The capacitance of earth of radius R treating as spherical conductor is

Options

A. `4piin_(0)R`
B. `4piR`
C. `in_(0)/R`
D. `(4piin_(0))/R`

Answer

To find the capacitance of the Earth, treating it as a spherical conductor, we can follow these steps:

### Step 1: Understand the Concept of Capacitance for a Spherical Conductor
The capacitance \( C \) of a spherical conductor can be derived from the general formula for spherical capacitors. For a spherical conductor with radius \( r \), the capacitance is given by the formula:

\[ C = \frac{4 \pi \epsilon_0}{\frac{1}{A} - \frac{1}{B}} \]

where \( A \) is the inner radius and \( B \) is the outer radius.

### Step 2: Define the Radii for the Earth
In this case, we treat the Earth as a solid sphere of radius \( r \). We can consider the outer radius \( B \) to be at infinity, which simplifies our calculations. Thus, we have:

- Inner radius \( A = r \)
- Outer radius \( B = \infty \)

### Step 3: Substitute the Values into the Capacitance Formula
Now, substituting the values into the capacitance formula:

\[ C = \frac{4 \pi \epsilon_0}{\frac{1}{r} - \frac{1}{\infty}} \]

Since \( \frac{1}{\infty} = 0 \), the formula simplifies to:

\[ C = \frac{4 \pi \epsilon_0}{\frac{1}{r}} = 4 \pi \epsilon_0 r \]

### Step 4: Final Expression for the Capacitance
Thus, the capacitance of the Earth, treating it as a spherical conductor, is:

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

### Step 5: Conclusion
This formula indicates that the capacitance of the Earth depends directly on its radius \( r \) and the permittivity of free space \( \epsilon_0 \).