At t =0, an inductor of zero resistance is joined to a cell of emf through a resistance. The current increases with a time constant $τ$ . After what time will the potential difference across the coil be equal to that across the resistance?

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$τ$

$τ l n 2$

$τ (1 - \ln 2)$

$\frac{τ}{l n 2}$

answer is B.

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Detailed Solution

$\begin{array}{l} i = i_{o} (1 - e^{- \frac{t}{τ}}) \\ \Rightarrow \frac{d i}{d t} = i_{o} \frac{1}{τ} e^{- \frac{t}{τ}} \\ \Rightarrow V_{L} = L \frac{d i}{d t} = i_{o} \frac{L}{τ} e^{- \frac{t}{τ}} = i_{o} R e^{- \frac{t}{τ}} \\ V_{R} = i R = R i_{o} (1 - e^{- \frac{t}{τ}}) \\ ∵ V_{L} = V_{R} \\ \Rightarrow i_{o} R e^{- \frac{t}{τ}} = R i_{o} (1 - e^{- \frac{t}{τ}}) \\ \Rightarrow 2 e^{- \frac{t}{τ}} = 1 \\ \Rightarrow t = τ l n (2) \end{array}$