A ray of light passes through an equilateral prism such that the angle of incidence is equal of emergence and later is equal to 3/4th the angle of prism. The angle of deviation is

Options

A. `45^(@)`
B. `20^(@)`
C. `39^(@)`
D. `30^(@)`

Answer

To solve the problem, we need to find the angle of deviation when a ray of light passes through an equilateral prism under the given conditions. Let's break down the steps:

### Step-by-Step Solution:

1. **Identify the Given Information:**
- The prism is equilateral, so the angle \( A \) of the prism is \( 60^\circ \).
- The angle of incidence \( i \) is equal to the angle of emergence \( e \).
- The angle of incidence and emergence is \( \frac{3}{4} \) of the angle of the prism.

2. **Calculate the Angle of Incidence and Emergence:**
- Since \( A = 60^\circ \), we can find \( i \) and \( e \):
\[ i = e = \frac{3}{4} \times A = \frac{3}{4} \times 60^\circ = 45^\circ \]

3. **Use the Relation for Minimum Deviation:**
- The relation for minimum deviation is given by:
\[ i + e = A + \delta_m \]
- Substituting the known values:
\[ 45^\circ + 45^\circ = 60^\circ + \delta_m \]

4. **Simplify the Equation:**
- Combine the left side:
\[ 90^\circ = 60^\circ + \delta_m \]

5. **Solve for the Angle of Deviation \( \delta_m \):**
- Rearranging gives:
\[ \delta_m = 90^\circ - 60^\circ = 30^\circ \]

6. **Conclusion:**
- The angle of deviation \( \delta_m \) is \( 30^\circ \).

### Final Answer:
The angle of deviation is \( 30^\circ \).