Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/451

Four conductors of great resistance may also be arranged as in Wheatstone's Bridge, and the bridge itself may consist of the electrodes of an electrometer instead of those of a galvanometer. The advantage of this method is that no permanent current is required to produce the deviation of the electrometer, whereas the galvanometer cannot be deflected unless a current passes through the wire.

354.] When the resistance of a conductor is so great that the current which can be sent through it by any available electromotive force is too small to be directly measured by a galvanometer, a condenser may be used in order to accumulate the electricity for a certain time, and then, by discharging the condenser through a galvanometer, the quantity accumulated may be estimated. This is Messrs. Bright and Clark's method of testing the joints of submarine cables.

355.] But the simplest method of measuring the resistance of such a conductor is to charge a condenser of great capacity and to connect its two surfaces with the electrodes of an electrometer and also with the extremities of the conductor. If $E$ is the diffference of potentials as shewn by the electrometer, $S$ the capacity of the condenser, and $Q$ the charge on either surface, $R$ the resistance of the conductor and $x$ the current in it, then, by the theory of condensers,

$Q=SE,$

By Ohm's Law,

$E=Rx$,

and by the definition of a current,

$x=-{\frac {dQ}{dt}}$.

Hence

$Q=RS{\frac {dQ}{dt}}$,

and

$Q=Q_{0}e^{-{\frac {t}{RS}}}$,

where $Q_{0}$ is the charge at first when $t=0$.

Similarly

$E=E_{0}e^{-{\frac {t}{RS}}}$

where $E_{0}$ is the original reading of the electrometer, and $E$ the same after a time $t$. From this we find

$R={\frac {t}{s\{\log _{e}E_{0}-\log _{e}E\}}}$,

which gives $R$ in absolute measure. In this expression a knowledge of the value of the unit of the electrometer scale is not required.