Question

In a series \(LCR\) circuit, resonance occurs when:

• A. \(X_L>X_C\)
• B. \(X_C>X_L\)
• C. \(X_L=X_C\)
• D. \(R=0\)

Hint:

At resonance: \[ Z=R \] and current becomes maximum in a series \(LCR\) circuit.

Answer 1

The Correct Option is C

Concept: In an \(LCR\) circuit: \[ X_L=\omega L \] and \[ X_C=\frac1{\omega C} \] At resonance: \[ X_L=X_C \] which makes impedance minimum and current maximum.

Step 1: Condition for resonance. At resonance: \[ \omega L=\frac1{\omega C} \] Thus: \[ X_L=X_C \]

Step 2: Effect at resonance. Net reactance becomes zero: \[ X=X_L-X_C=0 \] Therefore impedance: \[ Z=R \] becomes minimum.

Step 3: Final conclusion. \[ \boxed{ X_L=X_C } \]