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. `30^(@),sqrt2`
B. `45^(@),sqrt2`
C. `30^(@),(1)/(sqrt2)`
D. `45^(@),(1)/(sqrt2)`

Explanation

To solve the problem, we will follow these steps:

### Step 1: Identify the given values
- Angle of incidence (i) = 45°
- Angle of prism (A) = 60°

### Step 2: Use the formula for minimum angle of deviation
The formula for minimum angle of deviation (D) in a prism is given by:
\[ D = 2i - A \]

### Step 3: Substitute the known values into the formula
Substituting the values we have:
\[ D = 2(45°) - 60° \]
\[ D = 90° - 60° \]
\[ D = 30° \]

### Step 4: Calculate the refractive index (μ)
The formula for the refractive index of the prism material is given by:
\[ \mu = \frac{\sin\left(\frac{A + D}{2}\right)}{\sin\left(\frac{A}{2}\right)} \]

### Step 5: Substitute the values into the refractive index formula
First, we need to find \( \frac{A + D}{2} \) and \( \frac{A}{2} \):
- \( A + D = 60° + 30° = 90° \)
- \( \frac{A + D}{2} = \frac{90°}{2} = 45° \)
- \( \frac{A}{2} = \frac{60°}{2} = 30° \)

Now substituting into the refractive index formula:
\[ \mu = \frac{\sin(45°)}{\sin(30°)} \]

### Step 6: Calculate the sine values
- \( \sin(45°) = \frac{1}{\sqrt{2}} \)
- \( \sin(30°) = \frac{1}{2} \)

### Step 7: Substitute the sine values into the refractive index formula
\[ \mu = \frac{\frac{1}{\sqrt{2}}}{\frac{1}{2}} \]
\[ \mu = \frac{1}{\sqrt{2}} \times 2 \]
\[ \mu = \frac{2}{\sqrt{2}} \]
\[ \mu = \sqrt{2} \]

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

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