Page:Philosophical magazine 23 series 4.djvu/39

applied to Statical Electricity. 23

on each surface; then the capacity

$C={\frac {e}{\Psi _{1}-\Psi _{2}}}$

(138)

Within the dielectric we have the variation of $\Psi$ perpendicular to the surface

$={\frac {\Psi _{1}-\Psi _{2}}{\theta }}.$

Beyond either surface this variation is zero.

Hence by (115) applied at the surface, the electricity on unit of area is

${\frac {\Psi _{1}-\Psi _{2}}{4\pi E^{2}\theta }};$

(139)

and we deduce the whole capacity of the apparatus,

$C={\frac {S}{4\pi E^{2}\theta }};$

(140)

so that the quantity of electricity required to bring the one surface to a given tension varies directly as the surface, inversely as the thickness, and inversely as the square of E.

Now the coefficient of induction of dielectrics is deduced from the capacity of induction-apparatus formed of them; so that if D is that coefficient, D varies inversely as $E^{2}$, and is unity for air. Hence

$D={\frac {V^{2}}{V_{1}^{2}\mu }};$

(141)

where V and $V_{1}$ are the velocities of light in air and in the medium. Now if $i$ is the index of refraction, ${\frac {V}{V_{1}}}=i$, and

$D={\frac {i^{2}}{\mu }};$

(142)

so that the inductive power of a dielectric varies directly as the square of the index of refraction, and inversely as the magnetic inductive power.

In dense media, however, the optical, electric, and magnetic phenomena may be modified in different degrees by the particles of gross matter; and their mode of arrangement may influence these phenomena differently in different directions. The axes of optical, electric, and magnetic properties will probably coincide; but on account of the unknown and probably complicated nature of the reactions of the heavy particles on the ætherial medium, it may be impossible to discover any general numerical relations between the optical, electric, and magnetic ratios of these axes.

It seems probable, however, that the value of E, for any given