The line of sight distance between two antennas of heights 144 m and 324 m above earth surface is:
(Given \( \sqrt{20} = 4.4 \))
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( 105.6\,\mathrm{km} \)
An antenna of height \( h \) can reach up to its radio horizon, a distance \( \sqrt{2Rh} \) along the earth, where \( R \) is the radius of the earth. The maximum line of sight distance between two antennas is the sum of their two horizon distances.
\( \displaystyle d_M = \sqrt{2Rh_T} + \sqrt{2Rh_R} \)
Taking \( R = 6.4\times10^{6}\,\mathrm{m} \), the common factor is
\( \displaystyle \sqrt{2R} = \sqrt{1.28\times10^{7}} = \sqrt{64 \times 20 \times 10^{4}} = 8 \times 4.4 \times 10^{2} = 3520\,\mathrm{m^{1/2}} \)
\( \displaystyle \sqrt{144} = 12, \quad \sqrt{324} = 18 \)
\( \displaystyle d_M = 3520 \times (12 + 18) = 3520 \times 30 = 105600\,\mathrm{m} \)
\( d_M = 105.6\,\mathrm{km} \)
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.”