Question

The capacitive reactance offered by a capacitance of 10 microfarads to an alternating current of \( i = I_m \sin(100\pi t - \frac{\pi}{6}) \) amperes is

• A. 318.31 ohms
• B. 0.31831 ohm
• C. 31.831 ohms
• D. 3183.1 ohms

Hint:

To calculate capacitive reactance, use the formula \( X_C = \frac{1}{2\pi f C} \), where \( f \) is the frequency and \( C \) is the capacitance.

Answer 1

The Correct Option is A

Step 1: Use the formula for capacitive reactance.
The capacitive reactance \( X_C \) is given by the formula: \[ X_C = \frac{1}{2\pi f C}, \] where \( f \) is the frequency and \( C \) is the capacitance.
Step 2: Extract the frequency from the current equation.
The current equation is \( i = I_m \sin(100\pi t - \frac{\pi}{6}) \), so the frequency \( f \) is: \[ f = \frac{100\pi}{2\pi} = 50 \, \text{Hz}. \]
Step 3: Calculate the capacitive reactance.
Substitute \( f = 50 \, \text{Hz} \) and \( C = 10 \times 10^{-6} \, \text{F} \): \[ X_C = \frac{1}{2\pi \times 50 \times 10 \times 10^{-6}} = 318.31 \, \text{ohms}. \]
Step 4: Conclusion.
Thus, the capacitive reactance is 318.31 ohms, which corresponds to option (A).

Answer 2

The current is given as \( i = I_m\sin(100\pi t - \frac{\pi}{6}) \), so the angular frequency is read off directly as \( \omega = 100\pi \, \text{rad/s} \), without needing to first convert to the frequency \( f \) in Hz. The capacitive reactance formula in terms of \( \omega \) is: \[ X_C = \frac{1}{\omega C} = \frac{1}{100\pi \times 10\times10^{-6}}. \] Computing this: \( 100\pi \times 10\times10^{-6} = 100\pi \times 10^{-5} \approx 3.1416\times10^{-3} \), so \( X_C = \frac{1}{3.1416\times10^{-3}} \approx 318.31 \, \Omega \). Let's check each option.

1. 318.31 ohms: This matches the direct \( \omega \)-based calculation exactly.
2. 0.31831 ohm: This is smaller than the correct value by a factor of 1000, likely from a units slip treating the capacitance in farads as if it were in millifarads.
3. 31.831 ohms: This is smaller than the correct value by a factor of 10, inconsistent with the direct \( \omega \)-based computation.
4. 3183.1 ohms: This is larger than the correct value by a factor of 10, also inconsistent with the computation above.

Using the angular frequency directly from the given current expression confirms the capacitive reactance is 318.31 ohms.

Therefore, the correct answer is 318.31 ohms.