Alternating Current - Result Question 4

4. The instantaneous values of alternating current and voltages in a circuit are given as

$ \begin{aligned} & i=\frac{1}{\sqrt{2}} \sin (100 \pi t) \text{ amper } \\ & e=\frac{1}{\sqrt{2}} \sin (100 \pi t+\pi / 3) \text{ Volt } \end{aligned} $

The average power in Watts consumed in the circuit is :

(a) $\frac{1}{4}$

(b) $\frac{\sqrt{3}}{4}$

(c) $\frac{1}{2}$

(d) $\frac{1}{8}$

[2012M]

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Answer:

Correct Answer: 4. (d)

Solution:

  1. (d) The average power in the circuit where $\cos \phi=$ power factory

$

=V _{\text{rms }} \times I _{\text{rms }} \cos \phi$

$\phi=\pi / 3=$ phase difference $=\frac{180}{3}=60$

$V _{rms}=\frac{\frac{1}{\sqrt{2}}}{\sqrt{2}}=\frac{1}{2}$ volt

$I _{rms}=\frac{\frac{1}{\sqrt{2}}}{\sqrt{2}}=(\frac{1}{2}) A$

$\cos \phi=\cos \frac{\pi}{3}=\frac{1}{2}$

$

=\frac{1}{2} \times \frac{1}{2} \times \frac{1}{2}=\frac{1}{8} W$

The instantaneous power is the power in circuit at any instant of time. It is equal to the product of values of alternating voltage and alternating current at that time

$P _{\text{in }}=EI=(E_0 \sin \omega t)(I_0 \sin \omega t)$

(in non-inductive circuit)

$P _{\text{in }}=(E_0 \sin \omega t)(I_0 \sin \omega t \pm \theta)$

(in inductive circuit)