The reactance of a `25 muF` capacitor at the AC frequency of `4000Hz` is

Options

A. `5/pi``Omega`
B. `sqrt5/pi``Omega`
C. `10Omega`
D. `sqrt 10``Omega`

Explanation

To find the reactance of a capacitor, we can use the formula for capacitive reactance \( X_C \):

\[ X_C = \frac{1}{\omega C} \]

where:
- \( \omega \) is the angular frequency in radians per second,
- \( C \) is the capacitance in farads.

### Step 1: Calculate Angular Frequency \( \omega \)

The angular frequency \( \omega \) is given by the formula:

\[ \omega = 2 \pi f \]

where \( f \) is the frequency in hertz.

Given:
- \( f = 4000 \, \text{Hz} \)

Substituting the value of \( f \):

\[ \omega = 2 \pi \times 4000 \]

### Step 2: Calculate Capacitance \( C \)

The capacitance \( C \) is given as:

\[ C = 25 \, \mu F = 25 \times 10^{-6} \, F \]

### Step 3: Substitute \( \omega \) and \( C \) into the Reactance Formula

Now we can substitute \( \omega \) and \( C \) into the formula for \( X_C \):

\[ X_C = \frac{1}{\omega C} = \frac{1}{(2 \pi \times 4000) \times (25 \times 10^{-6})} \]

### Step 4: Calculate \( X_C \)

Calculating \( \omega \):

\[ \omega = 2 \pi \times 4000 \approx 25132.74 \, \text{rad/s} \]

Now substituting this value into the reactance formula:

\[ X_C = \frac{1}{25132.74 \times 25 \times 10^{-6}} \approx \frac{1}{0.628318} \approx 1.59155 \, \Omega \]

### Step 5: Final Calculation

To simplify further, we can express this in terms of \( \frac{5}{\pi} \):

\[ X_C \approx \frac{5}{\pi} \, \Omega \]

### Conclusion

Thus, the reactance of the \( 25 \mu F \) capacitor at an AC frequency of \( 4000 \, Hz \) is approximately:

\[ X_C \approx \frac{5}{\pi} \, \Omega \]