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]

Options

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

Answer

To find the range of communication for Line of Sight (LoS) mode between a transmitting antenna and a receiving antenna, we can use the formula:

\[ d = \sqrt{2 h_t R} + \sqrt{2 h_r R} \]

Where:
- \(d\) is the total range of communication,
- \(h_t\) is the height of the transmitting antenna,
- \(h_r\) is the height of the receiving antenna,
- \(R\) is the radius of the Earth.

Given:
- Height of transmitting antenna, \(h_t = 50 \, \text{m}\)
- Height of receiving antenna, \(h_r = 80 \, \text{m}\)
- Radius of the Earth, \(R = 6400 \, \text{km} = 6400 \times 10^3 \, \text{m}\)

### Step 1: Convert the radius of the Earth to meters
\[ R = 6400 \, \text{km} = 6400 \times 10^3 \, \text{m} \]

### Step 2: Calculate the first term \(\sqrt{2 h_t R}\)
\[ \sqrt{2 h_t R} = \sqrt{2 \times 50 \times (6400 \times 10^3)} \]
\[ = \sqrt{100 \times 6400 \times 10^3} = \sqrt{640000 \times 10^3} = \sqrt{6.4 \times 10^5 \times 10^3} = \sqrt{6.4 \times 10^8} \]
\[ = 8 \times 10^4 \, \text{m} \]

### Step 3: Calculate the second term \(\sqrt{2 h_r R}\)
\[ \sqrt{2 h_r R} = \sqrt{2 \times 80 \times (6400 \times 10^3)} \]
\[ = \sqrt{160 \times 6400 \times 10^3} = \sqrt{1024000 \times 10^3} = \sqrt{1.024 \times 10^6 \times 10^3} = \sqrt{1.024 \times 10^9} \]
\[ = 32 \times 10^4 \, \text{m} \]

### Step 4: Add both terms to find the total range \(d\)
\[ d = \sqrt{2 h_t R} + \sqrt{2 h_r R} = 8 \times 10^4 + 32 \times 10^4 = 40 \times 10^4 \, \text{m} \]
\[ = 4 \times 10^5 \, \text{m} \]

### Step 5: Convert the range from meters to kilometers
\[ d = \frac{4 \times 10^5}{1000} = 400 \, \text{km} \]

### Final Answer:
The range of communication for Line of Sight (LoS) mode is **400 km**.

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