An AC source with voltage $V=V_0\sin (\omega t)$ is connected to a series LCR circuit. If the frequency of the source is varied, the power factor of the circuit is $1/\sqrt{2}$ at two angular frequencies ${\omega }_1$ and ${\omega }_2$. Find the value of $|{\omega }_1-{\omega }_2|$.

Solution

1. The power factor is given by $$\begin{array}{rl}\cos \varphi & =R/Z\\ & =1/\sqrt{2}\end{array}$$, which implies $Z=\sqrt{2}R$.
2. Since $Z^2=R^2+X^2$, we have $2R^2=R^2+X^2$, meaning the net reactance $|\omega L-1/(\omega C)|=R$.
3. This results in two quadratic equations: ${\omega }^{2}L-\omega R-1/C=0$ and ${\omega }^{2}L+\omega R-1/C=0$.
4. The difference between the roots of these equations corresponds to the bandwidth of the circuit, which is given by $$\begin{array}{rl}\Delta \omega & =|{\omega }_1-{\omega }_2|\\ & =R/L\end{array}$$.
5. Hence the answer is (A).