The young's double slit experiment is done in a medium of refractive index `4//3`.A light of `600nm`wavelength is falling on the slits having `0.45` mm separation .The lower slit `S_(2)` is covered by a thin glass sheet of thickness `10.4mu m` and refractive index`1.5`.the intereference pattern is observed ona screen placed `1.5m` from the slits are shown

(a) Find the location of the central maximum (bright fringe with zero path difference)on the y-axis.
(b) Find the light intensity at point `O`relative to the maximum fringe intensity.
(c )Now,if `600nm`light is replaced by white light of range 400 to700 nm find the wavelength of the light that from maxima exactly point `O`.[All wavelengths in this problem are for the given medium of refractive index `4//3` .Ignore dispersion]

Answer

(i). 4.33m
(ii). `I=(3I_(max))/(4)`
(iii). 650nm;433.33 nm

Explanation

(i) location of central maximum on y-axis let the central maximum be obtained at a distance y below point O. it will be below O because glass sheet covers the lower slit.
`Deltax_(1)=S_(1)P-S_(2)P=(yd)/(D)`
Due to glass sheet `Deltax_(2)=((mu_(g))/(mu_(m))-1)t`
For zero path difference `Deltax_(1)=Deltax_(2)`
`implies(yd)/(B)=((mu_(g))/(mu_(m))-1)timpliesy=((mu_(g))/(mu_(m))-1)(tD)/(d)`
`impliesy=((3//2)/(4//3)-1)((10.4xx10^(-6))/(0.45xx10^(-3)))(1.5)`
`y=4.33xx10^(-3)m=4.33mm`
(ii) Light intensity at O, `Deltax_(1)=0,Deltax_(2)=((mu_(g))/(mu_(m))-1)t`
`therefore` Net phase difference `=(2pi)/(y)Deltax_(2)`
`thereforephi=(2pi)/(6xx10^(-7))((3//2)/(4//3)-1)(10.4xx10^(6))=(13pi)/(3)`
`thereforeI(phi)=I_(max)cos^(2)((phi)/(2))` or `(I(phi))/(I_(max))=(3)/(4)=0.75`
(iii). Wavelength of light that form maxima O. if 600 nm light is replaced by 400 to 700 nm light
At `O,Deltax=((mu_(g))/(mu_(m))-1)t`
For maximum intensity at O, `Deltax=nlamda`
where `n=1,2,3`...
`thereforelamda=(Deltax)/(1),(Deltax)/(2),(Deltax)/(3)`
`Deltax=((3//2)/(4//3)-1)(10.4xx10^(-6)m)` or `Deltax=1300nm`
For maximum intensity at O,
`lamda=1300nm,(1300)/(2)nm,(1300)/(2)nm,(1300)/(4)nm`
`lamda=1300nm,650nm,433.33nm` ....
or `lamda=6.5xx10^(-7)m,4.33xx10^(-7)m`