Page:Philosophical magazine 23 series 4.djvu/35

applied to Statical Electricity. 19

a hypothetically "perfect solid"^[1] in which

$5m=6\mu ,\,$

(110)

so that we must use equation (108).

Prop. XIV. — To correct the equations (9)^[2] of electric currents for the effect due to the elasticity of the medium.

We have seen that electromotive force and electric displacement are connected by equation (105). Differentiating this equation with respect to $t$, we find

${\frac {dR}{dt}}=-4\pi E^{2}{\frac {dh}{dt}},$

(111)

showing that when the electromotive force varies, the electric displacement also varies. But a variation of displacement is equivalent to a current, and this current must be taken into account in equations (9) and added to $r$. The three equations then become

$\left.{\begin{array}{l}p={\frac {1}{4\pi }}\left({\frac {d\gamma }{dy}}-{\frac {d\beta }{dz}}-{\frac {1}{E^{2}}}{\frac {dP}{dt}}\right),\\\\q={\frac {1}{4\pi }}\left({\frac {d\alpha }{dz}}-{\frac {d\gamma }{dx}}-{\frac {1}{E^{2}}}{\frac {dQ}{dt}}\right),\\\\r={\frac {1}{4\pi }}\left({\frac {d\beta }{dx}}-{\frac {d\alpha }{dy}}-{\frac {1}{E^{2}}}{\frac {dR}{dt}}\right),\end{array}}\right\}$

(112)

where p, q, r are the electric currents in the directions of x, y, and $z$; $\alpha ,\beta ,\gamma$ are the components of magnetic intensity; and P, Q, R are the electromotive forces. Now if $e$ be the quantity of free electricity in unit of volume, then the equation of continuity will be

${\frac {dp}{dx}}+{\frac {dq}{dy}}+{\frac {dr}{dz}}+{\frac {de}{dt}}=0.$

(113)

Differentiating (112) with respect to x, y, and z respectively, and substituting, we find

${\frac {de}{dt}}={\frac {1}{4\pi E^{2}}}{\frac {d}{dt}}\left({\frac {dP}{dx}}+{\frac {dQ}{dy}}+{\frac {dR}{dz}}\right);$

(114)

whence

$e={\frac {1}{4\pi E^{2}}}\left({\frac {dP}{dx}}+{\frac {dQ}{dy}}+{\frac {dR}{dz}}\right),$

(115)

the constant being omitted, because $e=0$ when there are no electromotive forces.

Prop. XV.— To find the force acting between two electrified bodies.

The energy in the medium arising from the electric displacements

1. See Rankine "On Elasticity," Camb. and Dub. Math. Journ. 1851.
2. Phil. Mag. March 1861.