Question

When an inductor and a resistor are connected in series to an AC source, the power factor of the circuit is \[ \frac{2}{\sqrt{13}}. \] If the same resistor and a capacitor are connected in series to the same AC source, then the power factor of the circuit is \[ \frac{1}{\sqrt2}. \] If these inductor, capacitor and resistor are connected in series to the same AC source, then the ratio of the resistance and impedance of the LCR circuit is

• A. \(1:\sqrt7\)
• B. \(2:\sqrt5\)
• C. \(2:\sqrt7\)
• D. \(1:\sqrt5\)

Hint:

For AC circuits, \[ \boxed{ \cos\phi=\frac{R}{Z}. } \] Also, \[ \boxed{ Z_{RL}=\sqrt{R^2+X_L^2},\qquad Z_{RC}=\sqrt{R^2+X_C^2}, } \] and for a series LCR circuit, \[ \boxed{ Z=\sqrt{R^2+(X_L-X_C)^2}. } \]

Answer 1

The Correct Option is B

Step 1: Find the inductive reactance. For the RL circuit, \[ \cos\phi = \frac{R}{\sqrt{R^2+X_L^2}} = \frac{2}{\sqrt{13}}. \] Squaring, \[ \frac{R^2}{R^2+X_L^2} = \frac{4}{13}. \] Hence, \[ 13R^2 = 4(R^2+X_L^2), \] \[ 9R^2 = 4X_L^2, \] \[ X_L=\frac{3R}{2}. \]

Step 2: Find the capacitive reactance. For the RC circuit, \[ \frac{R}{\sqrt{R^2+X_C^2}} = \frac{1}{\sqrt2}. \] Therefore, \[ 2R^2 = R^2+X_C^2, \] \[ X_C=R. \]

Step 3: Find the impedance of the LCR circuit. Net reactance is \[ X=X_L-X_C = \frac{3R}{2}-R = \frac{R}{2}. \] Hence, \[ Z = \sqrt{R^2+\left(\frac{R}{2}\right)^2} = \frac{R\sqrt5}{2}. \] Thus, \[ R:Z = R:\frac{R\sqrt5}{2} = 2:\sqrt5. \] Hence, \[ \boxed{2:\sqrt5}. \] Thus, \[ \boxed{(B)} \] is the correct answer.