Maxwell Versus Thomson

Maxwell’s investigation was published in 1856. In the same year, Thomson[13] put forward an alternative interpretation of magnetism.

He concluded, from a study of the magnetic rotation of the plane of polarization of light, that magnetism possesses a rotatory character. He suggested that the resultant angular momentum of the thermal motions of a body[14] might be taken as the measure of the magnetic moment.

He wrote:

“In the present state of science, it is impossible to decide:

• whether this matter is electricity or not
• whether it is a continuous fluid interpermeating the spaces between molecular nuclei, or is itself molecularly grouped.
• whether all matter is continuous, and molecular heterogeneousness consists in finite vortical or other relative motions of contiguous parts of a body”

The two interpretations of magnetism, in which the linear and rotatory characters respectively are attributed to it, occur frequently in the subsequent history of the subject.

The former was amplified in 1858, when Helmholtz published his researches[15] on vortex motion; for Helmholtz showed that if a magnetic field produced by electric currents is compared to the flow of an incompressible fluid, so that the magnetic vector is represented by the fluid velocity, then the electric currents correspond to the vortex-filaments in the fluid.

This analogy correlates many theorems in hydrodynamics and electricity; for instance, the theorem that a re-entrant vortex-filament is equivalent to a uniform distribution of doublets over any surface bounded by it, corresponds to Ampère’s theorem of the equivalence of electric currents and magnetic shells.

In his 1855 memoir, Maxwell had not attempted to construct a mechanical model of electrodynamic actions, but had expressed his inteution of doing so.

He wrote:

“By a careful study 16] of the laws of elastic solids, and of the motions of viscous fluids, I hope to discover a method of forming a mechanical conception of this electrotonic state adapted to general reasoning”.

In a footnote, he referred to the effort which Thomson had already made in this direction. Six years elapsed, however, before anything further on the subject was published.

In the meantime, Maxwell became Professor of Natural Philosophy in King’s College, London—a position in which he had opportunities of personal contact with Faraday, whom he had long reverenced. Faraday had now concluded the Experimental Researches, and was living in retirement at Hampton Court; but his thoughts frequently recurred to the great problem which he had brought so near to solution. It appears from his note-book that in 1857[17] he was speculating whether the velocity of propagation of magnetic action is of the same order as that of light, and whether it is affected by the susceptibility to induction of the bodies through which the action is transmitted.

The answer to this question was furnished in 1861-2, when Maxwell fulfilled his promise of devising a mechanical conception of the electromagnetic field.[18]

In the interval since the publication of his previous memoir Maxwell had become convinced by Thomson’s arguments that magnetism is in its nature rotatory. “The transference of electrolytes in fixed directions by the electric current, and the rotation of polarized light in fixed directions by magnetic force, are,” he wrote, “the facts the consideration of which has induced me to regard magnetism as a phenomenon of rotation, and electric currents as phenomena of translation.”

This conception of magnetism he brought into connexion with Faraday’s idea, that tubes of force tend to contract longitudinally and to expand laterally. Such a tendency may be attributed to centrifugal force, if it be assumed that each tube of force contains fluid which is in rotation about the axis of the tube. Accordingly Maxwell supposed that, in any magnetic field, the median whose vibrations constitute light is in rotation about the lines of magnetic force; each unit tube of force may for the present be pictured as an isolated vortex.

The energy of the motion per unit volume is proportional to μH2, where μ denotes the density of the medium, and H denotes the linear velocity at the circumference of each vortex. But, as we have seen,[19] Thomson had already shown that the energy of any magnetic field, whether produced by magnets or by electric currents, is

where the integration is taken over all space, and where u denotes the magnetic permeability, and H the magnetic force. It was therefore natural to identify the density of the medium at any place with the magnetic permeability, and the circumferential velocity of the vortices with the magnetic force.

But an objection to the proposed analogy now presents itself. Since two neighbouring vortices rotate in the same direction, the particles in the circumference of one vortex must be moving in the opposite direction to the particles contiguous to them in the circumference of the adjacent vortex; and it seems, therefore, as if the motion would be discontinuous. Maxwell escaped from this difficulty by imitating a well-known mechanical arrangement.

When it is desired that two wheels should revolve in the same sense, an “idle” wheel is inserted between them so as to be in gear with both. The model of the electromagnetic field to which Maxwell arrived by the introduction of this device greatly resembles that proposed by Bernoulli in 1736.[20]

He supposed a layer of particles, acting as idle wheels, to be interposed between each vortex and the next, and to roll without sliding on the vortices; so that each vortex tends to make the neighbouring vortices revolve in the same direction as itself. The particles were supposed to be not otherwise constrained, so that the velocity of the centre of any particle would be the mean of the circumferential velocities of the vortices between which it is placed. This condition yields (in suitable units) the analytical equation

where the vector l denotes the flux of the particles, so that its x-component ix, denotes the quantity of particles transferred in unit time across unit area perpendicular to the x-direction. On comparing this equation with that which represents Oersted’s discovery, it is seen that the flux l of the movable particles interposed between neighbouring vortices is the analogue of the electric current.

It will be noticed that in Maxwell’s model the relation between electric current and magnetic force is secured by a connexion which is not of a dynamical, but of a purely kinematical character. The above equation simply expresses the existence of certain non-holonomic constraints within the system.

If from any cause the rotatory velocity of some of the cellular vortices is altered, the disturbance will be propagated from that part of the model to all other parts, by the mutual action of the particles and vortices. This action is determined, as Maxwell showed, by the relation

which connects E, the force exerted on a unit quantity of particles at any place in consequence of the tangential action of the vortices, with Ḣ, the rate of change of velocity of the neighbouring vortices. It will be observed that this equation is not kinematical but dynamical. On comparing it with the electromagnetic equations

it is seen that E must be interpreted electromagnetically as the induced electromotive force. Thus the motion of the particles constitutes an electric current, the tangential force with which they are pressed by the matter of the vortex-cells constitutes electromotive force, and the pressure of the particles on each other may be taken to correspond to the tension or potential of the electricity.

The mechanism must next be extended so as to take account of the phenomena of electrostatics. For this purpose Maxwell assumed that the particles, when they are displaced from their equilibrium position in any direction, exert a tangential action on the elastic substance of the eclls; and that this gives rise to a distortion of the cells, which in turn calls into play a force arising from their elasticity, equal and opposite to the force which urges the particles away from the equilibrium position.

When the exciting force is removed, the cells recover their form, and the electricity returns to its former position. The state of the medium, in which the electric particles are displaced in a definite direction, is assumed to represent an electrostatic field.

Such a displacement does not itself con- stitute a current, because when it has attained a certain value it remains constant; but the variations of displacement are to be regarded as currents, in the positive or negative direction according as the displacement is increasing or diminishing.

The conception of the electrostatic state as a displacement of something from its equilibrium position was not altogether new, although it had not been previously presented in this form.