### Correct Option: A. When the inductive reactance equals the capacitive reactance #### Detailed Explanation: In a series RLC (Resistor, Inductor, Capacitor) circuit, resonance occurs when the inductive reactance (XL) and 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) Here, \( f \) is the frequency of the AC source, \( L \) is the inductance, and \( C \) is the capacitance. At resonance, the total impedance (Z) of the circuit is minimized, and it is purely resistive. This means that the circuit can draw maximum current from the power source. The formula for the total impedance in a series RLC circuit is: \[ Z = \sqrt{R^2 + (X_L - X_C)^2} \] When \( X_L = X_C \), the equation simplifies to: \[ Z = R \] This results in the maximum current flowing through the circuit, as per Ohm's Law: \[ I = \frac{V}{Z} \] Where \( I \) is the current, \( V \) is the voltage, and \( Z \) is the impedance. Since \( Z \) is minimized at resonance, the current \( I \) reaches its maximum value. #### Why Other Options Are Incorrect: **B. When the resistance is minimized** - While minimizing resistance can increase current, it does not guarantee resonance. Resonance specifically depends on the balance between inductive and capacitive reactance, not just the resistance. A circuit can have low resistance but still not be at resonance if \( X_L \) and \( X_C \) are not equal. **C. When the total voltage is at its peak** - The peak voltage does not necessarily indicate resonance. The voltage can be high without the circuit being at resonance. Resonance is defined by the relationship between reactances, not the voltage level. The current is maximized at resonance regardless of the voltage level. **D. When the frequency is twice the resonant frequency** - This statement is incorrect because resonance occurs at a specific frequency, known as the resonant frequency (\( f_0 \)). The resonant frequency is given by: \[ f_0 = \frac{1}{2\pi\sqrt{LC}} \] At twice the resonant frequency, the circuit would not be in resonance, and the current would not be maximized. ### Summary for Revision: - Resonance in a series RLC circuit occurs when inductive reactance equals capacitive reactance (\( X_L = X_C \)). - At resonance, the total impedance is minimized to just the resistance, allowing maximum current flow. - Other conditions like minimizing resistance or having peak voltage do not define resonance. - The resonant frequency is a specific value, not double the resonant frequency. Understanding these concepts is crucial for analyzing RLC circuits and predicting their behavior under different conditions.