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A series LCR circuit with inductance 10 H, capacitance $10\ \mu F$, resistance $50\ \Omega$ is connected to an ac source of voltage, $V=200\sin(100t)$ volt. If the resonant frequency of the LCR circuit is $\nu_0$ and the frequency of the ac source is $\nu$, then :
A
$\nu=100\ Hz;\ \nu_0=\frac{100}{\pi}\ Hz$
B
$\nu_0=\nu=50\ Hz$
C
$\nu_0=\nu=\frac{50}{\pi}\ Hz$
D
$\nu_0=\frac{50}{\pi}\ Hz,\ \nu=50\ Hz$
Detailed Solution
$V = 200\sin(100t)$, so $\omega = 100$ rad/s
Source frequency $\nu = \frac{\omega}{2\pi} = \frac{100}{2\pi} = \frac{50}{\pi}$ Hz
Resonant frequency $\nu_0 = \frac{1}{2\pi\sqrt{LC}} = \frac{1}{2\pi}\sqrt{\frac{1}{10 \times 10 \times 10^{-6}}} = \frac{50}{\pi}$ Hz
So $\nu_0 = \nu = \frac{50}{\pi}$ Hz