On Physical Lines of Force/Part I

On Physical Lines of Force → Part I

Philosophical Magazine, Series 4, Vol. XXI, No. 139, March 1861, pp. 161–175.

What Part I sets out to do

The magnetic forces were already known. What was not known was what, if anything, occupies the space in which they act. Part I asks a single question: what state of a medium would produce exactly the forces we observe, and nothing else?

The answer is arrived at in two moves.

First, by elimination. Maxwell reviews the ways a medium under magnetic influence has been conceived — as currents, as undulations, as displacement or strain, as pressure or stress — and settles on stress, because the mathematics of attraction resembles that of currents, conduction and elastic bodies, and does not resemble that of waves.

Second, by mechanism. The stress required is a peculiar one: a tension along each line of force, combined with a pressure across it. What in a mobile medium could produce unequal pressures in different directions? Maxwell's answer: the centrifugal force of eddies whose axes lie along the lines.

Tension along the lines of force, pressure across them

The argument in order

• The introductory discussion — what a line of force is; why action at a distance dissatisfies; the survey of rival hypotheses; the turn to stress; the hypothesis of molecular vortices
• Prop. I — that the pressure a vortex produces depends on its density and the square of its velocity, but not at all on its size. This is what licenses vortices of molecular scale.
• Prop. II — the stress of a vortex whose axis has direction-cosines l, m, n, resolved into three normal and three tangential stresses on the coordinate planes.
• Prop. III — the force on an element of the medium arising from the variation of that stress from point to point. The result, equations (12), (13), (14), contains five terms — and every one of them is a known force of magnetism or electromagnetism.
• The applications — a single magnetic pole; a straight cylindrical current; the attraction of two parallel currents. Including a prediction Maxwell offers for test.

The result to carry forward

The centre of Part I is Prop. III. Maxwell takes the divergence of his stress system and reads the answer term by term:

The first term of each expression refers to the force acting on magnetic poles. The second term to the action on bodies capable of magnetism by induction. The third and fourth terms to the force acting on electric currents. And the fifth to the effect of simple pressure.

One state of stress, differentiated, yields the whole of magnetostatics and the force on currents at once. That is the claim Part I establishes, and it is a strong one — strong enough that the rest of the paper is an attempt to keep it while fixing the mechanical difficulties it creates.

In the course of interpreting the third term Maxwell defines p, q, r as the electric current per unit area perpendicular to the axes, which gives him equations (9) — the relation between a current and the magnetic force circulating round it. He will use those equations, by number, in all three later Parts.

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

Provenance: the equation number is attested by Maxwell's own cross-references. The body above is a reconstruction in his notation — see Sources and editorial method.

What Part I leaves unanswered

Maxwell is candid at the opening of Part II about what he has and has not done. He has shown that a stress arranged according to the known laws of lines of force acts on magnets and currents exactly as direct action at a distance would. He has proposed vortices as the cause of that stress. He has not explained:

• how the vortices are set in rotation;
• why they arrange themselves according to the known laws;
• how neighbouring vortices, all turning the same way, can touch at all without grinding against each other.

The third of these is the mechanical scandal that drives Part II — and the fix for it turns out to be electricity.

Closing sentence of Part I

We shall next consider the nature of electric currents and electromotive forces in connexion with the theory of molecular vortices.

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Next: The introductory discussion · On to Part II

Plain-language versions: What is a line of force? · Why Maxwell wanted a mechanism · The sea of spinning cells · Why spinning makes a pull