View source for On Physical Lines of Force/Part III/Prop. XIII

''[[On Physical Lines of Force]] → [[On Physical Lines of Force/Part III|Part III]] → Prop. XIII''

== Maxwell's statement ==

<blockquote>
'''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''.
</blockquote>

== The result ==

Taking the deformation computed in [[On Physical Lines of Force/Part III/Prop. XII|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 [[On Physical Lines of Force/Part III/Conduction and induction|the introductory pages]]:

<math display="block">R \;=\; -\,4\pi E^{2}\,h \qquad\qquad\mathrm{(105)}</math>

where <math>h</math> is the electric displacement, <math>R</math> the electromotive force, and <math>E</math> 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 [[On Physical Lines of Force/Part III/Prop. XVI|Prop. XVI]] ("and by (108)"). The body of (105) above is the standard reconstruction. See [[On Physical Lines of Force/Sources and editorial method|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 <math>E^{2}</math>, not <math>E</math>, is what appears — which is why the ''square'' of the coefficient turns up in every consequence: in the force between charged bodies ([[On Physical Lines of Force/Part III/Prop. XV|Prop. XV]]), in the capacity of a jar ([[On Physical Lines of Force/Part III/Prop. XVII|Prop. XVII]]), and in the wave speed ([[On Physical Lines of Force/Part III/Prop. XVI|Prop. XVI]]).

In modern notation, with <math>\mathbf{D}=\varepsilon\mathbf{E}</math>, Maxwell's relation says

<math display="block">\varepsilon \;=\; \frac{1}{E^{2}}</math>

so his <math>E</math> is the reciprocal square root of the permittivity. It is the single most confusing symbol in the paper for a modern reader, because <math>E</math> now universally means the electric field, which for Maxwell is <math>P, Q, R</math>. See [[On Physical Lines of Force/Notation and concordance|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:

{| class="wikitable"
! quantity !! how it depends on E !! where
|-
| force between charges at fixed charge || as <math>E^{2}</math> || [[On Physical Lines of Force/Part III/Prop. XV|Prop. XV]]
|-
| force between bodies at fixed potential || as <math>1/E^{2}</math> || [[On Physical Lines of Force/Part III/Conduction and induction|introductory pages]]
|-
| capacity of a Leyden jar || as <math>1/E^{2}</math> || [[On Physical Lines of Force/Part III/Prop. XVII|Prop. XVII]]
|-
| velocity of transverse waves || as <math>E</math> || [[On Physical Lines of Force/Part III/Prop. XVI|Prop. XVI]]
|}

The last line is the one that changed physics. <math>E</math> 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.
* [[On Physical Lines of Force/Part I|Part I]] gave the '''density''' of the medium, in magnetic terms — the density of the vortex substance is <math>\mu</math>.
* An elastic medium of known stiffness and known density carries transverse waves at a speed <math>\sqrt{m/\rho}</math>.

Three lines. Maxwell writes them out in [[On Physical Lines of Force/Part III/Prop. XVI|Prop. XVI]] and the answer is the velocity of light.

----

''Next: [[On Physical Lines of Force/Part III/Prop. XIV|Prop. XIV — the displacement current]] · Previous: [[On Physical Lines of Force/Part III/Prop. XII|Prop. XII]]''