Question

If the diameter of the earth’s surface is \[128\times {{10}^{2}}\] km the its capacitance will be:
A. 711 \[\mu F\]
B. 331 \[\mu F\]
C. 221 \[\mu F\]
D. 111 \[\mu F\]

Answer 1

Hint: We are asked to find the capacitance of earth, if its diameter is given. For this, we can use the formula for capacitance of an isolated conducting sphere. Here, it is assumed that earth is a conducting sphere. Even though the general shape of earth is oval for calculation purposes we will consider it as a sphere. The permittivity of space is \[\varepsilon \] = \[8.854\times {{10}^{-12}}\]\[{{m}^{-3}}~k{{g}^{-1}}~{{s}^{4}}~{{A}^{2}}\].

Formula Used:
\[C=4\pi k\varepsilon r\]
Where,
k= relative permittivity
\[\varepsilon \]= permittivity of space
r= radius in meters.

Complete step by step answer:
A capacitor is a device that is used to store electric charge and energy. A single capacitor consists of two conductors separated by specific distance. The amount of charge stored in these capacitors determines the capacitance of these conductors. The unit of capacitance is Farad (F). It was named after physicist Michal Faraday.
We are asked to calculate the capacitance of earth
Therefore, we can assume that earth is an isolated conducting sphere
So, by using formula
\[C=4\pi k\varepsilon r\]
We can calculate the capacitance
We are given the radius of earth in km so during calculations appropriate conversion is needed.
Therefore,
After substituting the given values
We get,
\[C=4\pi \times 8.854\times {{10}^{-12}}\times \dfrac{128\times {{10}^{2}}\times {{10}^{3}}}{2}\] ………. (1km = 1000 m)
On solving above equation
We get,
\[C=711\mu F\]
Therefore, the correct answer is option A.

Additional Information:
In this question the value of permittivity of space and relative permittivity was not provided. Students must remember the value of permittivity of space as it is a physical constant and used in many calculations.

Note:
In this case we assumed earth as an isolated conducting sphere. We know that the capacitor is a pair of conductors carrying equal and opposite charges. For an isolated conductor it is considered that the other capacitor is at infinite distance.