Lens Set

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A camera with a 50 mm lens set at f /8 aperture has an opening 6.25 mm in diameter.?

A camera with a 50 mm lens set at f /8 aperture has an opening 6.25 mm in diameter.

(a) Suppose this lens acts like a slit 6.25 mm wide. For light with λ = 580 nm, what is the resolution of the lens - the distance from the middle of the central bright band to the first-order dark band? The film is 50.0 mm from the lens.
____mm

(b) The owner of a camera needs to decide which film to buy for it. The expensive one, called fine-grained film, has 200 grains/mm. The less costly, coarse-grained film has only 50 grains/mm. What is the width of the central bright band of each film?

200 grains/mm
____mm

50 grains/mm
____mm

The lens with the slit produces a pattern given by

I(x) = Io sin(x d / lambda f)^2 / (x d / lambda f)^2

where d is the aperture width, x is the position from the center of the pattern, f = 50mm, and lambda = 580nm. This is assuming that there is a rectangular slit; if you have a circular aperture then the answer requires the use of a Bessel-Sinc function, but you specified a rectangular slit in your problem.

Then the size of the central bright band is given by

xo d / lambda f = pi (this is the distance from the middle to the first-order dark band.

thus

xo = pi lambda f / d

and then the width of the first bright band

w = 2xo = 2 pi lambda f / d = 2 (3.14) (580E-9) (50E-3) / 6.25E-3
~ 30 microns

This translates into a 1/0.03 lines per mm or 33 lines per mm. Then an expensive film of 200 grains/mm is unnecessary as the camera can only resolve about 33 lines per mm. So I would suggest to stick to the 50 grains/mm coarse-grained film.

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