### Correct Option: B. 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 in RLC Circuits**: - The total impedance (Z) in a series RLC circuit is given by the formula: \[ Z = R + j(X_L - X_C) \] where: - \( R \) is the resistance, - \( X_L = \omega L \) is the inductive reactance, - \( X_C = \frac{1}{\omega C} \) is the capacitive reactance, - \( j \) is the imaginary unit. 2. **Resonance Condition**: - At resonance, the condition \( X_L = X_C \) holds true. This means: \[ \omega L = \frac{1}{\omega C} \] - When this condition is satisfied, the reactances cancel each other out: \[ X_L - X_C = 0 \] - Therefore, the impedance simplifies to: \[ Z = R + j(0) = R \] - This indicates that at resonance, the impedance of the circuit is purely resistive and equals 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, this is the minimum value of impedance that the circuit can achieve. In other words, the circuit allows maximum current to flow at this frequency because the impedance is at its lowest. ### Why Other Options Are Incorrect: - **Option A: The impedance is at its maximum value.** - This is incorrect because, at resonance, the impedance is not at its maximum. Instead, it is at its minimum value, which allows maximum current to flow through the circuit. - **Option C: The impedance equals the resistance only.** - While this statement is true at resonance, it does not fully capture the essence of the question regarding the nature of the impedance. The correct answer emphasizes that the impedance is at its minimum value, which is equal to the resistance. - **Option D: The impedance is purely capacitive.** - This is incorrect because, at resonance, the impedance is purely resistive (equal to \( R \)) and not capacitive. The capacitive reactance is canceled out by the inductive reactance, leading to a purely resistive impedance. ### 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 equal to the resistance \( R \), which is the minimum impedance the circuit can have. - This minimum impedance allows for maximum current flow in the circuit. - Understanding the resonance condition is crucial for analyzing RLC circuits and their behavior at different frequencies.