**Correct Option: C. The inductive reactance must equal the capacitive reactance.** ### Detailed Explanation: In a series RLC (Resistor, Inductor, Capacitor) circuit, resonance occurs when the circuit can oscillate at its natural frequency with maximum amplitude. This phenomenon is crucial in many applications, such as radio transmitters and receivers, where specific frequencies need to be amplified. 1. **Understanding Reactance:** - **Inductive Reactance (X_L)**: This is the opposition to the change of current by an inductor and is given by the formula: \[ X_L = 2\pi f L \] where \( f \) is the frequency of the AC source and \( L \) is the inductance. - **Capacitive Reactance (X_C)**: This is the opposition to the change of voltage by a capacitor and is given by the formula: \[ X_C = \frac{1}{2\pi f C} \] where \( C \) is the capacitance. 2. **Condition for Resonance:** - For resonance to occur in a series RLC circuit, the inductive reactance must equal the capacitive reactance: \[ X_L = X_C \] - This means: \[ 2\pi f L = \frac{1}{2\pi f C} \] - Rearranging this equation allows us to find the resonant frequency (\( f_0 \)): \[ f_0 = \frac{1}{2\pi \sqrt{LC}} \] - At this frequency, the total impedance of the circuit is minimized, and the circuit can draw maximum current. 3. **Why Option C is Correct:** - When \( X_L = X_C \), the reactive components cancel each other out, leading to a purely resistive impedance at resonance. This results in maximum current flow through the circuit, which is the hallmark of resonance. ### Why the Other Options are Incorrect: - **Option A: The inductive reactance must be greater than the capacitive reactance.** - This is incorrect because if \( X_L > X_C \), the circuit is inductively dominated, and the total impedance will be greater than the resistance, leading to lower current. Resonance requires equality, not inequality. - **Option B: The total impedance must be at its maximum value.** - This is also incorrect. At resonance, the total impedance is at its minimum value (equal to the resistance \( R \)). If the impedance were at its maximum, it would indicate that the circuit is not resonating and is instead experiencing higher opposition to current flow. - **Option D: The resistance must be zero.** - While a lower resistance can lead to higher quality factor (Q) and sharper resonance, it is not a requirement for resonance itself. Resonance can occur with any finite resistance; it just affects the sharpness of the resonance peak. ### Summary of Key Points: - Resonance in a series RLC circuit occurs when inductive reactance equals capacitive reactance (\( X_L = X_C \)). - The resonant frequency can be calculated using \( f_0 = \frac{1}{2\pi \sqrt{LC}} \). - At resonance, the total impedance is minimized, leading to maximum current flow. - Other conditions, such as having zero resistance or maximum impedance, do not satisfy the resonance condition. This understanding is crucial for analyzing and designing circuits that operate at specific frequencies, ensuring optimal performance in various applications.