Question

A thin hollow prism of refracting angle ${{3}^{\circ }}$, filled with water gives a deviation of ${{1}^{\circ }}$. The refractive index of water is?
$\begin{align}
  & \text{A}\text{. }1.59 \\
 & \text{B}\text{. }1.33 \\
 & \text{C}\text{. }1.46 \\
 & \text{D}\text{. }1.51 \\
\end{align}$

Answer 1

Hint: In a hollow prism, we can fill the cavity by adding any gas or other liquids. The refractive index of the fluid becomes the refractive index of the prism and we can apply the equations of the angle of deviation and refracting angle of the prism in this case. It should be kept in mind that the given prism is a thin hollow prism, or a small angled prism, and equations should be applied accordingly.
Formula used:
For refraction through a small angled prism:
$\delta =\left( \mu -1 \right)A$

Complete step by step solution:
An optical prism is defined as a transparent optical element with flat and polished surfaces that refract light. The two surfaces in a prism cannot be parallel to each other, one surface must be angled.
A hollow prism is a prism made using a glass plate with the prism’s centre part as empty. This cavity can be filled with gas or other liquids. Thus, a hollow prism is also called a fluid prism.

For refraction through a small angled prism, the angle of deviation is given by:
$\delta =\left( \mu -1 \right)A$
Where,
$\delta $is the angle of deviation
$\mu $is the refractive index
$A$is the refracting angle of prism
The angle of deviation in a small angled prism is directly proportional to the refracting angle of the prism. Also, the angle of deviation increases as the refractive index of the prism, or the fluid in case of hollow prism, increases.
We are given a thin hollow prism of refracting angle${{3}^{\circ }}$, filled with water and it gives a deviation of${{1}^{\circ }}$.
We have,
$\begin{align}
  & A={{3}^{\circ }} \\
 & \delta ={{1}^{\circ }} \\
\end{align}$
We have to calculate the refractive index of water, say $\mu $
For small angle of prism$\left( A<{{10}^{\circ }} \right)$
Relation between deviation angle and angle of prism (refracting angle) is given as,
$\delta =\left( \mu -1 \right)A$
Or,
$\begin{align}
  & \mu -1=\dfrac{\delta }{A} \\
 & \mu =1+\dfrac{\delta }{A} \\
\end{align}$
Putting values of $\delta $and $A$
$\begin{align}
  & A={{3}^{\circ }} \\
 & \delta ={{1}^{\circ }} \\
\end{align}$
$\begin{align}
  & \mu =1+\dfrac{1}{3}=\dfrac{4}{3} \\
 & \mu =1.33 \\
\end{align}$
The refractive index of water is$1.33$
Hence, the correct option is B.

Note:
In case of a hollow or fluid prism, the refractive index of prism is actually the refractive index of the fluid which is filled inside the prism. The above mentioned relation between angle of deviation and refracting angle of prism is valid only for the small angled prism, also called thin prism.