The maximum range for the tropospheric transmission of radio wave of wavelength 3m using the transmitting antenna and receiving antenna of heights 100 m and 60 m respectively is

Options

A. 8m
B. 800m
C. 8km
D. 80km

Explanation

To find the maximum range for the tropospheric transmission of a radio wave with a wavelength of 3 meters, using a transmitting antenna height of 100 meters and a receiving antenna height of 60 meters, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the given values**:
- Wavelength (\( \lambda \)) = 3 m
- Height of transmitting antenna (\( h_t \)) = 100 m
- Height of receiving antenna (\( h_r \)) = 60 m

2. **Use the formula for maximum range**:
The formula for the maximum range \( D \) for tropospheric transmission is given by:
\[ D = 4 \sqrt{h_t \cdot h_r} \]
where \( h_t \) is the height of the transmitting antenna and \( h_r \) is the height of the receiving antenna.

3. **Calculate the square root**:
First, calculate \( h_t \cdot h_r \):
\[ h_t \cdot h_r = 100 \cdot 60 = 6000 \, \text{m}^2 \]
Now, take the square root:
\[ \sqrt{6000} \approx 77.46 \, \text{m} \]

4. **Calculate the maximum range**:
Now, substitute back into the formula:
\[ D = 4 \cdot 77.46 \approx 309.84 \, \text{m} \]

5. **Convert to kilometers**:
To convert meters to kilometers, divide by 1000:
\[ D \approx 0.30984 \, \text{km} \]

6. **Final calculation**:
Since the question seems to imply a larger range, we should check the calculation again:
\[ D = 4 \cdot \sqrt{100 \cdot 60} = 4 \cdot 77.46 \approx 309.84 \, \text{m} \text{ or } 0.31 \, \text{km} \]

However, the correct formula for maximum range in terms of the height of antennas and wavelength is:
\[ D = \frac{4 \cdot h_t \cdot h_r}{\lambda} \]
Substituting the values:
\[ D = \frac{4 \cdot 100 \cdot 60}{3} = \frac{24000}{3} = 8000 \, \text{m} = 8 \, \text{km} \]

### Conclusion:
Thus, the maximum range for the tropospheric transmission of the radio wave is **8 km**.

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