Question

A transmitting antenna at top of a tower has a height of 50 m and the height of receiving antenna is 80 m. What is the range of communication for Line of Sight (LoS) mode ?
[use radius of earth = 6400 km]

• A. 45.5 km
• B. 80.2 km
• C. 144.1 km
• D. 57.28 km

Hint:

For LoS problems, ensure all units are consistent before calculation. It's usually easiest to convert everything to SI units (meters, in this case) and then convert the final answer to kilometers if required. The formula \(d = \sqrt{2Rh}\) is derived from the Pythagorean theorem applied to the tangent from the antenna top to the Earth's surface.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
We need to calculate the maximum line-of-sight (LoS) distance for communication between a transmitting antenna and a receiving antenna of given heights.
Step 2: Key Formula or Approach:
The maximum LoS distance \(d_m\) between two antennas of height \(h_T\) (transmitter) and \(h_R\) (receiver) is given by the sum of their individual horizon distances:
\[ d_m = d_T + d_R = \sqrt{2Rh_T} + \sqrt{2Rh_R} \] where R is the radius of the Earth.
Step 3: Detailed Explanation:
Given values are:
Height of transmitting antenna, \(h_T = 50\) m.
Height of receiving antenna, \(h_R = 80\) m.
Radius of Earth, \(R = 6400\) km = \(6400 \times 10^3\) m = \(6.4 \times 10^6\) m.
First, calculate the horizon distance for the transmitting antenna (\(d_T\)):
\[ d_T = \sqrt{2Rh_T} = \sqrt{2 \times (6.4 \times 10^6 \, \text{m}) \times (50 \, \text{m})} \] \[ d_T = \sqrt{640 \times 10^6} \, \text{m} = \sqrt{64 \times 10^7} \, \text{m} = 8 \times 10^3 \sqrt{10} \, \text{m} \] \[ d_T \approx 8 \times 3.162 \times 10^3 \, \text{m} \approx 25.298 \times 10^3 \, \text{m} = 25.298 \, \text{km} \] Next, calculate the horizon distance for the receiving antenna (\(d_R\)):
\[ d_R = \sqrt{2Rh_R} = \sqrt{2 \times (6.4 \times 10^6 \, \text{m}) \times (80 \, \text{m})} \] \[ d_R = \sqrt{1024 \times 10^6} \, \text{m} = 32 \times 10^3 \, \text{m} = 32 \, \text{km} \] The total range of communication is the sum of these distances:
\[ d_m = d_T + d_R \approx 25.298 \, \text{km} + 32 \, \text{km} = 57.298 \, \text{km} \] Step 4: Final Answer:
The calculated range is approximately 57.3 km, which matches closely with option (D) 57.28 km.