Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/312

in the position of the suspended disk is most important when $D$ is small, Sir W. Thomson prefers to make all his measurements depend on differences of the electromotive force $V$. Thus, if $V$ and $V'$ are two potentials, and $D$ and $D'$ the corresponding distances,

$V-V'=(D-D'){\sqrt {\frac {8\pi gW}{A}}}$

For instance, in order to measure the electromotive force of a galvanic battery, two electrometers are used.

By means of a condenser, kept charged if necessary by a replenisher, the lower disk of the principal electrometer is maintained at a constant potential. This is tested by connecting the lower disk of the principal electrometer with the lower disk of a secondary electrometer, the suspended disk of which is connected with the earth. The distance between the disks of the secondary electrometer and the force required to bring the suspended disk to its sighted position being constant, if we raise the potential of the condenser till the secondary electrometer is in its sighted position, we know that the potential of the lower disk of the principal electrometer exceeds that of the earth by a constant quantity which we may call $V$.

If we now connect the positive electrode of the battery to earth, and connect the suspended disk of the principal electrometer to the negative electrode, the difference of potentials between the disks will be $V+v$, if $v$ is the electromotive force of the battery. Let $D$ be the reading of the micrometer in this case, and let $D'$ be the reading when the suspended disk is connected with earth, then

$v=(D-D'){\sqrt {\frac {8\pi gW}{A}}}$

In this way a small electromotive force $v$ may be measured by the electrometer with the disks at conveniently measurable distances. When the distance is too small a small change of absolute distance makes a great change in the force, since the

and in the expression for the attraction we must substitute for $A$, the area of the disk, the corrected quantity

$A={\frac {1}{2}}\pi \left\{R^{2}+R'^{2}-\left(R'^{2}-R^{2}\right){\frac {\alpha }{D+\alpha }}+8(R+R')(D'-D)\log _{e}{\frac {4\pi (R+R')}{D'-D}}\right\}$

where

$R$ = radius of suspended disk,

$R'$ = radius of aperture in the guard-ring,

$D$ = distance between fixed and suspended disks,

$D'$ = distance between fixed disk and guard-ring,

$\alpha =0.220635(R'-R)$.

When $\alpha$ is small compared with $D$ we may neglect the second term, and when $D'-D$ is small we may neglect the last term.