Question

A transmitting antenna has a height of 320 m and that of receiving antenna is 2000 m. The maximum distance between them for satisfactory communication in line of sight mode is 'd'. The value of 'd' is _________ km. (Radius of Earth = 6400 km)

Hint:

In LOS communication problems, the most common mistake is unit inconsistency. Always convert all lengths (heights, radius) to the same unit (usually kilometers) before plugging them into the formula \(d = \sqrt{2Rh}\).

Answer 1

Correct Answer: 224

Step 1: Understanding the Question:
This problem deals with line-of-sight (LOS) communication between two antennas of given heights. We need to find the maximum possible separation between them for a signal to be received, taking into account the curvature of the Earth.
Step 2: Key Formula or Approach:
The maximum line-of-sight distance (\(d_{max}\)) between a transmitting antenna of height \(h_T\) and a receiving antenna of height \(h_R\) is the sum of their individual radio horizons. The formula is: \[ d_{max} = d_T + d_R = \sqrt{2Rh_T} + \sqrt{2Rh_R} \] where \(R\) is the radius of the Earth. It's important to use consistent units for all quantities. Since the answer is required in km, it's convenient to use R in km and the heights in km.
Step 3: Detailed Explanation:
Given values: Height of transmitting antenna, \(h_T = 320 \text{ m} = 0.32 \text{ km}\). Height of receiving antenna, \(h_R = 2000 \text{ m} = 2.0 \text{ km}\). Radius of Earth, \(R = 6400 \text{ km}\).
Now, let's calculate the radio horizon for each antenna.
Radio horizon for transmitting antenna: \[ d_T = \sqrt{2Rh_T} = \sqrt{2 \times 6400 \text{ km} \times 0.32 \text{ km}} \] \[ d_T = \sqrt{12800 \times 0.32} = \sqrt{4096} = 64 \text{ km} \] Radio horizon for receiving antenna: \[ d_R = \sqrt{2Rh_R} = \sqrt{2 \times 6400 \text{ km} \times 2.0 \text{ km}} \] \[ d_R = \sqrt{12800 \times 2} = \sqrt{25600} = 160 \text{ km} \] The maximum line-of-sight distance is the sum of these two distances: \[ d_{max} = d_T + d_R = 64 \text{ km} + 160 \text{ km} = 224 \text{ km} \] Step 4: Final Answer:
The value of 'd' is 224 km.