Question

Considering earth as a metallic sphere, its capacitance would be nearly __________$\mu F.$ (Radius of earth $ = 6400{\text{ km, }}{ \in _0} = 8.85 \times {10^{ - 12}}$ SI unit)
$
{\text{A) 71}} \\
 {\text{B) 7}}{\text{.11}} \\
 {\text{C) 711}} \\
 {\text{D) 7}}{\text{.1}} \times {\text{1}}{{\text{0}}^5} \\
 $

Answer 1

Hint: Capacitance is the ability of the capacitor to collect and store energy in the form of the electrical charge. Capacitors are an electronic component which stores energy with the charge and potential difference across the terminals which are available in many sizes and shapes. To find the capacitance of the metallic sphere, use formula$c = 4\pi { \in _0}R$. Place the given values and find the unknown term.

Complete step by step answer:
Given that, Radius of the earth, $R = 6400{\text{ km}}$
Convert the given unit of radius in MKS system (Meter Kilogram Second) system.
Since, one kilometre is equal to thousand meters.
$
R = 6400km \\
\implies R = 6400 \times 1000m \\
 $
${ \in _0} = 8.85 \times {10^{ - 12}}{\text{ SI unit}}$
According to the formula, the capacitance for the metallic sphere is $c = 4\pi { \in _0}R$
Place the given values in the above equations.
$c = 4 \times 3.14 \times 8.85 \times {10^{ - 12}} \times 6400 \times 1000$
Simplify the left hand side of the equation and find the value for the capacitance.
\[c = 4 \times \dfrac{{3.14}}{{{{10}^{12}}}} \times 8.85 \times 6400 \times 1000\] (By the law of exponents, when the exponents with negative powers are moved to denominator from the numerator becomes positive and vice-versa)
\[c = \dfrac{{7113.984 \times {{10}^5}}}{{{{10}^{12}}}}\]
Use, the law of exponents – the exponents with the same base and in division, then powers are subtracted)
$
c = \dfrac{{7113.984}}{{{{10}^{12 - 5}}}} \\
\implies c = \dfrac{{7113.984}}{{{{10}^7}}} \\
 $
Now, as per the required answer is in micro farad, convert the above expression in that form –
$c = \dfrac{{7113.984}}{{{{10}^6} \times 10}}$
Take denominator to the numerator to make the given exponent in micro terms
$
c = \left( {\dfrac{{7113.984}}{{10}}} \right) \times {10^{ - 6}} \\
\implies c = 711.3 \times {10^{ - 6}} \\
\therefore c = 711\mu F \\
 $
Therefore, the required answer is – the capacitance would be nearly \[711\mu F\]

So, the correct answer is “Option C”.

Note:
The capacitance value of the capacitor is measured in farads (F) behind the name of the Physicist Michael Faraday. Since farad is the large quantity of the capacitance, most of the house-hold appliances use a fraction of farads and generally from microfarads to pico-farad.