While studying the decay of current through $L R$ circuit, the current at any time $t,$ when maximum current is $I_{0}$ is given by

I_{0} ⎛ ⎜ ⎝ 1 - e^{\frac{t}{R C}} ⎞ ⎟ ⎠

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I_{0} ⎛ ⎜ ⎝ 1 - e^{\frac{- L}{R}} ⎞ ⎟ ⎠

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I_{0} e^{- \frac{R t}{L}}

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I_{0} e^{- \frac{L t}{R}}

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Solution

The correct option is D $I_{0} e^{- \frac{R t}{L}}$
The emf equation for current decay $L R$ circuit is given by
$L \frac{d I}{d t} + R I = 0$
or $\frac{d I}{I} = - \frac{R}{L} d t$
Integrating, $\int_{I_{0}}^{I} \frac{d I}{I} = - \frac{R}{L} \int_{0}^{t} d t$
or $l n I - l n I_{0} = \frac{- R t}{L}$
or $l n (\frac{I}{I_{0}}) = \frac{- R}{t} L$
or $\frac{I}{I_{0}} = e^{- \frac{R t}{L}}$
or $I = I_{0} e^{- \frac{R t}{L}}$

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