Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/401

at every point. We may suppose electromotive forces introduced for this purpose acting perpendicular to the surface of the disk.

The resistance within the wire will be the same as before, but in the electrode the rate of generation of heat will be the surface-integral of the product of the flow into the potential. The rate of flow at any point is ${\frac {C}{\pi a^{2}}},$ and the potential is the same as that of an electrified surface whose surface-density is $\sigma ,$ where

$2\pi \sigma ={\frac {C\rho '}{\pi a^{2}}},$

(20)

$\rho '$ being the specific resistance.

We have therefore to determine the potential energy of the electrification of the disk with the uniform surface-density $\sigma .$

The potential at the edge of a disk of uniform density $\sigma$ is easily found to be $4a\sigma .$ The work done in adding a strip of breadth $da$ at the circumference of the disk is $2\pi a\sigma \,da\cdot 4a\sigma ,$ and the whole potential energy of the disk is the integral of this,

${\mbox{or}}\quad \quad P={\frac {8\pi }{3}}a^{3}\sigma ^{2}.$

(21)

In the case of electrical conduction the rate at which work is done in the electrode whose resistance is $R'$ is

$C^{2}R'={\frac {4\pi }{\rho '}}P,$

(22)

whence, by (20) and (21),

$R'={\frac {8\rho '}{3\pi ^{2}a}},$

and the correction to be added to the length of the cylinder is

${\frac {\rho '}{\rho }}{\frac {8}{3\pi }}a,$

this correction being greater than the true value. The true correction to be added to the length is therefore ${\frac {\rho '}{\rho }}an,$ where $n$ is a number lying between ${\frac {\pi }{4}}$ and ${\frac {8}{3\pi }}$ or between 0.785 and 0.849.

Mr. Strutt, by a second approximation, has reduced the superior limit of $n$ to 0.8282.