Question

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)}

• A. \( \frac{2\pi\varepsilon_0 V}{A} \)
• B. \( \frac{8\pi\varepsilon_0 V}{A} \)
• C. \( \frac{12\pi\varepsilon_0 V}{A} \)
• D. \( \frac{4\pi\varepsilon_0 V}{A} \)

Hint:

Isolated Sphere: $C = 4\pi\varepsilon_0 R$.

Answer 1

The Correct Option is C

Step 1: Geometric Relations
For a sphere: Volume $V = \frac{4}{3}\pi R^3$ and Surface Area $A = 4\pi R^2$.
Dividing these: $\frac{V}{A} = \frac{R}{3} \implies R = \frac{3V}{A}$.
Step 2: Capacitance Formula
Capacitance of an isolated spherical conductor: $C = 4\pi\varepsilon_0 R$.
Step 3: Substitution
$C = 4\pi\varepsilon_0 \left( \frac{3V}{A} \right) = \frac{12\pi\varepsilon_0 V}{A}$.
Final Answer: (C)