By what angle should the second polarized sheet be rotated relative to the first to reduce the transmitted intensity to one-half
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Let us start from considering monochromatic light as an incidence on the film of a thickness t whose material has an index of refraction n determined by their respective properties.
From this point of view part of the light will be reflated and the other will be transmitted to the thin film. That additional distance traveled by the ray that was reflected from the bottom will be twice the thickness of the thin film at the point where the light strikes. Therefore, this relation of phase differences and additional distance can be expressed mathematically as
We are given the second smallest nonzero thickness at which destructive interference occurs.
This corresponds to, m = 2, therefore
The index of refraction of soap is given, then
Combining the results of all steps we get
Rearranging, we find