Angular width of central maximum in the Fraunhofer diffraction pattern of a slit is measured. The slit is illuminated by light of wavelength `6000Å`. When the slit is illuminated by light of another wavelength, the angular width decreases by `30%`. Calculate the wavelength of this light. The same decrease in the angular width of central maximum is obtained when the original apparatus is immersed in a liquid. Find the refractive index of the liquid.

Explanation

Angular width of central maximum `=2lambda/a`
`lambda_(1)=600nm, theta_(1)=(2lambda_(1))/a` ....(i)
`theta_(2)=theta_(1)xx0.7=(2lambda_(2))/a` ....(ii)
On diving Eq. (i) by Eq. (ii), we get
`1/(0.7)=lambda_(1)/lambda_(2)=600/lambda_(2)`
Wavelength `lambda_(2)=420nm`
When immersed in liquid `lambda_(2)=lambda_(1)//mu`
`theta_(1)=theta_(1)xx0.7`
`rArr (2lambda_(1)//mu)/a= (2lambda_(1))/axx0.7`
On diving Eq. (i) by Eq. (ii) we get
`1/(0.7)=mu`
Refractive index of the liquid `mu=10/7=1.42`