Question

If the refractive index of the material of a prism is $cot\bigg(\frac{A}{2}\bigg)$, where A is the angle of the prism, then the angle of minimum deviation will be:

• A. $π-3A$
• B. $π-2A$
• C. $A$
• D. $\frac{A}{2}$

Answer 1

Answer: (B)

$π-2A$

Answer 2

To find the angle of minimum deviation $\delta_{\text{min}}$:

Step 1. Given Relation:
$\cot \frac{A}{2} = \frac{\sin \frac{A + \delta_{\text{min}}}{2}}{\sin \frac{A}{2}}$

Step 2. Rearrange and Simplify: Take the cosine of both sides:
$\cos \frac{A}{2} = \sin \frac{A + \delta_{\text{min}}}{2}$

Step 3. Solve for $\delta_{\text{min}}$: Equate the arguments, giving:
$\frac{A + \delta_{\text{min}}}{2} = \frac{\pi}{2} - \frac{A}{2}$

- Solving, we get:
$\delta_{\text{min}} = \pi - 2A$

Thus, the angle of minimum deviation is $\pi - 2A$.