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A thin flake of mica (n = 1.58) is used to cover one slit of a double-slit interference arrangement. The central point on the viewing screen is now occupied by what had been the seventh bright side fringe (m = 7). If λ = 550 nm, what is the thickness of the mica?

I understand that the seventh bright side fringe means there is a path difference of 7λ, but what does that look like?

Imagine the wall and slits being vertical in the page, and the top slit having the flake. Would the whole pattern then be shifted upward and be darker?

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    $\begingroup$ The mike flake increases the path length for light that passes through it. If you imagine an infinitesimally thin flake, then draw some diagrams to see what happens to the fringes if you gradually increase the thickness of the flake, you may be able to figure this out. $\endgroup$ – S. McGrew May 6 at 0:01
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As pointed out by S.McGrew, the mica flake increases the path length for light passing through it owing to the fact that light travels slower in mica as compared to vacuum.

It is already mentioned that the mica flake is 'thin', so we can can make some reasonable approximations to arrive at the answer. Also, as nothing has been specifically mentioned, we can assume that there is no loss of energy as the light passes through the flake. So no, the pattern would not get darker, its intensity remains the same.

Yes, the whole pattern would be shifted upwards because if the added path length which the light has to travel. Now you for solving the question, you need to get an expression for the extra path length. This can be done with the concept of 'optical path', the distance which light would have traveled in the same time, had it been in vacuum. You can derive it to be the geometrical path times the refractive index of the medium.(Try deriving it.) If the refractive index is n, the extra path length would be $= nt-t$ , where t is the thickness. Now you add thid term to the expression for path difference and equate it too 7λ and solve the equation to get the value of t.

P.S. : Keep in mind that we have assumed the mica flake to be 'thin'.

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