If a prism having refractive index `sqrt 2` has angle of minimum deviation equal to the angle of refraction of the prism, then the angle of refraction of the prism is:

Options

A. `30^(@)`
B. `45^(@)`
C. `60^(@)`
D. `90^(@)`

Explanation

To solve the problem, we need to find the angle of refraction of the prism given that the refractive index (μ) is √2 and that the angle of minimum deviation (δ_min) is equal to the angle of refraction (r) of the prism.

### Step-by-Step Solution:

1. **Understanding the Condition of Minimum Deviation**:
At the condition of minimum deviation, the formula relating the refractive index (μ), the angle of minimum deviation (δ_min), and the angle of the prism (A) is given by:
\[ \mu = \frac{\sin\left(\frac{\delta_{min} + A}{2}\right)}{\sin\left(\frac{A}{2}\right)} \]

2. **Substituting Given Values**:
We know that:
- μ = √2
- δ_min = r (angle of refraction)
Therefore, we can rewrite the equation as:
\[ \sqrt{2} = \frac{\sin\left(\frac{r + A}{2}\right)}{\sin\left(\frac{A}{2}\right)} \]

3. **Rearranging the Equation**:
Rearranging gives:
\[ \sin\left(\frac{r + A}{2}\right) = \sqrt{2} \sin\left(\frac{A}{2}\right) \]

4. **Using the Double Angle Identity**:
We can use the identity for sine:
\[ \sin(2x) = 2 \sin(x) \cos(x) \]
So, we can express:
\[ \sin\left(\frac{r + A}{2}\right) = 2 \sin\left(\frac{A}{4}\right) \cos\left(\frac{A}{4}\right) \]

5. **Setting Up the Equation**:
Now, we can set up the equation:
\[ \sqrt{2} \sin\left(\frac{A}{2}\right) = 2 \sin\left(\frac{A}{4}\right) \cos\left(\frac{A}{4}\right) \]

6. **Using Trigonometric Identities**:
We know that:
\[ \sin\left(\frac{A}{2}\right) = 2 \sin\left(\frac{A}{4}\right) \cos\left(\frac{A}{4}\right) \]
Therefore, we can simplify:
\[ \sqrt{2} \cdot 2 \sin\left(\frac{A}{4}\right) \cos\left(\frac{A}{4}\right) = 2 \sin\left(\frac{A}{4}\right) \cos\left(\frac{A}{4}\right) \]

7. **Solving for A**:
Dividing both sides by \(2 \sin\left(\frac{A}{4}\right)\) (assuming it is not zero):
\[ \sqrt{2} = 1 \implies A = 90^\circ \]

8. **Conclusion**:
Therefore, the angle of refraction (r) is equal to the angle of the prism (A), which is:
\[ r = 90^\circ \]

### Final Answer:
The angle of refraction of the prism is **90 degrees**.