On Physical Lines of Force/Part IV

On Physical Lines of Force → Part IV

Philosophical Magazine, Series 4, Vol. XXIII, No. 152, February 1862, pp. 85–95.

What Part IV is about

Faraday had discovered in 1845 that when plane-polarized light travels along the lines of magnetic force through a transparent substance, the plane of polarization is turned. Part IV asks what Maxwell's spinning cells predict about that, and answers with six proportionalities.

But the physics is not the main reason Part IV is worth reading. Its opening pages contain the clearest statement Maxwell ever made of why he built the model the way he did — why magnetism is treated as rotation and electric currents as translation, rather than the other way round.

The opening: one mathematical form, many phenomena

THE connexion between the distribution of lines of magnetic force and that of electric currents may be completely expressed by saying that the work done on a unit of imaginary magnetic matter, when carried round any closed curve, is proportional to the quantity of electricity which passes through the closed curve. The mathematical form of this law may be expressed as in equations (9), which I here repeat, where [α, β, γ] are the rectangular components of magnetic intensity, and p, q, r are the rectangular components of steady electric currents …

The same mathematical connexion is found between other sets of phenomena in physical science.

(1) If [α, β, γ] represent displacements, velocities, or forces, then p, q, r will be rotatory displacements, velocities of rotation, or moments of couples producing rotation, in the elementary portions of the mass.

(2) If [α, β, γ] represent rotatory displacements in a uniform and continuous substance, then p, q, r represent the relative linear displacement of a particle with respect to those in its immediate neighbourhood. See a paper by Prof. W. Thomson "On a Mechanical Representation of Electric, Magnetic, and Galvanic Forces," Camb. and Dublin Math. Journ. Jan. 1847.

(3) If [α, β, γ] represent the rotatory velocities of vortices whose centres are fixed, then p, q, r represent the velocities with which loose particles placed between them would be carried along. See the second part of this paper (Phil. Mag. April 1861).

The relation between circulation and enclosed current

Three physically distinct systems, all obeying the same relation between a vector and its curl. Maxwell is pointing out that the mathematics does not, on its own, tell you which of the two quantities is the rotatory one.

It appears from all these instances that the connexion between magnetism and electricity has the same mathematical form as that between certain pairs of phenomena, of which one has a linear and the other a rotatory character.

The fork in the road

The alternatives are named, with their proponents.

Professor Challis conceives magnetism to consist in currents of a fluid whose direction corresponds with that of the lines of magnetic force; and electric currents, on this theory, are accompanied by, if not dependent on, a rotatory motion of the fluid about the axes of the current. Professor Helmholtz has investigated the motion of an incompressible fluid, and has conceived lines drawn so as to correspond at every point with the instantaneous axis of rotation of the fluid there. He has pointed out that the lines of fluid motion are arranged according to the same laws with respect to the lines of rotation, as those by which the lines of magnetic force are arranged with respect to electric currents. On the other hand, in this paper I have regarded magnetism as a phenomenon of rotation, and electric currents as consisting of the actual translation of particles, thus assuming the inverse of the relation between the two sets of phenomena.

So there were two live options in 1861, and Maxwell took the one opposite to Challis and Helmholtz. Part IV exists to justify the choice.

The principle he appeals to

Now it seems natural to suppose that all the direct effects of any cause which is itself of a longitudinal character, must be themselves longitudinal, and that the direct effects of a rotatory cause must be themselves rotatory. A motion of translation along an axis cannot produce a rotation about that axis unless it meets with some special mechanism, like that of a screw, which connects a motion in a given direction along the axis with a rotation in a given direction round it; and a motion of rotation, though it may produce tension along the axis, cannot of itself produce a current in one direction along the axis rather than the other.

Rotatory causes have rotatory effects; longitudinal causes have longitudinal effects. Absent a screw, the two do not convert into each other. So: look at the effects, and read off the character of the cause. That is the whole argument, and the rest of the opening applies it.

The argument in order

• Why magnetism must be rotatory — the evidence from electrolysis and from Faraday's rotation; Thomson's symmetry argument
• The six results, (A) to (F) — what the vortex model predicts about the amount of rotation, and how it fared against Verdet's measurements
• What became of the vortices — the 1865 Dynamical Theory, the 1873 Treatise, and how historians read the retreat

A note on this Part's text

Part IV is the least well established of the four in this edition. Whether it contains numbered Propositions could not be determined — proposition numbering certainly reaches XVII at the end of Part III, but no Prop. XVIII could be confirmed. No numbered equation from Part IV was recovered, and the final paragraph of the paper could not be obtained at all. See Sources and editorial method.

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Plain-language version: Magnetism twists light