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Chapter 32: Problem 57

A proposed "star wars" antimissile laser is to focus \(2.8-\mu \mathrm{m}-\) wavelength infrared light to a 50 -cm-diameter spot on a missile 2500 km distant. Find the minimum diameter for a concave mirror that can achieve this spot size, given the diffraction limit. (Your answer suggests one of many technical difficulties faced by antimissile defense systems.)

Short Answer

Expert verified
The minimum diameter for the mirror that can achieve the given spot size, considering the diffraction limit, is approximately \(16.94 m\).

Step by step solution

01

Understanding and Writing Down the Given Values

We need to first understand and write down the given inputs. The wavelength of the light, \(\lambda\), is given as \(2.8 \mu m\), the distance to the missile, \(d\), is given as \(2500 km\), and the diameter of the focused spot, \(D\), is given as \(50 cm\).
02

Convert All lengths to Same Unit (Meters)

In order to proceed with the calculation, we have to make sure that all the measurements are in the same unit. We will use meters (m) as our unit of measurement. So, \(\lambda = 2.8 \mu m = 2.8 x 10^-6 m\), \(d = 2500 km = 2.5 x 10^6 m\), and \(D = 50 cm = 0.5 m\).
03

Apply the Diffraction Limit Formula

The diffraction limit formula for a circular aperture is given by \(\Theta = 1.22 \times \frac{\lambda}{D}\) where \(\Theta\) is the angle of the divergence of the focused beam. Substituting the given values, \(\Theta = 1.22 \times \frac{2.8 x 10^-6}{0.5}\). Simplifying, we find that \(\Theta \approx 6.776 x 10^{-6} radians\).
04

Finding the Diameter of the Mirror

If the divergence angle \(\Theta\) and distance \(d\) are known, we can find the diameter of the mirror needed by using the formula \(D_{mirror} = d \times \Theta\). Substituting the calculated and given values, \(D_{mirror} = 2.5 x 10^{6} \times 6.776 x 10^{-6}\). Thus, the diameter of the mirror needed is approximately \(16.94 m\).

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