A satellite is orbiting around the earth in an orbitin equatorial plane of radius `2R_(e)` where `R_(e)` is the radius of earth. Find the area on earth, this satellite covers for communication purpose in its complete revolution.

Explanation

As shown in figure when statelite S revolves, it covers a complete circular belt on earth's surface for communication. If the colatitude of the farthest point on surface upto which singals can be received (point P) is `theta` then we have
`sin theta = (R_(e))/(2R_(e)) = (1)/(2)` or `theta = (pi)/(6)`

During revolution satellite leaves two spherical patches 1 and 2 on earth surface at north and south poles where no signals can be transmitted due to curvature. The areas of these patches can be obtained by solid angles. The solid angle subtended by a patch on earth's centre is
`Omega = 2pi(1- cos theta) = pi(2 - sqrt(3)) st`.
Area of patch 1 and 2 is
`A_(P) = Omega R_(e)^(2) = pi(2-sqrt(3))R_(e)^(2)`
Thus total area on earth's surface to which communication can be made is
`A_(C ) = 4pi R_(e)^(2) - 2A_(P)`
`= 4pi R_(e)^(2) - 2pi(2-sqrt(3))R_(e)^(2)`
`= 2piR_(e)^(2)(2-2+sqrt(3)) = 2sqrt(3) pi R_(e)^(2)`