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. $\frac{3i_0}{2}$
• C. $\sqrt{\frac{i_0}{2}}$
• D. $\frac{3}{4}\sqrt{i_0}$

Answer 1

Answer: (A)

As, $i=i_0(t/\tau )$
$i^2=\frac{\underset{0}{\overset{\tau }{\int }}{i}^{2}dt}{\tau }=\frac{\underset{0}{\overset{\tau }{\int }}{i}_0^2(t/\tau {)}^{2}dt}{\tau }$
$=\frac{i_0^2}{{\tau }^3}\underset{0}{\overset{\tau }{\int }}{t}^{2}dt=\frac{i_0^2}{{\tau }^3}\times \frac{{\tau }^3}{3}=\frac{i_0^2}{3}$
Thus, $i_{\text{rms }}=\sqrt{i^2}=\sqrt{\frac{i_0^2}{3}}=\frac{i_0}{\sqrt{3}}$