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

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

Maxwell's statement

Prop. XIII. — To find the relation between electromotive force and electric displacement when a uniform electromotive force R acts parallel to the axis of z.

The result

Taking the deformation computed in Prop. XII and integrating the displacement of the particles over the whole surface of the cell — equation (101) — Maxwell arrives at the relation he had announced informally in the introductory pages:

$$R=-4\pi E^{2}h\mathrm{(}\mathrm{105}\mathrm{)}$$

where $h$ is the electric displacement, $R$ the electromotive force, and $E$ a coefficient depending on the nature of the dielectric.

Equation (108) then records the value of the coefficient adopted — the "perfect solid" case, with the 6-to-5 ratio of the elastic moduli.

Provenance: the numbers (101), (105) and (108) are attested. Maxwell cites (105) by number on p. 19: "electromotive force and electric displacement are connected by equation (105)"; and (108) twice, on p. 19 and again in Prop. XVI ("and by (108)"). The body of (105) above is the standard reconstruction. See Sources.

Reading the equation

The minus sign is the elasticity: the medium pushes back against the displacement, as a spring does. The proportionality is Hooke's law for the cells. And $E^2$, not $E$, is what appears — which is why the square of the coefficient turns up in every consequence: in the force between charged bodies (Prop. XV), in the capacity of a jar (Prop. XVII), and in the wave speed (Prop. XVI).

In modern notation, with $𝐃=\epsilon 𝐄$, Maxwell's relation says

$$\epsilon =\frac{1}{E^2}$$

so his $E$ is the reciprocal square root of the permittivity. It is the single most confusing symbol in the paper for a modern reader, because $E$ now universally means the electric field, which for Maxwell is $P,Q,R$. See the concordance.

Why the whole result hangs on one constant

Equation (105) is the bridge between the mechanical model and electrostatics, and it is a bridge with a single plank. Everything about static electricity in this theory follows from it:

quantity	how it depends on E	where
force between charges at fixed charge	as $E^2$	Prop. XV
force between bodies at fixed potential	as $1/E^2$	introductory pages
capacity of a Leyden jar	as $1/E^2$	Prop. XVII
velocity of transverse waves	as $E$	Prop. XVI

The last line is the one that changed physics. $E$ has the dimensions of a velocity, because it converts between the electrostatic and the electromagnetic measure of a quantity of electricity — and Weber and Kohlrausch had measured it.

The step from statics to waves

It is worth being clear about how short the step is from here.

• Prop. XIII gives the elasticity of the medium, in electrical terms.
• Part I gave the density of the medium, in magnetic terms — the density of the vortex substance is $\mu$.
• An elastic medium of known stiffness and known density carries transverse waves at a speed $\sqrt{m/\rho }$.

Three lines. Maxwell writes them out in Prop. XVI and the answer is the velocity of light.

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Next: Prop. XIV — the displacement current · Previous: Prop. XII