Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/160

the outside of that surface will be equal to the potential at the same point due to that part of the original system which was on the inside of the surface, and the potential due to the electrified surface at any point on the inside added to that due to the part of the original system on the outside will be equal to $C$, the potential of the surface.

For let us alter the original system as follows :

Let us leave everything the same on the outside of the surface, but on the inside let us make $V_{2}$ everywhere equal to $C$, and let us do away with the electrified system $E_{2}$ on the inside of the surface, and substitute for it a surface-density $\rho$ at every point of the surface $S$, such that

$R=4\pi \sigma .\,$

(3)

Then this new arrangement will satisfy the characteristics of $V$ at every point.

For on the outside of the surface both the distribution of electricity and the value of $V$ are unaltered, therefore, since $V$ originally satisfied Laplace’s equation, it will still satisfy it.

On the inside $V$ is constant and $\rho$ zero. These values of $V$ and $\rho$ also satisfy the characteristic equations.

At the surface itself, if $V_{1}$ is the potential at any point on the outside and $V_{2}$ that on the inside, then, if l, m, n are the direction-cosines of the normal to the surface reckoned outwards,

$l{\frac {dV_{1}}{dx}}+m{\frac {dV_{1}}{dy}}+n{\frac {dV_{1}}{dz}}=-R=-4\pi \sigma ;$

(4)

and on the inside the derivatives of $V$ vanish, so that the superficial characteristic

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

(5)

is satisfied at every point of the surface.

Hence the new distribution of potential, in which it has the old value on the outside of the surface and a constant value on the inside, is consistent with the new distribution of electricity, in which the electricity in the space within the surface is removed and a distribution of electricity on the surface is substituted for it. Also, since the original value of $V_{1}$ vanishes at infinity, the new value, which is the same outside the surface, also fulfils this condition, and therefore the new value of $V$ is the sole and only value of $V$ belonging to the new arrangement of electricity.