A series L - C - R circuit containing 5 . 0H inductor, 80 capacitor...

A series

L - C - R

circuit containing

5 . 0H

inductor,

80\muF

capacitor and

40Ω

resistor is connected to

230V

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:

Options

A $25rad{s}^{- 1}\text{and} 75 rad{s}^{- 1}$
B $50rad{s}^{- 1}\text{and}25rad{s}^{- 1}$
C $46rad{s}^{- 1}\text{and}54rad{s}^{- 1}$
D $42rad{s}^{- 1}\text{and}58rad{s}^{- 1}$

Correct answer

C. $46rad{s}^{- 1}\text{and}54rad{s}^{- 1}$

Step-by-step solution

$L = 5H, C = 80\muF, R = 40\omega; V = 230V$

Resonance angular frequency $= {\omega}_{0} = \dfrac{1}{\sqrt{LC}} = \dfrac{1}{\sqrt{5 \times \left(80 \times {10}^{- 6}\right)}} = 50rad{s}^{- 1}$

Half power angular frequency $= {\omega}_{0} \pm \dfrac{R}{2L} = 50 \pm \dfrac{40}{2 \times 5} = 50 \pm 4 = 54\text{or}46$

We get,

${\omega}_{1} = 46rad{s}^{- 1}$

And ${\omega}_{2} = 54rad{s}^{- 1}$

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