Question

In an AC circuit, an inductor, a capacitor, and a resistor are connected in series with \( X_L = R = X_C \). The impedance of this circuit is:

• A. \( 2R^2 \)
• B. Zero
• C. \( R \)
• D. \( R\sqrt{2} \)

Hint:

When inductive and capacitive reactances are equal in a series circuit, the impedance is purely resistive.

Answer 1

The Correct Option is C

{Using Impedance Formula} 
\[ Z = \sqrt{(X_L - X_C)^2 + R^2} \] Since \( X_L = X_C \), \[ Z = \sqrt{(R - R)^2 + R^2} = \sqrt{0 + R^2} = R \] Thus, the correct answer is \( R \). 
 

Answer 2

Step 1: Understand the components in the series AC circuit
The circuit contains:
- An inductor with reactance \( X_L \)
- A capacitor with reactance \( X_C \)
- A resistor with resistance \( R \)
All connected in series.

Step 2: Total impedance in an RLC series circuit
The total impedance \( Z \) of a series RLC circuit is given by:
\[ Z = \sqrt{R^2 + (X_L - X_C)^2} \]

Step 3: Use the given condition
It is given that:
\[ X_L = X_C = R \]
So the difference \( X_L - X_C = 0 \)

Step 4: Substitute into the impedance formula
\[ Z = \sqrt{R^2 + (0)^2} = \sqrt{R^2} = R \]

Step 5: Final Answer
The impedance of the circuit is:
R