Question

Prism angle of a prism is $A$ and angle of minimum deviation is equal to prism angle. Refractive index of the material of prism will be:

• A. $2 \sin \dfrac{A}{2}$
• B. $2 \tan \dfrac{A}{2}$
• C. $2 \cos \dfrac{A}{2}$
• D. $\cot \dfrac{A}{2}$

Hint:

For prism refractive index: $\mu = \dfrac{\sin\left(\frac{A+D_m}{2}\right)}{\sin\left(\frac{A}{2}\right)}$. Substituting $D_m = A$ gives $\mu = 2 \cos \dfrac{A}{2}$.

Answer 1

The Correct Option is A

Step 1: Prism formula. Refractive index of prism material is given by: \[ \mu = \frac{\sin\left(\frac{A + D_m}{2}\right)}{\sin\left(\frac{A}{2}\right)} \] where $A$ = prism angle and $D_m$ = angle of minimum deviation. Step 2: Given condition. Here, $D_m = A$. \[ \mu = \frac{\sin\left(\frac{A + A}{2}\right)}{\sin\left(\frac{A}{2}\right)} = \frac{\sin(A)}{\sin\left(\frac{A}{2}\right)}. \] Step 3: Simplify using identity. \[ \sin(A) = 2 \sin\left(\frac{A}{2}\right)\cos\left(\frac{A}{2}\right). \] \[ \mu = \frac{2 \sin\left(\frac{A}{2}\right)\cos\left(\frac{A}{2}\right)}{\sin\left(\frac{A}{2}\right)} = 2 \cos\left(\frac{A}{2}\right). \] Wait carefully — correction! Actually: \[ \mu = \frac{\sin(A)}{\sin(A/2)} = 2 \cos(A/2). \] But from given options, the correct simplification matches (C). Correct Answer: (C) $2 \cos \dfrac{A}{2}$