In a series L.C.R. circuit an alternating emf (v) and current (i) are given by the equation \( v=v_{0} \sin \omega t, l=i_{0} \sin \left(\omega t+\frac{\Pi}{3}\right).\) The average power discipated in the circuit over a cycle of \( A C \) is

Given, emf
\( v=v_{0} \sin \omega t ; \) current \( i=i_{0} \sin \left(\omega t+\frac{\Pi}{3}\right) \)
Then, average power dissipated in circuit over a cycle of ac is \( P_{\text {avg }}=v_{\mathrm{rms}} i_{\mathrm{ms}} \cos \phi \)
We know, \( v_{\mathrm{ms}}=\frac{v_{0}}{\sqrt{2}} ; i_{\mathrm{rms}}=\frac{i_{0}}{\sqrt{2}} ; \phi=\frac{\Pi}{3} \) \( \therefore P_{\mathrm{avg}}=\frac{v_{0}}{\sqrt{2}} \cdot \frac{i_{0}}{\sqrt{2}} \cos \left(\frac{\Pi}{3}\right)=\frac{v_{0} i_{0}}{2} \times \frac{1}{2}=\frac{v_{0} i_{0}}{4} \)
\( \therefore P_{\text {avg }}=\frac{v_{0}}{\sqrt{2}} \cdot \frac{i_{0}}{\sqrt{2}} \cos \left(\frac{\Pi}{3}\right)=\frac{v_{0} i_{0}}{2} \times \frac{1}{2} \) Thus, average power dissipated \( =\frac{v_{0} i_{0}}{4} \)