Question

A transmitting antenna at the top of a tower has a height of \(50 \mathrm{~m}\) and the height of the receiving antenna is \(32 \mathrm{~m}\). The maximum distance between them for satisfactory communication in line of sight mode is Given radius of earth \(\mathrm{R}=6400 \mathrm{~km}\) (A) \(25.29 \times 10^{3} \mathrm{~km}\) (B) \(20.23 \times 10^{3} \mathrm{~km}\) (C) \(45.5 \mathrm{~km}\) (D) None of these.

Short answer

The maximum distance between the transmitting and receiving antennas for satisfactory communication in line of sight mode is 13.12 km, which is not among the given options. Thus, the correct answer is (D) None of these.

Step-by-step solution

Understand the line of sight distance formula

The line of sight distance between two points on the Earth's surface is given by the formula: \[d = \sqrt{2h_{1}R} + \sqrt{2h_{2}R}\] Where \(d\) is the line of sight distance, \(h_{1}\) and \(h_{2}\) are the heights of the two antennas, and \(R\) is the radius of the Earth.

Substitute the given values into the formula

We have the height of the transmitting antenna \(h_{1} = 50\mathrm{~m}\), the height of the receiving antenna \(h_{2} = 32\mathrm{~m}\), and the radius of the Earth \(R = 6400 \mathrm{~km}\). Convert the radius of Earth to meters: \(R = 6400 \times 10^3 \mathrm{~m}\) Next, substitute these values into the formula: \[d = \sqrt{2(50 \mathrm{~m})(6400 \times 10^3 \mathrm{~m})} + \sqrt{2(32 \mathrm{~m})(6400 \times 10^3 \mathrm{~m})}\]

Calculate the line of sight distance

Now, we can calculate the value of \(d\): \[d = \sqrt{2(50)(6400 \times 10^3)} + \sqrt{2(32)(6400 \times 10^3)}\] \[d = \sqrt{(100)(6400 \times 10^3)} + \sqrt{(64)(6400 \times 10^3)}\] \[d = \sqrt{(10^5)(64 \times 10^3)} + \sqrt{(64^2)(10^3)}\] \[d = 8000+ 5120\] \[d = 13120 \mathrm{~m}\]

Compare the calculated distance with the choices provided

Now, let's convert the calculated distance to kilometers: \[d = 13.12 \mathrm{~km}\] Compare this result to the choices provided: (A) \(25.29 \times 10^3 \mathrm{~km}\) (B) \(20.23 \times 10^3 \mathrm{~km}\) (C) \(45.5 \mathrm{~km}\) (D) None of these Our calculated distance of \(13.12 \mathrm{~km}\) is not equal to any of the given choices. Thus, the correct answer is: (D) None of these.

Key concepts

Antenna Height

Antenna height is a crucial factor in line of sight communication. It refers to the vertical distance from the base to the top of the antenna. The higher the antenna, the further the line of sight distance it can achieve. This is because a higher antenna can send and receive signals over a larger area. In our exercise, the transmitting antenna is 50 meters tall and the receiving antenna is 32 meters tall.
The height of the antenna directly influences the area that the signals can cover. Taller antennas are particularly advantageous in areas with obstacles such as buildings or trees. Their increased height allows the signals to travel over these obstacles more easily, reducing signal obstruction and improving communication effectiveness.
When calculating the line of sight distance, these antenna heights come into play, multiplying the reach through the square root formula that takes into account the Earth's curvature and the antenna's vertical reach.

Radius of Earth

Understanding the radius of the Earth is essential for calculations involving distances on the planet's surface. The Earth's radius is approximately 6400 kilometers, or 6400 x 10^3 meters. This measurement is vital when using the line of sight distance formula to calculate maximum communication reach.
The radius is based on the assumption of the Earth being a perfect sphere, which simplifies our calculations. While this is not entirely accurate, as the Earth is slightly oblate, the approximation is sufficient for most engineering and communication purposes. It's important to note here that this specific value provides a standard for calculations to ensure consistency across different problems and scenarios.
In the mathematical formula, the radius aids in determining how far the signal can propagate along the Earth's surface before it goes below the horizon. The value is part of the square root calculation that estimates the effective distance of communication between two elevations or heights.

Distance Calculation

The calculation of the maximum line of sight distance involves a specific formula that takes into account the radius of the Earth and the heights of both antennas. The formula used is:\[ d = \sqrt{2h_{1}R} + \sqrt{2h_{2}R} \]Here, \(d\) represents the line of sight distance, while \(h_1\) and \(h_2\) denote the heights of the transmitting and receiving antennas, respectively. \(R\) stands for the Earth's radius.
Substituting our values into this formula helps determine the theoretical maximum distance that line of sight communication can cover. For instance, with \(h_1 = 50\) meters and \(h_2 = 32\) meters and the Earth’s radius as \(6400 \times 10^3\) meters, you will compute:1. Calculate \(\sqrt{2(50)(6400 \times 10^3)}\)2. Calculate \(\sqrt{2(32)(6400 \times 10^3)}\)3. Sum both calculated values.The final answer needs to be compared against given options to determine the closest or exact match. This calculation helps in scenarios like setting up communications systems, ensuring they are within feasible distances without signal loss over the Earth's curvature.