Question 91: Ray Optics

Question

As shown in the diagram, when the incident ray is parallel to base of the prism, the emergent ray grazes along the second surface.If refractive index of the material of prism is $\sqrt{2}$, the angle $	heta$ of prism is.

Accepted Answer

The incident ray is parallel to the base where the base angle is $45^{\circ}$.
If the ray is parallel to the base, the angle of incidence i with the normal to the first surface is also $45^{\circ}$
Let the refractive index of the prism be $\mu=\sqrt{2}$ and air be $\mu_{	ext {air }}=1$.
Applying Snell's Law at the first surface :
$$ \mu_{\mathrm{air}} \sin (\mathrm{i})=\mu \sin \left(\mathrm{r}_1ight) $$
$$\Rightarrow $$ $1 	imes \sin \left(45^{\circ}ight)=\sqrt{2} 	imes \sin \left(\mathrm{r}_1ight)$
$$\Rightarrow $$ $\frac{1}{\sqrt{2}}=\sqrt{2} \sin \left(\mathrm{r}_1ight)$
$$\Rightarrow $$ $\sin \left(r_1ight)=\frac{1}{2}$
$$\Rightarrow $$ $$ r_1=30^{\circ} $$
It is given that the emergent ray grazes along the second surface. This means the angle of emergence e is $90^{\circ}$.
This occurs when the angle of incidence inside the prism at the second surface $\left(\mathrm{r}_2ight)$ is equal to the critical angle (C).
Applying Snell's Law at the second surface :
$$ \mu \sin \left(\mathrm{r}_2ight)=\mu_{\mathrm{air}} \sin \left(90^{\circ}ight) $$
$$\Rightarrow $$ $\sqrt{2} 	imes \sin \left(\mathrm{r}_2ight)=1 	imes 1$
$$\Rightarrow $$ $\sin \left(r_2ight)=\frac{1}{\sqrt{2}}$
$$\Rightarrow $$ $$ \mathrm{r}_2=45^{\circ} $$
For any prism, the relationship between the apex angle A and the internal refracted angles is :
$$ \mathrm{A}=\mathrm{r}_1+\mathrm{r}_2 $$
$$\Rightarrow $$ $$ \mathrm{A}=30^{\circ}+45^{\circ}=75^{\circ} $$
Sum of angles in a triangle is $180^{\circ}$ :
$$ 45^{\circ}+	heta+\mathrm{A}=180^{\circ} $$
$$\Rightarrow $$ $45^{\circ}+	heta+75^{\circ}=180^{\circ}$
$$\Rightarrow $$ $120^{\circ}+	heta=180^{\circ}$
$$\Rightarrow $$ $$ 	heta=60^{\circ} $$
So, the angle $	heta$ of prism is $60^{\circ}$.
Hence, the correct option is (C).

Answer

$75^{\circ}$

Answer

$90^{\circ}$

Answer

$60^{\circ}$

Answer

$45^{\circ}$

Ray Optics

As shown in the diagram, when the incident ray is parallel to base of the prism, the emergent ray grazes along the second surface.If refractive index of the material of prism is \sqrt{2}, the angle \theta of prism is.

Choose the correct answer: