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 solve the problem, we need to find the refractive index of the prism given the angle of incidence, the angle of the prism, and the angle of deviation. Here’s a step-by-step solution:

### Step 1: Identify the given values
- Angle of incidence (i) = 60°
- Angle of prism (A) = 30°
- Angle of deviation (D) = 30° (since the emergent ray makes an angle of 30° with the incident ray)

### Step 2: Use the formula relating the angles
The formula that relates the angles in a prism is:
\[ A + D = i + e \]
Where:
- A = angle of prism
- D = angle of deviation
- i = angle of incidence
- e = angle of emergence

### Step 3: Substitute the known values into the formula
Substituting the known values into the equation:
\[ 30° + 30° = 60° + e \]
This simplifies to:
\[ 60° = 60° + e \]

### Step 4: Solve for the angle of emergence (e)
From the equation:
\[ e = 60° - 60° = 0° \]

### Step 5: Use the relation between the angles of refraction and the angle of prism
We know that:
\[ r_1 + r_2 = A \]
Where:
- \( r_1 \) = angle of refraction at the first face
- \( r_2 \) = angle of refraction at the second face

Since \( e = 0° \), it implies that \( r_2 = 0° \). Thus:
\[ r_1 + 0° = 30° \]
So, \( r_1 = 30° \).

### Step 6: Apply Snell's law at the first face
Using Snell's law at the first face of the prism:
\[ n = \frac{\sin(i)}{\sin(r_1)} \]
Where \( n \) is the refractive index of the prism.

Substituting the known values:
\[ n = \frac{\sin(60°)}{\sin(30°)} \]

### Step 7: Calculate the sine values
We know:
- \( \sin(60°) = \frac{\sqrt{3}}{2} \)
- \( \sin(30°) = \frac{1}{2} \)

### Step 8: Substitute the sine values into the equation
Now substituting the sine values:
\[ n = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} \]
This simplifies to:
\[ n = \sqrt{3} \]

### Conclusion
The refractive index of the prism is:
\[ n \approx 1.732 \]