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(C) From the circuit diagram,we have resistance $R = 10 \ \Omega$,inductance $L = 0.1 \ H$,and angular frequency $\omega = 100 \ rad/s$.The power fact

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$500$

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(C) From the circuit diagram,we have resistance $R = 10 \ \Omega$,inductance $L = 0.1 \ H$,and angular frequency $\omega = 100 \ rad/s$.The power factor is given by $\cos \phi = \frac{R}{Z} = \frac{R}{\sqrt{R^2 + (X_L - X_C)^2}} = \frac{1}{\sqrt{2}}$.This implies $Z = R\sqrt{2}$,so $R^2 + (X_L - X_C)^2 = 2R^2$,which simplifies to $(X_L - X_C)^2 = R^2$.Thus,$X_L - X_C = \pm R$.Given $X_L = \omega L = 100 	imes 0.1 = 10 \ \Omega$.Case $1$: $10 - X_C = 10 \implies X_C = 0$ (not possible for finite $C$).Case $2$: $10 - X_C = -10 \implies X_C = 20 \ \Omega$.Since $X_C = \frac{1}{\omega C}$,we have $20 = \frac{1}{100 	imes C}$.$C = \frac{1}{2000} \ F = 0.5 	imes 10^{-3} \ F = 500 \ \mu F$.

The power factor of the given $LCR$ circuit is $1/\sqrt{2}$. Find the capacitance $C$ of the circuit in $\mu F$.

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