On Physical Lines of Force/Part II

On Physical Lines of Force → Part II

Philosophical Magazine, Series 4, Vol. XXI, No. 140 (April 1861), pp. 281–291; No. 141 (May 1861), pp. 338–348.

Where Part I left the argument

Maxwell opens by taking stock, and the stock-taking is unusually precise about what has been shown and what has been supposed.

WE have already shown that all the forces acting between magnets, substances capable of magnetic induction, and electric currents, may be mechanically accounted for on the supposition that the surrounding medium is put into such a state that at every point the pressures are different in different directions, the direction of least pressure being that of the observed lines of force, and the difference of greatest and least pressures being proportional to the square of the intensity of the force at that point.

Such a state of stress, if assumed to exist in the medium, and to be arranged according to the known laws regulating lines of force, will act upon the magnets, currents, &c. in the field with precisely the same resultant forces as those calculated on the ordinary hypothesis of direct action at a distance. This is true independently of any particular theory as to the cause of this state of stress, or the mode in which it can be sustained in the medium. We have therefore a satisfactory answer to the question, "Is there any mechanical hypothesis as to the condition of the medium indicated by lines of force, by which the observed resultant forces may be accounted for?" The answer is, the lines of force indicate the direction of minimum pressure at every point of the medium.

Then the second question, and the ranking of confidence that follows it:

The second question must be, "What is the mechanical cause of this difference of pressure in different directions?" We have supposed, in the first part of this paper, that this difference of pressures is caused by molecular vortices, having their axes parallel to the lines of force.

We also assumed, perfectly arbitrarily, that the direction of these vortices is such that, on looking along a line of force from south to north, we should see the vortices revolving in the direction of the hands of a watch.

We found that the velocity of the circumference of each vortex must be proportional to the intensity of the magnetic force, and that the density of the substance of the vortex must be proportional to the capacity of the medium for magnetic induction.

The questions Part II must answer

We have as yet given no answers to the questions, "How are these vortices set in rotation?" and "Why are they arranged according to the known laws of lines of force about magnets and currents?" These questions are certainly of a higher order of difficulty than either of the former; and I wish to separate the suggestions I may offer by way of provisional answer to them, from the mechanical deductions which resolved the first question, and the hypothesis of vortices which gave a probable answer to the second.

Read that sentence slowly, because it is a three-level epistemology stated in a single breath:

level	what it covers	Maxwell's word for it
1	the stress calculation of Part I	mechanical deductions — settled
2	that vortices cause the stress	a probable answer
3	the mechanism about to be introduced	a provisional suggestion

Very few authors label their own confidence this carefully, and fewer still keep the labels straight afterwards. Maxwell does.

And then the admission of ignorance that makes the whole enterprise honest:

We have, in fact, now come to inquire into the physical connexion of these vortices with electric currents, while we are still in doubt as to the nature of electricity, whether it is one substance, two substances, or not a substance at all, or in what way it differs from matter, and how it is connected with it.

We know that the lines of force are affected by electric currents, and we know the distribution of those lines about a current; so that from the force we can determine the amount of the current. Assuming that our explanation of the lines of force by molecular vortices is correct, why does a particular distribution of vortices indicate an electric current? A satisfactory answer to this question would lead us a long way towards that of a very important one, "What is an electric current?"

The argument in order

• The mechanism — the difficulty of neighbouring vortices, and the idle wheel that resolves it
• Prop. IV — the motion of a layer of particles separating two vortices
• Prop. V — the transference of particles across unit area. Equation (33) turns out to be equation (9).
• Prop. VI — the energy of a portion of the medium due to the motion of its vortices
• Prop. VII — the energy spent on a vortex by the layer of particles around it
• Prop. VIII — the relation between changing motion and the forces on the particles. Equations (54) to (58): induction.
• Props. IX and X — an element of the medium strained and rotated
• Prop. XI — the electromotive force in a moving body. Equation (77).
• Induced currents — the physical account, and the anchor-ring experiment

Vortex cells and the particles between them

What Part II achieves

Three things, in ascending order of importance.

1. It removes the mechanical scandal. Vortices all turning the same way cannot touch. A layer of rolling particles between them can make them turn the same way, and does so with a known piece of engineering — the idle wheel of a gear train.
2. It makes the current equation a result rather than a definition. In Part I, equations (9) were written down so that Prop. III would come out right. In Prop. V the same relation is derived from counting particles across a surface. Maxwell's own remark, when he arrives: equation (33) "will be identical with the first of equations (9)".
3. It gets induction from inertia. Vortices have mass; mass resists being started and stopped; the resistance is felt by the particles; and particles in motion are a current. Faraday's law appears in Prop. VIII as a statement about the bookkeeping of energy in a mechanical system.

And what it costs

The price is a piece of machinery Maxwell himself will not defend as real. His verdict, given near the end of Part II:

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.

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Next: The mechanism · Previous: Part I · On to Part III

Plain-language versions: What is an electric current? · Why a changing magnet makes a current