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  4. In a series LCR circuit an alternating emf (v) and current (i) are given by the equation v = v0 sin ω t, i = i0 sin ( ω t + (π/3) ) The average power dissipated in the circuit over a cycle of AC is

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Q. 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

KCETKCET 2016Alternating Current

Solution:

$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} $
$= \frac{V_{0} I_{0}}{2} \times \frac{1}{2}=\frac{V_{0} I_{0}}{4}$