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)

$\frac{V_0 i_0}{4}$

Answer 2

$V=V_{0} \sin\, \omega t$
$I=I_{0} \sin \left(\omega t+\frac{\pi}{3}\right)$
The average dissipated power in AC circuit.
$P= V_{ rms } I_{ rms } \cos \phi$
Here, $V_{ rms }= \frac{V_{0}}{\sqrt{2}}$
$I_{ rms }=\frac{I_{0}}{\sqrt{2}}$
$\Rightarrow \phi=\frac{\pi}{3}$
$P= \frac{V_{0}}{\sqrt{2}} \times \frac{I_{0}}{\sqrt{2}} \times \cos \frac{\pi}{3}$
$=\frac{V_{0} I_{0}}{2} \times \cos\, 60^{\circ}$
$= \frac{V_{0} I_{0}}{2} \times \frac{1}{2}=\frac{V_{0} I_{0}}{4}$