In the figure shown, a parallel beam of light is incident on the plane of the slits of a Young’s double slit experiment. Light incident on the slit, S 1 passes through a medium of variable refractive index μ = 1 + ax(where ‘x’ is the distance from the plane of slits as shown), upto a distance ‘ λ ’ before falling on S 1 . Rest of the space is filled with air. If at ‘O’ a minima is formed, then the minimum value of the positive constant a (in terms of λ and wavelength ‘ λ ’ in air) is :
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When light passes through a medium of refractive index μ , the optical path it travels is ( μ t).
Therefore, before reaching O light through S 1 travels ( μ l + b) distance while that through S 2 travels a distance ( l + b)
i.e. : path difference = ( μ l + b) – ( l + b) = ( μ – 1) l .
For a small element 'dx' path difference Δ x = [(1 + ax) – 1] dx = ax dx
For the whole length ;
Δ x = 
For a minima to be at 'O'.
Δ x = (2n + 1) 
i.e. : 
= (2n + 1) 
.
For minimum 'a'; n = 0
⇒ 
⇒ a = 
Ans.
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