Q.Write the relation between the height of a TV antenna and the maximum range up to which signals transmitted by the antenna can be received. How is this expression modified in the case of line of sight communication by space waves ? In which range of frequencies, is this mode of communication used ?

The key idea is that the curvature of the Earth limits the line-of-sight distance for a transmitting antenna of height $h_T$.

Reasoning:

1. For a TV antenna of height $h_T$ (in metres), the maximum range $d$ (in kilometres) is given by the formula derived from geometry of the Earth's curvature:

where $R$ is the radius of the Earth (≈ 6400 km). This gives $d \approx \sqrt{2 \times 6400 \times h_T} = 80\sqrt{h_T}$ km (with $h_T$ in km) or $d \approx 4\sqrt{2h_T}$ km (with $h_T$ in metres).

2. For line-of-sight (LOS) communication by space waves, both the transmitting and receiving antennas have heights $h_T$ and $h_R$ respectively. The maximum range becomes the sum of their individual horizons:

$d = \sqrt{2 R h_T} + \sqrt{2 R h_R}$ …

The core idea is simple: radio waves travel in straight lines, but the Earth is curved. So the maximum distance a signal can reach is limited by the horizon — the point where the line from the antenna just grazes the Earth’s surface. This is pure geometry, not wave physics.

Imagine a tall tower of height $h$ on a spherical Earth of radius $R$. The line of sight from the top of the tower to the horizon is tangent to the Earth’s surface. That tangent line, the Earth’s radius to the point of tangency, and the line from the tower’s top to the Earth’s centre form a right-angled triangle. The distance from the tower base to the horizon is the range $d$.

1. Set up the geometry. Let $R$ be the Earth’s radius (≈ 6400 km). The tower height $h$ is tiny compared to $R$ (typically tens to hundreds of metres). The distance from the Earth’s centre to the tower top is $R + h$. The line of sight to the horizon is tangent, so it meets the Earth’s surface at a right angle. By Pythagoras:

2. Simplify for $h \ll R$. Expand: $R^2 + 2Rh + h^2 = R^2 + d^2$ → $d^2 = 2Rh + h^2$. Since $h^2$ is negligible compared to $2Rh$ (e.g., $h = 100$ m gives $h^2 = 10^4$ while $2Rh \approx 1.28 \times 10^9$), we get:

This is the maximum range for a single antenna.

3. Now modify for line-of-sight (LOS) space wave communication. In LOS communication, you have both a transmitting antenna (height $h_T$) and a receiving antenna (height $h_R$). The signal can travel from the transmitter’s horizon to the receiver’s horizon. The total range is the sum of the two individual horizon distances:

This is the maximum straight-line distance over which the two antennas can “see” each other, assuming no obstacles. …