A ray of light traveling in a trasparent medium falls on a surface separating the medium from air, at an angle of incidence of `45^(@)`. The ray undergoes total internal reflection. If n is the refractive index of the medium with respect to air, select the possible values of n from the following.

Options

A. 1.3
B. 1.4
C. 1.5
D. 1.6

Answer

To solve the problem, we need to determine the refractive index \( n \) of the medium with respect to air, given that a ray of light is incident at an angle of \( 45^\circ \) and undergoes total internal reflection (TIR).

### Step-by-Step Solution:

1. **Understanding Total Internal Reflection (TIR)**:
- Total internal reflection occurs when light attempts to move from a denser medium to a less dense medium (in this case, from the transparent medium to air) and the angle of incidence exceeds the critical angle.

2. **Finding the Critical Angle**:
- The critical angle \( \theta_c \) can be calculated using the formula:
\[ \sin \theta_c = \frac{1}{n} \]
- Here, \( n \) is the refractive index of the medium with respect to air.

3. **Using the Given Angle of Incidence**:
- We are given that the angle of incidence \( \theta_i = 45^\circ \).
- For TIR to occur, the angle of incidence must be greater than the critical angle:
\[ \theta_i > \theta_c \]
- Therefore, we need:
\[ 45^\circ > \theta_c \]

4. **Calculating the Minimum Refractive Index**:
- From the critical angle formula, we can express \( \theta_c \) in terms of \( n \):
\[ \theta_c = \sin^{-1}\left(\frac{1}{n}\right) \]
- For \( 45^\circ \) to be greater than \( \theta_c \):
\[ \sin 45^\circ > \frac{1}{n} \]
- Since \( \sin 45^\circ = \frac{\sqrt{2}}{2} \), we can write:
\[ \frac{\sqrt{2}}{2} > \frac{1}{n} \]

5. **Rearranging the Inequality**:
- Rearranging the inequality gives:
\[ n > \frac{2}{\sqrt{2}} = \sqrt{2} \approx 1.41 \]

6. **Conclusion**:
- Thus, the refractive index \( n \) must be greater than \( 1.41 \) for total internal reflection to occur at an angle of incidence of \( 45^\circ \).

### Possible Values of \( n \):
- From the options provided, any value of \( n \) greater than \( 1.41 \) (like \( 1.5 \) or \( 1.6 \)) would satisfy the condition for TIR. Values such as \( 1.3 \) and \( 1.4 \) would not satisfy the condition.