Question

The refracting angle of a prism is A and the refractive index of the material of the prism is $\cot \dfrac{A}{2}$. The angle of minimum deviation of the prism is:
A) $\pi + 2A$
B) $\pi - 2A$
C) $\dfrac{\pi }{2} + A$
D) $\dfrac{\pi }{2} - A$

Answer 1

Prism is a portion of refracting surface, bounded by three plane surfaces meeting along straight lines. Refractive index of the material of the prism is given by:
$\mu = \dfrac{{\sin \dfrac{{(A + {\delta _m})}}{2}}}{{\sin \dfrac{A}{2}}}$ ($\delta$is the angle of minimum deviation, A is the summation of the angle of refraction)
We are being given the refractive index in the question, using above relation we will calculate the angle of minimum deviation.

Complete step by step solution:
Let us discuss Prism first in more details and then we will calculate the angle of minimum deviation. A Prism is the portion of transparent refracting medium bounded by two plane surfaces meeting each other along a straight edge.
When refraction through Prism takes place a ray which passes through a prism, the sum of the angle of prism and the angle of deviation is equal to the sum of the angle of incidence and the angle emergence.
Now we the calculate the angle of minimum deviation:
In the question we are being provided that refractive index of the medium is given as:
$\mu = \cot \dfrac{A}{2}$ ...................(1)
In general refractive index of the prism is given by:
$\mu = \dfrac{{\sin \dfrac{{(A + {\delta _m})}}{2}}}{{\sin \dfrac{A}{2}}}$...............(2)
We will equate the two equations because both are refractive index:
$\Rightarrow \dfrac{{\sin \dfrac{{(A + {\delta _m})}}{2}}}{{\sin \dfrac{A}{2}}} = \cot \dfrac{A}{2}$
$\Rightarrow \dfrac{{\sin \dfrac{{(A + {\delta _m})}}{2}}}{{\sin \dfrac{A}{2}}} = \dfrac{{\cos \dfrac{A}{2}}}{{\sin \dfrac{A}{2}}}$(cotA is equal to the ratio of cosine to sine)
$\Rightarrow \sin \dfrac{{(A + {\delta _m})}}{2} = \cos \dfrac{A}{2}$(We have cancelled the common term of sinA/2)
$\Rightarrow \sin \dfrac{{(A + {\delta _m})}}{2} = \sin (\dfrac{\pi }{2} - \dfrac{A}{2})$
$\Rightarrow \dfrac{{A + {\delta _m}}}{2} = (\dfrac{\pi }{2} - \dfrac{A}{2})$ (removed the sine term)
$\Rightarrow {\delta _m} = \pi - 2A$

Thus, option (B) is correct.

Note: Prism has many applications such as: used in Telescopes, periscopes and microscopes, scientists also use prisms in experiments that help them study of physical optics uses the wave nature of light to understand such phenomenon of interference patterns caused by light waves passing through diffraction gratings and spectroscopy.