As shown in the figure, a battery of emf $$\varepsilon$$ is connected to an inductor L and resistance R in series. The switch is closed at $$t = 0$$. The total charge that flows from the battery, between $$t = 0$$ and $$t = t_c$$ ($$t_c$$ is the time constant of the circuit) is:

Time-dependent current during growth phase: $$i(t) = \frac{\varepsilon}{R} \left(1 - e^{-\frac{t}{\tau_c}}\right)$$

Circuit time constant definition: $$\tau_c = \frac{L}{R}$$

$$q = \int_{0}^{\tau_c} i(t) \, dt = \int_{0}^{\tau_c} \frac{\varepsilon}{R} \left(1 - e^{-\frac{t}{\tau_c}}\right) dt$$

$$q = \frac{\varepsilon}{R} \left[ t + \tau_c e^{-\frac{t}{\tau_c}} \right]_0^{\tau_c} = \frac{\varepsilon}{R} \left[ \left(\tau_c + \tau_c e^{-1}\right) - (0 + \tau_c) \right]$$

$$q = \frac{\varepsilon}{R} \cdot \frac{\tau_c}{e}$$

$$q = \frac{\varepsilon}{R \cdot e} \left(\frac{L}{R}\right) = \frac{\varepsilon L}{e R^2}$$