Question

In a series $LCR$ circuit an alternating $emf(v)$ and current $(i)$ are given by the equation
$v=v_0\sin \omega t,i=i_0\sin \left(\omega t+\frac{\pi }{3}\right)$
The average power dissipated in the circuit over a cycle of $AC$ is

• A. $\frac{V_0{i}_0}{2}$
• B. $\frac{V_0{i}_0}{4}$
• C. $\frac{\sqrt{3}}{2}{V}_0{i}_0$
• D. Zero

Answer 1

Answer: (B)

$V=V_0\sin \omega t$
$I=I_0\sin \left(\omega t+\frac{\pi }{3}\right)$
The average dissipated power in AC circuit.
$P=V_{rms}{I}_{rms}\cos \varphi$
Here, $V_{rms}=\frac{V_0}{\sqrt{2}}$
$I_{rms}=\frac{I_0}{\sqrt{2}}$
$\Rightarrow \varphi =\frac{\pi }{3}$
$P=\frac{V_0}{\sqrt{2}}\times \frac{I_0}{\sqrt{2}}\times \cos \frac{\pi }{3}$
$=\frac{V_0{I}_0}{2}\times \cos 60^{\circ }$
$=\frac{V_0{I}_0}{2}\times \frac{1}{2}=\frac{V_0{I}_0}{4}$