Clerk Maxwell's electromagnetic theory

by H. A. Lorentz

In modern theory—I allude to the theory of relativity—one has found good reasons for combining into one unity, which we call the stress-energy-tensor, all these quantities of which I have spoken, viz. the energy, the flow of energy, Maxwell's stresses and the electromagnetic momentum. In Einstein's theory of gravitation this tensor determines the gravitation field that is produced by an electromagnetic system and in virtue of which such a system has an influence on the motion of material particles, unfortunately much too small to be observed.

I should be led too far astray if I dwelt on these questions, but what I want to point out is this, that we could never have gone so far if we had contented ourselves with the actions at a distance, if we had not fixed our attention on the intervening medium, localizing the energy in it and considering it as the seat of momenta and stresses which manifest themselves in the observed motions of bodies. All these modern ideas have their origin in Maxwell's work.

We are also concerned with a stress-energy-tensor, similar to the electromagnetic one, when we consider a system of material particles, whether unconnected like the molecules of a gas, or held together by internal forces as in an elastic body or a fluid. The question naturally arises: are these stress-energy-tensors, the electromagnetic and, let me say, the material one, wholly independent or can one be reduced to the other? One has often tried to do so and, more particularly, to imagine electromagnetic phenomena as produced by some invisible mechanism moving according to the laws of dynamics.

This was a favourite idea of Maxwell's and one of his most brilliant chapters is devoted to the dynamical theory of electromagnetism. It is the more important because it shows that such a theory can be developed on very general lines, it not being necessary to make definite assumptions regarding the underlying mechanism. Maxwell showed that in the case of linear circuits carrying electric currents we can account for the ponderomotive forces and for the phenomena of self and mutual induction by Lagrange's or Hamilton's equations of motion, provided that we introduce, besides the coordinates which determine the positions of the material circuits, a certain number of new coordinates, one for each circuit, the velocities belonging to these coordinates for the several circuits being proportional to the current intensities. In fact, the new or "internal" coordinate for each circuit represents the total quantity of electricity that has traversed some section since a fixed instant that is chosen as the origin of time. When the internal coordinates are given this meaning, the magnetic energy becomes the kinetic energy of the system, whereas the electric energy has to be identified with the potential energy and is comparable to the energy of deformation of an elastic body.

While he was applying the laws of dynamics in this very general way, Maxwell was led to discuss certain phenomena that might perhaps be expected to exist and some of which have been actually observed in our days though Maxwell was not able to detect them with the means at his disposal.

In the theory of dynamical systems there are as many velocities as there are coordinates and the kinetic energy is a homogeneous quadratic function of these velocities, in which in general not only the squares but also the products of the velocities appear. When we have one or more circuits carrying electric currents, we can distinguish in the kinetic energy one part that depends on the material velocities only, and this is the kinetic energy of ordinary mechanics, and a second part containing only the velocities corresponding to the internal coordinates; this is the magnetic energy that manifests itself in so many ways. Now, is this all? There would certainly be a third part of the kinetic energy if an electric current consisted in a real motion of some substance along the conducting wire, for if the wire were moving, say in the direction of its length, with the velocity v and if v' were the internal velocity proportional to the current, the total velocity of the moving substance would be v + v' and in its square we should have the term 2vv'. One is led to a similar conclusion on other less simple assumptions and so, independently of an special conception, the question arises whether any part of the kinetic energy consists of products of ordinary velocities and strengths of electric currents. Maxwell thinks this question to be of great importance and deems it "desirable that experiments should be made on the subject with great care."

He then proceeds to examine different ways in which the terms in question might be made to reveal themselves, the first of which he explains as follows:

If any part of the kinetic energy depends on the product of an ordinary velocity and the strength of a current, it will probably be most easily observed when the velocity and the current are in the same or in opposite directions. We therefore take a circular coil of a great many windings, and suspend it by a fine vertical wire, so that its windings are horizontal, and the coil is capable of rotating about a vertical axis, either in the same direction as the current in the coil, or in the opposite direction.

We shall suppose the current to be conveyed into the coil by means of the suspending wire, and, after passing round the windings, to complete its circuit by passing downwards through a wire in the same line with the suspending wire and dipping into a cup of mercury. A vertical mirror is attached to the coil to detect any motion in azimuth.

Now let a current be made to pass through the coil in the direction N.E.S.W. If electricity were a fluid like water, flowing along the wire, then, at the moment of starting the current, and as long as its velocity is increasing, a force would require to be supplied to produce the angular momentum of the fluid in passing round the coil, and as this must be supplied by the elasticity of the suspending wire, the coil would at first rotate in the opposite direction or W.S.E.N., and this would be detected by means of the mirror. On stopping the current there would be another movement of the mirror, this time in the same direction as that of the current.

It does not appear that Maxwell actually tried the experiment; he only says: "no phenomenon of this kind has yet been observed."

Now, if for Maxwell's coil we substitute a rod of iron, the magnetization and demagnetization of which are comparable to the starting and stopping of a current in the coil, we have exactly the Richardson-Einstein-de Haas effect that was really observed by Einstein and de Haas and by some other physicists. You know that it amounts to this, that a cylindrical rod of iron suspended in a vertical direction is set rotating, with a sudden jerk, when it is rapidly magnetized or demagnetized. When the magnetization is periodically reversed, the rod is made to oscillate and the amplitude of this motion may be increased by adjusting the frequency of the reversals to that of the free oscillations of the rod.