First slide

Power in a.c. circuit

Question

An alternating e.m.f. of angular frequency $ω$ is applied across an inductance. The instantaneous power developed in the circuit has an angular frequency

Moderate

A

$\frac{ω}{4}$
B

$\frac{ω}{2}$
C

$ω$
D

$2 ω$

Solution

The instantaneous values of e.m.f. and current in inductive circuit are given by $E = E_{0} sinωt$ and $i = i_{0} \sin (ωt - \frac{π}{2})$ respectively.
So, $P_{inst} = Ei = E_{0} \sin ωt \times i_{0} \sin (ωt - \frac{π}{2})$
$= E_{0} i_{0} \sin ωt (\sin ωt \cos \frac{π}{2} - \cos ωt \sin \frac{π}{2})$
$= E_{0} i_{0} \sin ωt \cos ωt$
$= \frac{1}{2} E_{0} i_{0} \sin 2 ωt (\sin 2 ωt = 2 \sin ωt \cos ωt)$
Hence, angular frequency of instantaneous power is
$2 ω .$

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