On Physical Lines of Force/Part IV/The six results

On Physical Lines of Force → Part IV → The six results

Maxwell's statement of the answer

In the following investigation, I have found that the only effect which the rotation of the vortices will have on the light will be to make the plane of polarization rotate in the same direction as the vortices, through an angle proportional —

(A) to the thickness of the substance, (B) to the resolved part of the magnetic force parallel to the ray, (C) to the index of refraction of the ray, (D) inversely to the square of the wave-length in air, (E) to the mean radius of the vortices, (F) to the capacity for magnetic induction.

Six proportionalities, of which two were already confirmed, two were testable, one was a slight correction to received opinion, and one was unmeasurable except by this route.

What each one says

	the rotation is proportional to	status in 1862
(A)	the thickness of the substance	confirmed by Verdet
(B)	the component of the magnetic force along the ray	confirmed by Verdet
(C)	the index of refraction	Maxwell's own correction to received opinion
(D)	the inverse square of the wavelength in air	approximately right, and known to be imperfect
(E)	the mean radius of the vortices	not otherwise measurable
(F)	the capacity for magnetic induction	testable in principle

Verdet's confirmations, and Maxwell's correction

A and B have been fully investigated by M. Verdet, who has shown that the rotation is strictly proportional to the thickness and to the magnetizing force, and that, when the ray is inclined to the magnetizing force, the rotation is as the cosine of that inclination. D has been supposed to give the true relation between the rotation of different rays; but it is probable that C must be taken into account in an accurate statement of the phenomena: The rotation varies, not exactly inversely as the square of the wave-length, but a little faster; so that for the highly refrangible rays the rotation is greater than that given by this law, but more nearly as the index of refraction divided by the square of the wave-length.

Two things are happening in that passage.

Verdet had done the work on A and B, and Maxwell says so plainly. The cosine dependence on inclination is exactly what "the resolved part of the magnetic force parallel to the ray" means, so (B) is confirmed in its precise form and not merely in outline.

On (C) and (D), Maxwell is correcting the received account with his own model. The accepted rule was that the rotation goes as the inverse square of the wavelength. Observation showed it going a little faster than that at the blue end of the spectrum. Maxwell's model says the rule should be the index of refraction divided by the square of the wavelength — and since the index of refraction rises toward the blue, that predicts precisely the observed excess.

This is the one place in Part IV where the vortex model produces a correction to an existing empirical rule and the correction points the right way. In modern terms, the quantity in question is the Verdet constant, and its wavelength dependence does involve dispersion in essentially the manner Maxwell indicates.

Why magnetism turns the plane of polarization

Result (E): measuring the unmeasurable

The relation (E) between the amount of rotation and the size of the vortices shows that different substances may differ in rotating power independently of any observable difference in other respects. We know nothing of the absolute size of the vortices; and on our hypothesis the optical phenomena are probably the only data for determining their relative size in different substances.

This is a striking piece of scientific bookkeeping, and it closes a loop opened in Prop. I.

Prop. I proved that the magnetic behaviour of the medium is independent of vortex size — which is what allowed Maxwell to say "molecular" without saying how small. The parameter dropped out of Parts I, II and III entirely.

Here it reappears. Magneto-optical rotation is the one phenomenon in the paper that depends on the size of the vortices. So the model, having declined to specify the parameter, now offers the only means of measuring it — and Maxwell notes that this explains something otherwise puzzling: two substances alike in every measurable respect may still differ in rotating power, because their vortices differ in size.

It is a proper prediction with a proper observational consequence, and it is the kind of thing that makes a model worth having even when it is wrong.

Where the recoverable text ends

On our theory, the direction of the rotation of the plane of polarization depends on that of the mean moment of momenta, or angular momentum, of the molecular vortices …

The transcription available for this edition breaks off here, mid-sentence. The mathematical investigation that follows, its equations, and the final paragraph of the entire paper could not be obtained. See Sources and editorial method.

What can be said is that the mechanism is the one Thomson's argument requires: the two circularly polarized components travel at different speeds because one circulates with the vortices and the other against them; a difference in speed is a difference in accumulated phase; and a phase difference between two counter-rotating circular components is exactly a rotation of the plane of the polarization they combine to make. The modern treatment is on the Faraday rotation page.

What Part IV amounts to

Judged as physics, Part IV is the weakest of the four. Its central mechanism — that light and magnetism are vibrations and rotations of one and the same matter — did not survive, and the modern account of Faraday rotation runs through the response of bound electrons in a magnetic field, not through the angular momentum of the ether.

Judged as reasoning, it is the most revealing. It is where Maxwell explains why he built what he built, credits Thomson with the argument that supports it, corrects an empirical rule and gets the correction right, and identifies the single observable that could measure the one parameter his theory had left free.

And the physical idea at its centre — that magneto-optical rotation shows something in the field is genuinely turning — was right. Magnetism is associated with angular momentum. It is the angular momentum of electrons rather than of cells of ether, but the character of the thing was correctly read off from Faraday's experiment, in 1862, by an argument about the symmetry of a helix.

---

Next: What became of the vortices · Previous: Why magnetism must be rotatory