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Q.

An e.m.f. of 15 volt is applied in a circuit containing 5 henry inductance and 10 ohm resistance. The ratio of the currents at time $t = \infty$ and at t = 1 second is

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a

$1 - e^{- 1}$

b

$\frac{e^{1 / 2}}{e^{1 / 2} - 1}$

c

$e^{- 1}$

d

$\frac{e^{2}}{e^{2} - 1}$

answer is B.

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Detailed Solution

$i = i_{0} (1 - e^{- Rt / L})$

$\begin{array}{l} i_{0} = \frac{E}{R} (Steady current) when t = \infty \\ i_{\infty} = \frac{E}{R} (1 - e^{- \infty}) = \frac{15}{10} = 1 .5 \\ i_{1} = 1 .5 (1 - e^{- R / L}) = 1 .5 (1 - e^{- 2}) \Rightarrow \frac{i_{\infty}}{i_{1}} = \frac{1}{1 - e^{- 2}} = \frac{e^{2}}{e^{2} - 1} \end{array}$

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