A ray of light is incident at an angle of `60^(@)` on the face of a prism with an angle of `60^(@)`. Then the refractive index of the material of the prism is (the prism is in minimum deviation position)

Options

A. 1.414
B. 1.623
C. 1.524
D. 1.732

Answer

To find the refractive index of the material of the prism when the prism is in the minimum deviation position, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Angle of incidence (I) = 60°
- Angle of the prism (A) = 60°
- The prism is in minimum deviation position.

2. **Use the Formula for Minimum Deviation:**
- The formula for minimum deviation (Δ) in terms of angle of incidence (I) and angle of the prism (A) is:
\[ \Delta_{min} = 2I - A \]

3. **Substitute the Known Values:**
- Substitute I = 60° and A = 60° into the formula:
\[ \Delta_{min} = 2 \times 60° - 60° = 120° - 60° = 60° \]

4. **Use the Refractive Index Formula:**
- The formula relating the refractive index (μ) to the angles is:
\[ \mu = \frac{\sin\left(\frac{A + \Delta_{min}}{2}\right)}{\sin\left(\frac{A}{2}\right)} \]

5. **Calculate the Angles:**
- Calculate \( \frac{A + \Delta_{min}}{2} \):
\[ \frac{A + \Delta_{min}}{2} = \frac{60° + 60°}{2} = \frac{120°}{2} = 60° \]
- Calculate \( \frac{A}{2} \):
\[ \frac{A}{2} = \frac{60°}{2} = 30° \]

6. **Substitute into the Refractive Index Formula:**
- Substitute the calculated angles into the refractive index formula:
\[ \mu = \frac{\sin(60°)}{\sin(30°)} \]

7. **Use Known Values of Sine:**
- We know that:
- \( \sin(60°) = \frac{\sqrt{3}}{2} \)
- \( \sin(30°) = \frac{1}{2} \)

8. **Calculate the Refractive Index:**
- Substitute the sine values:
\[ \mu = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \frac{\sqrt{3}}{2} \times \frac{2}{1} = \sqrt{3} \]

### Final Answer:
The refractive index of the material of the prism is \( \sqrt{3} \).

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