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

Dear Student,

Please find below the answer to the asked query:

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 R_{1} and R_{2}. This is due to curvature of earth.

Clearly, OR_{1} and OR_{2} are the maximum distances, from the antenna, upto which the transmission signal can be received.

Let OR_{1} = OR_{2} = 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 CR_{2}P, CP^{2} = CR_{2}^{2 }+ PR_{2}^{2}^{ }…. (1)

In right – angled triangle POR_{2}, PR_{2}^{2} = h^{2} + d^{2}

[Note that $\angle PO{R}_{2}$ can be taken as right angle]

Also, CR_{2} = CO = R and CP = R + h

From Equation (1) (R + h)^{2 }= R^{2} + (h^{2} + d^{2})

R^{2} + h^{2} + 2Rh = R^{2} + h^{2} + d^{2} or $d=\sqrt{2Rh}$

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.

Hope this information will clear your doubts about the topic.

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Regards

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