A resistor of $50 \Omega$, inductor of self inductance $\left(\frac{3}{\pi^2}\right) \mathrm{H}$ and a…
A resistor of $50 \Omega$, inductor of self inductance $\left(\frac{3}{\pi^2}\right) \mathrm{H}$ and a capacitor of unknown capacity are connected in series to an a.c. source of 100 V and 50 Hz . When the voltage and current are in phase, the value of capacitance is (nearly)
$0.66 \times 10^{-4} \mathrm{~F}$
$0.33 \times 10^{-4} \mathrm{~F}$
$0.66 \times 10^{-2} \mathrm{~F}$
$0.33 \times 10^{-2} \mathrm{~F}$
Solution
Given: $\mathrm{L}=\frac{3}{\pi^2} \mathrm{H}$ and $\mathrm{f}=50 \mathrm{~Hz}$
As voltage and current are in phase, the circuit is a resonant circuit.
$\begin{aligned}
& \therefore \quad \mathrm{X}_{\mathrm{C}}=\mathrm{X}_{\mathrm{L}} \\
& \frac{1}{\omega \mathrm{C}}=\omega \mathrm{L} \\
& \mathrm{C}=\frac{1}{\omega^2 \mathrm{~L}}=\frac{1}{4 \pi^2 \mathrm{f}^2 \mathrm{~L}} \\
&=\frac{1}{4 \pi^2 \times(50)^2 \times \frac{3}{\pi^2}}=0.33 \times 10^{-4} \mathrm{~F}
\end{aligned}$
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