by Brian Clegg
Apart from the theoretical definition defined from the spinning coil, the King’s team put together a ‘B.A. standard resistor’ design. This was a rather magnificent construction, first completed in 1865, consisting of a coil of platinum/silver alloy wire covered in silk for insulation and then wrapped around a hollow brass core. The whole thing was then coated in paraffin wax, with thick copper wires to link it to the circuit. To ensure a constant temperature, the resistor was suspended in a water bath. While a standard resistor could not be just put alongside another resistor for comparison by eye, as was the case with a standard distance rule, a simple piece of equipment known as a Wheatstone bridge made it possible to use the standard resistor to calibrate others.
What was particularly impressive was the forward-looking nature of the British Association committee involved in devising these units. At a time when most scientific work was mired in the clumsy Imperial units, the electrical units were based on the metric system. This meant that when the other scientific units were switched over to metric, there was no need to redefine the electrical standards. For example, an amp (electrical current) times a volt (electrical potential) was the unit of electrical power. At the time, mechanical power would have been measured in horsepower or foot pounds per minute. But when the metric system was fully adopted internationally in 1921, the unit of mechanical power was a watt – exactly equal to an amp times a volt.
Electromagnetism without visible support
It’s possible that this refocusing on the practical side of electromagnetism was what pushed Maxwell to think again about his remarkable achievement in modelling the phenomenon. Although his mechanically-based model was remarkably effective, he understood why others found that it depended too strongly on analogy and he wanted to strengthen the theory’s mathematical standing by a kind of scientific magic trick of removing the mechanical foundation and leaving the mathematics holding itself up by its own bootstraps.
Perhaps surprisingly, Maxwell focused on electricity and magnetism, rather than digging deeper into his theoretical basis for light as an electromagnetic wave. Although colour and vision would remain an interest throughout his life, he intentionally limited his theoretical developments on how light worked, perhaps because he felt that he was far closer to providing greater insights into the fundamentals of electricity and magnetism. He would never apply his theoretical approach fully to familiar behaviours of light, avoiding the whole business of how light and matter interacted†† other than making a few initial notes on reflection and refraction, based on some work by the French physicist Jules Jamin. He commented, for example, ‘In my book I did not attempt to discuss reflexion at all. I found that the propagation of light in a magnetised medium was a hard enough subject.’
The move from mechanical model to a purely mathematical one was a remarkably original approach – arguably Maxwell’s greatest work of genius – and would provide the basis for modelling taken by the majority of physics theory right up to the present day. In a Royal Institution debate in 2004, four proponents put forward different names for individuals who could arguably be called the first scientist. I was one of these debaters, championing the thirteenth-century friar Roger Bacon. But another held out Maxwell to be the first. One of his arguments was facile – that the term ‘scientist’ was not brought into use until 1834, so no one working earlier could be one. But his other argument was that Maxwell was the first scientist in the modern sense because he moved from trying to establish the ‘true nature’ of physical reality to mathematically modelling it.‡‡
It is interesting to speculate whether Maxwell was influenced philosophically in this move by the work of the German philosopher Immanuel Kant. Certainly, Maxwell would have heard about Kant’s work in his university philosophy classes. Kant distinguished the phenomenal world – what we can experience – from the noumenal world – the actual underlying reality, which he called in German das Ding an sich (the thing itself). Kant suggested there was no point trying to know the reality – we could at best work with our interpretations of phenomena. Where most of Maxwell’s contemporaries still believed they could discover the truth that lay beneath, Maxwell’s approach of moving to a purely mathematical model seemed to reflect Kant’s dismissal of such attempts.
Nearly 100 years earlier, the Italian-born French mathematician Joseph-Louis Lagrange had taken the traditional mechanics based on Newton’s work and transformed it mathematically into what is now called Lagrangian mechanics. This centres on a mathematical function called the Lagrangian, which pulls together all the information about the movements of the bodies in a system into a single structure. In mathematics, a function is a compact way of referring to one or more equations which take one set of values and change them into something else. It’s like a mathematical sausage machine.
A very simple function might be one that takes a number and does something to it, for example, producing the square of that number. It would usually be written as f(x) – pronounced ‘f of x’ – and we could say in this case, for example, that f(5)=25. Mathematical functions proved a very powerful mechanism both in physics and later in computing, where functions are commonly used to provide operational modules which can perform the same operations on differing inputs inside a computer program. The Lagrangian consists of a set of equations based on differential calculus that link the velocity, momentum and kinetic energy of a body.
Although the route to developing the Lagrangian involved thinking about actual physical processes, once the function is established and found to match what is observed it can be considered totally detached from any analogy. Rather than relying on a mechanical model, this kind of function is a purely mathematical model. It is a black box where the user provides certain inputs, ‘turns the handle’ and gets the outputs. If what comes out matches observation, the function can be used without any idea of how the system it is modelling actually works.
