The figure shows the path of a ray passing through an equiangular prism PQR. It is incident on face PR at angle slightly greater than the critical angle for total internal reflection . If the angle `alpha` shown in the figure is `30^(@)` . Calculate the R.I. of the material of the prism.

Answer

`((7)/(3))^(1//2)`

Explanation

For critical angle
`mu "sin"(90-r) = 1. "sin"90`
`"cosr" = (1)/(mu)`
`90 - (120 -r) = (r-30)`
From snell's law
`mu "sin"(r- 30^(@)) = 1 "sin"30^(@)`
`mu["sinr" "cos"30^(@) - "cos"30^(@) - "cos"r "sin" 30^(@)] = (1)/(2)`
`mu[(sqrt(mu^(2)-1))/(mu) xx (sqrt(3))/(2)- (1)/(mu) xx (1)/(2)] = (1)/(2)`
`sqrt(u^(2) - 1) xx (sqrt(3))/(2) = 1 implies mu^(2) - 1 = (4)/(3) implies mu ((7)/(3))^(1//2)`