The reactance of a capacitor of capcitance `C muF` for an a.c. of frequency `400/(pi)` Hz is `25 Omega`. What is the value of C?

Options

A. `25 mu F`
B. `50 mu F`
C. `75 mu F`
D. `90 mu F`

Explanation

To find the value of the capacitance \( C \) in microfarads, we can follow these steps:

### Step 1: Understand the relationship between reactance, frequency, and capacitance
The reactance \( X_C \) of a capacitor in an AC circuit is given by the formula:
\[ X_C = \frac{1}{\omega C} \]
where \( \omega \) is the angular frequency and \( C \) is the capacitance.

### Step 2: Calculate the angular frequency \( \omega \)
The angular frequency \( \omega \) is related to the frequency \( f \) by the formula:
\[ \omega = 2\pi f \]
Given that the frequency \( f \) is \( \frac{400}{\pi} \) Hz, we can substitute this value into the equation:
\[ \omega = 2\pi \left(\frac{400}{\pi}\right) = 800 \text{ rad/s} \]

### Step 3: Substitute known values into the reactance formula
We know that the reactance \( X_C \) is given as \( 25 \, \Omega \). Now we can rearrange the reactance formula to solve for \( C \):
\[ C = \frac{1}{\omega X_C} \]
Substituting the values of \( \omega \) and \( X_C \):
\[ C = \frac{1}{800 \times 25} \]

### Step 4: Calculate the capacitance \( C \)
Now we can compute \( C \):
\[ C = \frac{1}{20000} = 5 \times 10^{-5} \text{ F} \]

### Step 5: Convert capacitance to microfarads
To convert from farads to microfarads, we multiply by \( 10^6 \):
\[ C = 5 \times 10^{-5} \text{ F} \times 10^6 = 50 \, \mu F \]

### Final Answer
Thus, the value of the capacitance \( C \) is:
\[ C = 50 \, \mu F \]

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