A circular radar antenna on a Coast Guard ship has a diameter of 2.10 m and radiates at a frequency of 11.0 GHz. Two small boats are located 8.00 km away from the ship. How close together could the boats be and still be detected as two objects

Given Information:

Frequency = f = 11 Ghz = 11x10⁹ Hz

Distance between boats = L = 8 km = 8000 m

Diameter of radar antenna = D = 2.10 m

Required Information:

Distance between boats = d = ?

Answer:

Distance between boats = d = 126.65 m

Explanation:

The minimum distance between the two boats that will result in the detection by the radar antenna is given by

d = Ltan (θmin)

Where θmin is given by

θmin = sin⁻¹ (1.22*λ/D)

Where D is the diameter of the radar antenna and λ is the wavelength given by

λ = c/f

Where c is the speed of light

λ = 3x10⁸/11x10⁹

λ = 0.02727 m

θmin = sin⁻¹ (1.22 * (0.02727) / 2.10)

θmin = 0.907°

Therefore, the minimum distance between the two boats so that the radar antenna may detect is

d = 8000*tan (0.907)

d = 126.65 m