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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:
A
50 rad/s and 25 rad/s
B
46 rad/s and 54 rad/s
C
42 rad/s and 58 rad/s
D
25 rad/s and 75 rad/s
Detailed Solution
Resonant angular frequency: $\omega_0 = \frac{1}{\sqrt{LC}} = \frac{1}{\sqrt{5 \times 80 \times 10^{-6}}} = 50$ rad/s
Bandwidth: $\Delta\omega = \frac{R}{L} = \frac{40}{5} = 8$ rad/s
Half-power frequencies: $\omega_0 \pm \frac{\Delta\omega}{2} = 50 \pm 4$
So 46 rad/s and 54 rad/s.