Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/444

external field are so arranged as to neutralize each other's electric fields outside the molecule. For simplicity we may suppose that in each molecule only one corpuscle, of charge e, is capable of being displaced from its position; it follows from what has been assumed that the other corpuscles in the molecule exert the same electrostatic action as a charge – e situated at the original position of this corpuscle. Thus if e is displaced to an adjacent position, the entire molecule becomes equivalent to an electric doublet, whose moment is measured by the product of e and the displacement of e. The molecules in unit volume, taken together, will in this way give rise to a (vector) electric moment per unit volume, P, which may be compared to the (vector) intensity of magnetization in Poisson's theory of magnetism.^[1] As in that theory, we may replace the doublet-distribution P of the scalar quantity ρ by a volume-distribution of ρ, determined by the equation^[2]

${\bar {\rho }}=-{\text{div }}\mathbf {P}$.

This represents the part of ${\bar {\rho }}$ due to the dielectric molecules.

Moreover, the scalar quantity ρw_x, has also a doublet-distribution, to which the same theorem may be applied; the average value of the part of ρw_x, due to dielectric molecules, is therefore determined by the equation

${\bar {\rho w_{x}}}=-{\text{div }}(w_{x}\mathbf {P} )=-w_{x}{\text{div }}\mathbf {P} -(\mathbf {P.} \nabla )w_{x}$,

or

${\bar {\rho \mathbf {w} }}=-{\text{div }}\mathbf {P.w} -(\mathbf {P.} \nabla )\mathbf {w}$.

We have now to find that part of ${\bar {\rho \mathbf {u} }}$ which is due to dielectric molecules. For a single doublet of moment p we have, by differentiation,

$\textstyle \iiint \rho \mathbf {u} \ dx\ dy\ dz=d\mathbf {p} /dt$,

where the integration is taken throughout the molecule; so that

$\textstyle \iiint \rho \mathbf {u} \ dx\ dy\ dz=(d/dt)(V\mathbf {P} )$,

where the integration is taken throughout a volume V, which

1. Cf. p. 64.
2. We assume all transitions gradual, so as to avoid surface-distributions.