First slide

Power in a.c. circuit

Question

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

Moderate

A

$\frac{\omega }{4}$
B

$\frac{\omega }{2}$
C

ω
D

2ω

Solution

The instantaneous values of emf and current in inductive circuit are given by
$E = {E_o}\sin \omega t\,\,and\,\,i = {i_o}\sin \left( {\omega t - \frac{\pi }{2}} \right)\,\,respectively.$
$so,\,\,{P_{inst}} = Ei = {E_0}\sin \omega t \times {i_0}\sin \left( {\omega t - \frac{\pi }{2}} \right)$
$= {E_0}{i_0}\sin \omega t\left( {\sin \omega t\cos \frac{\pi }{2} - \cos \omega t\sin \frac{\pi }{2}} \right)$
$= {E_0}{i_0}\sin \omega t\;\cos \omega t$
$= \frac{1}{2}{E_0}{i_0}\sin 2\omega t$ $(\sin 2\omega t = 2\sin \omega t\;\cos \omega t)$
Hence, angular frequency of instantaneous power is 2ω .

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