At what angle should a ray of light be incident on the face of a prism of refracting angle 60° so that it just suffers total internal reflection at the other face? The refractive index of the material of the prism is 1.524.

#### Solution

The incident, refracted, and emergent rays associated with a glass prism ABC are shown in the given figure.

Angle of prism, ∠*A* = 60°

Refractive index of the prism, *µ* = 1.524

`i_1` = Incident angle

`r_1` = Refracted angle

`r_2` = Angle of incidence at the face AC

*e* = Emergent angle = 90°

According to Snell’s law, for face AC, we can have:

`(sin e)/(sin r_2) = mu`

`sin r_2 = 1/mu xx sin 90^@`

`= 1/1.524 = 0.6562`

`:. r_2 = sin^(-1) 0.6562 ~~ 41^@`

It is clear from the figure that angle `A = r_1 + r_2`

`:. r_1 = A - r_2 = 60 - 41 = 19^@`

According to Snell’s law, we have the relation:

`mu = (sin i_1)/(sin r_1)`

`sin i_1 = mu sin r_1`

`= 1.524 xx sin 19^@ = 0.496`

`:. i_1 = 29.75^@`

Hence, the angle of incidence is 29.75°.