Question

The 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 (\(ε_0\) = permittivity of free space)

• A. \(2πε_0V/A\)
• B. \(4πε_0V/A\)
• C. \(8πε_0V/A\)
• D. \(12πε_0V/A\)

Hint:

Use C = 4 pi eps0 R and express R through V and A.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
A charged conducting sphere of radius \(R\) has capacitance \(C=4\pi\varepsilon_0R\).

Step 2: Express R using V and A:
\(V=\dfrac43\pi R^3\) and \(A=4\pi R^2\). So \(\dfrac VA=\dfrac{R}{3}\), giving \(R=\dfrac{3V}{A}\).

Step 3: Substitute:
\(C=4\pi\varepsilon_0\cdot\dfrac{3V}{A}=\dfrac{12\pi\varepsilon_0V}{A}\). Option D.

Step 4: Why the other options are wrong.
Options A, B and C come from using \(R=\dfrac{V}{2A}\), \(R=\dfrac VA\) and \(R=\dfrac{2V}{A}\) instead of \(R=\dfrac{3V}{A}\). Note that \(V/A=R/3\), not \(R\).

Final Answer:
The capacitance is 12 pi eps0 V / A. \[ \boxed{\text{(D) }\dfrac{12\pi\varepsilon_0V}{A}} \]