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Question

With the help of a labelled diagram, show that fringe separation β in Young's double slit experiment is given by:
β=λDd
where the terms have their usual meaning

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Solution

Let S be the monochromatic source of light and let S1,S2 be the two parallel narrow slits close to each other. According to Huygen's principle S1 and S2 act as new wave source from which secondary wave lets originate. Let S1A is the perpendicular drawn on S2P from S1. Path difference is given by
S2A=S2PS1P
In right angled S1S2Asinθ=S2AS1S2
In right angled PCOtanθ=POCO
If θ is small; then S2AS1S2=OPCO
S2Ad=xD
So, Path difference =S2A=xdD
The intensity of light will be maximum at those places where the path difference between the interfering waves is zero or integral multiple of λ.
So, xdD=mλ
or x=mDλd
625769_597296_ans_68d77bc87f9c4d429422db4d50c9c5df.png

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