Question

Power dissipated in an LCR series circuit connected to an ac source of emf $\epsilon$ is

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

Answer 1

The correct option is D $\frac{{\epsilon }^{2}R}{[R^2+⟮\omega L-\frac{1}{\omega C}{⟯}^2]}$

$X_1=wL$

$X_2=\frac{1}{wc}$

Impedance of $Rt\left(z\right)=\sqrt{R^2+(X_1-X_2{)}^2}$

$\Rightarrow z=\sqrt{R^2+{(wL-\frac{1}{wc})}^2}$

$\Rightarrow$ Current $I$;

$\left(I\right)=\frac{ϵ}{z}=\frac{ϵ}{\sqrt{R^2+{(wL-\frac{1}{wc})}^2}}$

Heat dissipated across a resistor of resistance $R$

$H=I^{2}R$

$\Rightarrow H=\frac{{ϵ}^2}{z^2}{R}^{}=\frac{{ϵ}^{2}R}{[R^2+{(wL-\frac{1}{wc})}^2]}$