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 having heights \( h_T \) and \( h_R \) above the Earth, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Geometry**:
- We have two antennas: one at height \( h_T \) (transmitter) and the other at height \( h_R \) (receiver).
- The Earth can be approximated as a sphere with radius \( R \).

2. **Line of Sight Distance**:
- The maximum line-of-sight distance \( d_M \) is the sum of the distances from each antenna to the tangent point on the Earth's surface.

3. **Using the Right Triangle**:
- For the transmitting antenna, we can form a right triangle where:
- The height of the antenna \( h_T \) is one side.
- The radius of the Earth \( R \) is the other side.
- The line of sight distance \( d_T \) is the hypotenuse.
- By the Pythagorean theorem:
\[ d_T^2 = R^2 + h_T^2 \]
- Thus, we can express \( d_T \) as:
\[ d_T = \sqrt{R^2 + h_T^2} \]

4. **Similar Calculation for the Receiver**:
- Similarly, for the receiving antenna:
\[ d_R = \sqrt{R^2 + h_R^2} \]

5. **Total Line of Sight Distance**:
- The total line-of-sight distance \( d_M \) is the sum of \( d_T \) and \( d_R \):
\[ d_M = d_T + d_R = \sqrt{R^2 + h_T^2} + \sqrt{R^2 + h_R^2} \]

6. **Simplifying the Expression**:
- If we consider small heights compared to the radius of the Earth, we can use the approximation:
\[ d_T \approx \sqrt{2Rh_T} \quad \text{and} \quad d_R \approx \sqrt{2Rh_R} \]
- Therefore, the maximum line-of-sight distance can be approximated as:
\[ d_M \approx \sqrt{2Rh_T} + \sqrt{2Rh_R} \]

### Final Formula:
Thus, the maximum line-of-sight distance \( d_M \) between two antennas is given by:
\[ d_M \approx \sqrt{2Rh_T} + \sqrt{2Rh_R} \]