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Question
Consider the arrangement shown in the figure. The distance D is large compared to the separation d between the slits.
- Find the minimum value of d so that there is a dark fringe at O.
- Suppose d has this value. Find the distance x at which the next bright fringe is formed.
- Find the fringe-width.
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Solution
From the figure,
Δx = AMO − ANO
Here AM = MO = √[D2 + d2] AND AN = NO = D
=> Δx = 2(AM − AN) = 2 {√[D2 + d2] - D}
For minima at O,
2 {√[D2 + d2] − D} = (n +(1/2)) λ
Solving above equation for d, we get
d = √[Dλ/2]
(b) width of the dark fringe = w = Dλ/d
Now, the location x is given by
x = Dλ/[2√(Dλ/2)]
=> x = d
(c) As x = w/2
=> w = 2x = 2d
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