Question

In a double-slit experiment, the optical path difference between the waves coming from two coherent sources at a point P on one side of the central bright is 7.5 µm and that at a point Q on the other side of the central bright fringe and 1.8 µm. How many bright and dark fringes are observed between points P and Q if the wavelength of light used is 600 nm?

Answer 1

Data: Δl₁ = 7.5 × 10⁻⁶ m, Δl₂ = 1.8 × 10⁻⁶ m, λ = 6 × 10⁻⁷ m

For Point P: Let p be an integer such that p`λ/2` = Δl₁

∴ `p = (2Δl_1)/λ`

= `(2 xx 7.5 xx 10^-6)/(6 xx 10^-7)`

= `150/6`

= 25

∴ The path difference Δl₁ is an odd integral multiple of λ/2: Δl₁ = (2m - 1) `λ/2`, where m is an integer,

∴ 2m − 1 = 25

∴ m = 13

∴ Point P is at the centre of the 13th dark band.

For point Q: Let q be an integer such that q `λ/2` = Δl₂

∴ q = `(2Δl_2)/λ`

= `(2 xx 1.8 xx 10^-6)/(6 xx 10^-7)`

= `36/6`

= 6

∴ The path difference Δl₂ is an even integral multiple of `λ/2`: Δl₂ = (2n)  `λ/2`, where n is an integer,

∴ 2n = 6

∴ n = 3

∴ Point Q is at the centre of the 3rd bright band. Between points P and Q, excluding the respective bands at P and Q, the number of dark bands = 12 + 3 + 15 and the number of bright bands (including the central bright band) = 12 + 2 + 1 = 15.