On Physical Lines of Force/Part I/Prop. II

On Physical Lines of Force → Part I → Prop. II

Maxwell's statement

Prop. II. — If the direction-cosines of the axes of the vortices with respect to the axes of x, y, and z be l, m, and n, to find the normal and tangential stresses on the coordinate planes.

The decomposition

Maxwell first resolves the state of the medium into two pieces, which is the whole trick of the proposition:

a simple hydrostatic pressure acting in all directions, and a simple tension … acting along the axis of stress

Resolving the stress on an element of the medium

A hydrostatic pressure is isotropic and contributes the same amount to every normal stress and nothing to any tangential stress. The tension is the interesting part: it acts along one direction only, the axis of the vortices, and it is that single directed tension which carries all the magnetic behaviour.

If the magnetic force has components $\alpha ,\beta ,\gamma$ along the coordinate axes, then the axis direction-cosines are

$$l=\frac{\alpha }{\sqrt{{\alpha }^2+{\beta }^2+{\gamma }^2}},m=\frac{\beta }{\sqrt{{\alpha }^2+{\beta }^2+{\gamma }^2}},n=\frac{\gamma }{\sqrt{{\alpha }^2+{\beta }^2+{\gamma }^2}}$$

and the tension along that axis is proportional to $\mu ({\alpha }^2+{\beta }^2+{\gamma }^2)$ — the density of the vortex substance times the square of its circumferential velocity, exactly as Prop. I requires.

The six stresses

Projecting a tension $T$ along the direction $(l,m,n)$ onto the coordinate planes, and adding a uniform pressure $p_0$, gives three normal and three tangential stresses:

$$p_{xx}=\frac{\mu }{4\pi }{\alpha }^2-p_0,p_{yy}=\frac{\mu }{4\pi }{\beta }^2-p_0,p_{zz}=\frac{\mu }{4\pi }{\gamma }^2-p_0$$

$$p_{xy}=\frac{\mu }{4\pi }\alpha \beta ,p_{yz}=\frac{\mu }{4\pi }\beta \gamma ,p_{zx}=\frac{\mu }{4\pi }\gamma \alpha$$

Provenance: the printed equation numbers for Prop. II could not be established, and these bodies are reconstructions in Maxwell's notation. The content is standard and is corroborated by Maxwell's own term-by-term gloss on the result of Prop. III. See Sources.

Read the pair of lines together and a pattern appears: every component has the form

$$p_{ij}=\frac{\mu }{4\pi }{H}_i{H}_j-p_0{\delta }_{ij}$$

What this is, in modern terms

This is the Maxwell stress tensor of the magnetic field, and Part I is where it enters physics. In modern notation, with $𝐁=\mu 𝐇$,

$$T_{ij}=\frac{1}{4\pi }(B_i{H}_j-\frac{1}{2}{\delta }_{ij}𝐁\cdot 𝐇)$$

the $-\frac{1}{2}{\delta }_{ij}𝐁\cdot 𝐇$ being Maxwell's hydrostatic term, once the part of $p_0$ that comes from the vortex motion is separated from the part that is ordinary fluid pressure.

The physical content is the sentence that survives every later reformulation: along a line of force the medium is in tension; across it, in compression. The tensor is just that statement written six times.

See the modern derivation for the conversion in full, including where the 4π goes and why the momentum flux interpretation only became available later.

A note on what Maxwell has and has not proved

Nothing so far requires vortices. Prop. II would hold for any mechanism producing a tension along one axis and a pressure across it. Maxwell is careful about this at the opening of Part II:

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.

The stress result is secure. The vortices are the explanation offered for the stress, and are held to a lower standard — Maxwell calls the vortex hypothesis a "probable answer", against the "mechanical deductions" which he regards as settled.

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Next: Prop. III · Previous: Prop. I