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: Express radius from the area first:
$R=\sqrt{\dfrac{A}{4\pi}}$. Then $V=\dfrac43\pi R^3=\dfrac43\pi R\cdot\dfrac{A}{4\pi}=\dfrac{RA}{3}$.

Step 2: Solve for R:
$R=\dfrac{3V}{A}$.

Step 3: Capacitance:
$C=4\pi\varepsilon_0R=\dfrac{12\pi\varepsilon_0V}{A}$. Option D.

Final Answer:
V/A = R/3, so C = 4 pi eps0 R = 12 pi eps0 V/A. \[ \boxed{\text{(D) }\dfrac{12\pi\varepsilon_0V}{A}} \]