Question 167 of 190

In an $\text{A.C.}$ circuit, the instantaneous e.m.f. and current are given by, $E=100\sin 30t$$,I=20\sin \left(30t-\dfrac{\pi }{4}\right)$. In one cycle of $\text{A.C.}$, the average power consumed by the circuit (in watt) and the watt-less current (in ampere) are, respectively:

A$50, 0$B$50, 10$C$\dfrac{1000}{\sqrt{2}},10$D$\dfrac{50}{\sqrt{2}},0$

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Answer: (C) $\dfrac{1000}{\sqrt{2}},10$

Instantaneous emf and current is given by, $E=100\sin 30t$, $I=20\sin \left(30t-\dfrac{\pi }{4}\right)$.
Phase difference between emf and current is
$\phi =30t-\left(30t-\dfrac{\pi }{4}\right)=\dfrac{\pi }{4}$

Average power is given by,
${P}_{av}={E}_{rms}{I}_{rms}cos\phi$
$=\dfrac{100}{\sqrt{2}}\times \dfrac{20}{\sqrt{2}}\times \dfrac{1}{\sqrt{2}}$$=\dfrac{1000}{\sqrt{2}}$

Watt-less current is given by,
${I}_{WL}={I}_{rms}sin\phi$$=\dfrac{{I}_{o}}{\surd 2}\sin 45^{\circ}$$\Rightarrow {I}_{WL}=\dfrac{20}{\sqrt{2}}.\dfrac{1}{\sqrt{2}}=10A$.