Question

Two statements are given, one labelled Assertion (A) and the other labelled Reason (R). Select the correct answer from the codes (A), (B), (C), and (D) as given below.
Assertion (A): A series LCR circuit behaves as a pure resistive circuit at resonance.
Reason (R): At resonance, \( X_L = X_C \) gives \( \omega = \frac{1}{\sqrt{LC}} \).

• A. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
• B. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
• C. Assertion (A) is true, but Reason (R) is false.
• D. Both Assertion (A) and Reason (R) are false.

Hint:

At resonance in a series LCR circuit, \( X_L = X_C \), cancelling out the reactive parts, so only resistance remains in the impedance.

Answer 1

The Correct Option is A

In a series LCR circuit, resonance is achieved when inductive reactance ($X_L$) equals capacitive reactance ($X_C$), meaning $X_L = X_C$. This condition leads to: \[ \omega L = \frac{1}{\omega C} \Rightarrow \omega^2 = \frac{1}{LC} \Rightarrow \omega = \frac{1}{\sqrt{LC}} \] At this resonant frequency: - The net reactance, $X = X_L - X_C$, becomes 0. Consequently, the impedance, $Z$, equals the resistance, $R$ ($Z=R$). - Therefore, the circuit exhibits behavior characteristic of a pure resistive circuit during resonance. Based on these findings: - The assertion is valid. - The reason provided is also valid. - The reason accurately elucidates the assertion. Final answer: Option (A)