Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/197

If $E_{1}$ and $E_{2}$ are the charges on a portion of the two cylinders of length $l$, measured along the axis,

$E_{1}=2\pi al\sigma _{1}={\frac {1}{2}}{\frac {A-B}{\log {\frac {b}{a}}}}l=-E_{2},$

The capacity of a length $l$ of the interior cylinder is therefore

${\frac {1}{2}}{\frac {l}{\log {\frac {b}{a}}}}$

If the space between the cylinders is occupied by a dielectric of specific capacity $K$ instead of air, then the capacity of the inner cylinder is

${\frac {1}{2}}{\frac {lK}{\log {\frac {b}{a}}}}$

The energy of the electrical distribution on the part of the infinite cylinder which we have considered is

${\frac {1}{4}}{\frac {lK(A-B)^{2}}{\log {\frac {b}{a}}}}$

Fig. 5.

127.] Let there be two hollow cylindric conductors $A$ and $B$, Fig. 5, of indefinite length, having the axis of $x$ for their common axis, one on the positive and the other on the negative side of the origin, and separated by a short interval near the origin of co ordinates.

Let a hollow cylinder $C$ of length $2l$ be placed with its middle point at a distance $x$ on the positive side of the origin, so as to extend into both the hollow cylinders.

Let the potential of the positive hollow cylinder be $A$, that of the negative one $B$, and that of the internal one $C$, and let us put $\alpha$ for the capacity per unit of length of $C$ with respect to $A$, and $\beta$ for the same quantity with respect to $B$.

The capacities of the parts of the cylinders near the origin and near the ends of the inner cylinder will not be affected by the value of $x$ provided a considerable length of the inner cylinder enters each of the hollow cylinders. Near the ends of the hollow