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

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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

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

Fig. 19.

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