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 5.0 cm apart, and the spy satellite is at an altitude of 160 km, what must be the diameter of the camera’s aperture? (Assume light with a wavelength of 550 nm.)
Single-slit diffraction: wsinθ = mλ (where w is the width of the slit, m = 1, 2, 3, ... for destructive interference)
Small-angle approximation: sinθ ≈ tanθ = y/D (where y is the vertical height above the central axis, D is the distance between the slit(s) and the screen) for small θ
The Attempt at a Solution
Since the goal of the problem is to solve for w, I used the small-angle approximation to set up the following equation: mλ/w = y/D. I then rearranged the variables to isolate w, ending up with w = mλy/D. Now, I know that D = 160 * 103 m and λ = 500 * 10-9 m. I'm having trouble, however, figuring out what to do for m and y. Since the numbers on the plate are 5.0 cm apart, I'm assuming that, every 5.0 cm, we have constructive interference occurring. Since there will definitely be constructive interference along the central axis, I figured that the first-order minimum would occur halfway in-between the central bright fringe and the first-order bright fringe, which would give m = 1 and y = 2.5 cm. However, this does not give me the correct answer; if I try instead to use m = 1 and y = 5.0 cm, this does give me the right answer, but I can't come up with a sketch of the situation that would explain why. Some assistance with the intuition here would really be appreciated.
Thank you very much in advance for your help!