Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/315

 between $$A$$ and $$B$$. All these coefficients will in general vary with the position of $$C$$, and if $$C$$ is so arranged that the extremities of $$A$$ and $$B$$ are not near those of $$C$$ as long as the motion of $$C$$ is confined within certain limits, we may ascertain the form of these coefficients. If $$\theta $$ represents the deflexion of $$C$$ from $$A$$ towards $$B$$, then the part of the surface of $$A$$ opposed to $$C$$ will diminish as increases. Hence if $$A$$ is kept at potential 1 while $$B$$ and $$C$$ are kept at potential $$O$$, the charge on $$A$$ will be $$a=a_{0}-\alpha\theta $$, where $$a_{0} $$ and $$\alpha $$ are constants, and $$a$$ is the capacity of $$A$$.

If $$A$$ and $$B$$ are symmetrical, the capacity of $$B$$ is $$b=b_{0}+\alpha\theta $$.

The capacity of $$C$$ is not altered by the motion, for the only effect of the motion is to bring a different part of $$C$$ opposite to the interval between $$A$$ and $$B$$. Hence $$c=c_{0} $$.

The quantity of electricity induced on $$C$$ when $$B$$ is raised to potential unity is $$p=p_{0}-\alpha\theta $$.

The coefficient of induction between $$A$$ and $$C$$ is $$q=q_{0}+\alpha\theta $$.

The coefficient of induction between $$A$$ and $$B$$ is not altered by the motion of $$C$$, but remains $$r=r_{0} $$.

Hence the electrical energy of the system is

$Q=\frac{1}{2}A^{2}a+\frac{1}{2}B^{2}b+\frac{1}{2}C^{2}c+BCp+CAq+ABr $

and if $$\Theta $$ is the moment of the force tending to increase $$\theta $$,

$\begin{array}{ll} \Theta & =\frac{dQ}{d\theta},\ A,\ B,\ C\ \mathrm{being\ supposed\ constant,}\\ \\ & =\frac{1}{2}A^{2}\frac{da}{d\theta}+\frac{1}{2}B^{2}\frac{db}{d\theta}+\frac{1}{2}C^{2}\frac{dc}{d\theta}+BC\frac{dp}{d\theta}+CA\frac{dq}{d\theta}+AB\frac{dr}{d\theta},\\ \\ & =-\frac{1}{2}A^{2}\alpha+\frac{1}{2}B^{2}\alpha-BC\alpha+CA\alpha; \end{array} $

or

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

In the present form of Thomson's Quadrant Electrometer the conductors $$A$$ and $$B$$ are in the form of a cylindrical box completely divided into four quadrants, separately insulated, but joined by wires so that two opposite quadrants are connected with $$A$$ and the two others with $$B$$.

The conductor $$C$$ is suspended so as to be capable of turning about a vertical axis, and may consist of two opposite flat quadrantal arcs supported by their radii at their extremities. In the position of equilibrium these quadrants should be partly