Question

An antenna is mounted on a 400 m tall building. What will be the wavelength of signal that can be radiated effectively by the transmission tower upto a range of 44 km ?

• A. 37.8 m
• B. 75.6 m
• C. 302 m
• D. 605 m

Hint:

Recognize when a question's data seems inconsistent or when it might be flawed. In an exam, if you cannot find a clear path to a solution, it's best to check if you've missed a simple interpretation. If not, make an educated guess or move on. Problems like this are rare but can appear.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
The question asks for the wavelength (\(\lambda\)) of a signal that can be effectively radiated from an antenna on a 400 m building with a given range of 44 km. This question combines concepts of antenna theory and signal propagation.
Step 2: Key Formula or Approach:
There are two main physical principles mentioned: effective radiation and range.
1. Effective Radiation: For an antenna to radiate effectively, its physical size (L) should be comparable to the wavelength of the signal. A common relationship for a monopole antenna (like one on a building) is \( L = \lambda / 4 \) or \( L = \lambda / 2 \). Assuming the building height acts as the antenna length, \( h \approx L \).
2. Range of Transmission: For line-of-sight communication, the range \(d\) is related to the height of the transmitting antenna \(h_t\) and receiving antenna \(h_r\) by \( d = \sqrt{2Rh_t} + \sqrt{2Rh_r} \), where R is the radius of the Earth. This formula does not directly involve wavelength.
Step 3: Detailed Explanation:
Let's analyze the given data using the principles above.
- Antenna height, \( h = 400 \) m.
- Range, \( d = 44 \) km.
Analysis of Effective Radiation:
If we assume \(h\) is the antenna length, \(L = 400\) m.
- If \( L = \lambda/4 \), then \( \lambda = 4L = 4 \times 400 = 1600 \) m.
- If \( L = \lambda/2 \), then \( \lambda = 2L = 2 \times 400 = 800 \) m.
Neither of these values matches the options closely. The option 302 m is somewhat close to \( \lambda/2 \) if the effective length was different, and 605 m is not immediately obvious.
Analysis of Range:
The line-of-sight range for a 400 m tower to the horizon is
\( d_{LOS} = \sqrt{2Rh} \approx \sqrt{2 \times (6.4 \times 10^6 \text{ m}) \times 400 \text{ m}} \approx 71.5 \) km.
The given range of 44 km is well within the line-of-sight distance, so this information doesn't seem to constrain the wavelength in a standard way.
Conclusion based on Official Answer Key:
The question is ambiguously worded and does not seem to correspond to a standard, simple physics formula connecting height, range, and wavelength. The information provided appears either insufficient or contradictory. However, the official answer key indicates that the correct answer is 605 m. This result may be based on specific assumptions about the type of antenna, propagation mode, or may stem from an error in the question's data. Without further context or a specific non-standard formula, it's not possible to derive this answer from first principles.
Step 4: Final Answer:
Based on the provided official answer key, the wavelength is 605 m.