To determine the frequency at which a 10 H inductor has a reactance of 2000 Ω, we can use the formula for inductive reactance, which is given by: \[ X_L = 2\pi f L \] Where: - \(X_L\) is the inductive reactance (in ohms, Ω), - \(f\) is the frequency (in hertz, Hz), - \(L\) is the inductance (in henries, H). ### Step-by-Step Solution 1. **Identify the given values**: - Inductance, \(L = 10 \, \text{H}\) - Reactance, \(X_L = 2000 \, \Omega\) 2. **Substitute the known values into the formula**: \[ 2000 = 2\pi f (10) \] 3. **Simplify the equation**: \[ 2000 = 20\pi f \] 4. **Solve for frequency \(f\)**: \[ f = \frac{2000}{20\pi} \] \[ f = \frac{100}{\pi} \, \text{Hz} \] ### Conclusion The frequency at which a 10 H inductor has a reactance of 2000 Ω is: **C. \( \frac{100}{\pi} \, \text{Hz} \)** ### Explanation of Why the Answer is Correct The formula for inductive reactance \(X_L = 2\pi f L\) directly relates the frequency of the AC signal to the inductance and the reactance. By rearranging the formula to solve for frequency, we can find the specific frequency that results in the desired reactance. In this case, substituting the values for \(X_L\) and \(L\) allows us to isolate \(f\) and calculate it accurately. ### Explanation of Why Other Options are Incorrect - **Option A: \( \frac{\pi}{200} \, \text{Hz} \)** This value is too small. If we substitute it back into the reactance formula, it would yield a much lower reactance than 2000 Ω. - **Option B: \( \frac{\pi}{100} \, \text{Hz} \)** Similar to option A, this frequency would also produce a reactance that is significantly lower than 2000 Ω. - **Option D: \( 100\pi \, \text{Hz} \)** This frequency is too high. Substituting this value into the reactance formula would yield a reactance much greater than 2000 Ω. ### Common Pitfalls - **Forgetting to convert units**: Ensure that all units are consistent (e.g., henries for inductance and hertz for frequency). - **Misapplying the formula**: Remember that \(X_L\) is directly proportional to both frequency and inductance. If either value is miscalculated, the resulting frequency will be incorrect. ### Revision Summary - Inductive reactance is calculated using \(X_L = 2\pi f L\). - To find frequency, rearrange the formula to \(f = \frac{X_L}{2\pi L}\). - Substitute the values carefully to avoid calculation errors. - Always check if the calculated frequency makes sense in the context of the problem.