Question

The earth has volume $'V'$ and surface area $'A'$ then its capacitance would be

Answer 1

To solve the question, you need to consider the earth as a solid sphere and then use the formula of capacitance of a solid sphere, which is given as: $C = 4\pi {\varepsilon _o}R$ . You can find the value of earth’s radius using the given volume and surface area of the earth.

Complete step by step answer:
As explained in the hint section of the solution to the asked question, we need to consider earth as a solid sphere and then use the formula of the capacitance of a solid sphere to find out the capacitance of earth. But as we saw above, for this, we need to find the radius of the earth, which can be done using the given volume of earth and the surface area of the earth.
If the earth is considered to be a solid sphere, we can define its volume as:
Volume $= \dfrac{4}{3}\pi {R^3}$
Similarly, the surface area can be given as:
Area $= 4\pi {R^2}$
The question has told us that:
Volume $= V$ and,
Area $= A$
Hence, we can write:
$\implies$ $V = \dfrac{4}{3}\pi {R^3}$ and,
$\implies$ $A = 4\pi {R^2}$
Now, if we divide the volume by area, we get:
$\implies$ $\dfrac{R}{3} = \dfrac{V}{A}$
Or,
$\implies$ $R = \dfrac{{3V}}{A}$
Now, we have found out the value of the radius of earth, $R = \dfrac{{3V}}{A}$
All that is left to do is to substitute the found-out value of radius of earth in the formula of capacitance of a solid sphere, which is given as:
$\implies$ $C = 4\pi {\varepsilon _o}R$
Where, $C$ is the capacitance of the solid sphere
${\varepsilon _o}$ is the constant permittivity of free space or vacuum permittivity
$R$ is the radius of the given solid sphere
If we substitute the value of radius of earth, we get:
$C = 4\pi {\varepsilon _o} \times \dfrac{{3V}}{A} \\\ \Rightarrow C = 12\pi {\varepsilon _o}\dfrac{V}{A} \\\$
Hence, this is the value of the capacitance of the earth.

Note: The main consideration where many students make mistakes is that they may either consider the earth as a shell or as a hollow sphere, which is a completely wrong assumption in approaching solutions to such questions. Here we should assume that the second sphere(external sphere) is at infinite distance and hence only the radius of the internal sphere affects the value of capacitance.