derive an expression for the range of transmission via space wave from a transmitting antenna.

Dear Student,

The situation is shown in the figure below,

For large coverage, the transmission of signal is done from a tall antenna.
Consider an antenna OP of height h. The transmitting signals cannot be received beyond points R1 and R2. This is due to curvature of earth.
Clearly, OR1 and OR2 are the maximum distances, from the antenna, upto which the transmission signal can be received.
Let OR1 = OR2 = d [d is half the total range]
The relation between height h of the antenna and the maximum distance d upto which the signal can be received can be derived on the basis of geometrical considerations.
From right – angled triangle CR2P, CP2 = CR2+ PR22 …. (1)
In right – angled triangle POR2, PR22 = h2 + d2
[Note that  $\angle PO{R}_{2}$ can be taken as right angle]
Also, CR2 = CO = R and CP = R + h
From Equation (1) (R + h)= R2 + (h2 + d2

R2 + h2 + 2Rh = R2 + h2 + d2 or
It is clear from this equation that if h is large, d will be large. This explains as to why the television broadcasts are made from tall antennas.