White coherent light (400 nm-700 nm) is sent through the slits of a Young's double slit experiment. The separation between the slits is 05 mm and the screen is 50 cm away from the slits. There is a hole in the screen at a point 1.0 mm away (along the width of the fringes) from the central line. (a) Which wavelength(s) will be absent in the light coming from the hole ? (b) which wavelength(s) will have a strong intensity ?

Explanation

a. The absent wavelength will correspond to minima at position order of minima corresponding to `4000 Å,`
`y_(n) = ((2n - 1) D lambda)/(2d) implies (2n -1)`
`(y_(n) 2d)/(D lambda) implies n = (1)/(2) [(2d y_(n))/(D lambda) + 1 ]`
`n_(1) = (1)/(2) [(0.5 xx 10^(-3) xx 1 xx 10^(-3) xx 2)/(50 xx 10^(-2) xx 4000 xx 10^(-10)) + 1]`
`= (1)/(2) [(2 xx 10^(4))/(4000) + 1] = 3`
Order or minima corresponding to `7000 Å,`
`n_(2) = (1)/(2) [(2 xx 10^(4))/(7000) + 1] = 1.9`
Number of integers between `1.9` and `3.0` are 2 and 3.
Wavelength corresponding to `n = 2` is
`lambda_((2)) = (y_(n) 2d)/((2n - 1) D) = (1 xx 10^(-3) xx 2 xx 0.5 xx 10^(-3))/((2 xx 2 - 1) xx 50 xx 10^(-2))`
`y_(n) = (nD lambda)/(d) implies n = (y_(n) d)/(D lambda) = (1 xx 10^(-3) xx 0.5 xx 10^(-3))/(50 xx 10^(-2) xx lambda)`
`n_(4000) = (10^(-6))/(4000 xx 10^(-10)) = (10000)/(4000) = 2.5`
and `n_(7000) = (10000)/(7000) = 1.4`
Integer between `1.4` and `2.5` is 2.
`:. lambda_((2)) =(y _(n) d)/(nD) = (10^(-6))/(2) = 500 nm`
.