### Correct Option: A. The impedance is at its minimum value. ### Detailed Explanation: In a series RLC (Resistor, Inductor, Capacitor) circuit, resonance occurs at a specific frequency known as the resonant frequency. At this frequency, the inductive reactance (XL) and capacitive reactance (XC) are equal in magnitude but opposite in phase. This leads to a unique condition in the circuit that affects its impedance. 1. **Understanding Impedance**: - Impedance (Z) in an RLC circuit is a complex quantity that combines resistance (R) and reactance (X). It is given by the formula: \[ Z = R + j(X_L - X_C) \] - Here, \(X_L\) is the inductive reactance (\(X_L = \omega L\)), and \(X_C\) is the capacitive reactance (\(X_C = \frac{1}{\omega C}\)), where \(\omega\) is the angular frequency. 2. **Condition at Resonance**: - At resonance, the condition \(X_L = X_C\) holds true. This means: \[ X_L - X_C = 0 \] - Therefore, the impedance simplifies to: \[ Z = R + j(0) = R \] - This indicates that the impedance of the circuit at resonance is purely resistive and equal to the resistance \(R\). 3. **Minimum Impedance**: - Since the impedance at resonance is equal to the resistance \(R\) and there are no reactive components contributing to the impedance (as they cancel each other out), this is indeed the minimum value of impedance that the circuit can achieve. ### Why Other Options Are Incorrect: - **Option B: The impedance is at its maximum value.** - This is incorrect because at resonance, the reactances cancel each other out, leading to the lowest possible impedance, which is just the resistance \(R\). Maximum impedance would occur when either the inductive or capacitive reactance is at its highest, which is not the case at resonance. - **Option C: The impedance is equal to the resistance only.** - While this statement is true at resonance, it does not fully capture the essence of the question regarding the nature of impedance at resonance. The correct answer emphasizes that the impedance is at its minimum value, which is a more comprehensive understanding of the circuit behavior. - **Option D: The impedance is infinite.** - This is incorrect because infinite impedance would imply an open circuit condition, which is not the case in a resonant RLC circuit. At resonance, the circuit is fully operational with a finite impedance equal to the resistance. ### Summary of Key Points: - At resonance in a series RLC circuit, the inductive and capacitive reactances cancel each other out. - The impedance at resonance is purely resistive and equal to the resistance \(R\). - This condition represents the minimum impedance of the circuit. - Understanding resonance is crucial for analyzing RLC circuits in AC applications. By grasping these concepts, students can better understand the behavior of RLC circuits and their applications in various electrical systems.