The maximum distance between the transmitting and receiving antenna for a satisfactory communication in LOS mode is 48 km. If the height of the receiving antenna is 80 m, then height of the transmitting antenna is: (Given radius of earth = 6400 km)
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 20 m
In line-of-sight (LOS) communication the wave travels in a straight line, so the curvature of the earth limits the range. An antenna of height \( h \) can reach a point on the horizon at a distance \( \sqrt{2Rh} \), and the maximum line-of-sight distance between two antennas is the sum of their two horizon distances.
\( \displaystyle d_{M} = \sqrt{2Rh_{T}} + \sqrt{2Rh_{R}} \)
\( R = 6400\,\mathrm{km} = 6.4\times10^{6}\,\mathrm{m} \)
\( d_{M} = 48\,\mathrm{km} = 48000\,\mathrm{m} \)
For the receiving antenna,
\( \displaystyle \sqrt{2Rh_{R}} = \sqrt{2 \times 6.4\times10^{6} \times 80}\,\mathrm{m} = \sqrt{1.024\times10^{9}}\,\mathrm{m} = 32000\,\mathrm{m} \)
\( \displaystyle \sqrt{2Rh_{T}} = 48000 - 32000 = 16000\,\mathrm{m} \)
\( \displaystyle h_{T} = \frac{(16000)^{2}}{2 \times 6.4\times10^{6}}\,\mathrm{m} = \frac{2.56\times10^{8}}{1.28\times10^{7}}\,\mathrm{m} = 20\,\mathrm{m} \)
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 – “… 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.”