Question

The electric current in a circuit is given by $i=i_0\left(\frac{t}{\pi }\right)$ for some time. Calculate the $rms$ current for the period $t=0$ to $t=\pi$.

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

Answer 1

The correct option is C $\frac{i_0}{\sqrt{3}}$

The root mean square value of current is,

$i_{rms}=\sqrt{\frac{{\int }_0^{\pi }{i}^{2}dt}{{\int }_0^{\pi }dt}}$

$=\sqrt{\frac{1}{\pi }{\int }_0^{\pi }{i}_0^2{\left(\frac{t}{\pi }\right)}^{2}dt}$

$=\sqrt{\frac{i_0^2}{{\pi }^3}{\int }_0^{\pi }{t}^{2}dt}=\sqrt{\frac{i_0^2}{3}}$

$\Rightarrow i_{rms}=\frac{i_0}{\sqrt{3}}$

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