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Question

(a) In Young's double slit experiment, derive the condition for(i) constructive interference and(ii) destructive interference at a point on the screen.(b) A bream of light consisting of two wavelengths, 800nm and 600nm is used to obtain the interference fringes in a Young's double slit experiment on a screen placed 1.4m away. If the two slits are separated by 0.28nm, calculate the least distance from the central bright maximum where the bright fringes of the two wavelengths coincide.

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Solution

Young double slit experiment: Let S1 and S2 are the two narrow rectangular slits placed perpendicular to the plane of paper. Screen is placed on the perpendicular bisector ofS1S2 and is illuminated with monochromatic light.The slits are separated by a small distance d. A screen is placed at a distance D from S1S2. Consider a point P on the screen at a distance x from O. The path difference between the waves reaching P from S1 and S2 isP=S2P−S1P , draw a perpendicular S1N on S2PP=S2P−S1P=S2P−NP=S2NFrom the triangle S2NS2S1=sinθ∴P=S2N=S2S1sinθ=dsinθwhen θ is small, sinθ≃θ=tanθ=xD∴P=xdDFor constructive interference xdD=nλ;n=0,1,2.....Position of nth bright fringe Xn=nDλdwhen n=0 Xn= 0, central bright fringe is formed at O.For destructive interference ,xdD=(2n−1)λ2Xn=(2n+1)λD2Thus alternative bright fringe and dark fringe are formed on the screen.(b)Let the nth1 maximum corresponds to n1 and nth2 maximum corresponds n2, then n1λ1Dd=n2λ2Dd⟹n1n2=600800=34ymin=N1λ1Dd=3×800×10−9×1.40.28×10−3=12nm

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