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 band is 7.5 × 10^-6 m and that at a point Q on the other side of the central bright band is 1.8 × 10^-6 m. How many bright and dark bands are observed between points P and Q if the wavelength of light used is 6 × 10^-7 m ?

Answer 1

Data : ∆l₁ = 7.5 × 10^-6 m, 

∆l₂ = 1.8 × 10^-6 m 

λ = 6 × 10^-7 m

For point P : Let p \(\cfrac λ2\) = ∆l₁

The path difference ∆l is an odd integral multiple of λ/2 : ∆l₁ = (2m – 1) \(\cfrac λ2\), 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 \(\cfrac λ2\) = ∆l₂

The path difference ∆l₂ is an even integral multiple of \(\cfrac λ2\) : ∆l₂ = (2n) \(\cfrac λ2\), 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