A right angles prism `(45^(@),90^(@), 45^(@))` of refractive index n has a plate of refractive index `(n_(1)ltn)` cemented to its diagonal face. The assembley is in air. A ray is incident on AB.
a. Calculate the angle of incidence at AB for which the ray strikes the diagonal face at the critical angle.
b. Assuming `n=1.351`, calculate the angle of incidence at AB for which the refracted rey passes through the diagonal face undeviated.

Explanation

(a)
`(n_1 lt n)`
`nsinC=n_1sin90^@`
`sinC=n_1//n`
In `DeltaAPQ`,
`90^@-r+90^@-C+45^@=180^@`
`r+C=45^@impliesr=45^@-C`
Refraction at AB:
`1xxsini=nsinr`
`=nsin(45^@-C)`
`=nsin45^@cosC-cos45^@sinC`
`=n[1/sqrt2sqrt(1-n_1^2/n^2)-1/sqrt2n_1/n_2]`
`=1/sqrt2[sqrt(n^2-n_1^2)-n_1]`
`i=sin^-1[1/sqrt2{sqrt(n^2-n_1^2)-n_1}]`
(b) If refracted ray passes undeviated through AC, for this ray must be incident normally on AC i.e. `r^'=0`
`r+r^'=Aimpliesr=45^@`
`1xxsini=nsin45^@=(1.352)/(sqrt2)`
`i=sin^-1((1.352)/(sqrt2))=sin^-1(0.96)`