In the mathematical belfry
Maxwell, the regular churchgoer, felt that the ideal analogy for a Lagrangian approach (he might have dismissed mechanical models, but he still loved using them to explain things) was a belfry. It’s quite a lengthy quote, but it’s worth taking it slowly and absorbing it because with this simple illustration, Maxwell is showing how he brought modern physics into being.
We may regard this investigation as a mathematical illustration of the scientific principle that in the study of any complex object, we must fix our attention on those elements of it which we are able to observe and to cause to vary, and ignore those which we can neither observe nor cause to vary.§§
In an ordinary belfry, each bell has one rope which comes down through a hole in the floor¶¶ to the bell ringers’ room. But suppose that each rope, instead of acting on one bell, contributes to the motion of many pieces of machinery, and that the motion of each piece is determined not by the motion of one rope alone, but by that of several, and suppose, further, that all this machinery is silent and utterly unknown to the men on the ropes, who can only see as far as the holes in the floor above them.
Supposing all this, what is the scientific duty of the men below? They have full command of the ropes, but of nothing else. They can give each rope any position and any velocity, and they can estimate its momentum by stopping all the ropes at once, and feeling what sort of tug each rope gives. If they take the trouble to ascertain how much work they have to do in order to drag the ropes down to a given set of positions, and to express this in terms of these positions, they have found the potential energy of the system in terms of the known co‑ordinates. If they then find the tug on any one rope arising from a velocity equal to unity|||| communicated to itself or to any other rope, they can express the kinetic energy in terms of the co‑ordinates and velocities.
These data are sufficient to determine the motion of every one of the ropes when it and all the others are acted on by any given forces. This is all that the men at the ropes can ever know. If the machinery above has more degrees of freedom*** than there are ropes, th
e co-ordinates which express these degrees of freedom must be ignored. There is no help for it.
Because Maxwell’s earlier model was mechanical, it ought to be capable of being represented in a Lagrangian form, which would enable Maxwell to then ditch the cells and spheres, leaving them above the ceiling of the virtual bell ringers’ chamber and simply dealing with the mathematical formulae that were the equivalent of the bell ropes. This was anything but a trivial task. He needed to stretch the mathematics of the time to enable it to cope with the added complexities of modelling electromagnetism. His work crucially depended on the ability to think of energy – potential and kinetic in the mathematical model – and on keeping the concepts of energy while moving to the different frame of electromagnetism.
It didn’t help that many of the quantities to be dealt with were vectors – as we have seen (page 69), Thomson had previously introduced Maxwell to using vector mathematics in the earlier fluid model, but dealing with a mix of quantities in the Lagrangian framework, some vectors with size and direction, such as the strengths of the fields, others, such as electrical charge, scalars with just size, made the mathematics significantly more challenging.
Nevertheless, Maxwell achieved his goal, and by December 1864 he was able to present to the Royal Society his groundbreaking new mathematical model of electromagnetism, which he wrote up the next year in the seven-part A Dynamical Theory of the Electromagnetic Field.
A new physics
Generally speaking, there are two possible reactions to a truly novel theory. Either everyone is bowled over by its clarity – or they are baffled by the novelty. Maxwell’s theory was very much of the second kind.††† His audience at the Royal Society were appreciative, but simply could not grasp what he was suggesting. Up to this point, physics had largely been a discipline that was about experiment and philosophical theory, with the minimum of mathematics that was required to do the job. Now that maths was taking the lead, many in the audience were simply incapable of keeping up.
The difficulty of grasping the theory was not just a matter of making it accessible to the general public. The audience at the Royal Society included many of the leading physicists of the day. William Thomson, for example, admitted that he never came close to understanding Maxwell’s theory. His was the last generation of physicists who could become leading figures without a strong grasp of high-level mathematics.
The reaction, and also the difficulty of explaining a strongly mathematical piece of physics to the general public, was beautifully assessed by Michael Faraday a few years earlier in 1857 in a letter to Maxwell. Faraday wrote:
There is one thing I would be glad to ask you. When a mathematician engaged in investigating physical actions and results has arrived at his conclusions, may they not be expressed in common language as fully, clearly and definitely as in mathematical formulae? If so, would it not be a great boon to such as I to express them so? – translating them out of their hieroglyphics, that we also might work on them by experiment. I think it must be so, because I have always found that you could convey to me a perfectly clear idea of your conclusions, which, though they may give me no full understanding of the steps of your process, give me the results neither above nor below the truth, and so clear in character that I can think and work from them. If this be possible, would it not be a good thing if mathematicians, working on these subjects, were to give us the results in this popular, useful working state, as well as in that which is their own and proper to them?
In effect, Faraday was arguing for the kind of lay summary of papers that is only now becoming widely accepted as a requirement – and to some extent prefigures the success of popular science writing, which became increasingly important as Maxwell’s strongly mathematical approach took over physics. As ever, Faraday was a man of vision.
