An inductor resistance battery circuit is switched on at t = 0. If the emf of the battery is E, find the charge which passes through the battery during one time constant $\tau$.

Current at time r is given by

$i=i_0\left(1-e^{-t/\tau }\right) where i_0=E/R$

The charge passed through the battery during the period t to t + dt is idt. Thus, the total charge passed during time 0 to $\tau$ is

$Q=\underset{0}{\overset{\tau }{\int }}idt=i_0\underset{0}{\overset{\tau }{\int }}(1-e^{-t/\tau })dt$

$=i_0{\left[t+\frac{{\displaystyle e^{-t/\tau }}}{{\displaystyle \tau }}\right]}_0^{\tau }=i_0[\tau +\tau (e^{-1}-1)]$

$=\frac{i_0\tau }{e}=\frac{E}{R}\frac{\tau }{e}$