On Physical Lines of Force/Part II/The mechanism of idle wheels
On Physical Lines of Force → Part II → The mechanism
The difficulty
I have found great difficulty in conceiving of the existence of vortices in a medium, side by side, revolving in the same direction about parallel axes. The contiguous portions of consecutive vortices must be moving in opposite directions; and it is difficult to understand how the motion of one part of the medium can coexist with, and even produce, an opposite motion of a part in contact with it.
The objection is elementary and fatal. Two wheels side by side, both turning clockwise: where they touch, one surface is moving up and the other down. They cannot both be right. Either the vortices are not in contact, or they do not all turn the same way, or something sits between them.
The fix
The only conception which has at all aided me in conceiving of this kind of motion is that of the vortices being separated by a layer of particles, revolving each on its own axis in the opposite direction to that of the vortices, so that the contiguous surfaces of the particles and of the vortices have the same motion.
In mechanism, when two wheels are intended to revolve in the same direction, a wheel is placed between them so as to be in gear with both, and this wheel is called an "idle wheel." The hypothesis about the vortices which I have to suggest is that a layer of particles, acting as idle wheels, is interposed between each vortex and the next, so that each vortex has a tendency to make the neighbouring vortices revolve in the same direction with itself.
This is engineering, not physics, and Maxwell says so — he reaches for the machine shop rather than for a differential equation. Then he notes the specific variant he needs:
In mechanism, the idle wheel is generally made to rotate about a fixed axle; but in epicyclic trains and other contrivances, as, for instance, in Siemens's governor for steam-engines, we find idle wheels whose centres are capable of motion. In all these cases the motion of the centre is the half sum of the motions of the circumferences of the wheels between which it is placed.
The last sentence is the one that does the work. An ordinary idle wheel spins on a fixed pin. Maxwell's must be free to travel, because it is the travelling that he is about to identify with electricity. And when its centre is free, its velocity is the half sum of the two surface velocities it engages. That rule is Prop. IV, and everything about electric currents in this paper follows from it.
Why the particles must roll without sliding
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.
Two conditions, doing two different jobs.
- Rolling without sliding is what makes the gearing exact. If the particles slipped, the contact velocities would not match, the half-sum rule of Prop. IV would fail, and the kinematics would collapse.
- Not touching each other keeps the particles from grinding among themselves.
Together they make the medium lossless inside a molecule. All electrical resistance is then confined to the passage of particles between molecules — which is where Joule heating comes from on this model. That is a genuine physical consequence, not a patch: the model predicts that resistance is a property of the transfer between molecular units and not of the transmission of magnetic action, which is why a magnetic field can exist indefinitely in a medium that is a perfect insulator.
The figure
Maxwell's own figure is a plane section. In his description:
Let AB, Pl. V. fig. 2, represent a current of electricity in the direction from A to B. Let the large spaces above and below AB represent the vortices, and let the small circles separating the vortices represent the layers of particles placed between them, which in our hypothesis represent electricity.
and then he sets it going:
Now let an electric current from left to right commence in AB. The row of vortices above AB will be set in motion in the opposite direction to that of a watch.
The rows above and below the current turn opposite ways, and the small circles between them turn as idlers. The figure is meant to be read dynamically, row by row, watching the motion propagate outward from the current — which is exactly the reading required for induced currents.
A second figure in the same plate, fig. 1, fixes the handedness: a voltaic circuit of copper and zinc, a line of magnetic force embracing it, and two vortices seated on that line.
Maxwell's assessment of his own machine
It is worth having the full passage in one place, because it is the most quoted and most misquoted thing in the paper.
The conception of a particle having its motion connected with that of a vortex by perfect rolling contact may appear somewhat awkward. I do not bring it forward as a mode of connexion existing in nature, or even as that which I would willingly assent to as an electrical hypothesis.
It is, however, a mode of connexion which is mechanically conceivable, and easily investigated, and it serves to bring out the actual mechanical connexions between the known electro-magnetic phenomena; so that I venture to say that any one who understands the provisional and temporary character of this hypothesis, will find himself rather helped than hindered by it in his search after the true interpretation of the phenomena.
The historian Bruce Hunt has pointed out that these words are routinely quoted as Maxwell disowning the whole vortex theory, and that this is a misreading: the "somewhat awkward" thing is the rolling contact between particle and vortex, not the vortices. Maxwell seems to have thought the vortices were probably real. The idle wheels he offered as a device.
The distinction matters for reading Part III. When the cells turn out to need elasticity, Maxwell treats that as a discovery about a real medium — and it is the discovery that produces the displacement current.
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Plain-language version: What is an electric current?