Question

In a series LCR circuit, a resistor of \( 300 \, \Omega \), a capacitor of \( 25 \, \text{nF} \), and an inductor of \( 100 \, \text{mH} \) are used. For maximum current in the circuit, the angular frequency of the AC source is -----\( \times 10^4 \) radians s\(^{-1}\).

Hint:

The angular frequency for resonance in an LCR circuit is given by \( \omega_0 = \frac{1}{\sqrt{LC}} \), where \( L \) is the inductance and \( C \) is the capacitance.

Answer 1

Correct Answer: 2

For maximum current in an LCR circuit, the condition of resonance must be satisfied. The resonance angular frequency \( \omega_0 \) is given by: \[ \omega_0 = \frac{1}{\sqrt{LC}}, \] where: - \( L = 100 \, \text{mH} = 100 \times 10^{-3} \, \text{H} \), - \( C = 25 \, \text{nF} = 25 \times 10^{-9} \, \text{F} \). Substitute the values into the formula: \[ \omega_0 = \frac{1}{\sqrt{(100 \times 10^{-3}) (25 \times 10^{-9})}} = \frac{1}{\sqrt{2.5 \times 10^{-12}}} = 3 \times 10^4 \, \text{rad/s}. \] Thus, the angular frequency is \( 3 \times 10^4 \, \text{rad/s} \).