all homes now is the humble thermostat). In 1868, Maxwell wrote a paper named On Governors which, with his usual thoroughness, gave the process a mathematical treatment. In fact he pointed out that, at least by his definition, Watt’s device wasn’t a governor at all.
In the paper, Maxwell made the distinction between a ‘moderator’, in which the correction that is applied grows with the excess speed, and a ‘governor’, which also involves the integral of the speed.‡‡ He showed mathematically that only a governor with this facility would exactly regulate the speed. A moderator, like Watt’s, would still provide negative feedback, but could not give the exact value required to balance out the error.
The mathematics Maxwell used here drew on some of his work on Saturn’s rings – getting a governor right was the same kind of problem, in the sense that it involved the stability of a system. Ironically, his work on stability in governors would be generalised by another Cambridge mathematician, Edward Routh, who won the Adams Prize (checked by Maxwell) for his work, just as Maxwell had for his analysis of the rings of Saturn.
Decades later, this paper by Maxwell would be recognised by the American mathematician Norbert Wiener, who in the 1940s originated the concept of cybernetics§§ – the use of systems with communication and feedback that would become important in control systems, engineering and computer science. Wiener considered Maxwell the father of automatic controls, providing the first steps in the development of the control systems theory that lies behind everything from cruise control on cars to the systems that keep nuclear power stations safe.
Thinking in four dimensions
Sometimes the work that Maxwell did would have greater impact than was perhaps even intended. In his Treatise on Electricity and Magnetism, he took the first major step that would enable Oliver Heaviside to produce the compact versions of his equations. This was because Maxwell had become interested in quaternions, a relatively obscure mathematical device introduced by the Irish mathematician Sir William Hamilton (not to be confused with his contemporary, the Scottish philosopher Sir William Hamilton, whose lectures Maxwell attended while at Edinburgh University).
By the 1860s, most scientists were comfortable with the concept of complex numbers, which incorporated values in two dimensions at once. A complex number is an ordinary number combined with an imaginary number – a multiple of i, the square root of –1. So, for example, a complex number might be 3 + 4i. These numbers can be manipulated just like an ordinary number, and map onto a two-dimensional graph with the real numbers on one axis and the imaginary numbers on the other.
Complex numbers proved highly useful when dealing with anything that has the form of a wave, which has both its position in a particular direction and an amplitude that varies with time. However, many physical processes take place not in the flat plane of graph paper but in three spatial dimensions. Quaternions provided a mechanism for dealing with values that had amplitude and a three-dimensional location. This was done by having effectively three different imaginary components, so a single quaternion might look like 3 + 4i + 2j + 6k.
Hamilton thought, correctly, that quaternions had the potential to revolutionise the mathematics of physical processes, but the approach proved difficult to handle and instead a different mechanism to deal with values that varied in multiple dimensions, vector analysis, was developed.
Maxwell provided the bridge between Hamilton and practicality – inspired by quaternions, he came up with the terms ‘convergence’, ‘gradient’ and ‘curl’ to represent different forms of quaternion operation which would be carried through into vector calculus, thou
gh ‘convergence’ was replaced by its opposite ‘divergence’, and the longer names shortened, ending up with the operations div, grad and curl (two of which are seen in Heaviside’s formulation of Maxwell’s equations on page 193).
We can see a typical Maxwellian delight in words in the way that he worked towards the term ‘curl’, first in a letter to Peter Tait, where it was just an alternative to ‘twist’, and then in a paper written for the London Mathematical Society in 1871 when he had finally settled on curl.
The letter begins by asking if Tait called the mathematical operator that would be known as del ‘Atled’ (it is an upside-down delta). He then wrote:
The scalar part I would call the Convergence of the vector function and the vector part I would call the twist of the vector function. (Here the word twist has nothing to do with a screw or helix. If the words turn or version would do they would be better than twist for twist suggests a screw.) Twirl is free from the screw motion and is sufficiently racy. Perhaps it is too dynamical for pure mathematicians so for Cayley’s¶¶ sake I might say Curl (after the fashion of Scroll).
Confusingly, after saying this, Maxwell goes on in the rest of his letter to refer to the vector part as the twirl.