A ray of light is incident on a glass sphere of refractive index `3//2` . What should be the angle of incidence so that the ray which enters the sphere does not comme out of the sphere?

Options

A. `tan^(-1)(2//3)`
B. `60^(@)`
C. `90^(@)`
D. `30^(@)`

Explanation

To solve the problem, we need to determine the angle of incidence at which a ray of light strikes a glass sphere with a refractive index of \( \frac{3}{2} \) such that it does not exit the sphere after entering.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We have a glass sphere with a refractive index \( n = \frac{3}{2} \).
- We need to find the angle of incidence \( I \) such that the light ray, after entering the sphere, does not exit.

2. **Critical Angle Calculation**:
- For total internal reflection to occur, the angle of incidence at the boundary must be greater than the critical angle \( \theta_C \).
- The critical angle can be calculated using the formula:
\[ \sin \theta_C = \frac{1}{n} \]
- Substituting the refractive index:
\[ \sin \theta_C = \frac{1}{\frac{3}{2}} = \frac{2}{3} \]

3. **Finding the Critical Angle**:
- To find \( \theta_C \), we take the inverse sine:
\[ \theta_C = \sin^{-1}\left(\frac{2}{3}\right) \]

4. **Applying Snell's Law**:
- When the ray enters the sphere, we apply Snell's law at the point of incidence:
\[ n_1 \sin I = n_2 \sin \theta_T \]
- Here, \( n_1 = 1 \) (air), \( n_2 = \frac{3}{2} \), and \( \theta_T \) is the angle of refraction inside the sphere.
- Rearranging gives:
\[ \sin I = \frac{3}{2} \sin \theta_T \]

5. **Condition for Total Internal Reflection**:
- For the ray to not exit the sphere, the angle \( \theta_T \) must equal \( \theta_C \).
- Therefore, substituting \( \theta_T = \theta_C \):
\[ \sin I = \frac{3}{2} \sin \theta_C \]
- Since \( \sin \theta_C = \frac{2}{3} \):
\[ \sin I = \frac{3}{2} \cdot \frac{2}{3} = 1 \]

6. **Finding the Angle of Incidence**:
- The only angle for which \( \sin I = 1 \) is:
\[ I = 90^\circ \]

### Conclusion:
The angle of incidence \( I \) should be \( 90^\circ \) for the ray of light to enter the glass sphere and not come out.