Question

A spy camera is said to be able to read the numbers on a car's license plate. If the numbers on the plate are 4.30 cm apart, and the spy satellite is at an altitude of 140 km, what must be the diameter of the camera's aperture? (Assume light with a wavelength of 600 nm.)

Answer 1

Answer:2.38 m

Explanation:

First convert all numbers to meter

4.30 cm ==> 0.43 m=y

140 km ==> 140000m=L

600 nm ==> 6E-7= Wavelength

y=Ltanθ ==>Y/L =tan^-1

Formula θ=1.22*(λ)$theta=\frac{1.22*wavelength{} }{D{} {}{}}$

$\frac{1.22*(6.0*10^{-7}) }{tan^{-1} (\frac{0.043m}{140000m} )}$

Plug in the numbers to find the angle

MAKE SURE YOUR CALCULATOR IS IN RADIAN!!!!

You should get 2.38325515m

Answer 2

Answer:

D = 2.38 m

Explanation:

This exercise is a diffraction problem where we must be able to separate the license plate numbers, so we must use a criterion to know when two light sources are separated, let's use the Rayleigh criterion, according to this criterion two light sources are separated if The maximum diffraction of a point coincides with the first minimum of the second point, so we can use the diffraction equation for a slit

         a sin θ  = m λ

Where the first minimum occurs for m = 1, as in these experiments the angle is very small, we can approximate the sine to the angle

           θ = λ / a

Also when we use a circular aperture instead of slits, we must use polar coordinates, which introduce a numerical constant

           θ = 1.22 λ / D

Where D is the circular tightness

       

Let's apply this equation to our case

         D = 1.22 λ /  θ

To calculate the angles let's use trigonometry

         tan  θ = y / x

          θ = tan⁻¹  y / x

          θ = tan⁻¹ (4.30 10⁻² / 140 10³)

          θ = tan⁻¹ (3.07 10⁻⁷)

          θ = 3.07 10⁻⁷ rad

Let's calculate

        D = 1.22 600 10⁻⁹ / 3.07 10⁻⁷

        D = 2.38 m