Question

When the receiving antenna is on the ground, the range of a transmitting antenna of height 980 m is (Radius of the earth = 6400 km)

• A. 56 km
• B. 112 km
• C. 72.4 km
• D. 224 km

Hint:

When using the formula \(d = \sqrt{2Rh}\), if you keep R in km and h in meters, a useful approximation is \(d(\text{km}) \approx \sqrt{2 \times 6400 \times h/1000} = \sqrt{12.8h}\). For this problem: \(d \approx \sqrt{12.8 \times 980} = \sqrt{12544} = 112\) km. This avoids dealing with large powers of 10.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This problem deals with the line-of-sight propagation of radio waves. The maximum distance a signal can travel from a transmitting antenna to the horizon (its range) is limited by the curvature of the Earth.
Step 2: Key Formula or Approach:
The formula for the maximum line-of-sight distance, or range (\(d_T\)), from a transmitting antenna of height \(h_T\) is given by:
\[ d_T = \sqrt{2 R_e h_T} \] where \(R_e\) is the radius of the Earth. If there is also a receiving antenna of height \(h_R\), the total range is \(d = \sqrt{2R_e h_T} + \sqrt{2R_e h_R}\). In this case, the receiving antenna is on the ground, so \(h_R = 0\), and the formula simplifies to just \(d_T\).
Step 3: Detailed Explanation:
First, ensure all units are consistent. It's best to convert everything to meters.
- Height of transmitting antenna, \(h_T = 980 \text{ m}\).
- Radius of the Earth, \(R_e = 6400 \text{ km} = 6400 \times 10^3 \text{ m} = 6.4 \times 10^6 \text{ m}\).
Now, substitute these values into the range formula:
\[ d_T = \sqrt{2 \times (6.4 \times 10^6 \text{ m}) \times (980 \text{ m})} \] \[ d_T = \sqrt{12.8 \times 10^6 \times 980} \] \[ d_T = \sqrt{12544 \times 10^6} \text{ m} \] \[ d_T = \sqrt{12544} \times \sqrt{10^6} \text{ m} \] To find \(\sqrt{12544}\), we can note that \(110^2 = 12100\) and the number ends in 4, so the root must end in 2 or 8. Let's try 112:
\[ 112^2 = (100+12)^2 = 10000 + 2(100)(12) + 144 = 10000 + 2400 + 144 = 12544 \] So, \(\sqrt{12544} = 112\).
\[ d_T = 112 \times 10^3 \text{ m} \] Finally, convert the range from meters to kilometers:
\[ d_T = 112 \text{ km} \] Step 4: Final Answer:
The range of the transmitting antenna is 112 km. Therefore, option (B) is correct.