**A very thin sheet of plastic (n=1.6) covers one slit of a double slit apparatus illuminated by 540nm light. The center point of the screen, rather than being a maximum, is dark. What is the minimum thickness of the plastic?**

To get a dark spot at the centre of the pattern, you need the light coming out the two spots to have a phase difference of 180^{o} (+/- n*360^{o}).

Until the light hits the plastic, the light heading towards the two slits is exactly in phase. However, with in the glass the light travels slower - ie it has a shorter wavelength λ_{plastic}=λ/n_{plastic}.

Lets say that the thickness of the plastic, L, is x*λ (x is not necessarily an integer).

If L is the minimum thickness that creates the dark spot, then L must x+1/2 of the shorter â€œin plasticâ€� wavelength

- L=xλ

L=(x+1/2)λ/n

xλ=(x+1/2)λ/n

x(1-1/n)=1/2n

x(n-1)=1/2

x=1/2(n-1)

L=xλ=λ/2(n-1) = 450nm

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