A circular radar antenna on a Coast Guard ship has a diameter of 2.10 m and radiates at a freque

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A circular radar antenna on a Coast Guard ship has a...

A circular radar antenna on a Coast Guard ship has a diameter of $2.10 \mathrm{m}$ and radiates at a frequency of $15.0 \mathrm{GHz}$ Two small boats are located $9.00 \mathrm{km}$ away from the ship. How close together could the boats be and still be detected as two objects?

Key Concepts

Electromagnetic Wave Frequency and Wavelength

Electromagnetic waves have an inverse relationship between frequency and wavelength, given by the equation wavelength = speed of light / frequency. This relationship is fundamental in determining the wavelength of the radar signal, which in turn affects its diffraction and resolving capabilities.

Aperture and Diffraction Limit

Any antenna or optical instrument with a finite aperture suffers from diffraction, which limits its ability to resolve closely spaced objects. The size of the aperture relative to the wavelength determines the spread of the diffraction pattern, introducing a fundamental limit on the system’s spatial resolution.

Rayleigh Criterion

The Rayleigh criterion provides a standard for determining the smallest angular separation at which two objects can be distinguished as separate. For a circular aperture, the minimum resolvable angle is approximately 1.22 times the wavelength divided by the aperture diameter. This concept is crucial for quantifying the resolution limit of the radar system.

Spatial Resolution and Small-Angle Approximation

Once the minimal angular separation is determined via the Rayleigh criterion, the corresponding spatial resolution (or the minimum separation distance detectable) at a given range can be approximated by multiplying this angle (in radians) by the distance to the objects. The small-angle approximation simplifies this conversion, making it essential in translating theoretical angular resolution into practical distance limits.

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