On Physical Lines of Force — Part III

Philosophical Magazine, Series 4, Vol. XXIII, No. 151, January 1862, pp. 12–24.

The printed heading:

III. On Physical Lines of Force. By J. C. Maxwell, F.R.S., Professor of Natural Philosophy in King's College, London. Part III.— The Theory of Molecular Vortices applied to Statical Electricity.

Eight months separate this instalment from the last. It contains the two results the paper is remembered for.

Maxwell's own recapitulation

IN the first part of this paper I have shown how the forces acting between magnets, electric currents, and matter capable of magnetic induction may be accounted for on the hypothesis of the magnetic field being occupied with innumerable vortices of revolving matter, their axes coinciding with the direction of the magnetic force at every point of the field.
The centrifugal force of these vortices produces pressures distributed in such a way that the final effect is a force identical in direction and magnitude with that which we observe.
In the second part I described the mechanism by which these rotations may be made to coexist, and to be distributed according to the known laws of magnetic lines of force.
I conceived the rotating matter to be the substance of certain cells, divided from each other by cell-walls composed of particles which are very small compared with the cells, and that it is by the motions of these particles, and their tangential action on the substance in the cells, that the rotation is communicated from one cell to another.

The new assumption, and its reason

Here is the sentence on which the rest of physics turns.

I have not attempted to explain this tangential action, but it is necessary to suppose, in order to account for the transmission of rotation from the exterior to the interior parts of each cell, that the substance in the cells possesses elasticity of figure, similar in kind, though different in degree, to that observed in solid bodies. The undulatory theory of light requires us to admit this kind of elasticity in the luminiferous medium, in order to account for transverse vibrations. We need not then be surprised if the magneto-electric medium possesses the same property.

Note the reason given. The cells must be springy because otherwise the outside of a cell could not drag the inside round with it. That is a purely mechanical repair to a purely mechanical difficulty. The remark about the luminiferous medium is offered only as reassurance that the assumption is not outlandish.

Everything that follows — the displacement current, the electromagnetic theory of light — is a consequence of a patch applied for an unrelated reason. It is the most productive afterthought in the history of the subject.

What the parts of the machine now mean

According to our theory, the particles which form the partitions between the cells constitute the matter of electricity. The motion of these particles constitutes an electric current; the tangential force with which the particles are pressed by the matter of the cells is electromotive force, and the pressure of the particles on each other corresponds to the tension or potential of the electricity.

mechanical part	electrical meaning
the particles between the cells	the matter of electricity
the motion of the particles	an electric current
the tangential force of the cells on the particles	electromotive force
the pressure of the particles on one another	electric tension, or potential

If we can now explain the condition of a body with respect to the surrounding medium when it is said to be "charged" with electricity, and account for the forces acting between electrified bodies, we shall have established a connexion between all the principal phenomena of electrical science.

On the range of electrical resistance

We know by experiment that electric tension is the same thing, whether observed in statical or in current electricity; so that an electromotive force produced by magnetism may be made to charge a Leyden jar, as is done by the coil machine.
When a difference of tension exists in different parts of any body, the electricity passes, or tends to pass, from places of greater to places of smaller tension. If the body is a conductor, an actual passage of electricity takes place; and if the difference of tensions is kept up, the current continues to flow with a velocity proportional inversely to the resistance, or directly to the conductivity of the body.
The electric resistance has a very wide range of values, that of the metals being the smallest, and that of glass being so great that a charge of electricity has been preserved in a glass vessel for years without penetrating the thickness of the glass.

The footnote to the last sentence credits Professor W. Thomson.

The argument in order

Conduction and induction — two independent qualities of bodies; the porous membrane and the elastic one; the definition of electric displacement
Prop. XII — the equilibrium of an elastic sphere under surface forces
Prop. XIII — electromotive force and displacement. Equations (101), (105), (108).
Prop. XIV — the displacement current. Equations (112) and (113).
Prop. XV — the force between two electrified bodies. Equations (116) to (125).
Prop. XVI — the velocity of transverse undulations, and the identification of light.
Prop. XVII — the capacity of a Leyden jar; dielectrics and refractive index; crystals

What Maxwell says he has done

From the introductory pages, before any of the propositions:

I have deduced from this result the relation between the statical and dynamical measures of electricity, and have shown, by a comparison of the electro-magnetic experiments of MM. Kohlrausch and Weber with the velocity of light as found by M. Fizeau, that the elasticity of the magnetic medium in air is the same as that of the luminiferous medium, if these two coexistent, coextensive, and equally elastic media are not rather one medium.

That last clause is one of the great sentences in physics, and it is delivered as a subordinate qualification in a paragraph of housekeeping. Two media that occupy the same space, have the same extent, and have the same elasticity — if they are not rather one medium.

A caution about the sphere

In the following investigation I have considered the relation between the displacement and the force producing it, on the supposition that the cells are spherical. The actual form of the cells probably does not differ from that of a sphere sufficiently to make much difference in the numerical result.

Maxwell knows the cells cannot really be spheres — spheres do not fill space — and says so, and estimates that it will not matter much. It is a good example of a habit worth imitating: name the idealisation, say why you can live with it, move on.

Next: Conduction and induction · Previous: Part II · On to Part IV

Plain-language versions: Why the cells had to be springy · The missing term · How Maxwell found the speed of light