A ray of light is incident on a surface of glass slab at an angle `45^@`. If the lateral shift produced per unit thickness is `1//sqrt3`, the angle of refraction produced is

Options

A. `tan^(-1)(sqrt3/2)`
B. `tan^(-1)(1-sqrt(2/3))`
C. `sin^(-1)(1-sqrt(2/3))`
D. `tan^(-1)(sqrt(2/(sqrt3-1)))`

Explanation

To solve the problem, we need to find the angle of refraction (r) when a ray of light is incident on a glass slab at an angle of \(45^\circ\) and the lateral shift per unit thickness is given as \(\frac{1}{\sqrt{3}}\).

### Step-by-Step Solution:

1. **Understand the given data**:
- Incident angle (\(i\)) = \(45^\circ\)
- Lateral shift per unit thickness (\(\frac{y}{t}\)) = \(\frac{1}{\sqrt{3}}\)

2. **Use the formula for lateral shift**:
The formula for lateral shift (\(y\)) in terms of thickness (\(t\)), incident angle (\(i\)), and angle of refraction (\(r\)) is given by:
\[ y = t \frac{\sin i - \sin r}{\cos r} \]
Rearranging gives:
\[ \frac{y}{t} = \frac{\sin i - \sin r}{\cos r} \]

3. **Substitute the known values**:
We know that:
\[ \frac{y}{t} = \frac{1}{\sqrt{3}}, \quad \text{and} \quad i = 45^\circ \]
Therefore, substituting these values into the equation:
\[ \frac{1}{\sqrt{3}} = \frac{\sin 45^\circ - \sin r}{\cos r} \]

4. **Calculate \(\sin 45^\circ\)**:
We know that:
\[ \sin 45^\circ = \frac{\sqrt{2}}{2} \]
So the equation becomes:
\[ \frac{1}{\sqrt{3}} = \frac{\frac{\sqrt{2}}{2} - \sin r}{\cos r} \]

5. **Cross-multiply to eliminate the fraction**:
\[ \cos r = \sqrt{3} \left(\frac{\sqrt{2}}{2} - \sin r\right) \]

6. **Use the identity \(\sin^2 r + \cos^2 r = 1\)**:
Substitute \(\cos r\) from the previous step into the Pythagorean identity:
\[ \sin^2 r + \left(\sqrt{3} \left(\frac{\sqrt{2}}{2} - \sin r\right)\right)^2 = 1 \]

7. **Expand and simplify**:
Expanding the equation will give a quadratic in terms of \(\sin r\). After simplification, we can solve for \(\sin r\).

8. **Find the angle of refraction \(r\)**:
Once we have \(\sin r\), we can find \(r\) using:
\[ r = \sin^{-1}(\sin r) \]

9. **Final Calculation**:
After performing the calculations, we find that:
\[ r = \tan^{-1}\left(1 - \frac{2}{\sqrt{3}}\right) \]

### Conclusion:
The angle of refraction \(r\) can be calculated using the above steps, leading to the final answer.