A ray of light is incident at `60^(@)` on one face of a prism of angle `30^(@)` and the emergent ray makes `30^(@)` with the incident ray. The refractive index of the prism is

Options

A. 1.732
B. 1.414
C. 1.5
D. 1.33

Explanation

To find the refractive index of the prism given the incident angle, prism angle, and the angle between the incident and emergent rays, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Given Values:**
- Angle of incidence (I) = 60°
- Angle of the prism (A) = 30°
- Angle of deviation (D) = 30° (since the emergent ray makes an angle of 30° with the incident ray)

2. **Use the Angle of Deviation Formula:**
The angle of deviation (D) is related to the angles of incidence (I), emergence (E), and the prism angle (A) by the formula:
\[ D = I + E - A \]
Rearranging this gives:
\[ E = D + A - I \]

3. **Substitute the Known Values:**
Substitute the known values into the equation:
\[ E = 30° + 30° - 60° = 0° \]
This means that the angle of emergence is 0°, indicating that the light exits the prism parallel to the base.

4. **Apply Snell's Law at the First Surface:**
At the first surface of the prism, we can apply Snell's law:
\[ n_1 \sin(I) = n_2 \sin(R_1) \]
Where:
- \( n_1 \) = 1 (refractive index of air)
- \( n_2 \) = refractive index of the prism (μ)
- \( R_1 \) = angle of refraction at the first surface

5. **Determine the Angle of Refraction (R1):**
Since the angle of emergence (E) is 0°, the angle of refraction at the second surface (R2) is also 0°. Therefore, using the prism angle relation:
\[ R_1 + R_2 = A \]
Substituting \( R_2 = 0° \):
\[ R_1 + 0° = 30° \]
Thus, \( R_1 = 30° \).

6. **Substitute Values into Snell's Law:**
Now we can substitute the values into Snell's law:
\[ 1 \cdot \sin(60°) = μ \cdot \sin(30°) \]
We know:
- \( \sin(60°) = \frac{\sqrt{3}}{2} \)
- \( \sin(30°) = \frac{1}{2} \)

Therefore, we have:
\[ \frac{\sqrt{3}}{2} = μ \cdot \frac{1}{2} \]

7. **Solve for the Refractive Index (μ):**
Rearranging gives:
\[ μ = \frac{\sqrt{3}}{2} \cdot 2 = \sqrt{3} \]

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

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