Question
In an $LR$ -circuit, time constant is that time in which current grows from zero to the value (where ${I_0}$ is the steady state current)
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Answer
(a) Current at any instant of time $t$ after closing an $L-R$ circuit is given by $I = {I_0}\left[ {1 - {e^{\frac{{ - R}}{L}t}}} \right]$ Time constant $t = \frac{L}{R}$
$\therefore \,\,I = {I_0}\left[ {1 - {e^{\frac{{ - R}}{L} \times \frac{L}{R}}}} \right] = {I_0}\,(1 - {e^{ - 1}}) = {I_0}\,\left( {1 - \frac{1}{e}} \right)$.
$ = {I_0}\,\left( {1 - \frac{1}{{2.718}}} \right) = 0.63{I_0} = 63\% $ of ${I_0}$
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