When light rays are incident on a prism at an angle of `45^(@)`, the minimum deviation is obtained. If refractive index of the material of prism is `sqrt(2)`, then the angle of prism will be

Options

A. `30^(@)`
B. `75^(@)`
C. `90^(@)`
D. `60^(@)`

Explanation

To find the angle of the prism when light rays are incident at an angle of \(45^\circ\) and the refractive index of the material of the prism is \(\sqrt{2}\), we can follow these steps:

### Step 1: Understand the relationship between the angles
In the case of minimum deviation, the angle of incidence \(i\) is equal to the angle of emergence \(e\). The relationship is given by:
\[ i = \frac{A + \delta_m}{2} \]
where \(A\) is the angle of the prism and \(\delta_m\) is the minimum deviation.

### Step 2: Substitute the known values
We know that \(i = 45^\circ\). Therefore, we can write:
\[ 45^\circ = \frac{A + \delta_m}{2} \]
This implies:
\[ A + \delta_m = 90^\circ \quad \text{(Equation 1)} \]

### Step 3: Use the refractive index formula
The refractive index \(\mu\) is given by:
\[ \mu = \frac{\sin(i + \frac{A}{2})}{\sin(\frac{A}{2})} \]
Substituting \(\mu = \sqrt{2}\) and \(i = 45^\circ\), we can rewrite the equation as:
\[ \sqrt{2} = \frac{\sin(45^\circ + \frac{A}{2})}{\sin(\frac{A}{2})} \]

### Step 4: Calculate \(\sin(45^\circ)\)
We know that:
\[ \sin(45^\circ) = \frac{1}{\sqrt{2}} \]
Thus, the equation becomes:
\[ \sqrt{2} = \frac{\frac{1}{\sqrt{2}} + \sin(\frac{A}{2})}{\sin(\frac{A}{2})} \]

### Step 5: Simplify the equation
Cross-multiplying gives:
\[ \sqrt{2} \sin(\frac{A}{2}) = \frac{1}{\sqrt{2}} + \sin(\frac{A}{2}) \]
Rearranging this gives:
\[ \sqrt{2} \sin(\frac{A}{2}) - \sin(\frac{A}{2}) = \frac{1}{\sqrt{2}} \]
Factoring out \(\sin(\frac{A}{2})\):
\[ (\sqrt{2} - 1) \sin(\frac{A}{2}) = \frac{1}{\sqrt{2}} \]

### Step 6: Solve for \(\sin(\frac{A}{2})\)
Thus,
\[ \sin(\frac{A}{2}) = \frac{1}{\sqrt{2}(\sqrt{2} - 1)} \]
To simplify \(\sqrt{2}(\sqrt{2} - 1)\):
\[ \sqrt{2}(\sqrt{2} - 1) = 2 - \sqrt{2} \]
So,
\[ \sin(\frac{A}{2}) = \frac{1}{2 - \sqrt{2}} \]

### Step 7: Find the angle \(\frac{A}{2}\)
Using the known values, we can find \(\frac{A}{2}\):
\[ \sin(\frac{A}{2}) = \frac{1}{2 - \sqrt{2}} \implies \frac{A}{2} = 30^\circ \]
Thus, multiplying by 2 gives:
\[ A = 60^\circ \]

### Conclusion
The angle of the prism \(A\) is \(60^\circ\).