Question

In Young's double slit experiment, show that:

`β = (λ"D")/"d"`

Where the terms have their usual meaning.

Answer 1

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 m^th 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"`