Question

The current decaying in a circuit has a function $i=I_0{e}^{-t/\tau }$ where $\tau$ is the time constant of the circuit. Find $rms$ value of the current for the period $t=0\text{to}t=\tau$.

Answer 1

We know that root mean square value of current is given as,

$I_{rms}^2=\frac{1}{T}{\int }_0^T{i}^{2}dt$

$\Rightarrow I_{rms}^2=\frac{1}{\tau }{\int }_0^{\tau }{I}_0^2{e}^{-2t/\tau }dt$

$=\frac{I_0^2}{\tau }{\left[\frac{e^{-2t/\tau }}{-2/\tau }\right]}_0^{\tau }=-\frac{I_0^2}{2}[e^{-2}-1]$

$=\frac{I_0^2}{2}(1-\frac{1}{e^2})$

$\therefore I_{rms}=\frac{I_0}{\sqrt{2}}\sqrt{1-\frac{1}{e^2}}$

Hence, $\left(A\right)$ is the correct option.