Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/220

In this way we may analyse Y_i into its component conjugate harmonics by means of a finite number of ascertained values at selected points on the sphere.

Application of Spherical Harmonic Analysis to the Determination of the Distribution of Electricity on Spherical and nearly Spherical Conductors under the Action of known External Electrical Forces.

146.] We shall suppose that every part of the electrified system which acts on the conductor is at a greater distance from the centre of the conductor than the most distant part of the conductor itself, or, if the conductor is spherical, than the radius of the sphere.

Then the potential of the external system, at points within this distance, may be expanded in a series of solid harmonics of positive degree

$V=A_{0}+A_{1}rY_{1}+\mathrm {etc} .+A_{i}Y_{i}r^{i}.$

(70)

The potential due to the conductor at points outside it may be expanded in a series of solid harmonics of the same type, but of negative degree

$U=B_{0}{\frac {1}{r}}+B_{1}Y_{1}{\frac {1}{r^{2}}}+\mathrm {etc} .+B_{i}Y_{i}{\frac {1}{r^{i+1}}}$

(71)

At the surface of the conductor the potential is constant and equal, say, to $C$. Let us first suppose the conductor spherical and of radius $a$. Then putting $r=a$, we have $U+V=C$, or, equating the coefficients of the different degrees,

${\begin{array}{l}B_{0}=a\left(C-A_{0}\right),\\B_{1}=-a^{3}A_{1},\\-\ -\ -\ -\ -\\B_{i}=-a^{2i+1}A_{i}.\end{array}}$

(72)

The total charge of electricity on the conductor is $B_{0}$.

The surface-density at any point of the sphere may be found from the equation

${\begin{array}{ll}4\pi \sigma &={\frac {dV}{dr}}-{\frac {dU}{dr}}\\\\&={\frac {B_{0}}{a^{2}}}-3a^{3}A_{1}rY_{1}-\mathrm {etc} .-(2i+1)a^{2i+1}A_{i}Y_{i}.\end{array}}$

(73)

Distribution of Electricity on a nearly Spherical Conductor.

Let the equation of the surface of the conductor be

$r=a(1+F)$