A radar for tracking aircraft broadcasts a 12 GHz microwave beam from a 2.0-m-diameter circular radar antenna. From a wave perspective, the antenna is a circular aperture through which the microwaves diffract. a. What is the diameter of the radar beam at a distance of 30 km? b. If the antenna emits 100 kW of power, what is the average microwave intensity at 30 km.
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Donald Cheek
For a circular aperture with diameter D, the diameter of the central maximum $w$ is:
$w=\frac{2.44\lambda L}{D}$
$\lambda$ is the wavelength and $L$ is a distance to the screen
Given: the frequency of the wave $f=12GHz$, the diameter of the aperture $D=2m$, The distance to the screen $L=30km$, the antenna emits $P=100kW$
$\lambda =\frac{c}{f}=\frac{3×{10}^{8}}{12×{10}^{9}}m=25mm$
Now, find the diameter of the radar bean:
$w=\frac{2.44\lambda L}{D}=\frac{2.44×25×{10}^{-3}×30×{10}^{3}}{2}m=915m$
The intensity of radiation I at $L=30km$ is given by:
$I=\frac{P}{{\left(\frac{w}{2}\right)}^{2}\pi }=\frac{100×{10}^{3}×4}{{\left(915\right)}^{2}×3.14}\frac{W}{{m}^{2}}=0.152\frac{W}{{m}^{2}}$
Thus, the answers are: (a) $915m$; (b) $0.152\frac{W}{{m}^{2}}$