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The instantaneous emf and current equations of an A.C. circuit are respectively $ e= 200 \sin (\omega t + \dfrac{\pi}{3}) $ and $ i=10 \sin \omega t $ . The average power consumed over one complete cycle is:
(A) $ 2000W $
(B) $ 1000W $
(C) $ 500 \mathrm{W} $
(D) $ 707W $

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Last updated date: 24th Jun 2024
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Answer
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Hint
According to the Faraday’s Law states that the instantaneous EMF (voltage) induced in a circuit equals the rate of change of magnetic flux through the circuit. The value of an alternating quantity at a particular instant is called instantaneous value. The graph of instantaneous values is plotted of an alternating quantity plotted against time is called waveform. Based on this concept we have to solve this question.

Complete step by step answer
The phase difference between current and emf is denoted as, $ \phi=\pi / 3 $
From the given data in the question, we get the peak value of the voltage $ \mathrm{V}_{\mathrm{o}}=200 $ volts
Also, we get the peak value of current $ \mathrm{i}_{0}=10 \mathrm{A} $
Therefore, we can calculate the average power that is consumed over the complete cycle with the equation,
 $ P_{a v g}=\dfrac{V_{o} i_{o}}{2} \cos \phi $
 $ \therefore \mathrm{P}_{\mathrm{avg}}=\dfrac{200 \times 10}{2} \cos \dfrac{\pi}{3}=\dfrac{200 \times 10}{2} \times 0.5 $
Equating $ P_{a v g}=\dfrac{V_{o} i_{o}}{2} \cos \phi $ , we get the results as,
 $ {{\text{P}}_{\text{avg}}}=500\text{W} $
Hence, the correct answer is Option (C).

Note
The maximum value attained by an alternating quantity during one cycle is called its peak value. It is also known as the maximum value or amplitude or crest value. The rms value is defined as the effective value of a varying voltage or current. It is the equivalent steady DC, which is constant value which gives the same effect. For instance, a lamp connected to a 6 V RMS AC supply will shine with the same brightness when connected to a steady 6 V DC supply.
We should know that attempts to find an average value of AC would directly provide us with the answer zero. Hence, the RMS values are used.