Part II — Prop. V

On Physical Lines of Force → Part II → Prop. V

Maxwell's statement

Prop. V. — To determine the whole amount of particles transferred across unit of area in the direction of x in unit of time.

This is the payoff proposition of Part II, and arguably of the whole paper up to this point.

The calculation

Take the expression from Prop. IV for the velocity of the particle layer on any surface separating two vortices. Multiply by the surface density of particles. Sum over every such surface within a volume of the medium, and ask how many particles cross unit area in unit time.

The integration is over closed surfaces, so the terms linear in the coordinates integrate to zero and only terms of the form $\int xdydz$ survive, contributing the enclosed volume. What comes out is equation (33).

Then comes the step that makes the proposition work. The surface density of the particles has not yet been fixed, and Maxwell fixes it — equation (34) — with a single purpose in mind:

If we make [equation (34)] then equation (33) will be identical with the first of equations (9), which give the relation between the quantity of an electric current and the intensity of the lines of force surrounding it.

$$4\pi p=\frac{d\gamma }{dy}-\frac{d\beta }{dz},4\pi q=\frac{d\alpha }{dz}-\frac{d\gamma }{dx},4\pi r=\frac{d\beta }{dx}-\frac{d\alpha }{dy}\mathrm{(}\mathrm{9}\mathrm{=}\mathrm{33}\mathrm{)}$$

Provenance: the equation numbers (33) and (34) are attested by Maxwell's own citations. The body above is the reconstruction of equations (9) given on the Prop. III page; the printed form of (33) and the value assigned to the surface density in (34) could not be recovered. See Sources.

Why this is more than bookkeeping

A sceptical reader will say: Maxwell chose the particle density in order to reproduce equations (9), so of course they came out. That objection misses what was and was not free.

• The numerical constant was free, and Maxwell fixed it. That much is fitting.
• The form was not free. The calculation could have produced anything — a divergence, a gradient, a second derivative, an expression involving the field itself rather than its variation. What it produced was the antisymmetric combination of first derivatives that Ampère's law requires, and it did so because of the difference structure inherited from Prop. IV.

One free constant fixed, and an entire differential relation reproduced. That is a genuine result, and it is the moment at which the mechanical model stops merely re-describing electromagnetism and starts predicting it.

Circulation round a closed curve equals the current enclosed

The identification

Now Maxwell says what an electric current is, on his hypothesis.

It appears therefore that, according to our hypothesis, an electric current is represented by the transference of the moveable particles interposed between the neighbouring vortices. We may conceive that these particles are very small compared with the size of a vortex, and that the mass of all the particles together is inappreciable compared with that of the vortices, and that a great many vortices, with their surrounding particles, are contained in a single complete molecule of the medium. The particles must be conceived to roll without sliding between the vortices which they separate, and not to touch each other, so that, as long as they remain within the same complete molecule, there is no loss of energy by resistance. When, however, there is a general transference of particles in one direction, they must pass from one molecule to another, and in doing so, may experience resistance, so as to waste electrical energy and generate heat.

Three quantitative commitments are made in that passage, all of them the kind that keep a model honest:

• the particles are small compared with a vortex;
• their total mass is negligible compared with the vortices' — which is why they never appear in the energy calculation of Prop. VI;
• many vortices and their particles make up one "complete molecule" of the medium.

Why the lines of force arrange themselves

Maxwell closes the proposition by answering the second of the two questions he set at the start of Part II — why the vortices arrange themselves according to the known laws.

Now let us suppose the vortices arranged in a medium in any arbitrary manner. The quantities [the derivatives], &c. will then in general have values, so that there will at first be electrical currents in the medium. These will be opposed by the electrical resistance of the medium; so that, unless they are kept up by a continuous supply of force, they will quickly disappear, and we shall then have [the derivatives vanishing]; that is, [the magnetic force] will be a complete differential (see equations (15) and (16)); so that our hypothesis accounts for the distribution of the lines of force.

The argument is neat. Arrange the vortices carelessly and their rotation varies from place to place; by Prop. IV that variation is a current; the medium's resistance kills the current; and the arrangement left over is the one in which the variation has vanished — that is, the one in which the magnetic force is derivable from a potential. The observed distribution of lines of force is the equilibrium state of the machine.

What it means today

In modern notation equation (9) is

$$\mathrm{\nabla }\times 𝐇=4\pi 𝐉$$

which is Ampère's circuital law. What Prop. V provides is a mechanical reason for it: current is the drift of the idle wheels, and idle wheels drift only where neighbouring rotations differ.

It is worth noting what is missing. Take the divergence of both sides: the left is identically zero, so this equation demands $\mathrm{\nabla }\cdot 𝐉=0$ — current can never accumulate anywhere. That is true of steady currents and false in general, and it is the defect Prop. XIV will repair with the displacement current. Maxwell does not notice it here; he arrives at it from the elasticity of the cells, by a completely different route.

See the modern derivation.

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Next: Prop. VI · Previous: Prop. IV