Question

From the graph of angle of deviation versus angle of incidence for an equilateral prism, the refractive index of material of prism is

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

Hint:

For an equilateral prism, if the minimum deviation is also equal to the prism angle (\( \delta_m = A = 60^\circ \)), the refractive index will always evaluate to \( \sqrt{3} \). Note that in this state, the angle of incidence \( i \) also equals \( 60^\circ \).

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The question asks us to calculate the refractive index of a prism's material based on a given graph of angle of deviation (\( \delta \)) vs. angle of incidence (\( i \)) for an equilateral prism.
Step 2: Key Formula or Approach:
The refractive index (\( n \)) of a prism can be determined using the prism formula:
\[ n = \frac{\sin\left(\frac{A + \delta_m}{2}\right)}{\sin\left(\frac{A}{2}\right)} \]
Where:
- \( A \) is the angle of the prism.
- \( \delta_m \) is the angle of minimum deviation.
Step 3: Detailed Explanation:
1. Determine the Prism Angle (\( A \)):
An "equilateral prism" has all its internal angles equal to \( 60^\circ \). Therefore, the angle of the prism \( A = 60^\circ \).
2. Identify the Minimum Deviation (\( \delta_m \)) from the graph:
The graph shows how the deviation changes with the angle of incidence. The lowest point on the U-shaped curve represents the condition of minimum deviation.
Looking at the graph, the coordinates of the minimum point are \( (i, \delta) = (60^\circ, 60^\circ) \).
Thus, the angle of minimum deviation \( \delta_m = 60^\circ \).
3. Calculation of Refractive Index (\( n \)):
Substitute the values \( A = 60^\circ \) and \( \delta_m = 60^\circ \) into the formula:
\[ n = \frac{\sin\left(\frac{60^\circ + 60^\circ}{2}\right)}{\sin\left(\frac{60^\circ}{2}\right)} \]
\[ n = \frac{\sin(60^\circ)}{\sin(30^\circ)} \]
Using known trigonometric values: \( \sin(60^\circ) = \frac{\sqrt{3}}{2} \) and \( \sin(30^\circ) = \frac{1}{2} \).
\[ n = \frac{\sqrt{3}/2}{1/2} = \sqrt{3} \]
Step 4: Final Answer:
The refractive index of the prism material is \( \sqrt{3} \), which corresponds to option (C).