The angle of incidence for a ray of light at a refracting surface of a prism is `45^(@)`. The angle of prism is `60^(@)`. If the ray suffers minimum deviation through the prism, the angle of minimum deviation and refractive index of the material of the prism respectively, are `:`

Options

A. `45^(@), (1)/(sqrt(2))`
B. `30^(@), sqrt(2)`
C. `45^(@), sqrt(2)`
D. `30^(@), (1)/(sqrt(2))`

Explanation

To solve the problem, we need to determine the angle of minimum deviation (Δm) and the refractive index (μ) of the prism given the angle of incidence (i) and the angle of the prism (A).

### Step-by-Step Solution:

1. **Given Values**:
- Angle of incidence (i) = 45°
- Angle of prism (A) = 60°

2. **Using the Formula for Minimum Deviation**:
When a ray of light passes through a prism and suffers minimum deviation, the relationship between the angle of prism (A), angle of incidence (i), and angle of minimum deviation (Δm) is given by:
\[ A + Δm = 2i \]

3. **Substituting the Known Values**:
Substitute the known values into the equation:
\[ 60° + Δm = 2 \times 45° \]
\[ 60° + Δm = 90° \]

4. **Solving for Δm**:
Rearranging the equation to find Δm:
\[ Δm = 90° - 60° = 30° \]

5. **Calculating the Refractive Index (μ)**:
The refractive index (μ) can be calculated using the formula:
\[ μ = \frac{\sin\left(\frac{A + Δm}{2}\right)}{\sin\left(\frac{A}{2}\right)} \]
First, calculate \( \frac{A + Δm}{2} \) and \( \frac{A}{2} \):
- \( A + Δm = 60° + 30° = 90° \)
- \( \frac{A + Δm}{2} = \frac{90°}{2} = 45° \)
- \( \frac{A}{2} = \frac{60°}{2} = 30° \)

6. **Substituting into the Refractive Index Formula**:
Now substitute these values into the formula:
\[ μ = \frac{\sin(45°)}{\sin(30°)} \]

7. **Using Known Values of Sine**:
- \( \sin(45°) = \frac{1}{\sqrt{2}} \)
- \( \sin(30°) = \frac{1}{2} \)

8. **Calculating μ**:
Substitute the sine values:
\[ μ = \frac{\frac{1}{\sqrt{2}}}{\frac{1}{2}} = \frac{1}{\sqrt{2}} \times 2 = \frac{2}{\sqrt{2}} = \sqrt{2} \]

### Final Results:
- Angle of minimum deviation (Δm) = 30°
- Refractive index (μ) = √2