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

Options

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

Explanation

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

### Step-by-Step Solution:

1. **Understand the formula for capacitance**:
The capacitance \( C \) of a metallic sphere of radius \( r \) is given by the formula:
\[ C = 4 \pi \epsilon_0 r \]
where \( \epsilon_0 \) is the permittivity of free space.

2. **Rearrange the formula to solve for radius \( r \)**:
We need to find \( r \), so we rearrange the formula:
\[ r = \frac{C}{4 \pi \epsilon_0} \]

3. **Substitute the known values**:
We know that the capacitance \( C \) is given as \( 1 \mu F = 1 \times 10^{-6} F \). The value of \( \epsilon_0 \) is approximately \( 8.85 \times 10^{-12} \, F/m \). Now substituting these values into the equation:
\[ r = \frac{1 \times 10^{-6}}{4 \pi (8.85 \times 10^{-12})} \]

4. **Calculate the denominator**:
First, calculate \( 4 \pi \epsilon_0 \):
\[ 4 \pi \epsilon_0 \approx 4 \times 3.14 \times 8.85 \times 10^{-12} \approx 1.112 \times 10^{-10} \, F/m \]

5. **Calculate the radius**:
Now substitute this value back into the equation for \( r \):
\[ r = \frac{1 \times 10^{-6}}{1.112 \times 10^{-10}} \approx 8.99 \times 10^{3} \, m \]

6. **Convert to kilometers**:
To convert meters to kilometers, divide by 1000:
\[ r \approx 8.99 \, km \]

### Final Answer:
The radius of the metallic sphere will be approximately **9 kilometers**.