A monochromatic light of `lambda=500 nm` is incident on two identical slits separated by a distance of `5xx10^(-4)m`. The interference pattern is seen on a screen placed at a distance of `1 m` from the plane of slits. A thin glass plate of thickness `1.5xx10^(-6)m` and refractive index `mu=1.5` is placed between one of the slits and the screen. Find the intensity at the centre of the screen if the intensity is `I_(0)` in the absence of the plate. Also find the lateral shift of the central maxima and number of fringes crossed through centre.

Explanation

Given `lambda=500nm d=5xx10^(-4)m D=1m`
`t=2.5xx10^(-6)m` and `mu=1.5`
Path difference due to plate
`Deltax=(mu-1)t=(1.5-1)xx2.5xx10^(-6)=1.25xx10^(-6)m`
Phase difference
`phi=(2pi)/lambdaDeltax=(2pi)/(500xx10^(-9))xx0.75xx10^(-6)=5pi`
Intensity `I=I_(0)cos^(2)(phi//2)=I_(0)cos^(2)(2.5pi)=0`
Lateral shift
`y_(0)=((mu-1)tD)/d=(1.25xx10^(-6)xx1)/(5xx10^(-4))`
`=0.25xx10^(-2)m=2.5mm`
Number of fringes crossing the centre
`y_(0)/beta=(((mu-1)tD)/d)/((Dlambda)/d)=((mu-1)t)/lambda=(1.25xx10^(-6))/(500xx10^(-9))=2.5`