Question

For the prism of refractive index 1.732, the angle of minimum deviation is equal to the angle of the prism. The angle of the prism is
A. ${80^0}$
B. ${70^0}$
C. ${60^0}$
D. ${50^0}$

Answer 1

Hint: The angle between its two lateral faces is called the angle of the prism or the prism angle. Use the expression for the refractive index of the prism in terms of angle of prism and angle of deviation. Substitute the angle of deviation equal to angle of prism and solve the equation to get the angle of prism.

Complete step by step answer:
We know that the refractive index $\left( \pi \right)$of a prism is related to the prism angle $\left( A \right)$and the minimum deviation angle $\left( {Dm} \right)$by the relation,
$\pi = \dfrac{{\sin \dfrac{{A + Dm}}{2}}}{{\sin \dfrac{A}{2}}}$
Given that the minimum deviation angle and the prism angle are equal. i.e., $A = Dm$
And, refractive index of the prism is $\pi = 1.732$
Thus,
$1.732 = \dfrac{{\sin A}}{{\sin \left( {A/2} \right)}}$

we can express $\sin A$, as $\sin A = 2\sin \left( {\dfrac{A}{2}} \right)\cos \left( {\dfrac{A}{2}} \right)$. Therefore, we can write the above equation as,
$ \Rightarrow 1.732 = \dfrac{{2\sin \left( {A/2} \right)\cos \left( {A/2} \right)}}{{\sin \left( {A/2} \right)}}$
$ \Rightarrow 1.732 = 2\cos \left( {A/2} \right)$
$ \Rightarrow \cos \left( {A/2} \right) = 0.866$
$ \therefore A = 60^\circ $

So, the correct answer is option C.

Additional Information:
The angle between its two lateral faces is called the angle of the prism or the prism angle. When the light ray is allowed to pass through the prism, it makes the emergent ray bend at an angle to the direction of the incident ray. This angle is called the angle of deviation for the prism.

The change in direction or bending of a light wave passing, from one transparent medium to another caused by the change in waves speed is the refraction. The extent of bending of light rays entering from one medium to another is the refractive index and is denoted by the $'n'$.It is represented as $n = \dfrac{c}{v}$, where, c is the speed of light in the vacuum and v is the speed of light in the given medium.

Note:Angle of prism is the angle between the two surfaces of the prism from which the light enters into the prism and from which the light goes out of refraction. The angle deviation of the light from the prism depends on the refractive index of the prism. Students often incorrectly express $\sin A$ as, $\sin A = 2\sin A\cos A$. The correct way to express $\sin A$ is given in the solution.