Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/122

whence

${\frac {d^{2}V'}{dx^{2}}}dx+{\mbox{etc.}}=Cl$,

(16)

by the first of equations (12).

Multiplying by $Rl$, and remembering that at the second surface

$Rl\,dx=h$

(17)

we find

${\frac {d^{2}V'}{dx^{2}}}h=CR\,l^{2}$.

(18)

Similarly

${\frac {d^{2}V'}{dy^{2}}}h=CR\,m^{2}$

(19)

and

${\frac {d^{2}V'}{dz^{2}}}h=CR\,n^{2}$

(20)

Adding

$\left({\frac {d^{2}V'}{dx^{2}}}+{\frac {d^{2}V'}{dy^{2}}}+{\frac {d^{2}V'}{dz^{2}}}\right)\,h=CR$;

(21)

but

${\frac {d^{2}V'}{dx^{2}}}+{\frac {d^{2}V'}{dy^{2}}}+{\frac {d^{2}V'}{dz^{2}}}=-4\pi \rho ^{'}$ and $h=vR$;

(22)

hence

$C=-4\pi \rho ^{'}v\,\!=-4\pi \sigma$;

(23)

where $\sigma$ is the surface-density; or, multiplying the equations (12) by $l,m,n$ respectively, and adding,

$l\left({\frac {dV_{2}}{dx}}-{\frac {dV_{1}}{dx}}\right)++m\left({\frac {dV_{2}}{dy}}-{\frac {dV_{1}}{dy}}\right)+n\left({\frac {dV_{2}}{dz}}-{\frac {dV_{1}}{dz}}\right)+4\pi \sigma =0$.

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This equation is called the characteristic equation of $V$ at a surface. This equation may also be written

${\frac {dV_{1}}{dv_{1}}}+{\frac {dV_{2}}{dv_{2}}}+4\pi \sigma =0$;

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where $v_{1},v_{2}$ are the normals to the surface drawn towards the first and the second medium respectively, and $V_{1},V_{2}$ the potentials at points on these normals. We may also write it

$R_{2}\cos \epsilon _{2}+R_{1}\cos \epsilon _{1}+4\pi \sigma =0\,\!$;

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where $R_{1},R_{2}$ are the resultant forces, and $\epsilon _{1},\epsilon _{2}$ the angles which they make with the normals drawn from the surface on either side.

79.] Let us next determine the total mechanical force acting on an element of the electrified surface.

The general expression for the force parallel to $x$ on an element whose volume is $dx\,dy\,dz$, and volume-density $\rho$, is

$dX=-{\frac {dV}{dx}}\rho \;dz\,dy\,dz$.

(27)