It wasn’t just the physicists who wrestled with Maxwell’s mathematics. Mathematicians also struggled to understand his work, because he used physical terms rather than familiar mathematical ones to describe what he was doing. The Serbian-American physicist Michael Pupin took a trip to Europe after graduation from his first degree in 1883 to try to get to grips with Maxwell’s theory. He started in Cambridge with the intention of speaking to Maxwell himself, not realising that Maxwell was by then dead. He could find no one in Cambridge who seemed capable of explaining the theory to him, getting a satisfactory explanation only when he travelled to Berlin and studied under Hermann von Helmholtz (who apparently did understand the theory).
It would be a long time before there was good experimental evidence that supported the way that Maxwell’s model went beyond what had previously been observed. Crucially, his concept of electromagnetic waves, though impressive in its coincidence with the speed of light, needed experimental verification – someone needed to generate waves from an electrical source and demonstrate them crossing space. It would take twenty years before Heinrich Hertz did this with the first artificially produced radio waves.
The beautiful equations
It didn’t help that Maxwell’s mathematical formulation was decidedly messy. There were a total of twenty equations to cover six different properties such as electrical current and magnetic field strength. The sheer compact power of what Maxwell had done did not become obvious until twenty years later, when the self-taught English electrical engineer and physicist Oliver Heaviside‡‡‡ (who had been influenced in his work by his uncle, Faraday’s friend Charles Wheatstone) used the relatively new mathematics of vector calculus to reformulate Maxwell’s equations as just four, stunningly compact lines of text.
These can be presented in a number of ways, depending on the units and whether they take into account a material other than a vacuum, but the simplest version looks like this:
Part of the compactness here is due to the use of ‘operator’ notation. An operator applies a mathematical procedure to every value in a set. So, for instance, if I make up an operator called T which adds 2 to a number and apply it to the positive counting numbers, often called the ‘natural numbers’, the result would be to produce the set 3, 4, 5, 6 … because I started with the natural numbers =1, 2, 3, 4 … and the operator T told me to add 2 to each of them.
The inverted delta operator in the compact version of Maxwell’s equations is usually known as ‘del’ these days, though in Maxwell’s time it was sometimes called ‘nabla’, a term for the symbol suggested to Peter Tait by the theologian William Robertson Smith. The word reflected its shape, deriving from the Ancient Greek term for a type of harp with the same rough outline. Maxwell was never comfortable with this odd-sounding word and regularly mocked it, for example using it to derive a nonsense word in a letter to Lewis Campbell: ‘This letter is called “Nabla”, and the investigation a Nablody.’ At one point, Maxwell wrote to Peter Tait, ‘what do you think of “space-variation” as the name of Nabla?’ We will stick to the modern term, del.
Del indicates differential equations being applied to what could be a whole range of values, either in traditional differential calculus – the sort Newton used – or the vector calculus which Maxwell needed to deal with changing quantities that had both size and direction. Here the dot after the del indicates a particular type of matrix mathematics. (A matrix is just a two-dimensional array of values.) This is imaginatively called the dot product, which produces the ‘divergence’ of a vector field, providing the values of the field at each point. When there is a cross after the del, as in the third and fourth equation, the operation is a ‘cross product’ producing the ‘curl’ of a vector field, which shows the rotation at each point.§§§
Between them, the four equations describe the key behaviours of electricity and magnetism. The first,
gives us Gauss’s law. This provides the relationship between the strength of the electrical field¶¶¶ on the left and the density of electrical charge on the right.
The second,
shows that the magnetic field has zero divergence, which amounts to saying that it is impossible to have
an isolated magnetic pole – they always come in pairs that cancel each other out.
The third,
is where we get Faraday’s induction explained, providing a mathematical relationship between a changing magnetic field (B) and the electrical field (E) that it produces.
And finally,
describes the way that an electrical field produces magnetism. Here H is the ‘magnetising field’, proportional to the magnetic field B, but varying depending on the medium, J reflects the electrical current and the D part of the equation deals with the changing electrical field. Variants on these last two equations combined give all that is needed to describe a wave of changing electric field producing changing magnetic field producing changing electric field and so on, running on at the speed of light.
There is no need to be a physicist or immersed in the workings of the mathematics to appreciate how remarkably compact and powerful is the stark beauty of these four equations (often found in variants on T-shirts), despite their ability to encompass all the phenomena of electromagnetism.
Getting away from it all
Einstein regarded Maxwell as one of the greatest physicists ever – and though Maxwell was probably a better teacher than Einstein, they both suffered from frustration at the way the workload and administration of academic life could eat into their time for getting on with the work that they loved. In Einstein’s case, the ideal solution came up with the opportunity to move to the Institute for Advanced Study (IAS) at Princeton in the United States.|||||| While some academics find that the almost monastic existence in such locations gets in the way of being able to develop new thinking, it was a comfortable workplace for Einstein.