A ray of light is incident on the face AB of a glass prism ABC having vertex anlge A equal to `30^(@)`. The face AC is silvered and a ray of light incident on the face AB retraces its path. If the refractive index of the material of prism is `sqrt(3)` find the angle of incidence on the face AB.

Options

A. `60^(@)`
B. `30^(@)`
C. `45^(@)`
D. `25^(@)`

Answer

To solve the problem, we need to find the angle of incidence on the face AB of a glass prism with a vertex angle A of 30 degrees, where the face AC is silvered, and the ray retraces its path. The refractive index of the prism is given as √3.

### Step-by-Step Solution:

1. **Understand the Geometry of the Prism:**
- The prism has a vertex angle A = 30°.
- The ray of light is incident on face AB and retraces its path after reflecting off the silvered face AC.

2. **Identify the Angles:**
- Let the angle of incidence on face AB be denoted as I.
- When the ray refracts at face AB, it will have an angle of refraction R.
- Since the face AC is silvered, the ray will reflect off this surface and retrace its path.

3. **Use the Geometry of Reflection:**
- When the ray retraces its path after reflecting off the silvered face, the angle of incidence at AC will be equal to the angle of reflection.
- The angle of incidence at AC can be calculated as follows:
- The total angle at point A is 180°.
- The angle at point A is divided into two parts: angle R (the angle of refraction at AB) and the angle of incidence at AC.
- Therefore, the angle at A can be expressed as:
\[ 180° = R + 90° + (30° - R) \]
This simplifies to:
\[ R + 90° + 30° - R = 180° \implies 90° + 30° = 180° \]
This confirms the angles are correct.

4. **Apply Snell's Law:**
- According to Snell's law:
\[ n = \frac{\sin I}{\sin R} \]
- Here, the refractive index (n) of the prism is given as √3, and R is the angle of refraction at face AB.
- Since the angle A is 30°, we can find R:
\[ R = 30° \quad \text{(because the ray retraces its path)} \]

5. **Substituting Values into Snell's Law:**
- Substitute the known values into Snell's law:
\[ \sqrt{3} = \frac{\sin I}{\sin 30°} \]
- Since \(\sin 30° = \frac{1}{2}\), we have:
\[ \sqrt{3} = \frac{\sin I}{\frac{1}{2}} \implies \sin I = \sqrt{3} \cdot \frac{1}{2} = \frac{\sqrt{3}}{2} \]

6. **Finding the Angle of Incidence I:**
- The angle I can be found using the inverse sine function:
\[ I = \sin^{-1}\left(\frac{\sqrt{3}}{2}\right) \]
- This gives us:
\[ I = 60° \]

### Final Answer:
The angle of incidence on the face AB is **60 degrees**.