In an ac circuit, an inductor, a capacitor and a resistor are connected in series with `X_(L) = R = X_(c)` . Impedance of this circuit is :

Options

A. zero
B. `2R^(2)`
C. `Rsqrt(2)`
D. R

Answer

To find the impedance of an AC circuit consisting of a resistor (R), an inductor (L), and a capacitor (C) connected in series, where the inductive reactance \(X_L\), the capacitive reactance \(X_C\), and the resistance \(R\) are all equal, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Reactances**:
Given that \(X_L = R = X_C\), we can denote this common value as \(X\). Thus, we have:
\[ X_L = X, \quad R = X, \quad X_C = X \]

2. **Write the Impedance Formula**:
The impedance \(Z\) of a series RLC circuit is given by the formula:
\[ Z = \sqrt{R^2 + (X_L - X_C)^2} \]

3. **Substitute the Values**:
Substituting \(X_L\) and \(X_C\) with \(X\):
\[ Z = \sqrt{R^2 + (X - X)^2} \]

4. **Simplify the Expression**:
Since \(X - X = 0\), the equation simplifies to:
\[ Z = \sqrt{R^2 + 0^2} = \sqrt{R^2} \]

5. **Final Result**:
Thus, the impedance \(Z\) is:
\[ Z = R \]

### Conclusion:
The impedance of the circuit is equal to the resistance \(R\).