A series LCR circuit with inductance 10 H, capacitance 10 μF, and resistance 50 Ω is connected to an AC source

Question

A series LCR circuit with inductance 10 H, capacitance 10 μF, and resistance 50 Ω is connected to an AC source of voltage $V = 200 \sin(100t)$ V. If the resonant frequency of the LCR circuit is $
u_0$ and the frequency of the AC source is $
u$, then:

Accepted Answer

According to the sources, the resonant frequency ($
u_0$) of a series LCR circuit is given by the formula $
u_0 = \frac{1}{2\pi\sqrt{LC}}$ . Given $L = 10$ H and $C = 10$ μF ($10 	imes 10^{-6}$ F), the calculation is: $
u_0 = \frac{1}{2\pi\sqrt{10 	imes 10 	imes 10^{-6}}} = \frac{1}{2\pi\sqrt{10^{-4}}} = \frac{1}{2\pi 	imes 10^{-2}} = \frac{100}{2\pi} = \frac{50}{\pi}$ Hz. For the AC source frequency ($
u$), we use the general voltage equation $V = V_m \sin(\omega t)$. Comparing this with $V = 200 \sin(100t)$, we find the angular frequency $\omega = 100$ rad/s. Since $\omega = 2\pi
u$, the source frequency is $
u = \frac{100}{2\pi} = \frac{50}{\pi}$ Hz. Therefore, $
u_0 = 
u = \frac{50}{\pi}$ Hz.

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