To determine the reactance of the inductor in this problem, we need to 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 Explanation 1. **Identify the Given Values**: - The inductance \(L\) is given as \(2 \, \text{H}\). - The frequency \(f\) is given as \(50/\pi \, \text{Hz}\). 2. **Substitute the Values into the Formula**: We can now substitute the values of \(L\) and \(f\) into the formula for inductive reactance: \[ X_L = 2\pi \left(\frac{50}{\pi}\right) (2) \] 3. **Simplify the Expression**: - First, simplify the frequency term: \[ X_L = 2\pi \cdot \frac{50}{\pi} \cdot 2 \] - The \(\pi\) in the numerator and denominator cancels out: \[ X_L = 2 \cdot 50 \cdot 2 \] - Now, calculate the multiplication: \[ X_L = 2 \cdot 50 = 100 \] \[ X_L = 100 \cdot 2 = 200 \, \Omega \] 4. **Final Result**: Thus, the reactance of the inductor is \(200 \, \Omega\). ### Explanation of the Options - **Option A: 200 Ω** - This is the correct answer as we calculated \(X_L\) to be \(200 \, \Omega\). - **Option B: 50 Ω** - This option is incorrect. It does not match our calculated value and does not take into account the inductance and frequency correctly. - **Option C: 100/π Ω** - This option is also incorrect. It seems to confuse the relationship between frequency and inductance. The reactance is not simply a division by \(\pi\). - **Option D: 25/π Ω** - This option is incorrect as well. Similar to option C, it does not reflect the correct calculation of inductive reactance and is not derived from the formula correctly. ### Common Pitfalls - **Forgetting to use the correct units**: Always ensure that the inductance is in henries and frequency in hertz. - **Misapplying the formula**: Ensure that you are using the correct formula for inductive reactance and substituting values accurately. - **Neglecting to simplify**: Sometimes, students forget to simplify expressions, leading to incorrect answers. ### Revision Summary - The formula for inductive reactance is \(X_L = 2\pi f L\). - Substitute the values of frequency and inductance carefully. - Simplify the expression step-by-step to avoid mistakes. - The correct answer for the reactance of a 2H inductor at \(50/\pi \, \text{Hz}\) is \(200 \, \Omega\).