Maxwell’s Equations

James Clerk Maxwell was born in 1831. He held chairs at Aberdeen and London and was the first head of the Cavendish Laboratory in Cambridge. He died at the early age of 48, leaving behind, as well as much other first-rate work in physics, something quite epoch-making, ‘Maxwell’s equations’, which predicted clearly that electromagnetic waves could exist, and that light was of this nature. He did not go on to generate radio waves: one wonders why, and why this had to wait for some decades. But the correct theory was all there, in two papers in the Philosophical Magazine published in the early 1860s.

The author of this book begins with the statement: ‘For physicists the name of James Clerk Maxwell ranks next to Newton and Einstein.’ Yet, he says, non-scientific people know little of Maxwell, and in this book he hopes to correct the injustice and to show the nature of his achievement and how his life, his work and his philosophy were always interwoven. So the reader, particularly the non-scientific reader, will pick up the book, asking: ‘Here is one of the creators of modern physics – can I understand what he did, and in what his greatness consisted?’

In physics, every physicist will have his own heroes. In my view, Newton stands apart from all the others. This does not mean that his intellectual capacity was necessarily greater than that of some of those who came later: but, standing at the beginning of the scientific age, he wrought the greatest transformations in our thinking. His concept of action at a distance, together with laws of motion which made the future calculable if the present was known, set the pattern of our thought for more than a century.

Nature and Nature’s laws lay hid in night;
God said ‘Let Newton be’ and all was light.

Professor Tolstoy believes that Maxwell’s greatest contribution was to replace Newton’s concept of action at a distance, the attraction or repulsion of one electric charge by another, by that of a ‘field’. This is not a very easy concept to communicate, though Professor Tolstoy makes a valiant attempt: the idea is that, at every point in space, a force would be exerted both on an electric charge if it were placed there, and on a magnet. This capacity to exert a force is called the ‘field’. The same ideas can be applied to gravitation: we now say, not so much that the sun attracts the earth, but that the earth is held in orbit by the sun’s gravitational field. For gravity the results are the same – but not in electromagnetism. Here, as the author of this book would agree, a great deal is owed to that earlier genius, Michael Faraday, whose ideas on ‘fields’ lead the way. But Faraday was no mathematician and Maxwell put it all into mathematical form.

What came out of this, in ‘Maxwell’s equations’, Tolstoy does not, in my view, fully describe. It was known that a current would generate a magnetic field: this is the principle on which the electric motor works. But when this was expressed in a mathematical form, it became absolutely clear that, if the current was changing from point to point along its path and so building up an electric charge, which in its turn produced an electric field, the rate of change of this field had to be added to the current: otherwise the equations did not make sense, so that no prediction could be made from them. This extra fictitious current was called the displacement current. I remember learning about Maxwell’s equations in my undergraduate days in Cambridge: Nature could not be otherwise, it was immediately and exquisitely clear. I remember only one other personal moment of illumination of this kind: when Dirac showed that the electron, if it was to be described by the theory of relativity as well as by the newly discovered quantum mechanics, could not be a point charge but must be a small magnet as well – a fact that had already been discovered by the painstaking methods of spectroscopy, but really did surprise physicists. Why was Nature not content with a simple point charge, and why did it have to be a magnet too? Dirac showed that it could not be otherwise.

When Maxwell had done with it, Nature still appeared thoroughly deterministic in the Newtonian sense, but to calculate the future one could have to know, not only the positions and velocities of all the particles in the universe, but also the values of the electromagnetic field everywhere. It took the formulation of quantum mechanics in 1925-27 to show that Nature was not like this, and that an element of chance lay deep in the laws of physics, though Einstein would never accept the idea, rejecting, as he said, a God who played dice. It is no reflection on the genuius of those who built up quantum mechanics in Göttingen, Copenhagen and elsewhere, and of Heisenberg for his statement of the Uncertainty Principle, to say that theirs was much more of a co-operative effort than could have been the case in Maxwell’s day. The Göttingen school under Born and that in Copenhagen under Bohr reinforced each other to such a degree that one cannot but feel that quantum mechanics would soon have surfaced had any one of these men not lived. This may even be true of most of Einstein’s work, with the notable exception of the General Theory, which gave the explanation of gravity and showed that Newton’s achievement was not quite complete.

So Professor Tolstoy is right, I think, in placing Maxwell next to Newton and Einstein among the greatest theoretical physicists. Whether he shows that his work and philosophy are deeply interconnected I am less certain. What surely led to Maxwell’s equations was, in addition to first-class mathematical talent, an exceptional capacity to shake off the received ideas of the time. I would add the ability to choose a really fruitful problem. The book recounts in a very readable way the main facts of Maxwell’s life: his base at Glenair in Scotland, how he lived there between professorships, his marriage – perhaps not entirely happy, though little is known of it because his first biographer and lifelong friend, the Reverend Lewis Campbell, chose not to write about it. He was throughout his life a practising Christian and did not share the scepticism which was already becoming common among scientists. That he shared the ideals of scientists is shown by the following extract from a letter to Campbell:

Now, my great plan, which was conceived of old, and quickens and kicks periodically, and is continually to make itself more obtrusive, is a plan of Search and Recovery, or Revision and Correction, of Inquisition and Execution. The Rule of the plan is to let nothing be wilfully left un-examined. Nothing is to be holy ground consecrated to Stationary Faith, whether positive or negative ... Never hide anything, be it weed or no, nor seem to wish it to be hidden. So shall all men passing by pluck up the weeds and brandish them in your face, or at least display them for your inspection ... Again I assert the Right to Trespass on any plot of Holy Ground which any man has set apart.

Tolstoy says that Maxwell’s religious convictions were so deeply embedded that he was unconscious of the inconsistency. Rather, Maxwell subscribed to the view that faith and intellect are separate activities of the mind. As he put it many years later in a draft letter: ‘I think that the results which each man arrives at in his attempts to harmonise his science with his Christianity ought not to be regarded as having any significance except to the man himself, and to him only for a time, and should not receive the stamp of society. For it is of the nature of science, especially of those branches of science which are spreading into unknown regions, to be continually ...’ Here the draft was left unfinished. It is tantalising not to know what this great man was going to write. But one can guess. And I cannot help concluding that it was his science, Maxwell’s equations, which influenced his religion, rather than the other way round.