Question

The angel of deviation by a prism is $({180^0} - 2A)$. It's critical angel will be
A. ${\sin ^{^{ - 1}}}(\tan \dfrac{A}{2})$
B. $(\cot \dfrac{A}{2})$
C. ${\cos ^{ - 1}}(\cot \dfrac{A}{2})$
D.${\cos ^{ - 1}}(\tan \dfrac{A}{2})$

Answer 1

The angle of deviation is the measure of the deviation of the incident ray from its actual path. The critical angle is the angle of incidence in the denser medium for which the angle of incidence is ${90^o}$in the rarer medium. The relationship between the refractive index and the sine of the critical angle has to be used here. Also, the refractive index can be written in terms of the angle of deviation.

Complete step by step solution:
The angle of deviation is $\left( {180 - 2A} \right)$. We have to calculate the critical angle.
We have the following relation to calculate the critical angle.
$\mu = \dfrac{1}{{\sin {i_c}}}$ (1)
Here, $\mu$ is the refractive index and ${i_c}$ is the critical angle.
Also, we have another relation for finding the refractive index.
$\mu = \dfrac{{\sin \left( {\dfrac{{A + {\delta _m}}}{2}} \right)}}{{\sin \dfrac{A}{2}}}$ (2)
Here, $A$ is the angle of the prism and ${\delta _m}$is the angle of deviation.
Now, equating equations (1) and (2).
$\dfrac{1}{{\sin {i_c}}} = \dfrac{{\sin \left( {\dfrac{{A + {\delta _m}}}{2}} \right)}}{{\sin \dfrac{A}{2}}}$
Let us now substitute the values in the above equation.
$\dfrac{1}{{\sin {i_c}}} = \dfrac{{\sin \left( {\dfrac{{A + \left( {180 - 2A} \right)}}{2}} \right)}}{{\sin \dfrac{A}{2}}} = \dfrac{{\cos \dfrac{A}{2}}}{{\sin \dfrac{A}{2}}}$
Let us simplify the above expression.
$\dfrac{1}{{\sin {i_c}}} = \cot \dfrac{A}{2} \Rightarrow \sin {i_c} = \dfrac{1}{{\cot \dfrac{A}{2}}} = \tan \dfrac{A}{2}$
On further simplifying to match the option we get the following.
${i_c} = {\sin ^{ - 1}}\left( {\tan \dfrac{A}{2}} \right)$
Hence, the correct option is (A) ${\sin ^{ - 1}}\left( {tan\dfrac{A}{2}} \right)$.

Note:
Prism is made of three transparent planes, of which one forms the base and the other two are inclined to each other. The angle at which these two surfaces meet is called the angle of the prism (A).
When the prism is adjusted to the minimum deviation, the angle of incidence is equal to the angle of emergence.
The refractive index is the ratio of the speed of light in a vacuum or air and the speed in that medium. Using Snell’s law we can calculate the refractive index if the angle of incidence and angle of refraction is known.