The height of transmitting antenna and receiving antenna are 45 m and 80 m. If the radius of earth is \( 6.4\times10^{6}\,\mathrm{m} \). Then the maximum distance between the two antennas for satisfactory communication in LOS mode is:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 56 Km
In line-of-sight (LOS) communication the maximum distance between a transmitting antenna of height \( h_T \) and a receiving antenna of height \( h_R \) is the sum of the distances to the horizon of the two antennas, \( d_M = \sqrt{2Rh_T}+\sqrt{2Rh_R} \).
\( \displaystyle \sqrt{2Rh_T} = \sqrt{2\times6.4\times10^{6}\times45} = \sqrt{576\times10^{6}} = 24000\,\mathrm{m} \)
\( \displaystyle \sqrt{2Rh_R} = \sqrt{2\times6.4\times10^{6}\times80} = \sqrt{1024\times10^{6}} = 32000\,\mathrm{m} \)
\( d_M = 24000\,\mathrm{m}+32000\,\mathrm{m} = 56000\,\mathrm{m} = 56\,\mathrm{km} \)
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 \( h_T \), then you can show that the distance to the horizon \( d_T \) is given as \( d_T = \sqrt{2Rh_T} \), where \( R \) is the radius of the earth (approximately 6400 km). \( d_T \) is also called the radio horizon of the transmitting antenna. With reference to Fig. 15.5 the maximum line-of-sight distance \( d_M \) between the two antennas having heights \( h_T \) and \( h_R \) above the earth is given by
\( \displaystyle d_M = \sqrt{2Rh_T}+\sqrt{2Rh_R} \qquad (15.1) \)
where \( h_R \) is the height of receiving antenna.”