Question

The electric current in a circuit is given by $i=i_{0}(t / \tau)$ for same time. The rms current for the period $t=0$ to $t=\tau$ is

• A. $\frac{i_{0}}{\sqrt{3}}$
• B. $\frac{3 i_{0}}{2}$
• C. $\sqrt{\frac{i_{0}}{2}}$
• D. $\frac{3}{4} \sqrt{i_{0}}$

Answer 1

Answer: (A)

As, $i =i_{0}(t / \tau)$
$i^{2} =\frac{\int\limits_{0}^{\tau} i^{2} d t}{\tau}=\frac{\int\limits_{0}^{\tau} i_{0}^{2}(t / \tau)^{2} d t}{\tau}$
$=\frac{i_{0}^{2}}{\tau^{3}} \int\limits_{0}^{\tau} t^{2} d t=\frac{i_{0}^{2}}{\tau^{3}} \times \frac{\tau^{3}}{3}=\frac{i_{0}^{2}}{3}$
Thus, $i_{ \text{rms }}=\sqrt{i^{2}}=\sqrt{\frac{i_{0}^{2}}{3}}=\frac{i_{0}}{\sqrt{3}}$