As shown in the figure, a battery of emf $\in$ 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:

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q = ∫ 0 T c i d t = ∈ R t - e - t T c - 1 T c 0 T c = ∈ R T c + T c e - 1 - T c = ∈ R × 1 e × L R q = ∈ L R 2 e

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