Question

The resistance \( R \), the inductive reactance \( X_L \), and the capacitive reactance \( X_C \) of a series LCR circuit connected to an ac source are related by \( R = 0.6 X_L = 3 X_C \). The impedance of the circuit is:

• A. \( \frac{R}{3} \)
• B. \( \frac{2R}{3} \)
• C. \( \frac{4R}{3} \)
• D. \( \frac{5R}{3} \)

Hint:

The impedance in a series LCR circuit is calculated using \( Z = \sqrt{R^2 + (X_L - X_C)^2} \), where \( X_L \) and \( X_C \) are the inductive and capacitive reactances, respectively.

Answer 1

The Correct Option is D

The impedance \( Z \) of the series LCR circuit is: \[ Z = \sqrt{R^2 + (X_L - X_C)^2} \] Given the relationships \( X_L = 0.6R \) and \( X_C = \frac{R}{3} \), substitute these into the formula: \[ Z = \sqrt{R^2 + (0.6R - \frac{R}{3})^2} \] Simplify the expression: \[ Z = \sqrt{R^2 + \left( \frac{1.8R - R}{3} \right)^2} \] Thus, the impedance is \( Z = \frac{5R}{3} \).