The maximum line-of -sight distance `d_(M)` between two antennas having heights `h_(T)` and `H_(R)` above the earth is

Options

A. `sqrt(R(h_(T)+h_(R)))`
B. `sqrt(2R//(h_(T)+h_(R)))`
C. `sqrt(Rh_(T))+sqrt(2Rh_(R))`
D. `sqrt(2Rh_(T))+sqrt(2Rh_(R))`

Explanation

To find the maximum line-of-sight distance \( d_M \) between two antennas with heights \( h_T \) (transmitting antenna) and \( h_R \) (receiving antenna) above the Earth, we can follow these steps:

### Step 1: Understand the Geometry
We need to visualize the scenario where two antennas are placed at heights \( h_T \) and \( h_R \) above the Earth's surface. The Earth is approximately spherical, and the line-of-sight distance between the two antennas can be affected by the curvature of the Earth.

### Step 2: Use the Pythagorean Theorem
For each antenna, we can apply the Pythagorean theorem. The distance from the center of the Earth to the top of each antenna can be represented as:
- For the transmitting antenna: \( R + h_T \)
- For the receiving antenna: \( R + h_R \)

Where \( R \) is the radius of the Earth.

### Step 3: Calculate the Line-of-Sight Distance
Using the Pythagorean theorem, we can derive the line-of-sight distance \( d_M \). The distance \( x \) from the transmitting antenna to the point where the line of sight touches the Earth can be expressed as:
\[ x_T = \sqrt{(R + h_T)^2 - R^2} \]
Simplifying this gives:
\[ x_T = \sqrt{2Rh_T + h_T^2} \]
Assuming \( h_T \) is much smaller than \( R \) (i.e., \( h_T \ll R \)), we can approximate:
\[ x_T \approx \sqrt{2Rh_T} \]

Similarly, for the receiving antenna, we have:
\[ x_R \approx \sqrt{2Rh_R} \]

### Step 4: Total Line-of-Sight Distance
The total maximum line-of-sight distance \( d_M \) between the two antennas is the sum of the distances from each antenna to the point where the line of sight touches the Earth:
\[ d_M = x_T + x_R \]
Substituting the approximations from above:
\[ d_M \approx \sqrt{2Rh_T} + \sqrt{2Rh_R} \]

### Step 5: Factor Out Common Terms
We can factor out \( \sqrt{2R} \):
\[ d_M \approx \sqrt{2R}( \sqrt{h_T} + \sqrt{h_R} ) \]

### Conclusion
Thus, the maximum line-of-sight distance \( d_M \) between the two antennas is given by:
\[ d_M \approx \sqrt{2R}( \sqrt{h_T} + \sqrt{h_R} ) \]