Question

Power dissipated in an $LCR$ series circuit connected to an a.c. source of emf $e$ is :

• A. $\frac{\varepsilon^{2} \sqrt{R^{2}+\left(L \omega-\frac{1}{C \omega}\right)^{2}}}{R}$
• B. $\varepsilon^{2}\left[R^{2}+\left(L \omega-\frac{1}{C \omega}\right)^{2}\right]$
• C. $\frac{\varepsilon^{2} R}{\sqrt{R^{2}+\left(L \omega-\frac{1}{C \omega}\right)^{2}}}$
• D. $\frac{\varepsilon^{2} R}{\left[R^{2}+\left(L \omega-\frac{1}{C \omega}\right)^{2}\right]}$

Answer 1

Answer: (D)

Average power, $P=E_{\text {r.m.s }} I_{\text {r.m.s }} \cos \phi$
$Z=\sqrt{R^{2}+\left(X_{L}-X_{C}\right)^{2}}, \cos \phi=\frac{R}{Z}$
But $I_{ \text{r . m . s }}=\frac{E_{\text{ r . m . s }}}{Z}, $
$\therefore P=E_{\text{ r . m . s }}^{2} \cdot \frac{R}{Z^{2}}$
$\therefore P =E_{\text{ r . m . s }}^{2} \frac{R}{\left\{R^{2}+\left(X_{L}-X_{C}\right)^{2}\right\}}$
$=\frac{\varepsilon^{2} R}{\left[R^{2}+\left(L \omega-\frac{1}{C \omega}\right)^{2}\right]}$