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 =∈ t 0 T c ​ ​ i d t = R ∈ ​ [ t − − T c ​ 1 ​ e − T c ​ t ​ ​ ] 0 T c ​ ​ = R ∈ ​ [ T c ​ + T c ​ e − 1 − T c ​ ] = R ∈ ​ × e 1 ​ × R L ​ q = R 2 e ∈ L ​

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