Question

The refracting angle of a prism is A, and refractive index of the material of the prism is cot $\left(\frac{A}{2}\right)$. The angle of minimum deviation is :

• A. 180⁰ + 2A
• B. 180⁰ – 3A
• C. 180⁰ – 2A
• D. 90⁰– A

Answer 1

Answer: (C)

. 180⁰ – 2A.

The refractive index of the material of a prism is given by,

$\mu =\frac{sin\left(\frac{A+\delta m}{2}\right)}{sin\frac{A}{2}}$

$\mu =cot\frac{A}{2}=\frac{sin\left(\frac{A+\delta m}{2}\right)}{sin\frac{A}{2}}$

$\therefore \frac{cos\left(\frac{A}{2}\right)}{sin\left(\frac{A}{2}\right)}=\frac{sin\left(\frac{A+\delta m}{2}\right)}{sin\frac{A}{2}}$
$\therefore cos\frac{A}{2}=sin\left(\frac{A+\delta m}{2}\right)$

$\therefore sin(90-\frac{A}{2})=sin\left(\frac{A+\delta m}{2}\right)$

$\therefore {90}^{\circ }-\frac{A}{2}=\frac{A+\delta m}{2}$
$\therefore {180}^{\circ }-A=A+\delta m$
$\therefore \delta m={180}^{\circ }-2A$