a prism of refractive index `sqrt2` refracting angle A produces minimum deviation `D_m` of a ray on one face at an angle of incidence `45^@` . The values of A and `D_m` are respectively.

Options

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

Answer

To solve the problem, we need to find the refracting angle \( A \) of the prism and the minimum deviation \( D_m \) when a ray of light passes through the prism at an angle of incidence of \( 45^\circ \) and the refractive index \( \mu \) of the prism is \( \sqrt{2} \).

### Step-by-Step Solution:

1. **Understanding the Minimum Deviation Condition**:
- At minimum deviation, the angle of incidence \( I \) is equal to the angle of emergence \( E \).
- Therefore, we have \( I = E \).

2. **Using the Prism Formula**:
- The relationship between the angles in a prism is given by:
\[ A = r_1 + r_2 \]
- At minimum deviation, \( r_1 = r_2 = \frac{A}{2} \).
- Thus, we can rewrite the equation as:
\[ A = 2r_1 \]

3. **Applying Snell's Law**:
- According to Snell's Law:
\[ \mu_1 \sin I = \mu_2 \sin r \]
- Here, \( \mu_1 = 1 \) (refractive index of air), \( \mu_2 = \sqrt{2} \), \( I = 45^\circ \), and \( r = r_1 \).
- Plugging in these values:
\[ 1 \cdot \sin(45^\circ) = \sqrt{2} \cdot \sin(r_1) \]
- Since \( \sin(45^\circ) = \frac{1}{\sqrt{2}} \), we have:
\[ \frac{1}{\sqrt{2}} = \sqrt{2} \cdot \sin(r_1) \]

4. **Solving for \( r_1 \)**:
- Rearranging the equation gives:
\[ \sin(r_1) = \frac{1}{2} \]
- Therefore, \( r_1 = \sin^{-1}\left(\frac{1}{2}\right) = 30^\circ \).

5. **Finding the Angle of Prism \( A \)**:
- Now substituting \( r_1 \) back into the equation for \( A \):
\[ A = 2r_1 = 2 \times 30^\circ = 60^\circ \]

6. **Calculating Minimum Deviation \( D_m \)**:
- The formula for the deviation \( D \) in a prism is:
\[ D = I + E - A \]
- For minimum deviation, since \( I = E \):
\[ D_m = 2I - A \]
- Substituting \( I = 45^\circ \) and \( A = 60^\circ \):
\[ D_m = 2 \times 45^\circ - 60^\circ = 90^\circ - 60^\circ = 30^\circ \]

### Final Answers:
- The refracting angle \( A \) is \( 60^\circ \).
- The minimum deviation \( D_m \) is \( 30^\circ \).