The frequency for which a 5`mu`F capacitor has a reactance of `1/1000 Omega`is given by

Options

A. `100/pi` MHz
B. `1000/pi` MHz
C. `1/1000 Hz`
D. 1000 Hz

Explanation

To find the frequency for which a 5 µF capacitor has a reactance of \( \frac{1}{1000} \, \Omega \), we can use the formula for capacitive reactance:

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

where:
- \( X_C \) is the capacitive reactance,
- \( \omega = 2\pi f \) (angular frequency),
- \( C \) is the capacitance.

### Step 1: Write the equation for capacitive reactance
We start with the formula for capacitive reactance:

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

### Step 2: Substitute the values
We know that \( X_C = \frac{1}{1000} \, \Omega \) and \( C = 5 \, \mu F = 5 \times 10^{-6} \, F \). Substituting these values into the equation gives:

\[ \frac{1}{1000} = \frac{1}{\omega \cdot 5 \times 10^{-6}} \]

### Step 3: Rearranging the equation
Rearranging the equation to solve for \( \omega \):

\[ \omega \cdot 5 \times 10^{-6} = 1000 \]

\[ \omega = \frac{1000}{5 \times 10^{-6}} \]

### Step 4: Calculate \( \omega \)
Calculating \( \omega \):

\[ \omega = \frac{1000}{5 \times 10^{-6}} = 200000000 \, \text{rad/s} \]

### Step 5: Convert \( \omega \) to frequency \( f \)
Now, we convert \( \omega \) to frequency \( f \) using the relation \( \omega = 2\pi f \):

\[ f = \frac{\omega}{2\pi} = \frac{200000000}{2\pi} \]

### Step 6: Calculate the frequency \( f \)
Calculating \( f \):

\[ f = \frac{200000000}{2 \times 3.14} \approx \frac{200000000}{6.28} \approx 31830988.18 \, \text{Hz} \]

### Step 7: Convert to MHz
To convert Hz to MHz:

\[ f \approx 31.83 \, \text{MHz} \]

### Final Answer
Thus, the frequency for which a 5 µF capacitor has a reactance of \( \frac{1}{1000} \, \Omega \) is approximately:

\[ f \approx 31.83 \, \text{MHz} \]