Light enters at an angle of incidence in a transparent rod of refractive index n. For what value of the refractive index of the material of the rod the light once entered into it will not leave it through its lateral face whatsoever be the value of angle of incidence.

Explanation

Let a light ray enters at A and refrected beam is AB. At the lateral face, the angle of incidence is `theta`. For no refraction at this face, `thetagtC`.
i.e., `sinthetagtsinC`
But `theta+r=90^(@)rArrtheta=90^(@)-r`

`sin(90^(@)-r)gtsin Cor cosrgtsinC`.....(i)
By Snell's law, `mu=(sini)/(sinr)rArrsinr=(sini)/mu`
`cosr=sqrt(1-sin^(2)r)=sqrt((1-(sin^(2)i)/(n^(2))))`
From Eq. (i), `sqrt(1-(sin^(2)i)/(n^(2)))gtsinC`
Also, sin C =`1/µrArr1-(sin^(2)i)/(n^(2))gt1/(mu^(2))ormu^(2)gtsin^(2)i+1`
The maximum value of sin i is l.
`mu^(2)gt2 or mugtsqrt2`