Question 776: Electrostats & Capacitance

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)

Accepted Answer

Step 1: Capacitance of a conducting isolated sphere
For a conducting sphere of radius $R$,
$ C = 4 \pi \varepsilon_0 R $
Step 2: Express $R$ in terms of $V$ (volume) and $A$ (surface area)
Volume of sphere:
$ V = \frac{4}{3} \pi R^3 $
Surface area of sphere:
$ A = 4 \pi R^2 $
We need to eliminate $R$.
Step 3: Relationship between $V$ and $A$
From $A = 4 \pi R^2 \implies R^2 = 	frac{A}{4\pi}$.
From $V = 	frac{4}{3}\pi R^3 \implies R^3 = 	frac{3V}{4\pi}$.
Now compute $R$ in terms of $V$ and $A$.
$ R = \frac{3V}{A} $
(Reason: $V = 	frac{4}{3}\pi R^3$ and $A = 4\pi R^2$. Divide:
$ \frac{V}{A} = \frac{	frac{4}{3}\pi R^3}{4\pi R^2} = \frac{R}{3} \implies R = \frac{3V}{A}. $)
Step 4: Substitute into capacitance formula
$ C = 4 \pi \varepsilon_0 R = 4 \pi \varepsilon_0 \cdot \frac{3V}{A} $
$ C = \frac{12 \pi \varepsilon_0 V}{A} $
Ã¢ÂÂ Correct Answer:
Option C: $	frac{12 \pi \varepsilon_0 V}{A}$

Answer

$\frac{2 \pi \varepsilon_0 V}{A}$

Answer

$\frac{8 \pi \varepsilon_0 \mathrm{~V}}{\mathrm{~A}}$

Answer

$\frac{12 \pi \varepsilon_0 \mathrm{~V}}{\mathrm{~A}}$

Answer

$\frac{4 \pi \varepsilon_0 V}{A}$

Electrostats & Capacitance

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)

Choose the correct answer: