A radar of power 1 kW and is operating at a frequency of 20 GHz. It is located at a height of 1 km on a mountain. The maximum distance up to which it can detect object located on the surface of earth (Radius of earth = \( 6.4\times10^{6}\,\mathrm{m} \)) is:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 113 km
Radar signals travel in straight lines, so the farthest object it can detect on the ground is at the line-of-sight distance (the horizon) of the antenna. For an antenna of height h above the ground, a tangent from its top to the earth of radius R gives \( (R+h)^{2} = R^{2} + d^{2} \), and for \( h \ll R \):
\( d = \sqrt{2Rh} \)
\( d = \sqrt{2\times6.4\times10^{6}\,\mathrm{m}\times1\times10^{3}\,\mathrm{m}} = \sqrt{1.28\times10^{10}}\,\mathrm{m} \)
\( d \approx 1.13\times10^{5}\,\mathrm{m} = 113\,\mathrm{km} \)
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.”