One face of prism of refracting angle `30^@` and refractive index `1.414` is silvered. At what angle must a ray of light fall on the unsilvered face so that it retraces its path out of the prism ?

Explanation

The ray diagram for the given situation is as given below. PQR is prism described in question.

AB is the incident ray at point B on face PQ of prism. Let i be the angle of incidence as shown in the figure. BC is the refracted ray inside prism at an angle r with the normal to the surface. To retrace its path back ray BC must fall perpendicularly on surface PR which is silvered. We can easily understand that angle PBC is `60^(@)` and hence angle r becomes `30^(@)` because r is measured from normal to surface. Now we can apply Snell.s law at point B as follows:
`musin theta="constant"`
`rArr" "1xxsin i=sqrt2=sinr`
`rArr" "1xxsini=sqrt2xxsin30^(@)`
`rArr" "sini=sqrt2xx(1)/(2)`
`rArr" "sini=(1)/(sqrt2)`
`rArr" "i=45^(@)`