A question about double-slit experiment with cover.?

Q: Why is the path difference d=(n-1)*thickness?

A: In the equation: path difference = m入, the "path difference" is the "optical path difference", NOT the "physical path difference".

The optical path = n x physical path
where n is the refractive index of the medium.
The optical path accounts for the slower speed of light in a medium. Thus there is a phase shift of light waves even if the physical paths of light through air and the medium are the same.

In air or vacuum, n = 1, the optical path = physical path

In your problem, the path difference between mica and air
= n入 - (1).入 = (n-1)入

Q: l/v = x/c x=n*thickness. i dont understand why is the time set to be the same?

A: Let x and l stand for the distances traveled by light in air and medium respectively in an arbitrary time t.
Hence, speed of light in air c = x/t
speed of light in medium v = l/t

Refractive index n = c/v = (x/t)/(l/t) = x/l
i.e. c/v = x/l
or l/v = x/c which is your equation
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You cannot use that relation because it compares the distances traveled by light in air and medium at the same time interval. The no. of wavelength in x and l are not the same as there is a phase difference between light traveled in air and medium. Clearly, x must be > l as light travels slower in medium.

The proof of optical path = n(physical path) is as follows:
Frequency in air f = c/入
In medium, f = v/入' = (c/n)/入' = c/(n入')
where 入' is the wavelength in medium due to lower speed of light.
Hence, c/入 = c/(n入')
i.e. 入 = n入'

The optical path is defined as the no. of wavelengths in the medium equals that in air (such that there will not be any phase shift).

But no. of wavelengths in air =La/入
No. of wavelength in medium = Lm/入'
where La is the length of air (optical path) that is equivalent to a length of Lm in medium (physical path) (i.e. in both lengths, they contain the same no. of wavelengths) .

Hence, La/入 = Lm/入'
i.e La/(n入') = Lm/入'
La/n = Lm
or La = n(Lm) [ optical path = n(physical path) ]