### Correct Option: A. \( X_L = X_C \) ### Detailed Explanation: In a tuned radio receiver, we often use a series circuit consisting of a resistor (R), an inductor (L), and a capacitor (C). The goal of this circuit is to achieve resonance, which is a condition where the circuit can efficiently select a specific frequency from a range of frequencies, such as those used in radio transmissions. **Resonance Condition:** For resonance to occur in an RLC series circuit, the inductive reactance (\( X_L \)) must equal the capacitive reactance (\( X_C \)). This is mathematically expressed as: \[ X_L = X_C \] **Definitions:** - **Inductive Reactance (\( X_L \))**: This is the opposition that an inductor offers to the flow of alternating current (AC) due to its inductance. It is given by the formula: \[ X_L = 2\pi f L \] where: - \( f \) is the frequency of the AC signal, - \( L \) is the inductance in henries (H). - **Capacitive Reactance (\( X_C \))**: This is the opposition that a capacitor offers to the flow of AC due to its capacitance. It is given by the formula: \[ X_C = \frac{1}{2\pi f C} \] where: - \( C \) is the capacitance in farads (F). ### Why Option A is Correct: At resonance, the circuit behaves as if it has no reactance, meaning the total impedance is minimized, and the circuit can draw maximum current. This condition is achieved when: \[ X_L = X_C \] This equality ensures that the energy stored in the inductor is equal to the energy stored in the capacitor, allowing for efficient energy transfer and resonance. ### Why the Other Options are Incorrect: - **Option B: \( X_L = \frac{1}{2}X_C \)** This suggests that the inductive reactance is half of the capacitive reactance. This condition does not satisfy the resonance condition and would lead to a situation where the circuit is not tuned to the desired frequency, resulting in poor performance. - **Option C: \( X_L = \frac{1}{X_C} \)** This implies that the inductive reactance is the reciprocal of the capacitive reactance. This relationship does not hold true in the context of resonance and would not provide a valid condition for tuning the circuit. - **Option D: \( X_L = 2X_C \)** This indicates that the inductive reactance is twice that of the capacitive reactance. Similar to the previous options, this does not satisfy the resonance condition and would lead to an imbalance in the circuit, preventing it from effectively tuning to the desired frequency. ### Summary of Key Points: - In a tuned RLC series circuit, resonance occurs when \( X_L = X_C \). - Inductive reactance (\( X_L \)) is given by \( 2\pi f L \) and capacitive reactance (\( X_C \)) is given by \( \frac{1}{2\pi f C} \). - Resonance allows maximum current flow and efficient energy transfer in the circuit. - Other options do not satisfy the resonance condition and would lead to ineffective tuning. This understanding is crucial for anyone studying radio frequency circuits and their applications in communication technology.