A ray of light incident on a glass sphere (refractive index `sqrt(3)` ) suffers total internal reflection before emerging out exactly parallel to the incident ray. The angle of incidence was

Options

A. `75^(@)`
B. `30^(@)`
C. `45^(@)`
D. `60^(@)`

Explanation

To solve the problem, we need to determine the angle of incidence (I) for a ray of light incident on a glass sphere with a refractive index of \(\sqrt{3}\), which undergoes total internal reflection and emerges parallel to the incident ray.

### Step-by-Step Solution:

1. **Understanding the Geometry**:
- A ray of light is incident on a glass sphere. When it enters the sphere, it refracts towards the normal due to the higher refractive index of glass compared to air.
- After some internal reflection, it emerges parallel to the incident ray.

2. **Identifying Angles**:
- Let the angle of incidence be \(I\) and the angle of refraction be \(R\).
- According to the problem, the ray undergoes total internal reflection, which means that the angle of incidence inside the sphere (when it reflects) is equal to the critical angle.

3. **Using Snell's Law**:
- Snell's Law states that:
\[ \frac{\sin I}{\sin R} = n \]
- Here, \(n\) is the refractive index of the glass sphere, which is given as \(\sqrt{3}\).

4. **Applying the Total Internal Reflection Condition**:
- For total internal reflection, we have:
\[ I = 2R \]
- This means the angle of incidence \(I\) is twice the angle of refraction \(R\).

5. **Substituting into Snell's Law**:
- We can substitute \(I = 2R\) into Snell's Law:
\[ \frac{\sin(2R)}{\sin R} = \sqrt{3} \]
- Using the double angle formula, \(\sin(2R) = 2 \sin R \cos R\), we get:
\[ \frac{2 \sin R \cos R}{\sin R} = \sqrt{3} \]
- Simplifying this gives:
\[ 2 \cos R = \sqrt{3} \]

6. **Solving for \(R\)**:
- Rearranging gives:
\[ \cos R = \frac{\sqrt{3}}{2} \]
- The angle \(R\) that satisfies this is:
\[ R = 30^\circ \]

7. **Finding the Angle of Incidence \(I\)**:
- Since \(I = 2R\):
\[ I = 2 \times 30^\circ = 60^\circ \]

### Final Answer:
The angle of incidence \(I\) is \(60^\circ\).

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