Question

The minimum number of geostationary satellites required to televise a programme all over the earth is ____
(A) 2
(B) 6
(C) 4
(D) 3

Answer 1

Hint
Geostationary satellites have the same angular velocity as earth. So, if one of the satellites covers a solid angle of ${120}^{\circ }$ on the center of the earth, then by taking the total solid angle to be ${360}^{\circ }$ , we can find the number of satellites which will be needed from the formula,
$\Rightarrow n={\displaystyle \frac{\text{total solid angle}}{\text{solid angle subtended by the satellite}}}$
where $n$ is the number of required satellites.

Complete step by step answer
The term geostationary means the fact that when an observer on the ground sees the satellite in the sky, it appears to be stationary. The geostationary satellites are those which orbits the earth by revolving in the same direction as the earth, which is from west to east. They are present at an altitude of 36,000 km directly above the earth’s equator or a distance of 42,400 km from the center of the earth and have an angular velocity which is same as that of the earth, which means that its orbit takes 24 hours, which is the same length of time as earth to rotate once on its axis.
Now, when a geostationary satellite is at a distance of approximately 42,400 km from the center of the earth, they form a solid angle at the center. We can calculate this solid angle by the formula,
 $\Rightarrow \mathrm{\Omega }=2\pi \left(1-\cos \theta \right)$
For a geostationary satellite, we know this solid angle to be approximately ${120}^{\circ }$ .
So to cover the total surface of the earth, the solid angle has to be ${360}^{\circ }$ .
So the number of satellites required, $n$ , will be given by
 $\Rightarrow n={\displaystyle \frac{\text{total solid angle}}{\text{solid angle subtended by the satellite}}}$
By substituting the values we have
 $\Rightarrow n={\displaystyle \frac{{360}^{\circ }}{{120}^{\circ }}}$
 $\Rightarrow n=3$
Therefore, the number of satellites will be three.
So the correct answer is option (D).

Note
We can solve this problem also by the percentage of area covered by a single geostationary satellite on the surface of the earth. As we know a single satellite covers an area of $40\mathrm{\%}$ of the total surface of the earth. Thus to cover the full $100\mathrm{\%}$ of the surface of the earth, we will need three of them.