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Question
In Young's double slit experiment, show that:
`β = (λ"D")/"d"`
Where the terms have their usual meaning.
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Solution
Assume that S is a light source that is receiving light from another light source. Two line sources, S1 and S2, are equally spaced apart from light source S.
Let S1S2 = d
Path difference between waves at point P,
S2P - S1P = S2A
ΔS1S2A and ΔPCO are similar to each other.
∴ `("S"_2"A")/("S"_1"S"_2) =("OP")/("CP")`
Since CO is very much greater than S1S2,
So CP ≈ CO
∴ `("S"_2"A")/("S"_1"S"_2) = ("OP")/("CO")`
`("S"_2"A")/"d" = x/"D"`
∴ Path difference = `x/"D"`
For bright fringes, = mλ
`x = m ("D"λ)/"d"`
For mth and (m + 1)th fringe,
`x_m = m ("D"λ)/"d"` and
`x_(m + 1) = (m + 1) ("D"λ)/"d"`
Fringe width, `β = x_(m + 1) - x_m`
= `(m + 1) ("D"λ)/"d" - m ("D"λ)/"d"`
`β = ("D"λ)/"d"`
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