A ray of light travelling in a transparent 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 value 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.

### Step-by-Step Solution:

1. **Understanding Total Internal Reflection**:
Total internal reflection occurs when a light ray travels from a denser medium to a less dense medium (in this case, from the transparent medium to air) and the angle of incidence is greater than the critical angle.

2. **Using Snell's Law**:
According to Snell's law, we have:
\[ n \sin(i) = 1 \sin(r) \]
where:
- \( n \) is the refractive index of the medium,
- \( i \) is the angle of incidence,
- \( r \) is the angle of refraction (which is \( 90^\circ \) at the critical angle).

3. **Critical Angle**:
The critical angle \( C \) can be defined as the angle of incidence at which the angle of refraction is \( 90^\circ \). Thus, we can write:
\[ n \sin(C) = 1 \sin(90^\circ) = 1 \]
This simplifies to:
\[ n = \frac{1}{\sin(C)} \]

4. **Condition for Total Internal Reflection**:
For total internal reflection to occur, the angle of incidence \( i \) must be greater than the critical angle \( C \):
\[ \sin(i) > \sin(C) \]
Since \( i = 45^\circ \), we have:
\[ \sin(45^\circ) = \frac{1}{\sqrt{2}} \]

5. **Relating \( n \) and \( C \)**:
From the earlier equation, we can express the condition for total internal reflection in terms of \( n \):
\[ \frac{1}{\sqrt{2}} > \sin(C) \]
Using the relationship \( n = \frac{1}{\sin(C)} \), we can rewrite the inequality:
\[ \frac{1}{\sqrt{2}} > \frac{1}{n} \]
This implies:
\[ n > \sqrt{2} \]

6. **Calculating \( \sqrt{2} \)**:
The value of \( \sqrt{2} \) is approximately \( 1.41 \). Therefore, we conclude that:
\[ n > 1.41 \]

7. **Selecting Possible Values**:
From the options provided, any value of \( n \) that is greater than \( 1.41 \) is acceptable. For example, \( 1.5 \) and \( 1.6 \) are both valid choices.

### Conclusion:
The possible values of \( n \) that satisfy the condition for total internal reflection are \( n > 1.41 \). Therefore, options \( 1.5 \) and \( 1.6 \) are correct.