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,

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

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

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



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