Physics - EMI/AC Question with Solution | TestHub

The current in the LR circuit at any time $t$ is
$I=\frac{E}{R}\left(1-e^{-\frac{Rt}{L}}\right)$

Where $E,R,L,I$ are the EMF, resistance, inductance and the current respectively.
$\therefore$ Charge flown through the circuit from time 0 to $\frac{L}{R}$ is
(charge) $\Delta q=\in t_0^{\frac{L}{R}}I.dt=\in t_0^{\frac{L}{R}}\frac{E}{R}\left(1-e^{-\frac{Rt}{L}}\right)dt$
$\Rightarrow \Delta q=\frac{E}{R}{\left[t+\frac{L}{R}{e}^{-\frac{Rt}{L}}\right]}_0^{\frac{L}{R}}$
$\Rightarrow \Delta q=\frac{E}{R}\left[\frac{L}{R}+\frac{L}{R}{e}^{-1}-\frac{L}{R}\right]$
$\Rightarrow \Delta q=\frac{E}{R}\times \frac{L}{eR}$
$\Rightarrow \Delta q=\frac{EL}{e{R}^2}=\frac{EL}{2.7R^2}$