An inductor-resistance -battery circuit is switched on at t = 0. If the emf of the battery in one time constant tau.

Explanation

The current at tiem it is given by `I = i_0(1-e^(t/tau)) where i_0 = epsilon/R`
The charge passed through the battery during the period t to t+dt is I dt
Thus, the total charge passed during o to tau is `Q = int_0^(tau) I dt = i_0 int_0^(tau) (1-e^(-t/tau)) dt = i_0[(t -(e^(-t/tau)/(-1/tau)]_0^(tau)` = i_0[tau+tau(e^(-1) -1)] = i_0 tau/e