Q.7 A 50-metre tall antenna transmits at 107 MHz (one of the FM radio broadcast frequencies).
Calculate the maximum distance from the antenna at which the transmitted signal can be heard.
Ignore atmospheric attenuation and give your answer correct to the nearest kilometer only. You are
given that the radius of the earth is 6400 km.
The maximum distance at which a 50-meter tall antenna transmitting at 107 MHz can be heard is 25 km, determined by the radio horizon limited by Earth’s curvature. This line-of-sight calculation ignores atmospheric attenuation as specified, using Earth’s radius of 6400 km.
Radio Horizon Formula
FM radio signals at 107 MHz propagate via line-of-sight (LOS), blocked by Earth’s curvature beyond the radio horizon. The formula for maximum distance d from a single antenna of height h is derived from geometry: d = √(2Rh), where R is Earth’s radius and h is antenna height in consistent units.
Convert values to meters: R = 6400 × 1000 = 6,400,000 m, h = 50 m. Then, d = √(2 × 6,400,000 × 50) = √640,000,000 ≈ 25,298 m, or 25 km when rounded to the nearest kilometer.
Step-by-Step Calculation
Start with the right triangle formed by Earth’s center, tangent point, and antenna top. The hypotenuse is R + h ≈ R (since h ≪ R), leading to d² + R² ≈ (R + h)², simplifying to d ≈ √(2Rh).
Plug in numbers: 2Rh = 2 × 6.4 × 10⁶ × 50 = 6.4 × 10⁸. Square root yields ~25,300 m. Divide by 1000 for km and round: 25 km.
This assumes ground-level receiver; actual range doubles if receiver antenna height is considered, but problem implies single antenna horizon.
Why Frequency Doesn’t Matter Here
107 MHz is VHF, confirming LOS dominance over ground waves or sky waves for FM broadcast. The problem ignores attenuation, so wavelength (~2.8 m) affects only confirmation of LOS mode, not distance.
Common Options Explained
- 25 km (Correct): Matches
√(2 × 6400 × 0.05)km direct calc (~25.3 km, rounds to 25). - 50 km: Wrong; perhaps confusing diameter or doubling without basis.
- 80 km: From taller antennas (e.g., 500 m yields ~80 km via same formula).
- 100 km: Overestimates, ignoring precise
√(2Rh)or using effective Earth radius factor (~4/3, but not specified).
