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Question: Answered & Verified by Expert
In an AC circuit the emf $(e)$ and the current (i) at any instant are given respectively by
$$
\begin{array}{l}
e=E_{0} \sin \omega t \\
i=I_{0} \sin (\omega t-\phi)
\end{array}
$$
The average power in the circuit over one cycle of AC is
PhysicsAlternating CurrentMHT CETMHT CET 2008
Options:
  • A $\frac{E_{0} I_{0}}{2}$
  • B $\frac{E_{0} I_{0}}{2} \sin \phi$
  • C $\frac{E_{0} I_{0}}{2} \cos \phi$
  • D $E_{0} I_{0}$
Solution:
1805 Upvotes Verified Answer
The correct answer is: $\frac{E_{0} I_{0}}{2} \cos \phi$
Power $=$ rate of work done in one complete cycle.
Or
$$
P_{\mathrm{av}}=\frac{W}{T}
$$
Or
$$
P_{\mathrm{av}}=\frac{\left(E_{0} I_{0} \cos \phi\right) T / 2}{T}
$$
or
$$
P_{\mathrm{av}}=\frac{E_{0} I_{0} \cos \phi}{2}
$$
where $\cos \phi$ is called the power factor of an $\mathrm{AC}$ circuit.

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