### Correct Option: C. The inductive reactance is equal to the capacitive reactance. ### Detailed Explanation: In an RLC series circuit, resonance occurs when the inductive reactance (XL) and the 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 AC source, - \( L \) is the inductance, - \( C \) is the capacitance. At resonance, the circuit behaves in a very specific way: 1. The total impedance (Z) of the circuit is minimized and is equal to the resistance (R) of the circuit. 2. The current in the circuit is at its maximum because the impedance is at its lowest. 3. The phase angle between the voltage and current is zero, meaning they are in phase. #### Why Option C is Correct: - When \( X_L = X_C \), the effects of the inductor and capacitor cancel each other out. This means that the reactive components do not contribute to the total impedance, allowing the circuit to draw maximum current from the source. - This condition is crucial for applications such as tuning circuits in radios, where you want to select a specific frequency. ### Why the Other Options are Incorrect: **A. The inductive reactance is greater than the capacitive reactance.** - This condition indicates that the circuit is inductively dominated, which means that the total impedance is higher than the resistance alone. The current would be lower, and resonance would not occur. Instead, the circuit would be in a state of lagging current. **B. The resistance is equal to the inductive reactance.** - This condition does not lead to resonance. While it may create a specific impedance condition, it does not satisfy the requirement that \( X_L \) must equal \( X_C \). The circuit would still have reactive components affecting the current and voltage phase relationship. **D. The total impedance is at its maximum.** - This statement is incorrect for resonance. At resonance, the total impedance is at its minimum (equal to the resistance). A maximum impedance would indicate that the circuit is not resonating and would have a higher reactance, leading to lower current flow. ### Formulas and Intermediate Steps: 1. **Inductive Reactance**: \[ X_L = 2\pi f L \] 2. **Capacitive Reactance**: \[ X_C = \frac{1}{2\pi f C} \] 3. **Condition for Resonance**: Set \( X_L = X_C \): \[ 2\pi f L = \frac{1}{2\pi f C} \] Rearranging gives: \[ f^2 = \frac{1}{(2\pi)^2 LC} \] Thus, the resonant frequency \( f_0 \) is: \[ f_0 = \frac{1}{2\pi \sqrt{LC}} \] ### Common Pitfalls: - Confusing the conditions for resonance with those for inductive or capacitive dominance. - Forgetting that at resonance, the impedance is purely resistive, and the phase angle is zero. - Miscalculating the resonant frequency by not using the correct formula. ### Revision Summary: - Resonance in an RLC circuit occurs when \( X_L = X_C \). - At resonance, the total impedance is minimized and equals the resistance. - The current is maximized, and the voltage and current are in phase. - The resonant frequency can be calculated using \( f_0 = \frac{1}{2\pi \sqrt{LC}} \).