Question

In a series $LCR$ circuit, the inductance $L$ is 10 mH, capacitance $C$ is 1µF and resistance $R$ is 100 $\Omega$. The frequency at which resonance occurs is $\frac{x}{\pi}$ Hz. Find $\frac{x}{100}$.

Answer 1

50

Answer 2

The resonant frequency ($f_r$) of a series LCR circuit is given by the formula: $f_r = \frac{1}{2\pi\sqrt{LC}}$

Given values: Inductance $L = 10 \text{ mH} = 10 \times 10^{-3} \text{ H} = 10^{-2} \text{ H}$ Capacitance $C = 1 \text{ µF} = 1 \times 10^{-6} \text{ F}$ Resistance $R = 100 \text{ } \Omega$ (Resistance is not required for calculating the resonant frequency).

Substitute the values of L and C into the formula: $f_r = \frac{1}{2\pi\sqrt{(10^{-2} \text{ H})(10^{-6} \text{ F})}}$ $f_r = \frac{1}{2\pi\sqrt{10^{-8}}}$ $f_r = \frac{1}{2\pi \times (10^{-8})^{1/2}}$ $f_r = \frac{1}{2\pi \times 10^{-4}}$ $f_r = \frac{10^4}{2\pi}$ $f_r = \frac{10000}{2\pi}$ $f_r = \frac{5000}{\pi} \text{ Hz}$

The problem states that the frequency at which resonance occurs is $\frac{x}{\pi}$ Hz. Comparing this with our calculated frequency: $\frac{x}{\pi} = \frac{5000}{\pi}$ From this, we find the value of x: $x = 5000$ The question asks to find the value of $\frac{x}{100}$: $\frac{x}{100} = \frac{5000}{100}$ $\frac{x}{100} = 50$