Question

An AC source of angular frequency $\omega$ is applied across an inductor. The instantaneous power developed in the circuit, has an angular frequency of,

• A. $\omega$
• B. $2\omega$
• C. $\frac{\omega }{2}$
• D. $\frac{\omega }{4}$

Answer 1

The correct option is B $2\omega$

Let us assume, instantaneous voltage is $E=E_0\sin (\omega t+\pi /2)=E_0\cos \left(\omega t\right)$.

For a purely inductive circuit, current lags behind the voltage by $\pi /2$.

So, instantaneous current,

$i=i_0\sin (\omega t+\pi /2-\pi /2)=i_0\sin \left(\omega t\right)$

Now, instantaneous power,

$P=Ei=E_0\cos \left(\omega t\right)\times i_0\sin \left(\omega t\right)$

$\Rightarrow P=\frac{E_0{i}_0}{2}\times 2\cos \left(\omega t\right)\sin \left(\omega t\right)$

$\Rightarrow P=\frac{E_0{i}_0}{2}\sin \left(2\omega t\right)$

The instantaneous power developed in the circuit, has an angular frequency of $2\omega$.

Hence, option $\left(B\right)$ is the correct answer.