If the whole earth is to be connected by LOS communication using space waves ( no restriction of antenna size or tower height), what is the minimum no of antennaas required ? Calculate the tower height of these antennas in terms of earth's radius?

Explanation

Consider the figure given below to solve this question

Suppose the height of tansmitting antenna or receiving antenna in order to cover the entire surface of earth through communication is `h_(t)` and radius of earth is R
Then, maximum distance
`d_(m)^(2) = (R + h_(t))^(2) + (R+h_(t))^(2)`
`=2(R+h_(t))^(2)`
`d_(m) = sqrt(2h_(t)R)+sqrt(2h_(t)R)=2sqrt(2h_(t)R)`
`therefore " "8h_(t)R=2(R + h_(t))^(2)`
`implies 4h_(t)R=R^(2)+2Rh_(t)+h_(t)^(2)`
`implies R^(2) - 2h_(t) R+ h_(t)^(2)=0`
`implies (R - h_(t))^(2)=0`
`implies R = h_(t)`
Since, space wave frequency is used so `lambda ltlt h_(t)`, hence only tower height is to be taken into consideration. In three dimensions of earth, 6 antenna towers of each of height `h_(t) = R` would be used to cover the entire surface of earth with communication programme.