Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/209

$V_{i}$ the potential outside it, then by making the surface-density $\sigma$ satisfy the characteristic equation

${\frac {dH_{i}}{dr}}-{\frac {dV_{i}}{dr}}+4\pi \sigma =0,$

(47)

we shall have a distribution of potential which satisfies all the conditions.

It is manifest that if $H_{i}$ and $V_{i}$ are derived from the same value of $Y_{i}$, the surface $H_{i}=V_{i}$ will be a spherical surface, and the surface-density will also be derived from the same value of $Y_{i}$.

Let $a$ be the radius of the sphere, and let

$H_{i}=Ar^{i}Y_{i},\ V_{i}=B{\frac {Y_{i}}{r^{i+1}}},\ \sigma =CY_{i}.$

(48)

Then at the surface of the sphere, where $r=a$,

$Aa^{i}={\frac {B}{a^{i+1}}},$

and

${\frac {dV}{dr}}-{\frac {dH}{dr}}=-4\pi \sigma ;$

or

$(i+1){\frac {B}{a^{i+2}}}+ia^{i-1}A=4\pi C;$

whence we find $H_{i}$ and $V_{i}$ in terms of $C$,

$H_{i}={\frac {4\pi C}{2i+1}}{\frac {r^{i}}{a^{i-1}}}Y_{i},\ V_{i}={\frac {4\pi C}{2i+1}}{\frac {a^{i+2}}{r^{i+1}}}Y_{i}.$

(49)

We have now obtained an electrified system in which the potential is everywhere finite and continuous. This system consists of a spherical surface of radius $a$, electrified so that the surface-density is everywhere $CY_{i}$, where $C$ is some constant density and $Y_{i}$ is a surface harmonic of degree $i$. The potential inside this sphere, arising from this electrification, is everywhere $H_{i}$, and the potential outside the sphere is $V_{i}$.

These values of the potential within and without the sphere might have been obtained in any given case by direct integration, but the labour would have been great and the result applicable only to the particular case.

135.] We shall next consider the action between a spherical surface, rigidly electrified according to a spherical harmonic, and an external electrified system which we shall call $E$.

Let $V$ be the potential at any point due to the system $E$, and $V_{i}$ that due to the spherical surface whose surface-density is $\sigma$.