A lens of focal length `1m` forms Fraunhofer diffraction pattern of a single slit of width `0.04 cm` in its focal plane. The incident light contains two wavwlength `lambda_(1)` and `lambda_(2)`. It is found that the fourth minimum crresponding to `lambda_(1)` and the fifth minimum corresponding to `lambda_(2)` occur at the same point `0.5 cm` from the central maximum. Compute `lambda_(1)` and `lambda_(2)`

Explanation

The angular position of nth minima
a `sin theta=nlambda`
`lambda_(1): a sin theta_(1) =4lambda_(1)` …(i)
`lambda_(2): a sin theta_(2) =5lambda_(2)` …(ii)
`theta_(1)=theta_(2)=theta`
a `sin theta=4lambda_(1)=5lambda_(2)` …(iii)
`y=ftheta impliestheta=y//f`, f:focal length of the lens
`y=0.5 cm=5xx10^(-3)m,f=1 m`
`theta=y/f=(5xx10^(-3))/1=0.005 rad`
`lambda_(1)=(a sintheta)/4=(atheta)/4=(0.04xx10^(-2)xx0.005)/4`
`=5xx10^(7)m =500 nm`
`lambda_(2)=(4lambda_(1))/5=(4xx500)/5=400 nm`