A resistor of 50 Omega,an inductor of self-in | vedclass

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(D) In an $LCR$ series circuit,the voltage and current are in phase at resonance.At resonance,the inductive reactance $(X_L)$ is equal to the capaciti

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(D) In an $LCR$ series circuit,the voltage and current are in phase at resonance.At resonance,the inductive reactance $(X_L)$ is equal to the capacitive reactance $(X_C)$.$X_L = X_C$$\omega L = \frac{1}{\omega C}$Given: $L = \frac{2}{\pi^2} 	ext{ H}$,$f = 50 	ext{ Hz}$.Angular frequency $\omega = 2\pi f = 2 	imes \pi 	imes 50 = 100\pi 	ext{ rad/s}$.Substituting the values:$100\pi 	imes \frac{2}{\pi^2} = \frac{1}{100\pi 	imes C}$$\frac{200}{\pi} = \frac{1}{100\pi 	imes C}$$C = \frac{1}{100\pi 	imes (200/\pi)} = \frac{1}{20000} 	ext{ F}$$C = 0.5 	imes 10^{-4} 	ext{ F} = 50 	imes 10^{-6} 	ext{ F} = 50 \mu 	ext{F}$.

$A$ resistor of $50 \Omega$,an inductor of self-inductance $(\frac{2}{\pi^2}) \text{ H}$,and a capacitor of unknown capacity are connected in series to an $A$.$C$. source of $100 \text{ V}, 50 \text{ Hz}$. When the voltage and current are in phase,the value of the capacitance is: (in $\mu \text{F}$)

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