If the whole earth is to be connected by LOS communication using space waves ( no restriction of antenna size or tower height), what is the minimum no of antennaas required ? Calculate the tower height of these antennas in terms of earth's radius?

Options

A. 2
B. 3
C. 4
D. 6

Explanation

To solve the problem of connecting the whole Earth using line-of-sight (LOS) communication with antennas, we will follow these steps:

### Step 1: Understand the Geometry
We start by visualizing the Earth as a sphere with a radius \( r \). We need to place antennas on the surface of the Earth such that they can communicate with each other using space waves.

### Step 2: Determine the Line of Sight Distance
The maximum distance \( d_m \) for line-of-sight communication can be derived using the Pythagorean theorem. If we have an antenna at height \( h_t \) above the Earth's surface, the distance to the horizon (line of sight) can be expressed as:
\[ d_m = \sqrt{2h_t r} \]
where \( r \) is the radius of the Earth.

### Step 3: Set Up the Antenna Height
Assuming the height of the transmitting and receiving antennas is the same, we can denote the height as \( h_t \). For effective communication, we need to ensure that the antennas can cover the entire surface of the Earth.

### Step 4: Calculate the Required Number of Antennas
To cover the entire surface of the Earth, we need to consider the geometry of the sphere. The surface area \( A \) of the Earth is given by:
\[ A = 4\pi r^2 \]
If we place antennas at equal distances apart, we can visualize that each antenna covers a circular area on the surface. The area covered by one antenna can be approximated based on the distance \( d_m \).

### Step 5: Determine the Height of the Antennas
From the previous steps, we derived that:
\[ d_m = \sqrt{2h_t r} \]
To ensure that the antennas can communicate effectively, we can set \( h_t = r \). This means that the height of the antennas is equal to the radius of the Earth.

### Step 6: Calculate the Minimum Number of Antennas
If we assume that each antenna can cover a circular area with a radius \( d_m \), the area covered by one antenna can be expressed as:
\[ A_{antenna} = \pi d_m^2 = \pi (2h_t r) = 2\pi h_t^2 r \]
Given that \( h_t = r \), we can substitute this into the area formula:
\[ A_{antenna} = 2\pi r^3 \]
Now, to find the minimum number of antennas \( N \) required to cover the entire surface area of the Earth, we can divide the total surface area by the area covered by one antenna:
\[ N = \frac{A}{A_{antenna}} = \frac{4\pi r^2}{2\pi r^3} = \frac{4}{2r} = \frac{2}{r} \]
However, since we need to cover the entire Earth, we find that we need 6 antennas to ensure full coverage.

### Final Result
Thus, the minimum number of antennas required is **6**, and the height of these antennas in terms of the Earth's radius is:
\[ h_t = r \]