Question

A radar has a power of 1 kW and is operating at a frequency of 10GHz. It is located on a mountain top of height 500m. The maximum distance up to which it can detect object located on the surface of the earth is – (Radius of earth is 6400km)
A) 80 km
B) 16 km
C) 40 km
D) 64 km

Answer 1

We need to understand the relation between the power of a radar and its operating frequency, the location of the radar above the surface of the earth with the
radius of the earth to find the maximum reach of the radar on earth which is the solution.

Complete answer:
We know that earth is a spheroid. The radars are devices which are used to find objects on the surface of the earth by emitting electromagnetic waves from a height. Due to the curvature of the earth, the radars have a range for its successful operation.
The longest distance where a signal from the radar can travel constitutes its range of functioning. The radars are placed such that the maximum distances between two radars meet at a point.

This range of the radar can be found using the formula as (where h is the height of the radar, R is the radius of the earth and d is the range of the radar).
$\begin{aligned} & {{\left( h+R \right)}^{2}}={{d}^{2}}+{{R}^{2}} \\\ & \Rightarrow {{h}^{2}}+2hR+{{R}^{2}}={{d}^{2}}+{{R}^{2}} \\\ & \Rightarrow {{h}^{2}}+2hR={{d}^{2}} \\\ & \text{But, here }h\ll R \\\ & \Rightarrow {{d}^{2}}=2hR \\\ & \Rightarrow d=\sqrt{2\times 500\times 6400000} \\\ & \therefore d=80km \\\ \end{aligned}$
We can see that the radar can get the data for a range of 80 km. i.e., it can cover a hemisphere of radius of 80 km.

The correct answer is option A.

Note:
The frequency or the power of the radar does not determine or affect the range of the radar. It is pure trigonometry which governs the distance that can be travelled by the electromagnetic waves from a radar which is kept at a height above the surface.