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`
For `lambda_(1), a sin theta_(1)=4 lambda_(1)`
For `lambda_(2), a sin theta_(2)=5lambda_(2)`
we have `theta_(1) = theta_(2)=theta`
`a sin theta=4lambda_(1)=5lambda_(2)`
`y=ftheta rArr theta=y//f [:. D=f]`
Separation `y=0.5 cm=5xx10^(-3)m` focal length `f=1` m
`theta = y/f=(5xx10^(-3))/1=0.005` rad
Wavelength `lambda_(1)=(a sin theta)/4=(atheta)/4`
`=(0.04xx10^(-2)xx0.005)/4`
`=5xx10^(-7)m=500nm`
Wavelength `lambda_(2)=(4lambda_(1))/5=(4xx500)/5=400nm`