First slide

Time -Varying current in L-R circuit

Question

An e.m.f. of 15 v is applied in a circuit containing 5 H inductance and $10 Ω$ resistance. The ratio of the currents at time $t = \infty$ and at t = 1 s is

Moderate

A

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

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

$1 - e^{- 1}$
D

$e^{- 1}$

Solution

$i = i_{0} (1 - e^{- Rt / L})$
$i_{0} = \frac{E}{R} (Steady current) when t = \infty$
$i_{\infty} = \frac{E}{R} (1 - e^{- \infty}) = \frac{5}{10} = 1.5$
$i_{1} = 1.5 (1 - e^{- R / L}) = 1.5 (1 - e^{- 2})$
$\Rightarrow \frac{i_{\infty}}{i_{1}} = \frac{1}{1 - e^{- 2}} = \frac{e^{2}}{e^{2} - 1}$

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