The current in a discharging LR circuit is given by $i = i_{0} e^{- t / τ}$ where $τ$ is the time constant of the circuit. Calculate the rms current for the period $t = 0 to t = τ$ .

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$\frac{i_{0}}{e} \sqrt{\frac{(e^{- 2} + 1)}{2}}$

$\frac{i_{0}}{e} \sqrt{\frac{(e + 2)}{3}}$

$\frac{i_{0}}{e} \sqrt{\frac{(e^{2} - 1)}{2}}$

$\frac{i_{0}}{e} \sqrt{\frac{(e^{2} - 2)}{2}}$

answer is A.

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Detailed Solution

$\begin{array}{l} I_{max} = \sqrt{\frac{\int_{0}^{τ} i_{0}^{2} e^{- \frac{2 t}{τ}} dt}{\int_{0}^{τ} dt}} = \sqrt{\frac{i_{0}^{2} (e^{\frac{- 2 τ}{τ}} - 1)}{(\frac{- 2}{τ}) τ}} \\ = \sqrt{\frac{i_{0}^{2}}{2} (1 - \frac{1}{e^{2}})} \end{array}$

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