If the minimum deviation produced is `30^@` for an equilateral prism, then angle of emergences

Options

A. `15^@`
B. `30^@`
C. `45^@`
D. `60^@`

Answer

To solve the problem of finding the angle of emergence for an equilateral prism with a minimum deviation of \(30^\circ\), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Minimum deviation (\( \delta_m \)) = \(30^\circ\)
- Angle of the prism (\(A\)) for an equilateral prism = \(60^\circ\)

2. **Use the Formula for Prism:**
The relationship between the angle of incidence (\(I\)), angle of emergence (\(E\)), angle of the prism (\(A\)), and minimum deviation (\(\delta_m\)) is given by the formula:
\[ I + E = A + \delta_m \]

3. **Recognize the Symmetry in Equilateral Prism:**
For an equilateral prism, the angle of incidence (\(I\)) is equal to the angle of emergence (\(E\)) when minimum deviation occurs. Therefore, we can set:
\[ I = E \]

4. **Substitute \(I\) with \(E\) in the Formula:**
Replacing \(I\) with \(E\) in the formula gives:
\[ E + E = A + \delta_m \]
This simplifies to:
\[ 2E = A + \delta_m \]

5. **Plug in the Values:**
Substitute the values of \(A\) and \(\delta_m\):
\[ 2E = 60^\circ + 30^\circ \]
\[ 2E = 90^\circ \]

6. **Solve for \(E\):**
Divide both sides by 2 to find \(E\):
\[ E = \frac{90^\circ}{2} = 45^\circ \]

7. **Conclusion:**
The angle of emergence (\(E\)) is \(45^\circ\).

### Final Answer:
The angle of emergence is \(45^\circ\).

---