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

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

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