Question

For a given series LCR circuit, it is found that maximum current is drawn when value of variable capacitance is $2.5 \, \text{nF}$. If resistance of $2000 \, \Omega$ and $100 \, \text{mH}$ inductor is being used in the given circuit. The frequency of AC source is ____ $\times \, 10^3 \, \text{Hz}$. (Given $\pi^2 = 10$)

Answer 1

Correct Answer: 10

For maximum current, the circuit must be in resonance.
The resonant frequency \( f_0 \) is given by:
\[f_0 = \frac{1}{2\pi\sqrt{L \cdot C}}\]
Substitute \( L = 100 \times 10^{-3} \, \text{H} \) and \( C = 2.5 \times 10^{-9} \, \text{F} \):
\[f_0 = \frac{1}{2\pi\sqrt{100 \times 10^{-3} \times 2.5 \times 10^{-9}}}\]
Simplify under the square root:
\[f_0 = \frac{1}{2\pi\sqrt{25 \times 10^{-11}}}\]
\[f_0 = \frac{1}{2\pi \times 5 \times 10^{-6}}\]
Using \( \pi^2 = 10 \):
\[f_0 = \frac{10^5 \sqrt{10}}{10}\]
\[f_0 = 10 \times 10^3 \, \text{Hz}\]