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

Options

A. sin`A/2`
B. cos`A/2`
C. cosec`A/2`
D. sec`A/2`

Answer

C

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
1sini=nsin`r_1`…(ii)
`A=r_1+r_2=r_1+C`…(iii)
From eqn. (ii) n is minimum when `r_1` is maximum , i.e., `r_1=C`. In this case `i=90^@`
From eqn. (iii) , A=2C or C=A/2
From eqn. (ii),
sin C=`1/n`, sin`A/2=1/n`
`therefore` n=cosec`A/2`