### Correct Option: A. f₀ = 1/(2π√(LC)) ### Detailed Explanation: In an RLC series circuit, resonance occurs when the inductive reactance (XL) and capacitive reactance (XC) are equal. At this point, the circuit can oscillate at its natural frequency, known as the resonance frequency (f₀). 1. **Understanding Inductive and Capacitive Reactance**: - The inductive reactance (XL) is given by the formula: \[ X_L = 2\pi f L \] where \( f \) is the frequency in hertz (Hz) and \( L \) is the inductance in henries (H). - The capacitive reactance (XC) is given by: \[ X_C = \frac{1}{2\pi f C} \] where \( C \) is the capacitance in farads (F). 2. **Setting Reactances Equal**: At resonance, we set \( X_L = X_C \): \[ 2\pi f L = \frac{1}{2\pi f C} \] 3. **Solving for Resonance Frequency (f₀)**: To find the resonance frequency, we can rearrange the equation: - Multiply both sides by \( 2\pi f C \): \[ 2\pi f L \cdot 2\pi f C = 1 \] - This simplifies to: \[ (2\pi f)^2 LC = 1 \] - Rearranging gives: \[ (2\pi f)^2 = \frac{1}{LC} \] - Taking the square root of both sides: \[ 2\pi f = \frac{1}{\sqrt{LC}} \] - Finally, solving for \( f \): \[ f = \frac{1}{2\pi\sqrt{LC}} \] - Thus, the resonance frequency \( f₀ \) is: \[ f₀ = \frac{1}{2\pi\sqrt{LC}} \] ### Why Other Options Are Incorrect: - **Option B: f₀ = 2π√(LC)**: - This option suggests that the resonance frequency is directly proportional to the square root of the product of inductance and capacitance, which is incorrect. The correct relationship involves the reciprocal of the square root, as derived above. - **Option C: f₀ = LC/(2π)**: - This option implies a linear relationship between frequency and the product of inductance and capacitance, which does not hold true in the context of resonance. The frequency is inversely related to the square root of the product of L and C, not directly proportional. - **Option D: f₀ = 1/(LC)**: - This option suggests that the frequency is inversely proportional to the product of inductance and capacitance. However, the correct relationship involves the square root of the product, not the product itself. ### Summary of Key Points: - The resonance frequency \( f₀ \) in an RLC series circuit is given by the formula \( f₀ = \frac{1}{2\pi\sqrt{LC}} \). - At resonance, the inductive and capacitive reactances are equal, allowing the circuit to oscillate at its natural frequency. - Understanding the relationships between inductance, capacitance, and frequency is crucial for analyzing RLC circuits. - Remember that resonance frequency is inversely related to the square root of the product of inductance and capacitance, not directly proportional.