Earth is assumed to be a charged conducting sphere having volume V and surface area A. The capacitance of the earth in free space is (\( \varepsilon_0 = \) permittivity of free space)}
Step 1: Understanding the Concept:
An isolated conducting sphere has a capacitance that depends only on its geometry (its radius) and the permittivity of the surrounding medium.
We need to express this capacitance in terms of the sphere's volume \( V \) and surface area \( A \). Step 2: Key Formula or Approach:
The capacitance \( C \) of an isolated spherical conductor of radius \( R \) is:
\[ C = 4\pi\varepsilon_0 R \]
The volume \( V \) of a sphere is:
\[ V = \frac{4}{3}\pi R^3 \]
The surface area \( A \) of a sphere is:
\[ A = 4\pi R^2 \]
Step 3: Detailed Explanation:
We want to replace the radius \( R \) in the capacitance formula with an expression involving \( V \) and \( A \).
Let's find the ratio of volume to surface area:
\[ \frac{V}{A} = \frac{\frac{4}{3}\pi R^3}{4\pi R^2} \]
Simplify the expression by canceling \( 4\pi \) and \( R^2 \):
\[ \frac{V}{A} = \frac{R}{3} \]
Now, rearrange this equation to solve for the radius \( R \):
\[ R = \frac{3V}{A} \]
Substitute this expression for \( R \) into the formula for capacitance:
\[ C = 4\pi\varepsilon_0 \left( \frac{3V}{A} \right) \]
Multiply the constants together:
\[ C = \frac{12\pi\varepsilon_0 V}{A} \]
This matches option (C). Step 4: Final Answer:
The capacitance of the earth is \( \frac{12\pi\varepsilon_0 V}{A} \).