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 emf and current in inductive circuit are given by
$E = E_{0} \sin ω t and i = i_{0} \sin (ω t - \frac{π}{2}) respectively. So, P_{inst} = E i = E_{0} \sin ω t \times i_{0} \sin (ω t - \frac{π}{2}) \begin{array}{l} = E_{0} i_{0} \sin ω t (\sin ω t \cos \frac{π}{2} - \cos ω t \sin \frac{π}{2}) \\ = E_{0} i_{0} \sin ω t \cos ω t \\ = \frac{1}{2} E_{0} i_{0} \sin 2 ω t (\sin 2 ω t = 2 \sin ω \cos ω t) \end{array}$
Hence, angular frequency of instantaneous power is 2 $ω$

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