The frequency for which a 5.0 muF capacitor has a reactance of `1000Omega` is given by

Options

A. (1000)/(pi) cycles//sec
B. (100)/(pi) cycles//sec
C. 200 cycle //sec
D. 5000 cycles //sec

Answer

To find the frequency for which a 5.0 µF capacitor has a reactance of 1000 Ω, we can follow these steps:

### Step 1: Understand the formula for capacitive reactance
The capacitive reactance (Xc) is given by the formula:
\[ X_c = \frac{1}{\omega C} \]
where:
- \( X_c \) is the capacitive reactance,
- \( \omega \) is the angular frequency in radians per second,
- \( C \) is the capacitance in farads.

### Step 2: Relate angular frequency to frequency
The angular frequency \( \omega \) can be expressed in terms of frequency \( f \) (in hertz) as:
\[ \omega = 2\pi f \]
Substituting this into the reactance formula gives:
\[ X_c = \frac{1}{2\pi f C} \]

### Step 3: Rearrange the formula to solve for frequency
Rearranging the formula to solve for frequency \( f \):
\[ f = \frac{1}{2\pi X_c C} \]

### Step 4: Substitute the known values
We know:
- \( X_c = 1000 \, \Omega \)
- \( C = 5.0 \, \mu F = 5.0 \times 10^{-6} \, F \)

Now substitute these values into the frequency formula:
\[ f = \frac{1}{2\pi \times 1000 \times 5.0 \times 10^{-6}} \]

### Step 5: Calculate the frequency
Now, calculate the frequency:
1. Calculate the denominator:
\[ 2\pi \times 1000 \times 5.0 \times 10^{-6} = 2\pi \times 0.005 = 0.0314159 \, \text{(approximately)} \]
2. Now take the reciprocal:
\[ f \approx \frac{1}{0.0314159} \approx 31.83 \, \text{Hz} \]

### Step 6: Final result
Thus, the frequency for which a 5.0 µF capacitor has a reactance of 1000 Ω is approximately:
\[ f \approx 31.83 \, \text{Hz} \]