### Correct Option: D. The impedance is at its minimum value and equals the resistance (R). #### Detailed Explanation: In a series RLC circuit, resonance occurs when the inductive reactance (XL) and capacitive reactance (XC) are equal. This condition can be expressed mathematically as: \[ X_L = X_C \] Where: - \( X_L = 2\pi f L \) (inductive reactance) - \( X_C = \frac{1}{2\pi f C} \) (capacitive reactance) - \( f \) is the frequency of the source, - \( L \) is the inductance, - \( C \) is the capacitance. At resonance, the total impedance (Z) of the circuit can be calculated using the formula: \[ Z = R + j(X_L - X_C) \] Where: - \( R \) is the resistance, - \( j \) is the imaginary unit, - \( X_L - X_C \) is the net reactance. Since at resonance \( X_L = X_C \), the net reactance becomes zero: \[ Z = R + j(0) = R \] This means that the impedance of the circuit at resonance is purely resistive and equal to the resistance \( R \). #### Why Option D is Correct: - At resonance, the impedance is minimized because the reactive components (inductive and capacitive) cancel each other out. Therefore, the only component contributing to the impedance is the resistance \( R \). This results in the impedance being at its minimum value, which is equal to \( R \). #### Why Other Options are Incorrect: **A. The impedance is equal to the resistance (R) only.** - This statement is partially correct but does not capture the essence of the condition at resonance. While it is true that the impedance equals the resistance at resonance, it does not mention that this is the minimum value of impedance, which is crucial for understanding the behavior of the circuit. **B. The impedance is at its maximum value.** - This statement is incorrect. At resonance, the impedance is not at its maximum; rather, it is at its minimum. The maximum impedance in a series RLC circuit occurs when either the inductive or capacitive reactance is at its peak, which is not the case at resonance. **C. The impedance is equal to zero.** - This statement is also incorrect. While the net reactance is zero at resonance, the impedance cannot be zero because there is always a resistance \( R \) present in the circuit. Therefore, the impedance is equal to \( R \), not zero. ### Summary: - At resonance in a series RLC circuit, the inductive and capacitive reactances cancel each other out. - The total impedance is minimized and equals the resistance \( R \). - The correct answer is that the impedance is at its minimum value and equals the resistance (Option D). - Understanding resonance is crucial for analyzing RLC circuits and their behavior in AC systems.