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 -

42 rad/s

and

58 rad/s

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25 rad/s

and

75 rad/s

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50 rad/s

and

25 rad/s

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46 rad/s

and

54 rad/s

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Solution

The correct option is D $46 rad/s$ and $54 rad/s$
The resonance frequency of LCR series circuit is given as,

$ω_{0} = \frac{1}{\sqrt{L C}} = \frac{1}{\sqrt{5 \times 80 \times 10^{- 6}}} = 50 rad/s$

Now, half power frequencies are given as,

$ω = ω_{0} \pm \frac{R}{2 L}$

So,

$ω_{L} = 50 - \frac{40}{2 \times 5} = 46 rad/s$

$ω_{H} = 50 + \frac{40}{2 \times 5} = 54 rad/s$

Hence, option $(D)$ is the correct answer.

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Similar questions

(1) Figure shows a series LCR circuit connected to a variable frequency $230 V$ source. $L = 5.0 H, C = 80 μ F, R = 40 Ω$ Determine the source frequency which drives the circuit in resonance.

(2) Figure shows a series LCR circuit connected to a variable frequency $230 V$ source. $L = 5.0 H, C = 80 μ F, R = 40 Ω$ Obtain the impedance of the circuit and the amplitude of current at the resonating frequency.

(3) Figure shows a series LCR circuit connected to a variable frequency $230 V$ source. $L = 5.0 H, C = 80 μ F, R = 40 Ω$ Determine the rms potential drops across the three elements of the circuit. Show that the potential drop across the LC combination is zero at the resonating frequency.