Ray Optics and Optical Instruments - Result Question 55

57. The refracting angle of a prism is ’ $A$ ‘, and refractive index of the material of the prism is $\cot (A / 2)$. The angle of minimum deviation is :

[2015]

(a) $180^{\circ}-2 A$

(b) $90^{\circ}-A$

(c) $180^{\circ}+2 A$

(d) $180^{\circ}-3 A$

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Answer:

Correct Answer: 57. (a)

Solution:

  1. (a) As we know, the refractive index of the material of the prism

$ \mu=\frac{\sin (\frac{\delta_m+A}{2})}{\sin (A / 2)} $

$ \begin{aligned} & \cot A / 2=\frac{\sin (\frac{A+\delta_m}{2})}{\sin A / 2}=\frac{\cos (A / 2)}{\sin (A / 2)} \\ & {[\because \mu=\cot (A / 2)]} \end{aligned} $

$\Rightarrow \quad \sin (\frac{\delta_m+A}{2})=\sin (90^{\circ}+A / 2)$

$\Rightarrow \delta _{\text{min }}=180^{\circ}-2 A$