Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/448

which vanishes by virtue of the principle of conservation of electricity. Thus

${\text{div }}\mathbf {S} =0,\qquad \qquad (V)$

or the total current is a circuital vector, Equations (I) to (V) are the fundamental equations of Lorentz' theory of electrons.

We have now to consider the relation by which the polarization P of dielectrics is determined. If the dielectric is moving with velocity w, the ponderomotive force on unit electric charge moving with it is (as in all theories)^[1]

$\mathbf {E} ^{\prime }=\mathbf {E} +[\mathbf {w.B} ].\qquad \qquad (1)$

In order to connect P with E′, it is necessary to consider the motion of the corpuscles. Let e denote tho charge and m the mass of a corpuscle, (ξ, η, ζ, ) its displacement from its position of equilibrium, k²(ξ, η, ζ, ) the restitutive force which retains it in the vicinity of this point; then the equations of motion of the corpuscle are

$m{\ddot {\xi }}+k^{2}\xi =eE_{x}^{\prime }$

and similar equations in η and ζ. When the corpuscle is set in motion by light of frequency n passing through the medium, the displacements and forces will be periodic functions of nt—say,

$\xi =Ae^{nt{\sqrt {-1}}},\qquad E_{x}^{\prime }=E_{0}e^{nt{\sqrt {-1}}}$

Substituting these values in the equations of motion, we obtain

$A(k^{2}-mn^{2})=eE_{0},\qquad {\text{and therefore}}\qquad \xi (k^{2}-mn^{2})=eE_{x}^{\prime }$.

Thus, if N denote the number of polarizable molecules per unit volume, the polarization is determined by the equation

$\mathbf {P} =Ne(\xi ,\eta ,\zeta )=Ne^{2}\mathbf {E} ^{\prime }/(k^{2}-mn^{2})$.

In the particular case in which the dielectric is at rest, this equation gives

$=(1/4\pi c^{2})\mathbf {E} +\mathbf {P} =(1/4\pi c^{2})\mathbf {E} +Ne^{2}\mathbf {E} /(k^{2}-mn^{2})$.

But, as we have seen^[2] D bears to E the ratio μ²/4πc², where μ

1. Cf. p. 365.
2. Cf. p. 281.