Question

An inductor of 10 mH, capacitor of 0.1 µF and a resistor of 100 Ω are connected in series across an a.c power supply 220 V, 70 Hz. The power factor of the given circuit is 0.5. The difference in the inductive reactance and capacitance reactance is \(\sqrt{3} a\) Ω. The value of \(a\) is ______.

Answer 1

Correct Answer: 100

Step 1: Understanding the Concept:
In an LCR series circuit, the power factor is defined as \(\cos \phi = R/Z\), where \(Z\) is the total impedance. The impedance \(Z\) is given by \(\sqrt{R^2 + (X_L - X_C)^2}\).

Step 2: Key Formula or Approach:
1. Power Factor \(\cos \phi = \frac{R}{Z} = 0.5\) 2. \(Z = \sqrt{R^2 + (X_L - X_C)^2}\) 3. Reactance difference \(\Delta X = |X_L - X_C|\)

Step 3: Detailed Explanation:
1. Since \(\cos \phi = 0.5 = 1/2\), we have \(Z = 2R\). 2. Substitute into the impedance formula: \[ (2R)^2 = R^2 + (X_L - X_C)^2 \] \[ 4R^2 = R^2 + (X_L - X_C)^2 \] \[ 3R^2 = (X_L - X_C)^2 \] 3. Taking the square root: \[ |X_L - X_C| = \sqrt{3}R \] 4. Given \(R = 100 \, \Omega\): \[ |X_L - X_C| = \sqrt{3} \times 100 \] 5. Comparing with the given form \(\sqrt{3} a\): \[ \sqrt{3} a = \sqrt{3} \times 100 \implies a = 100 \]

Step 4: Final Answer:
The value of \(a\) is 100.