What should be the minimum value of refractive index of a prism, refracting angle A, so that there is no emergent ray irrespective of the angle of incidence?

Explanation

If the ray just emerges from face AC,
`e=90^@` and `r_(2)=C` (i)
From Snell's law at face AB, we have
`1 sin i=nsin r_(1)` (ii)
`A=r_(1)+r_(2)=r_(1)+C` (iii)
From Eq. (ii), n is minimum when `r_(1)` is maximum, i.e., `r_(2)=C` . In this case, `i=90^(@)`
From Equ. (iii),
`A=2C` or `C=A//2`
As `sinC =(1)/(n)rArrsin(A)/(2)=(1)/(n)`
`n=cosec(A)/(2)`