A series LCR circuit containing 5.0 H inductor, 80 capacitor and 40 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 fr

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Solution

Q.
A series LCR circuit containing 5.0 H inductor, 80 $μ F$ capacitor and 40 $Ω$ 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 [2021]

1 42 rad/s and 58 rad/s

2 25 rad/s and 75 rad/s

3 50 rad/s and 25 rad/s

4 46 rad/s and 54 rad/s

Ans.
(4)

$ω_{r} = \frac{1}{\sqrt{L C}} = \frac{1}{\sqrt{5 \times 80 \times 10^{- 6}}} = 50$ $rad/s$

Power transferred = half of power at resonance

So, frequencies at which power transferred is half $= ω_{r} \pm Δ ω \Rightarrow Δ ω = \frac{R}{2 L} = \frac{40}{2 \times 5} = 4$ $rad/s$

So, range is $ω_{r} \pm Δ ω = 50 \pm 4 =$ 54 rad/s and 46 rad/s

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