Question

Calculate the $rms$ value of the $emf$ for a cycle, whose time variation is shown in figure.

• A. $V_0$
• B. $\frac{V_0}{2}$
• C. $\frac{2V_0}{3}$
• D. $\frac{V_0}{\sqrt{2}}$

Answer 1

The correct option is D $\frac{V_0}{\sqrt{2}}$

The variation of voltage with respect to time $t$ in one cycle according to the given graph is,

$V=\{\begin{array}{ccc}& V_0& 0<t<T/2\\ & 0& T/2<t<T\end{array}$

So, root mean square value of voltage,

$V_{rms}^2=\frac{1}{T}{\int }_0^T{V}^{2}dt$

$=\frac{1}{T}{\int }_0^{T/2}{V}_0^{2}dt$

$=\frac{V_0^2}{T}\times \frac{T}{2}=\frac{V_0^2}{2}$

So, $V_{rms}=\sqrt{\frac{V_0^2}{2}}=\frac{V_0}{\sqrt{2}}$

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