Question

A glass prism has refractive index 1.5 and the refracting angle is ${{90}^{0}}$ . If a ray falls on it at an angle of incidence of ${{30}^{0}}$ , then angle of emergence will be:
(A) ${{60}^{0}}$
(B) ${{30}^{0}}$
(C) ${{45}^{0}}$
(D) The ray will not emerge out of the prism

Answer 1

Hint: We will first calculate the angle of deviation of the incident ray on the prism using its formula which is a product of refracting angle and refractive index minus refracting angle. Then, we shall use this value to calculate the angle of emergence of the incident ray. Once we get that, we will be able to comment if the ray will emerge or not from the prism.

Complete answer:
Let us first assign some useful terms that we will use in our solution.
$\delta :$ let this term denote the angle of deviation of the incident light ray.
$\mu :$ let this term denote the refractive index of the prism whose value is given to be 1.5
$A:$ let this term denote the refracting angle of the prism. Its value is given to be ${{90}^{0}}$
${{i}_{1}}:$ let it be the angle of incidence of the light ray and its value is given as ${{30}^{0}}$
${{i}_{2}}:$ let this be the angle of emergence
Now, we will first calculate the angle of deviation of the incident light ray. We can do so using the following formula:
$\Rightarrow \delta =(\mu -1)A$
Putting the values of all the known terms, we get:
$\begin{align}
  & \Rightarrow \delta =(1.5-1){{90}^{0}} \\
 & \Rightarrow \delta ={{45}^{0}} \\
\end{align}$
Now, we will calculate the angle of emergence of the light ray using the formula:
$\Rightarrow \delta =({{i}_{1}}+{{i}_{2}})-A$
$\Rightarrow {{i}_{2}}=\delta -{{i}_{1}}+A$
Putting the values of all the known terms in the above equation, we get the value of angle of emergence as:
$\begin{align}
  & \Rightarrow {{i}_{2}}=45-30+90 \\
 & \therefore {{i}_{2}}={{105}^{0}} \\
\end{align}$
Therefore, angle of emergence has an inclination with the second surface equal to:
$\begin{align}
  & =90-105 \\
 & =-{{15}^{0}} \\
\end{align}$
Since, this angle comes out to be negative it means there will be total internal refraction at the second surface.
Hence, the ray will not emerge out of the prism.

Hence, option (D) is the correct option.

Note:
Here, we used the basic formulas of a triangular prism to get our solution and these should be remembered thoroughly. It should also be noted that the cuts (carats) of a precious ornamental stone are the representation of extent of the total internal reflection in the stone. The more the total internal reflection, the costlier the stones.