Q 12-07-011NEETNEET 2021Top questionHard
A series LCR circuit containing $5.0$ H inductor, $80\ \mu$F capacitor and $40\ \Omega$ resistor is connected to $230$ V variable frequency ac source. The angular frequencies of the source at which power transferred to the circuit is half the power at the resonant angular frequency are likely to be :
Answer: (D) $46$ rad/s and $54$ rad/s
Concept. The half-power frequencies are $\omega_0 \pm \dfrac{R}{2L}$, with $\omega_0 = \dfrac{1}{\sqrt{LC}}$.
Step 1. Resonance.
$$\omega_0 = \frac{1}{\sqrt{5 \times 80 \times 10^{-6}}} = 50\ \text{rad/s}$$
Step 2. Half-width. $\dfrac{R}{2L} = \dfrac{40}{10} = 4$ rad/s.
Step 3. Frequencies. $46$ and $54$ rad/s.
Answer. $46$ rad/s and $54$ rad/s (Option D).
Solution by Sreeraj P, M.Sc Physics