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

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 $\pi^2 = 10$:
$f_0 = \frac{10^5 \sqrt{10}}{10}$
$f_0 = 10 \times 10^3 \, \text{Hz}$