Question

If the heights of transmitting and the receiving antennas are each equal to $h$, the maximum line-of-sight distance between them is (R is the radius of earth)

• A. $\sqrt{2Rh}$
• B. $\sqrt{4Rh}$
• C. $\sqrt{6Rh}$
• D. $\sqrt{8Rh}$
• E. $\sqrt{Rh}$

Hint:

Always remember to add the distances for both antennas. The distance $\sqrt{2Rh}$ is only for a single antenna reaching the Earth's horizon.

Answer 1

The Correct Option is D

Concept: Line-of-sight (LOS) communication is limited by the curvature of the Earth. The maximum distance ($d$) an antenna of height $h$ can transmit to the horizon is determined by geometry.
• The horizon distance for an antenna is given by: $d = \sqrt{2Rh}$.

Step 1: Apply the formula to both antennas. The total maximum distance between a transmitting antenna ($h_t$) and a receiving antenna ($h_r$) is the sum of their individual horizon distances: \[ D_{max} = \sqrt{2Rh_t} + \sqrt{2Rh_r} \]

Step 2: Substitute $h$ and simplify. Given $h_t = h_r = h$: \[ D_{max} = \sqrt{2Rh} + \sqrt{2Rh} = 2\sqrt{2Rh} \] To bring the "2" inside the square root, we square it: \[ D_{max} = \sqrt{4 \times 2Rh} = \sqrt{8Rh} \]