403.
ON THE ELECTRICAL CAPACITY OF APPROXIMATE SPHERES AND CYLINDERS.
[Philosophical Magazine, Vol. xxxi. pp. 177 — 186, March 1916.]
MANY years ago I had occasion to calculate these capacities* so far as to include the squares of small quantities, but only the results were recorded. Recently, in endeavouring to extend them, I had a little difficulty in retracing the steps, especially in the case of the cylinder. The present communication gives the argument from the beginning. It may be well to remark at the outset that there is an important difference between the two cases. The capacity of a sphere situated in the open is finite, being equal to the radius. But when we come to the cylinder, supposed to be entirely isolated, we have to recognize that the capacity reckoned per unit length is infinitely small If a be the radius of the cylinder and 6 that of a coaxal enveloping case at potential zero, the capacity of a length I isf
..*'. i
log Wo) '
which diminishes without limit as b is increased. For clearness it may be well to retain the enveloping case in the first instance.
In the intervening space we may take for the potential in terms of the usual polar coordinates
(f> = H0 log (r/6) + Hjr-1 cos (0 - ej + K{r cos (6 - e/) + . . .
+ Hnr~n cos (7i0 - en) + Knrn cos (nO - en'). Since <£ = 0 when r = b,
e ' — f TT — — TT h~zn
^n — kn> -°-« — -LJ-<iiu >
and
............... (1)
* "On the Equilibrium of Liquid Conducting Masses charged with Electricity," Phil. Mag. Vol. xiv. p. 184 (1882) ; Scientific Papers, Vol. n. p. 130. t Maxwell's Electricity, § 126.t with the walls of the tube is of unit specific resistance.