On Physical Lines of Force/Part III/Prop. XV

On Physical Lines of Force → Part III → Prop. XV

Maxwell's statement

Prop. XV. — To find the force acting between two electrified bodies.

Printed on pp. 19–21, and carrying equations (116) to (125).

The method

Maxwell does not compute the force by adding up actions at a distance. He computes the energy stored in the medium by the displacements of its cells, and gets the force by asking how that energy changes when the bodies move.

The calculation integrates the energy over all space, integrates by parts, and invokes Green's Essay on Electricity, p. 10 — which is the identity now known as Green's theorem, allowing a volume integral over the whole field to be turned into a surface integral plus a term concentrated on the charges.

What emerges is Coulomb's inverse-square law.

Provenance: equation numbers (116) to (125) are attested as belonging to Prop. XV, along with the citations of Green. Their printed bodies could not be recovered, and none is reproduced here. Equation (115) is cited on p. 20 ("or by (115)") but its content is not established. See Sources.

Why this is not circular

It would be fair to ask whether Maxwell has done anything but recover the law he started from. He has, and the difference is worth spelling out.

Coulomb's law states that the force between two charges falls off as the inverse square. It says nothing about where the energy resides. On the action-at-a-distance view the energy is a property of the pair of bodies, a function of their separation and nothing else.

Maxwell's calculation puts the energy in the space between them, distributed with a definite density at every point, stored as the strain of the cells. He then recovers the inverse square from that distribution. That is not the same statement, and it makes different predictions — it says the force depends on what the intervening space is made of, which is the prediction about turpentine.

It appears also from Prop. XV. that the attraction between two electrified bodies depends on the value of E², and that therefore it would be less in turpentine than in air, if the quantity of electricity in each body remains the same. If, however, the potentials of the two bodies were given, the attraction between them would vary inversely as E², and would be greater in turpentine than in air.

The constant that matters

The other product of Prop. XV is the one Maxwell carries forward. The calculation fixes the constant relating the two measures of a quantity of electricity — the electrostatic measure, defined by the force between charges at rest, and the electromagnetic measure, defined by the force between currents.

That constant is his $E$, and it has the dimensions of a velocity. Maxwell says as much in the introduction to Part III:

I have deduced from this result the relation between the statical and dynamical measures of electricity …

Nothing in either definition involves motion. A torsion balance measuring the repulsion of two charged pith balls, and a current balance measuring the attraction of two wires, between them define a speed. Why should they?

The coincidence that settled it

Prop. XVI answers the question, and the answer is that the speed is the speed at which disturbances travel through the medium — which is to say, the speed of light. But the dimensional fact is settled here, in a proposition about the force between two charged bodies, and it is the reason a measurement made by Weber and Kohlrausch in 1856 with no optical apparatus whatever could yield the velocity of light.

See the modern treatment of the unit ratio for why, in modern language, $1/\sqrt{{\epsilon }_0{\mu }_0}$ has to be a velocity and has to be $c$.

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Next: Prop. XVI — the velocity of transverse undulations · Previous: Prop. XIV