The earth has volume 'V' and surface area 'A'. What is the capacitance of the earth?

Options

A. `4piepsi_(0)(A)/(V)`
B. `4piepsi_(0)(V)/(A)`
C. `12piepsi_(0)(V)/(A)`
D. `12piepsi_(0)(A)/(V)`

Explanation

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

### Step 1: Assume the Earth as a spherical conductor
We start by assuming that the Earth can be modeled as a spherical conductor. Let the radius of the Earth be \( R \).

### Step 2: Write the formulas for volume and surface area
For a sphere, the volume \( V \) and surface area \( A \) are given by the following formulas:
- Volume:
\[ V = \frac{4}{3} \pi R^3 \]
- Surface Area:
\[ A = 4 \pi R^2 \]

### Step 3: Relate volume and surface area to the radius
We can express the radius \( R \) in terms of volume \( V \) and surface area \( A \). From the surface area formula, we can rearrange it to find \( R \):
\[ R^2 = \frac{A}{4\pi} \implies R = \sqrt{\frac{A}{4\pi}} \]

### Step 4: Substitute \( R \) into the volume formula
We can also express \( V \) in terms of \( R \):
\[ V = \frac{4}{3} \pi R^3 \]
Now, substituting \( R \) from the surface area into the volume formula:
\[ V = \frac{4}{3} \pi \left( \sqrt{\frac{A}{4\pi}} \right)^3 \]
Calculating this gives:
\[ V = \frac{4}{3} \pi \cdot \frac{A^{3/2}}{(4\pi)^{3/2}} = \frac{4}{3} \cdot \frac{A^{3/2}}{8\sqrt{\pi^3}} = \frac{A^{3/2}}{6\sqrt{\pi}} \]

### Step 5: Find the capacitance of the Earth
The capacitance \( C \) of a spherical conductor is given by:
\[ C = 4 \pi \epsilon_0 R \]
Substituting \( R \) from our earlier expression:
\[ C = 4 \pi \epsilon_0 \left( \sqrt{\frac{A}{4\pi}} \right) \]
This simplifies to:
\[ C = 4 \pi \epsilon_0 \cdot \frac{\sqrt{A}}{2\sqrt{\pi}} = 2 \epsilon_0 \sqrt{A \pi} \]

### Step 6: Express capacitance in terms of \( V \) and \( A \)
Now, we can express \( R \) in terms of \( V \) and \( A \) using the earlier derived relationships. We know:
\[ R = \frac{3V}{A} \]
Substituting this into the capacitance formula:
\[ C = 4 \pi \epsilon_0 \left( \frac{3V}{A} \right) = \frac{12 \pi \epsilon_0 V}{A} \]

### Final Result
Thus, the capacitance of the Earth is given by:
\[ C = 12 \pi \epsilon_0 \frac{V}{A} \]