Question

A series LCR circuit containing a 5.0 H inductor, 80 µF capacitor and 40 Ω resistor is connected to a 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. 42 rad/s and 58 rad/s
• B. 25 rad/s and 75 rad/s
• C. 50 rad/s and 25 rad/s
• D. 46 rad/s and 54 rad/s

Answer 1

The Correct Option is D

To find the angular frequencies at which the power transferred to a series LCR circuit is half of that at the resonant frequency, we can follow these steps: 

The resonant angular frequency \(\omega_0\) in a series LCR circuit is given by:

\(\omega_0 = \frac{1}{\sqrt{LC}}\)

Where \(L\) is the inductance and \(C\) is the capacitance.

Given \(L = 5.0 \text{ H}\) and \(C = 80 \mu\text{F} = 80 \times 10^{-6} \text{ F}\),

we calculate \(\omega_0\) as follows:

\(\omega_0 = \frac{1}{\sqrt{5 \times 80 \times 10^{-6}}} = \frac{1}{\sqrt{0.0004}} = \frac{1}{0.02} = 50 \, \text{rad/s}\)

At resonance, the impedance \(Z\) is purely resistive, so the power transferred at resonance \(P_0\) can be expressed as:

\(P_0 = \frac{V^2}{2R}\)

Where \(V\) is the source voltage and \(R\) is the resistance. Power at the half-power points is:

\(P = \frac{P_0}{2} = \frac{V^2}{4R}\)

Impedance at half-power points and the relation with resonance frequency gives the condition:

\(R^2 = (\omega L - \frac{1}{\omega C})^2\)

This leads to a quadratic equation for \(\omega\):

\(R = |\omega L - \frac{1}{\omega C}|\)

Plugging in the values:\(\omega L = 40 \omega\)and \(\frac{1}{\omega C} = \frac{1}{80 \times 10^{-6} \omega}\), we solve

\(40 = 40 \omega - \frac{1}{80 \times 10^{-6} \omega}\) or \(40 = \frac{1}{80 \times 10^{-6} \omega} - 40 \omega\)

This simplifies to a quadratic equation leading us to find the roots:

\(\omega_1 = 46 \text{ rad/s} \quad \text{and} \quad \omega_2 = 54 \text{ rad/s}\)

Therefore, the angular frequencies of the source at which power transferred to the circuit is half the power at the resonant angular frequency are 46 rad/s and 54 rad/s.

The correct answer is 46 rad/s and 54 rad/s.