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

Options

A. (1000pi) 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 use the formula for capacitive reactance:

### Step 1: Write the formula for capacitive reactance
The capacitive reactance (X_C) is given by the formula:
\[ X_C = \frac{1}{2 \pi f C} \]
where:
- \(X_C\) is the capacitive reactance,
- \(f\) is the frequency in hertz,
- \(C\) is the capacitance in farads.

### Step 2: Substitute the known values
We know that:
- \(X_C = 1000 \, \Omega\)
- \(C = 5.0 \, \mu F = 5.0 \times 10^{-6} \, F\)

Substituting these values into the formula, we have:
\[ 1000 = \frac{1}{2 \pi f (5.0 \times 10^{-6})} \]

### Step 3: Rearrange the formula to solve for frequency (f)
To find \(f\), we can rearrange the equation:
\[ f = \frac{1}{2 \pi C X_C} \]

### Step 4: Substitute the values into the rearranged formula
Now substituting the values of \(C\) and \(X_C\):
\[ f = \frac{1}{2 \pi (5.0 \times 10^{-6}) (1000)} \]

### Step 5: Calculate the frequency
Calculating the denominator:
\[ 2 \pi (5.0 \times 10^{-6}) (1000) = 2 \pi \times 5.0 \times 10^{-3} = 10 \pi \times 10^{-3} \]
Now substituting this back into the equation for \(f\):
\[ f = \frac{1}{10 \pi \times 10^{-3}} = \frac{100}{\pi} \, Hz \]

### Final Answer
Thus, the frequency for which the 5.0 µF capacitor has a reactance of 1000 Ω is:
\[ f = \frac{100}{\pi} \, Hz \]