Question

In Young’s double-slit experiment, show that: 

`beta = (lambda "D")/"d"` where the terms have their usual meaning.

Answer 1

d = Distance between the slits

D = Distance between slit and screen

'P' is the position of m^th 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

∴ x_m = `("m"lambda"D")/"d"`

∴ x_m+1 = (m + 1) `(lambda"D")/"d"`

∴ Fringe width β = [(m + 1)-m] `(lambda"D")/"d"`

β = `(lambda"D")/"d"`