The earth has volume V and surface area A then capacitance would be

Options

A. `4 pi epsilon_(0)(A)/(V)`
B. `2 pi epsilon_(0)(A)/(V)`
C. `12 pi epsilon_(0)(V)/(A)`
D. `12 pi epsilon_(0)(A)/(V)`

Answer

To find the capacitance of the Earth given its volume \( V \) and surface area \( A \), we can follow these steps:

### Step-by-Step Solution:

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

2. **Relate Volume and Surface Area to Radius**:
The volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \]
The surface area \( A \) of a sphere is given by:
\[ A = 4 \pi r^2 \]

3. **Express Radius in Terms of Volume and Surface Area**:
To find the radius \( r \) in terms of \( V \) and \( A \), we can divide the volume formula by the surface area formula:
\[ \frac{V}{A} = \frac{\frac{4}{3} \pi r^3}{4 \pi r^2} \]
Simplifying this, we get:
\[ \frac{V}{A} = \frac{r}{3} \]
Thus, we can express \( r \) as:
\[ r = \frac{3V}{A} \]

4. **Substitute \( r \) into the Capacitance Formula**:
Now, substitute \( r \) back into the capacitance formula:
\[ C = 4 \pi \epsilon_0 r = 4 \pi \epsilon_0 \left(\frac{3V}{A}\right) \]
This simplifies to:
\[ C = \frac{12 \pi \epsilon_0 V}{A} \]

5. **Final Result**:
Therefore, the capacitance \( C \) of the Earth in terms of its volume \( V \) and surface area \( A \) is:
\[ C = \frac{12 \pi \epsilon_0 V}{A} \]