Part II — Props. IX and X

On Physical Lines of Force → Part II → Props. IX and X

Maxwell's statements

Prop. IX. — To find the variations of α, β, γ in the parallelopiped when x becomes x(1+ε), y becomes y(1+ε′), and z becomes z(1+ε″), the volume of the figure remaining the same.

Prop. X. — To find the variations of α, β, γ due to a rotation about the axis of x from 0 to θ, a rotation about the axis of y from 0 to θ′, and a rotation about the axis of z from 0 to θ″.

Provenance: these two statements rest on a single extraction and were not independently corroborated. See Sources.

What the pair is for

Everything so far has treated the medium as sitting still while the vortices change their speed. Prop. XI will ask what happens in a body that is moving — a wire dragged through a magnetic field, say. Before that question can be answered, Maxwell needs to know how the magnetic state of a small element of the medium changes when the element itself is carried along, stretched, squashed and turned.

Any small deformation can be decomposed into three pieces:

1. a translation, which changes nothing about the element's internal state;
2. a pure strain — stretching along the three axes with the volume preserved. This is Prop. IX;
3. a rotation about the three axes. This is Prop. X.

Take the two together and you have the complete change in $\alpha ,\beta ,\gamma$ produced by carrying an element about and deforming it. Prop. XI then adds the change produced by the electromotive forces, and the sum is the total rate of change.

Prop. IX: the strain

Stretch the element by $1+\epsilon$ along $x$, $1+{\epsilon }^{\prime }$ along $y$, $1+{\epsilon }^{″}$ along $z$, with the volume held constant.

The magnetic induction behaves as a flux through an area. Stretch an element along $x$ and the cross-section perpendicular to $x$ shrinks in proportion, so the same number of lines threads a smaller area and $\mu \alpha$ rises. Written out to first order:

$$\delta (\mu \alpha )=\mu \alpha \epsilon ,\delta (\mu \beta )=\mu \beta {\epsilon }^{\prime },\delta (\mu \gamma )=\mu \gamma {\epsilon }^{″}$$

Provenance: this expression is a reconstruction of the proposition's content; the printed form and its equation numbers could not be established. See Sources.

This is the statement that the magnetic lines are frozen into the medium — stretch the medium along a line of force and the line stretches with it. That property is exactly what makes the vortex picture coherent: a vortex is a piece of the medium in rotation, so carrying the medium about carries the vortex, axis and all.

Prop. X: the rotation

Rotate the element and the axes of the vortices rotate with it. For small rotations $\theta ,{\theta }^{\prime },{\theta }^{″}$ about the three axes, the components transform as any vector does:

$$\delta \alpha =\gamma {\theta }^{\prime }-\beta {\theta }^{″},\delta \beta =\alpha {\theta }^{″}-\gamma \theta ,\delta \gamma =\beta \theta -\alpha {\theta }^{\prime }$$

—— which is the cross product of the rotation with the field.

Why Maxwell needs this and Faraday did not

Faraday's law as an experimenter states it concerns a circuit, and it does not much matter whether the flux through the circuit changes because the magnet moved or because the circuit did. As a statement about a field at a point, the two cases look quite different, and Maxwell is now committed to statements about points.

The distinction is one that his notation hides and his prose carries. He writes $d/dt$ for both

• the rate of change at a fixed point in space, and
• the rate of change following a moving element of the medium.

Props. IX and X supply the difference between the two. In modern language they compute the convective terms in the material derivative

$$\frac{D}{Dt}=\frac{\partial }{\partial t}+(𝐯\cdot \mathrm{\nabla })$$

together with the strain and rotation contributions that arise because $𝐁$ is a flux density carried by the medium rather than a scalar painted on it. The whole apparatus is the same one used in fluid mechanics for a vector field frozen into a moving fluid, and it is why the magnetic field in a moving conductor obeys the equation it does.

The result to carry forward

Combine Props. IX and X and you have the change in the magnetic state of an element from motion alone. Add the change from electromotive force, already given by Prop. VIII equation (54), and the sum is the complete account. Maxwell states the plan at the head of Prop. XI:

The variation of the velocity of the vortices in a moving element is due to two causes — the action of the electromotive forces, and the change of form and position of the element.

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