In the given figure, the faces of prism `ABCD` made of glass with a refractive index `n` form dihedral angles `/_A=90^(@),/_B75^(@),,/_C=135^(@) & /_D=60^(@)` (The Abbe's prism) .A beam of light falls on face `AB` `&` after total internal reflection from face `BC` escapes through face `AD`.Find the range of `n` and angle of incidence `alpha` of the beam onto face `AB`, if a beam that has passed through the prism in this mahner is perpedicular to the incident beam.

Answer

`sqrt2 lt nlt2, and 45^(@)ltalpha le 90^(@)`

Explanation

For the figure since incidence angle at face `AB` is `alpha` and refraction angle is `r` , condition that emergent ray is perpndicular to incidence ray `rArr` angle of emergence at face `QD` is `alpha`
`rArr` incidence angle at face `QD` is `r.`
In quadrlateral `PQDC 15^(@)+r+135^(@)+r+90=180^(@)`
Now, `75^(@)-r=45^(@)gtcrArr(1)/(sqrt2)gt(10/(n)rArrngtsqrt2`
Also, `sin alpha =n sin r =n sin 30^(@)=(n)/(2)`
`rArr n=2sin alpha rArr (n)/(2)le1rArrnle2`
`therefore sqrt2lenle2` and `(sin alpha)/(sin r)=n rArrsqrt2 le'2sin alphale2`
`rArr (1)sqrt2)ltsin alpha le1 rArr45^(@)ltalphale90^(@)`.