Question

The least angle of deviation for a glass prism is equal to its refracting angle. The refractive index of glass is $1.5$ The angle of the prism is:
A. $2{\cos ^{ - 1}}(\dfrac{3}{4})$
B. ${\sin ^{ - 1}}(\dfrac{3}{4})$
C. ${\sin ^{ - 1}}(\dfrac{3}{2})$
D. ${\cos ^{ - 1}}(\dfrac{3}{2})$

Answer 1

A glass prism is an optic device having three planes, one is called the base and the other two are inclined to each other at some angle called the angle of prism. When white light is passed through a prism, it disperses the light into different colors. A rainbow is the result of the dispersion of light by the water droplets present in the atmosphere.

Complete step by step solution:
The relation between refractive index and angle of deviation is given by the following formula.
$\mu = \dfrac{{\sin \left( {\dfrac{{A - \left( {180 - {\delta _m}} \right)}}{2}} \right)}}{{\sin \dfrac{A}{2}}}$
Here, $\mu$ is the refractive index, $A$ is the angle of the prism, and ${\delta _m}$ is the angle of deviation.
It is given in the question that the least angle of deviation is equal to its refracting angle.
$A = {\delta _m}$
Let us put this value in the above equation and we get the following.
$\mu = \dfrac{{\sin \left( {\dfrac{{A + A}}{2}} \right)}}{{\sin \dfrac{A}{2}}} \Rightarrow \mu = \dfrac{{\sin A}}{{\sin \dfrac{A}{2}}}$
Let us further simplify it.
$\mu = \dfrac{{2\sin \dfrac{A}{2}\cos \dfrac{A}{2}}}{{\sin \dfrac{A}{2}}} = 2\cos \dfrac{A}{2}$
Now, let us simplify and find the value of $A$ in the above expression.
$\cos \dfrac{A}{2} = \dfrac{\mu }{2} \Rightarrow A = 2{\cos ^{ - 1}}\dfrac{\mu }{2}$
A Refractive index is given in the question $1.5$ , but this value is in the above expression.
$A = 2{\cos ^{ - 1}}\dfrac{{1.5}}{2} = 2{\cos ^{ - 1}}\left( {\dfrac{3}{4}} \right)$
Hence, the correct option is (A) $2{\cos ^{ - 1}}\left( {\dfrac{3}{4}} \right)$.

Note:
Dispersion occurs because the different colors of light travel at different speeds in the glass.
When white light enters the prism, the angle of emergence of different colors of light is different as the bending angle is different for a different color of light and thus we can see all the components of white light separately.