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 of finding the refracting angle \( A \) and the minimum deviation \( D_m \) for a prism with a refractive index of \( \sqrt{2} \) and an angle of incidence of \( 45^\circ \), we will follow these steps:

### Step 1: Use Snell's Law
According to Snell's law, we have:

\[ \frac{\sin i}{\sin r} = \mu \]

Where:
- \( i \) is the angle of incidence,
- \( r \) is the angle of refraction,
- \( \mu \) is the refractive index.

### Step 2: Substitute Known Values
Given:
- \( \mu = \sqrt{2} \)
- \( i = 45^\circ \)

Substituting these values into Snell's law:

\[ \frac{\sin 45^\circ}{\sin r} = \sqrt{2} \]

### Step 3: Calculate \( \sin 45^\circ \)
We know that:

\[ \sin 45^\circ = \frac{1}{\sqrt{2}} \]

Substituting this into the equation gives:

\[ \frac{\frac{1}{\sqrt{2}}}{\sin r} = \sqrt{2} \]

### Step 4: Solve for \( \sin r \)
Cross-multiplying gives:

\[ \frac{1}{\sqrt{2}} = \sqrt{2} \cdot \sin r \]

Thus,

\[ \sin r = \frac{1}{2} \]

### Step 5: Find \( r \)
From the sine value, we can find \( r \):

\[ r = \sin^{-1}\left(\frac{1}{2}\right) = 30^\circ \]

### Step 6: Relate \( r \) to \( A \)
For a prism, the relationship between the angle of refraction \( r \) and the refracting angle \( A \) is given by:

\[ r = \frac{A}{2} \]

Substituting \( r = 30^\circ \):

\[ 30^\circ = \frac{A}{2} \]

### Step 7: Solve for \( A \)
Multiplying both sides by 2 gives:

\[ A = 60^\circ \]

### Step 8: Find Minimum Deviation \( D_m \)
The relationship between the angle of incidence \( i \), the refracting angle \( A \), and the minimum deviation \( D_m \) is:

\[ i = \frac{A}{2} + \frac{D_m}{2} \]

Substituting \( i = 45^\circ \) and \( A = 60^\circ \):

\[ 45^\circ = \frac{60^\circ}{2} + \frac{D_m}{2} \]

### Step 9: Simplify and Solve for \( D_m \)
This simplifies to:

\[ 45^\circ = 30^\circ + \frac{D_m}{2} \]

Subtracting \( 30^\circ \) from both sides gives:

\[ 15^\circ = \frac{D_m}{2} \]

Multiplying by 2 results in:

\[ D_m = 30^\circ \]

### Final Answers
Thus, the values of \( A \) and \( D_m \) are:

\[ A = 60^\circ, \quad D_m = 30^\circ \]

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