**Correct Option: A. The impedance is equal to the resistance of the circuit.** ### 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 means that they cancel each other out. 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. **At Resonance:** - At resonance, the condition \( X_L = X_C \) holds true. Therefore, the impedance simplifies to: \[ Z = R + j(0) = R \] - This means that the total impedance of the circuit at resonance is purely resistive and equal to the resistance \( R \) of the circuit. 3. **Physical Interpretation:** - At resonance, the energy oscillates between the inductor and capacitor without any loss, leading to maximum current flow in the circuit. The impedance being equal to the resistance indicates that the circuit behaves like a purely resistive circuit at this frequency. ### Why Other Options Are Incorrect: **B. The impedance is minimized and equal to zero.** - This statement is incorrect because while the impedance is minimized at resonance, it is not zero. The impedance is equal to the resistance \( R \), which is a positive value. A zero impedance would imply a short circuit, which is not the case in a resonant RLC circuit. **C. The impedance is equal to the inductive reactance.** - This statement is incorrect because at resonance, the inductive reactance \( X_L \) is equal to the capacitive reactance \( X_C \), and they cancel each other out. Therefore, the impedance cannot be equal to the inductive reactance; it is simply equal to the resistance \( R \). **D. The impedance is maximized and equal to the capacitive reactance.** - This statement is also incorrect. At resonance, the impedance is not maximized; rather, it is minimized and equal to the resistance \( R \). The capacitive reactance \( X_C \) does not determine the impedance at resonance; instead, it is the balance between \( X_L \) and \( X_C \) that leads to the cancellation of reactance. ### Summary of Key Points: - At resonance in a series RLC circuit, the impedance is equal to the resistance \( R \). - The inductive and capacitive reactances cancel each other out, resulting in a purely resistive impedance. - The circuit allows maximum current flow at the resonant frequency due to this condition. - Understanding the behavior of reactance and impedance at resonance is crucial for analyzing RLC circuits effectively.