Questions

A ray OP of monochromatic light is incident on the face AB of prism ABCD near vertex B at an incident angle of 60º as shown in figure. If the refractive index of the material of the prism is $\sqrt{3}$ , which of the following is(are) correct ?

a

The ray gets totally internally reflected at face CD.

b

The ray comes out through face AD

c

The angle between the incident ray and the emergent ray is 90º

d

The angle between the incident ray and the emergent ray is 120º

detailed solution

Correct option is A

Applying Snell’s law at face AB, we get

$\begin{array}{l}1\times \mathrm{sin}{60}^{\circ}=\sqrt{3}\times \mathrm{sin}\mathrm{r}\\ \frac{\sqrt{3}}{2}=\sqrt{3}\mathrm{sin}\mathrm{r}\text{or}\mathrm{sin}\mathrm{r}=\frac{1}{2}\\ \mathrm{r}={\mathrm{sin}}^{-1}\left(\frac{1}{2}\right)={30}^{\circ}\\ \mathrm{sin}{\mathrm{\theta}}_{\mathrm{c}}=\frac{1}{\sqrt{3}}\end{array}$

(where, ${\mathrm{\theta}}_{\mathrm{c}}$ is the critical angle)

${\mathrm{\theta}}_{\mathrm{c}}={\mathrm{sin}}^{-1}\left(\frac{1}{\sqrt{3}}\right)\text{or}{\mathrm{\theta}}_{\mathrm{c}}\approx {35}^{\circ}$

From geometry, angle of incidence at Q is 45º. At face CD angle of incidence is greater than ${\mathrm{\theta}}_{\mathrm{c}}$. Therefore, the rays gets totally internally reflected at face CD. Hence, option (a) is correct. From geometry, angle of incidence at R is 30º. At face AD angle of incidence is less than ${\mathrm{\theta}}_{\mathrm{c}}$So, the ray comes through face AD. Hence option (b) is correct. Applying Snell’s law at face AD, we get$\mathrm{e}={\mathrm{sin}}^{-1}\left(\frac{\sqrt{3}}{2}\right)={60}^{\circ}$

The angle between the incident ray and the emergent ray is 90º.

Hence, option (c) is correct and option (d) is incorrect

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