Question

Consider the earth to be a sphere, having charge \(Q\). Let \(R\), \(V\) and \(A\) be its radius, volume and area respectively. The capacitance of the earth is [\(\varepsilon _{0}\) = permittivity of free space]

• A. \(4\pi \varepsilon _{0}\frac{V}{A}\)
• B. \(4\pi \varepsilon _{0}\frac{A}{V}\)
• C. \(12\pi \varepsilon _{0}\frac{A}{V}\)
• D. \(12\pi \varepsilon _{0}\frac{V}{A}\)

Hint:

The capacitance of a sphere is 4 pi epsilon naught R. Express R using the volume and the surface area.

Answer 1

The Correct Option is D

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

Step 2: Express R
\(V=\dfrac{4}{3}\pi R^{3}\) and \(A=4\pi R^{2}\), so \(\dfrac{V}{A}=\dfrac{R}{3}\), that is \(R=\dfrac{3V}{A}\).

Step 3: Calculation
\(C=4\pi\varepsilon_0\times\dfrac{3V}{A}=12\pi\varepsilon_0\dfrac{V}{A}\)

Step 4: Compare the options
This is the last option.

Final Answer:
The capacitance is 12 pi epsilon naught times V over A. \[ \boxed{\text{(D) }12\pi\varepsilon_{0}\dfrac{V}{A}} \]