### 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) At resonance, the total impedance (Z) of the circuit is minimized, and the circuit behaves as if it were purely resistive. This leads to maximum current flowing through the circuit, as the impedance is at its lowest value. **Step-by-Step Explanation:** 1. **Understanding Reactance:** - Inductive reactance (\(X_L\)) increases with frequency, meaning that as the frequency of the source increases, the opposition to current flow due to the inductor increases. - Capacitive reactance (\(X_C\)) decreases with frequency, meaning that as the frequency increases, the opposition to current flow due to the capacitor decreases. 2. **Condition for Resonance:** - At resonance, the two reactances are equal: \(X_L = X_C\). This means that the inductive and capacitive effects cancel each other out. - The formula for resonance frequency (\(f_0\)) can be derived from the condition \(X_L = X_C\): \[ 2\pi f_0 L = \frac{1}{2\pi f_0 C} \] Rearranging gives: \[ f_0 = \frac{1}{2\pi \sqrt{LC}} \] 3. **Implications of Resonance:** - When resonance occurs, the impedance of the circuit is minimized to just the resistance (R), leading to maximum current according to Ohm's Law: \[ I = \frac{V}{R} \] - Here, \(I\) is the current, \(V\) is the voltage across the circuit, and \(R\) is the resistance. 4. **Energy Transfer:** - At resonance, energy oscillates between the inductor and capacitor, allowing for efficient energy transfer and maximum current flow. #### Why Other Options Are Incorrect: - **Option B: When the resistance is at its maximum value** - This option is incorrect because maximum resistance would increase the total impedance of the circuit, thereby reducing the current. Resonance is not about maximizing resistance but rather about balancing reactances. - **Option C: When the voltage source is at its peak** - While having a peak voltage can lead to a higher instantaneous current, it does not define the condition for resonance. Resonance is specifically about the relationship between inductive and capacitive reactance, not the voltage level. - **Option D: When the frequency of the source is twice the natural frequency of the circuit** - This option is incorrect because resonance occurs at the natural frequency (\(f_0\)), not at double that frequency. At double the natural frequency, the circuit would not be in resonance, and the current would not be maximized. ### Revision Summary: - Resonance in a series RLC circuit occurs when inductive reactance equals capacitive reactance (\(X_L = X_C\)). - The resonance frequency can be calculated using \(f_0 = \frac{1}{2\pi \sqrt{LC}}\). - At resonance, the impedance is minimized to just the resistance, leading to maximum current flow. - Other options do not correctly describe the conditions for resonance, focusing instead on resistance or voltage levels.