The capacitance of a metallic sphere is `1 muF`, if its radius is

Options

A. 9 km
B. 10 m
C. 1.11 m
D. 1.11 cm

Answer

To find the radius of a metallic sphere given its capacitance, we can use the formula for the capacitance of a sphere:

\[ C = 4 \pi \epsilon_0 R \]

Where:
- \( C \) is the capacitance,
- \( \epsilon_0 \) is the permittivity of free space (\( \epsilon_0 \approx 8.85 \times 10^{-12} \, \text{F/m} \)),
- \( R \) is the radius of the sphere.

Given that the capacitance \( C \) is \( 1 \, \mu F = 1 \times 10^{-6} \, F \), we can rearrange the formula to solve for \( R \):

\[ R = \frac{C}{4 \pi \epsilon_0} \]

Now, substituting the values into the equation:

1. **Substituting the values**:
\[ R = \frac{1 \times 10^{-6}}{4 \pi (8.85 \times 10^{-12})} \]

2. **Calculating the denominator**:
\[ 4 \pi \epsilon_0 = 4 \times 3.14 \times (8.85 \times 10^{-12}) \approx 1.112 \times 10^{-10} \]

3. **Calculating \( R \)**:
\[ R = \frac{1 \times 10^{-6}}{1.112 \times 10^{-10}} \approx 8.99 \times 10^{3} \, \text{m} \]

4. **Converting to kilometers**:
\[ R \approx 8.99 \, \text{km} \]

Thus, the radius of the metallic sphere is approximately \( 8.99 \, \text{km} \).