Part II — Prop. XI

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

Maxwell's statement

Prop. XI. — To find the electromotive forces in a moving body.

And his opening sentence, which is the whole plan of the proposition:

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.

The assembly

Everything needed is already in hand.

• The change from electromotive force — Prop. VIII, equation (54).
• The change from deformation and rotation of the moving element — Props. IX and X.

Add them, require that the total account be consistent, and the electromotive force acquires a term it did not have before. The result is equation (77), the climax of Part II:

$$P=\mu (\gamma \frac{dy}{dt}-\beta \frac{dz}{dt})-\frac{dF}{dt}-\frac{d\psi }{dx}\mathrm{(}\mathrm{77}\mathrm{)}$$

$$Q=\mu (\alpha \frac{dz}{dt}-\gamma \frac{dx}{dt})-\frac{dG}{dt}-\frac{d\psi }{dy}$$

$$R=\mu (\beta \frac{dx}{dt}-\alpha \frac{dy}{dt})-\frac{dH}{dt}-\frac{d\psi }{dz}$$

Provenance: the equation number (77) is attested and independently corroborated as the complete electromotive-force relation including the velocity-dependent term. The body is a reconstruction in Maxwell's notation. The numbering between (58) and (77) could not be resolved. See Sources.

Reading the three terms

term	mechanically	modern form	name today
$\mu (\gamma \dot{y}-\beta \dot{z})$	the element is carried through the field	$𝐯\times 𝐁$	the motional term
$-dF/dt$	the electromagnetic momentum of the medium is changing	$-\partial 𝐀/\partial t$	the induced term
$-d\psi /dx$	the particles press on one another	$-\mathrm{\nabla }\phi$	the electrostatic term

In vector notation the whole of (77) is

$$𝐄=𝐯\times 𝐁-\frac{\partial 𝐀}{\partial t}-\mathrm{\nabla }\phi$$

What the first term is

$𝐯\times 𝐁$ is the magnetic part of what is now called the Lorentz force — the force felt by a charge moving through a magnetic field. It appears here, in 1861, thirty-odd years before Lorentz, and it appears not as a postulate about charges but as a consequence of carrying a piece of the medium about.

The distinction Maxwell is making is one that would trouble physics for the next forty years. Consider a magnet and a loop of wire, in relative motion:

• Move the loop, hold the magnet still. The element of the medium is transported through a field that is not changing. The first term of (77) supplies the electromotive force.
• Move the magnet, hold the loop still. The field at the loop is now changing in time. The second term supplies it.

The measured current is the same in both cases. But the explanation on this model is different in the two cases, and that asymmetry is exactly what Einstein opens the 1905 relativity paper by complaining about — his first paragraph is about a magnet and a conductor, and about a theory that gives two accounts of one phenomenon. Special relativity dissolves the distinction by making $𝐄$ and $𝐁$ two aspects of one object seen from different frames.

Maxwell could not have that resolution. What he had instead was a medium — and in a medium there is a fact of the matter about what is moving, so two accounts are not a defect but a feature. It is worth appreciating that the model behaves entirely correctly here; the trouble is philosophical, not numerical, and it took a change in the concept of space to remove it.

The state of the theory at the end of Part II

result	where	modern form
currents and the curl of the magnetic force	Prop. V, eq. (33) = (9)	$\mathrm{\nabla }\times 𝐇=4\pi 𝐉$
induction	Prop. VIII, eq. (54)	$\mathrm{\nabla }\times 𝐄=-\partial 𝐁/\partial t$
no magnetic sources	Prop. VIII, eq. (56)	$\mathrm{\nabla }\cdot 𝐁=0$
the vector potential	Prop. VIII, eq. (55)	$𝐁=\mathrm{\nabla }\times 𝐀$
electromotive force, complete	Prop. XI, eq. (77)	$𝐄=𝐯\times 𝐁-\partial 𝐀/\partial t-\mathrm{\nabla }\phi$

Everything on that list was already known experimentally. Part II is a triumph of explanation, not of prediction: it accounts mechanically for laws that Ørsted, Ampère and Faraday had established. Nothing in it would have surprised a well-read physicist of 1861 except the manner of its arrival.

That changes in Part III.

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Next: Induced currents · Previous: Props. IX and X · On to Part III

Modern derivation: The motional term and the Lorentz force