Question

The angle of minimum deviation for a prism of angle \(A\) is \(180 - 2A\). The refractive index is

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

Hint:

Use \(\sin(90^\circ - \theta) = \cos\theta\) and \(\cos\theta/\sin\theta = \cot\theta\) to simplify the expression.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
\(n = \dfrac{\sin\left(\dfrac{A + D_m}{2}\right)}{\sin\left(\dfrac{A}{2}\right)}\).
Step 2: Detailed Explanation:
\(D_m = 180 - 2A\)
\[ \frac{A + D_m}{2} = \frac{A + 180 - 2A}{2} = \frac{180 - A}{2} = 90 - \frac{A}{2} \] \[ n = \frac{\sin\left(90 - \frac{A}{2}\right)}{\sin\frac{A}{2}} = \frac{\cos\frac{A}{2}}{\sin\frac{A}{2}} = \cot\frac{A}{2} \]
Step 3: Final Answer:
Refractive index \(n = \mathbf{\cot\dfrac{A}{2}}\).