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Power dissipated in an LCR series circuit connected to an a.c. source of emf $\varepsilon$ is
A
$\varepsilon^2R\Big/\sqrt{R^2 + \left(L\omega - \frac{1}{C\omega}\right)^2}$
B
$\varepsilon^2R\Big/\left[R^2 + \left(L\omega - \frac{1}{C\omega}\right)^2\right]$
C
$\varepsilon^2\sqrt{R^2 + \left(L\omega - \frac{1}{C\omega}\right)^2}\Big/R$
D
$\varepsilon^2\left[R^2 + \left(L\omega - \frac{1}{C\omega}\right)^2\right]\Big/R$
Detailed Solution
Impedance of a series LCR circuit: $Z = \sqrt{R^2 + \left(L\omega - \frac{1}{C\omega}\right)^2}$
rms current: $I = \frac{\varepsilon}{Z}$
Power is dissipated only in the resistance: $P = I^2R$ (equivalently $P = \varepsilon I\cos\phi$ with $\cos\phi = \frac{R}{Z}$)
$P = \frac{\varepsilon^2R}{Z^2}$
$P = \frac{\varepsilon^2R}{R^2 + \left(L\omega - \frac{1}{C\omega}\right)^2}$
rms current: $I = \frac{\varepsilon}{Z}$
Power is dissipated only in the resistance: $P = I^2R$ (equivalently $P = \varepsilon I\cos\phi$ with $\cos\phi = \frac{R}{Z}$)
$P = \frac{\varepsilon^2R}{Z^2}$
$P = \frac{\varepsilon^2R}{R^2 + \left(L\omega - \frac{1}{C\omega}\right)^2}$