Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/409

 

resistance of the rest of the system between the electrodes of the condenser, and $E$ the electromotive force due to the connexions with the batter.

Hence

$Q=\left(Q_{0}+EC\right)e^{-{\frac {t}{R_{2}C}}}-EC$,

(7)

where $Q_{0}$ is the initial value of $Q$.

If $\tau$ is the time during which contact is maintained during each discharge, the quantity in each discharge is

$Q=2EC{\frac {1-e^{-{\frac {\tau }{R_{2}C}}}}{1+e^{-{\frac {\tau }{R_{2}C}}}}}$.

(8)

By making $c$ and $\gamma$ in equation (4) large compared with $\beta$, $a$, or $\alpha$, the time represented by $R_{2}C$ may be made so small compared with $\tau$, that in calculating the value of the exponential expression we may use the value of $C$ in equation (5). We thus find

${\frac {\tau }{R_{2}C}}=2{\frac {R_{1}+R_{2}}{R_{2}}}{\frac {\tau }{T}}$,

(9)

where $R_{1}$ is the resistance which must be substituted for the condenser to produce an equivalent effect. $R_{2}$ is the resistance of the rest of the system, $T$ is the interval between the beginning of a discharge and the beginning of the next discharge, and $\tau$ is the duration of contact for each discharge. We thus obtain for the corrected value of $C$ in electromagnetic measure

$C={\frac {1}{2}}{\frac {T}{R_{1}+R_{2}}}{\frac {1+e^{-2{\frac {R_{1}+R_{2}}{R_{2}}}{\frac {\tau }{T}}}}{1-e^{-2{\frac {R_{1}+R_{2}}{R_{2}}}{\frac {\tau }{T}}}}}$.

(10)

 

IV. Comparison of the Electrostatic Capacity of a Condenser with the Electromagnetic Capacity of Self-induction of a Coil.

Fig. 64. 778.] If two points of a conducting circuit, between which the resistance is $R$, are connected with the electrodes of a condenser whose capacity is $C$, then, when an electromotive force acts on the circuit, part of the current, instead of passing through the resistance $R$, will be employed in charging the condenser. The current through $R$ will therefore rise to its final value from zero in a gradual manner. It appears from the mathematical theory that the manner in which the current through