A equilateral glass prism has a refractive index `sqrt(2)`. A light ray is incident at `45^(@)` on one face. Total deviation of ray is `nxx15^(@)` where n is

Answer

To solve the problem, we need to calculate the total deviation of a light ray passing through an equilateral prism with a refractive index of \(\sqrt{2}\) and an incident angle of \(45^\circ\).

### Step-by-Step Solution:

1. **Understand the Prism Geometry**:
- An equilateral prism has angles of \(60^\circ\) each.
- The light ray is incident at \(45^\circ\) on one face of the prism.

2. **Identify the Angles**:
- Let \(i\) be the angle of incidence, which is \(45^\circ\).
- The angle of the prism \(A\) is \(60^\circ\).
- The angle of refraction \(r_1\) at the first face can be calculated using Snell's Law:
\[ n_1 \sin(i) = n_2 \sin(r_1) \]
where \(n_1 = 1\) (air) and \(n_2 = \sqrt{2}\) (glass).

3. **Calculate the Angle of Refraction**:
- Using Snell's Law:
\[ \sin(45^\circ) = \sqrt{2} \sin(r_1) \]
\[ \frac{1}{\sqrt{2}} = \sqrt{2} \sin(r_1) \]
\[ \sin(r_1) = \frac{1}{2} \implies r_1 = 30^\circ \]

4. **Determine the Angle of Refraction at the Second Face**:
- The light ray travels inside the prism and hits the second face. The angle of incidence at the second face \(i_2\) can be found as:
\[ i_2 = A - r_1 = 60^\circ - 30^\circ = 30^\circ \]
- Now apply Snell's Law again at the second face:
\[ \sqrt{2} \sin(30^\circ) = 1 \sin(r_2) \]
\[ \sqrt{2} \cdot \frac{1}{2} = \sin(r_2) \]
\[ \sin(r_2) = \frac{\sqrt{2}}{2} \implies r_2 = 45^\circ \]

5. **Calculate the Total Deviation**:
- The total deviation \(\delta\) can be calculated using:
\[ \delta = i + e - A \]
where \(e\) is the angle of emergence (which is equal to \(r_2\)).
- Thus:
\[ \delta = 45^\circ + 45^\circ - 60^\circ = 30^\circ \]

6. **Relate Total Deviation to Given Expression**:
- The problem states that the total deviation is \(n \times 15^\circ\).
- We have found that \(\delta = 30^\circ\):
\[ 30^\circ = n \times 15^\circ \]
- Solving for \(n\):
\[ n = \frac{30}{15} = 2 \]

### Final Answer:
The value of \(n\) is \(2\).