A prism is shown in the figure with prism angle `75^(@)` and refractive index `sqrt(3)`. A light ray incidents on a surface at incident angle `theta`. Other face is coated with a medium of refractive index n. for ` theta le 60^(@)` ray suffers total internal reflection find value of `n^(2)`.

Answer

`3/2`

Explanation

For T.I.R at coated
`sin c=n/(sqrt(3))=sinc`
Applying snell's law at first surface
`sin theta=sqrt(3) sin(75-c)`
for limiting conduction, at `theta=60^(@)`
`sin60=sqrt(3) sin (75-c)`
`(sqrt(3))/2=sqrt(3) sin (75-c)`
`1/2=sin(75-c)rArr sin30 =sin(75-c)`
`30=75-c rArr c=45^(@)`
`n/(sqrt(3))=1/(sqrt(2))rArr n^(2)= 3/2=1.50`