In an `LCR` circuit `R=100 Omega`. 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 impedance of the circuit is

Options

A. `50 Omega`
B. `100 Omega`
C. `200 Omega`
D. `400 Omega`

Explanation

To find the impedance of the LCR circuit given the conditions in the problem, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Circuit Configuration**:
The circuit consists of a resistor (R), inductor (L), and capacitor (C). We know that R = 100 Ω.

2. **Condition When Capacitor is Removed**:
When the capacitor is removed, the circuit consists of R and L. The current lags the voltage by \( \frac{\pi}{3} \) radians (or 60 degrees). This indicates that the circuit is inductive.

3. **Relate Phase Angle to Impedance**:
The phase angle \( \phi \) in an RL circuit can be expressed as:
\[ \tan(\phi) = \frac{X_L}{R} \]
where \( X_L \) is the inductive reactance. For \( \phi = \frac{\pi}{3} \):
\[ \tan\left(\frac{\pi}{3}\right) = \sqrt{3} \]
Therefore, we can write:
\[ \sqrt{3} = \frac{X_L}{100} \]
From this, we find:
\[ X_L = 100\sqrt{3} \quad \text{(Equation 1)} \]

4. **Condition When Inductor is Removed**:
When the inductor is removed, the circuit consists of R and C. The current leads the voltage by \( \frac{\pi}{3} \) radians (or 60 degrees). This indicates that the circuit is capacitive.

5. **Relate Phase Angle to Capacitive Reactance**:
The phase angle \( \phi \) in an RC circuit can be expressed as:
\[ \tan(\phi) = \frac{X_C}{R} \]
where \( X_C \) is the capacitive reactance. For \( \phi = \frac{\pi}{3} \):
\[ \tan\left(\frac{\pi}{3}\right) = \sqrt{3} \]
Therefore, we can write:
\[ \sqrt{3} = \frac{X_C}{100} \]
From this, we find:
\[ X_C = 100\sqrt{3} \quad \text{(Equation 2)} \]

6. **Equate Inductive and Capacitive Reactance**:
From Equations 1 and 2, we have:
\[ X_L = X_C \]
This implies:
\[ 100\sqrt{3} = 100\sqrt{3} \]
which is consistent.

7. **Calculate the Impedance**:
The total impedance \( Z \) of the LCR circuit is given by:
\[ Z = \sqrt{R^2 + (X_L - X_C)^2} \]
Since \( X_L = X_C \), we have:
\[ Z = \sqrt{R^2 + 0^2} = \sqrt{R^2} = R \]
Therefore:
\[ Z = 100 \, \Omega \]

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