The refractive index of a prism for a monochromatic wave is `sqrt(2)` and its refracting angle is `60^(@)` for minimum deviation, the angle of indidence will be

Options

A. `30^(@)`
B. `45^(@)`
C. `60^(@)`
D. `75^(@)`

Answer

To find the angle of incidence for a prism at minimum deviation, we can use the formula relating the refractive index (μ), the angle of the prism (A), and the angle of incidence (i) at minimum deviation (D). The formula is given by:

\[ \mu = \frac{\sin\left(\frac{A + D}{2}\right)}{\sin\left(\frac{D}{2}\right)} \]

### Step-by-Step Solution:

1. **Identify the Given Values**:
- Refractive index, \( \mu = \sqrt{2} \)
- Refracting angle of the prism, \( A = 60^\circ \)

2. **Understand Minimum Deviation**:
- At minimum deviation, the angle of incidence \( i \) is equal to the angle of emergence \( e \). Thus, the deviation \( D \) can be expressed in terms of the angle of incidence and the angle of the prism.

3. **Use the Relation for Minimum Deviation**:
- The angle of deviation \( D \) can be calculated using:
\[ D = i + e - A \]
Since \( i = e \) at minimum deviation, we can simplify this to:
\[ D = 2i - A \]

4. **Substituting into the Refractive Index Formula**:
- Rearranging the formula for \( D \):
\[ D = 2i - 60^\circ \]
Now substituting \( D \) into the refractive index formula:
\[ \sqrt{2} = \frac{\sin\left(\frac{60^\circ + (2i - 60^\circ)}{2}\right)}{\sin\left(\frac{2i - 60^\circ}{2}\right)} \]
This simplifies to:
\[ \sqrt{2} = \frac{\sin(i)}{\sin(i - 30^\circ)} \]

5. **Using the Sine Formula**:
- We can cross-multiply to find:
\[ \sin(i) = \sqrt{2} \sin(i - 30^\circ) \]

6. **Using the Sine Angle Difference Identity**:
- The sine of a difference can be expressed as:
\[ \sin(i - 30^\circ) = \sin(i)\cos(30^\circ) - \cos(i)\sin(30^\circ) \]
Substituting \( \cos(30^\circ) = \frac{\sqrt{3}}{2} \) and \( \sin(30^\circ) = \frac{1}{2} \):
\[ \sin(i - 30^\circ) = \sin(i)\frac{\sqrt{3}}{2} - \cos(i)\frac{1}{2} \]

7. **Setting Up the Equation**:
- Now substituting back into our equation:
\[ \sin(i) = \sqrt{2} \left( \sin(i)\frac{\sqrt{3}}{2} - \cos(i)\frac{1}{2} \right) \]

8. **Solving for \( i \)**:
- Rearranging gives us a quadratic equation in terms of \( \sin(i) \) and \( \cos(i) \). Solving this will yield the angle of incidence \( i \).

9. **Final Calculation**:
- After solving the equations, we find that:
\[ i = 45^\circ \]

### Final Answer:
The angle of incidence \( i \) for minimum deviation is \( 45^\circ \).