Question

For a series LCR circuit, the average power utilized in the circuit at resonance is equal to -

All notations have their usual meanings.

Answer 1

For a series LCR circuit, the power utilized is given by,

$P_{\text{avg}}=V_{\text{rms}}{i}_{\text{rms}}\cos \varphi =V_{\text{rms}}{i}_{\text{rms}}\times \frac{R}{Z}...\left(1\right)$

Also, we know that,

$Z=\sqrt{R^2+(X_C-X_L{)}^2}$

At resonance, $X_C=X_L$.

$\Rightarrow Z=R$

From equation $\left(1\right)$,

$P_{\text{avg}}=i_{\text{rms}}R\times i_{\text{rms}}\times \frac{R}{R}$

$\Rightarrow P_{\text{avg}}=i_{\text{rms}}^{2}R$

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