Question

Angle of minimum deviation is equal to half of the angle of prism in an equilateral prism. The refractive index of the prism is _____.

• A. \(1.5\)
• B. \( \sqrt{3} \)
• C. \( \sqrt{2} \)
• D. \(1.65\)

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We need to find the refractive index (\(\mu\)) of an equilateral prism given a relationship between the angle of prism (\(A\)) and the angle of minimum deviation (\(\delta_m\)).
Step 2: Key Formula or Approach:
Prism formula for refractive index:
\[ \mu = \frac{\sin\left(\frac{A + \delta_m}{2}\right)}{\sin\left(\frac{A}{2}\right)} \]
Step 3: Detailed Explanation:
1. For an equilateral prism, the angle of the prism \(A = 60^{\circ}\).
2. According to the problem, \(\delta_m = \frac{A}{2} = \frac{60^{\circ}}{2} = 30^{\circ}\).
3. Substituting these values into the prism formula:
\[ \mu = \frac{\sin\left(\frac{60^{\circ} + 30^{\circ}}{2}\right)}{\sin\left(\frac{60^{\circ}}{2}\right)} = \frac{\sin(45^{\circ})}{\sin(30^{\circ})} \]
4. Using trigonometric values \(\sin(45^{\circ}) = \frac{1}{\sqrt{2}}\) and \(\sin(30^{\circ}) = \frac{1}{2}\):
\[ \mu = \frac{1/\sqrt{2}}{1/2} = \frac{2}{\sqrt{2}} = \sqrt{2} \]
Step 4: Final Answer:
The refractive index of the material of the prism is \(\sqrt{2} \approx 1.414\).