Question

The refracting angle of a prism is A, and refractive index of the material of the prism is $\cot \left( {A/2} \right).$ The angle of minimum deviation is:
A.${90^0} - A$
B.${180^0} + 2A$
C.${180^0} - 3A$
D.${180^0} - 2A$

Answer 1

This is the special case of prism in which we have to calculate the minimum deviation in this case angle of incidence is equal to the angle of emergence .We will use a formula for calculating the minimum deviation .
Formula used:
$\mu = \dfrac{{\sin \left( {\dfrac{{{\delta _m} + A}}{2}} \right)}}{{\sin \dfrac{A}{2}}}$
Where
$\mu =$ Refractive index of the prism
${\delta _m} =$ Angle of minimum deviation
$A =$ Angle of prism
It is given in the question that the refractive index of the prism is $\cot \left( {A/2} \right).$ And the angle of the prism is A.
As we know that refractive index is given by
$\mu = \dfrac{{\sin \left( {\dfrac{{{\delta _m} + A}}{2}} \right)}}{{\sin \dfrac{A}{2}}}$
Now, substituting these values in the above formula
$\Rightarrow \cot \dfrac{A}{2} = \dfrac{{\sin \left( {\dfrac{{{\delta _m} + A}}{2}} \right)}}{{\sin \dfrac{A}{2}}} \\\ \Rightarrow \dfrac{{\cos \dfrac{A}{2}}}{{\sin \dfrac{A}{2}}} = \dfrac{{\sin \left( {\dfrac{{{\delta _m} + A}}{2}} \right)}}{{\sin \dfrac{A}{2}}} \\\ \Rightarrow \cos \dfrac{A}{2} = \sin \left( {\dfrac{{{\delta _m} + A}}{2}} \right) \\\$
After further simplification, we will get
$\Rightarrow \sin \left( {\dfrac{\pi }{2} - \dfrac{A}{2}} \right) = \sin \left( {\dfrac{{{\delta _m} + A}}{2}} \right) \\\ \Rightarrow \dfrac{\pi }{2} - \dfrac{A}{2} = \dfrac{{{\delta _m} + A}}{2} \\\ \Rightarrow \pi - 2A = {\delta _m} \\\ \Rightarrow {\delta _m} = \pi - 2A \\\$

Hence the correct option is “D”.

Additional Information:
Refractive Index (Index of Refraction) is a value measured from the ratio of light velocity in a vacuum to that of a second greater density medium. A prism is an optical translucent element with smooth, polished surfaces refracting the light. At least one surface must be angled and elements with two parallel surfaces are not prisms. The typical geometrical shape of an optical prism is that of a triangular prism with a triangular base and rectangular sides.

Note:
If this prism would have been in water, oil or any other medium then we would have to take the relative refractive index of prism with respect to that water or oil. If light inside the prism strikes one of the surfaces at a sufficiently steep angle, there is total inner reflection and all of the light is reflected