Question

Given below are two statements: one is labeled as Statement (1) and the other is labeled as Statement (2).
Statement (1): Maximum power is dissipated in a circuit containing an inductor, a ca pacitor and a resistor connected in series with an AC source, when resonance occurs
Statement (2): Maximum power is dissipated in a circuit containing pure resistor due to zero phase difference between current and voltage.
In the light of the above statements, choose the correct option:

• A. (1) and (2) both are correct
• B. (1) is correct but (2) is incorrect
• C. (1) is incorrect but (2) is correct
• D. Both (1) and (2) are incorrect

Hint:

Recall the behavior of RLC circuits at resonance and the power dissipation in resistive circuits. At resonance in an RLC circuit, the impedance is minimum, leading to maximum current and power dissipation. In a purely resistive AC cir cuit, the current and voltage are in phase, resulting in maximum power dissipa tion.

Answer 1

The Correct Option is A

Analysis of Statements on RLC Circuits and Power Dissipation:

Statement I:

In a series RLC circuit, resonance occurs when:

\( X_L = X_C \), where \( X_L = \omega L \) and \( X_C = \frac{1}{\omega C} \).

At resonance:

• The impedance (\( Z \)) becomes purely resistive (\( Z = R \)).
• The impedance is at its minimum, leading to the maximum current in the circuit.
• The average power dissipated in the circuit is given by:

\[ P_{\text{avg}} = I_{\text{rms}}^2 R \]

Since \( I_{\text{rms}} \) is maximum at resonance, the power dissipated is also maximum. Therefore, Statement I is true.

Statement II:

In a purely resistive circuit:

• The current and voltage are in phase, meaning the phase difference (\( \phi \)) is zero.
• The average power dissipated in a resistor is given by:

\[ P_{\text{avg}} = V_{\text{rms}} I_{\text{rms}} \cos\phi \]

Since \( \phi = 0 \), \( \cos\phi = 1 \), so:

\[ P_{\text{avg}} = V_{\text{rms}} I_{\text{rms}} \]

The power dissipation is maximum. Therefore, Statement II is true.

Conclusion:

Both Statement I and Statement II are true.

Answer 2