In an `LCR` circuit `R=100 ohm`. When capacitance `C` is removed, the current lags behind the voltage by `pi//3`. When inductance `L` is removed, the current leads the voltage by `pi//3`. The impedence of the circuit is

Options

A. 50ohm
B. 100ohm
C. 200ohm
D. 400ohm

Answer

To solve the problem, we will analyze the given conditions in the LCR circuit and use the relationships between resistance, inductance, capacitance, and impedance.

### Step-by-Step Solution:

1. **Understanding the Circuit Conditions**:
- The circuit has a resistance \( R = 100 \, \Omega \).
- When the capacitance \( C \) is removed, the circuit behaves as an RL circuit, and the current lags the voltage by \( \frac{\pi}{3} \).
- When the inductance \( L \) is removed, the circuit behaves as a RC circuit, and the current leads the voltage by \( \frac{\pi}{3} \).

2. **Using the Phase Angle in RL Circuit**:
- For the RL circuit (when \( C \) is removed), the phase angle \( \phi \) is given by:
\[ \tan(\phi) = \frac{X_L}{R} \]
- Here, \( \phi = \frac{\pi}{3} \) (which is \( 60^\circ \)).
- Therefore:
\[ \tan\left(\frac{\pi}{3}\right) = \sqrt{3} = \frac{X_L}{100} \]
- From this, we can find \( X_L \):
\[ X_L = 100 \sqrt{3} \, \Omega \]

3. **Using the Phase Angle in RC Circuit**:
- For the RC circuit (when \( L \) is removed), the phase angle \( \phi \) is given by:
\[ \tan(\phi) = \frac{X_C}{R} \]
- Again, \( \phi = \frac{\pi}{3} \) (which is \( 60^\circ \)).
- Therefore:
\[ \tan\left(\frac{\pi}{3}\right) = \sqrt{3} = \frac{X_C}{100} \]
- From this, we can find \( X_C \):
\[ X_C = 100 \sqrt{3} \, \Omega \]

4. **Equating Inductive and Capacitive Reactance**:
- From the two equations derived, we have:
\[ X_L = X_C \]
- Thus:
\[ 100 \sqrt{3} = 100 \sqrt{3} \quad \text{(which is consistent)} \]

5. **Calculating the Impedance**:
- The total impedance \( Z \) of the LCR circuit can be calculated using the formula:
\[ Z = \sqrt{R^2 + (X_L - X_C)^2} \]
- Since \( X_L = X_C \), we have:
\[ Z = \sqrt{R^2 + 0} = R \]
- Therefore:
\[ Z = 100 \, \Omega \]

### Conclusion:
The impedance of the circuit is \( 100 \, \Omega \).