Question

Which of the following is true in a series LCR circuit, at \(X_L=X_C\)?
A. Total impedance is equal to resistance \(R\),
B. Current in circuit is in phase with applied emf,
C. Current in circuit leads the emf by \(\dfrac{\pi}{2}\),
D. Current in circuit lags the emf by \(\dfrac{\pi}{2}\), E. Current may lead or lag the emf by \(\dfrac{\pi}{4}\). Choose the correct answer from the options given below:

• A. A, B Only
• B. A, B, D Only
• C. A, C, E Only
• D. A, D Only

Hint:

At resonance in a series LCR circuit, \(X_L=X_C\), impedance is minimum and equal to \(R\), and current is in phase with voltage.

Answer 1

The Correct Option is A

Concept:
In a series LCR circuit, resonance occurs when: \[ X_L=X_C \]

Step 1: Net reactance at resonance. \[ X=X_L-X_C \] At resonance: \[ X_L=X_C \] \[ X=0 \]

Step 2: Total impedance. \[ Z=\sqrt{R^2+(X_L-X_C)^2} \] \[ Z=\sqrt{R^2+0^2} \] \[ Z=R \] So, \[ A \text{ is correct} \]

Step 3: Phase relation.
At resonance, the circuit behaves like a purely resistive circuit. Therefore, current and applied emf are in the same phase. \[ B \text{ is correct} \]

Step 4: Check remaining statements.
Since current is in phase with emf, it neither leads nor lags by \(\frac{\pi}{2}\) or \(\frac{\pi}{4}\). \[ C,D,E \text{ are incorrect} \] Thus, correct statements are: \[ A,B \] \[ \therefore \text{Correct Answer is (A)} \]