The line of sight distance for two antennas of height 100 m and 400 m above earth surface is approximately. (Given \( R = 6400\,\mathrm{km},\ \sqrt{20}=4.5 \))
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 108 km
The maximum line-of-sight distance between two antennas of heights \( h_1 \) and \( h_2 \) is the sum of their horizon distances, each of which is \( \sqrt{2Rh} \).
\( d = \sqrt{2Rh_1}+\sqrt{2Rh_2} \)
Work in km: \( R = 6400\,\mathrm{km} \), \( h_1 = 100\,\mathrm{m} = 0.1\,\mathrm{km} \), \( h_2 = 400\,\mathrm{m} = 0.4\,\mathrm{km} \).
\( \sqrt{2Rh_1} = \sqrt{2\times6400\times0.1} = \sqrt{1280} = \sqrt{64\times20} = 8\sqrt{20} = 8\times4.5 = 36\,\mathrm{km} \)
\( \sqrt{2Rh_2} = \sqrt{2\times6400\times0.4} = \sqrt{5120} = \sqrt{256\times20} = 16\sqrt{20} = 16\times4.5 = 72\,\mathrm{km} \)
\( d = 36\,\mathrm{km}+72\,\mathrm{km} = 108\,\mathrm{km} \)
Note: This question is from the pre-rationalised NCERT book (2022-23). The topic was removed from the latest NCERT book (2026-27), and it also appears to be out of the CUET 2026 syllabus.
NCERT (Pre-rationalised edition, 2022-23):
Space Wave – “… If the transmitting antenna is at a height hT, then you can show that the distance to the horizon dT is given as \( d_T = \sqrt{2Rh_T} \), where R is the radius of the earth (approximately 6400 km). dT is also called the radio horizon of the transmitting antenna. With reference to Fig. 15.5 the maximum line-of-sight distance dM between the two antennas having heights hT and hR above the earth is given by
\( \displaystyle d_M = \sqrt{2Rh_T} + \sqrt{2Rh_R} \qquad (15.1) \)
where hR is the height of receiving antenna.”