Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/198

cylinders, and near the ends of the inner cylinder, there will be distributions of electricity which we are not yet able to calculate, but the distribution near the origin will not be altered by the motion of the inner cylinder provided neither of its ends comes near the origin, and the distributions at the ends of the inner cylinder will move with it, so that the only effect of the motion will be to increase or diminish the length of those parts of the inner cylinder where the distribution is similar to that on an infinite cylinder.

Hence the whole energy of the system will be, so far as it depends on $x$,

$Q={\frac {1}{2}}\alpha (l+x)(C-A)^{2}+{\frac {1}{2}}\beta (l-x)(C-B)^{2}+$ quantities independent of $x$;

and the resultant force parallel to the axis of the cylinders will be

$X={\frac {dQ}{dx}}={\frac {1}{2}}\alpha (C-A)^{2}-{\frac {1}{2}}\beta (C-B)^{2}.$

If the cylinders $A$ and $B$ are of equal section, $\alpha =\beta$ and

$X=\alpha (B-A)\left(C-{\frac {1}{2}}(A+B)\right)$

It appears, therefore, that there is a constant force acting on the inner cylinder tending to draw it into that one of the outer cylinders from which its potential differs most.

If $C$ be numerically large and $A+B$ comparatively small, then the force is approximately

$X=\alpha (B-A)C;$

so that the difference of the potentials of the two cylinders can be measured if we can measure $X$, and the delicacy of the measurement will be increased by raising $C$, the potential of the inner cylinder.

This principle in a modified form is adopted in Thomson’s Quadrant Electrometer, Art. 219.

The same arrangement of three cylinders may be used as a measure of capacity by connecting $B$ and $C$. If the potential of $A$ is zero, and that of $B$ and $C$ is $V$, then the quantity of electricity on $A$ will be

$E_{3}=\left(q_{13}+\alpha (l+x)\right)V;$

so that by moving $C$ to the right till $x$ becomes $x+\xi$ the capacity of the cylinder becomes increased by the definite quantity $\alpha \xi$, where

$\alpha ={\frac {1}{2\log {\frac {b}{a}}}},$

$a$ and $b$ being the radii of the opposed cylindric surfaces.