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Question
In Young’s double-slit experiment, show that:
`beta = (lambda "D")/"d"` where the terms have their usual meaning.
Solution
d = Distance between the slits
D = Distance between slit and screen
'P' is the position of mth order bright fringe.
From diagram, the path difference p is
S2P - S1P = S2P - AP
= S2A
`"sin"theta = ("S"_2"A")/("S"_1"S"_2)` [From Δ S1AS2]
`"tan" theta = "OP"/"CO"` [From Δ POC]
θ → 0 ∴ sin θ ≅ tan θ
∴ `("S"_2"A")/("S"_1"S"_2) = "OP"/"OC"`
`therefore ("S"_2"A")/"d" = "x"/"D"`
`therefore "S"_2"A" = "x d"/"D"`
or p = `"x" "d"/"D"`
∴ For bright fringe,
`"x d"/"D" = "m" lambda`, where m is an integer
∴ xm = `("m"lambda"D")/"d"`
∴ xm+1 = (m + 1) `(lambda"D")/"d"`
∴ Fringe width β = [(m + 1)-m] `(lambda"D")/"d"`
β = `(lambda"D")/"d"`
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