Choose the most appropriate graph of `i_(min)v//s A`, for a light ray which is incident on one face of a prism and is coming out from the opposite face (without `TIR`). (The symbols have their standard meanings).

Options

A. A.
B. B.
C. C.
D. D.

Explanation

To solve the problem of determining the appropriate graph of `i_(min)v//s A` for a light ray incident on a prism, we need to analyze the relationship between the prism angle (A) and the minimum angle of incidence (i_min) under the condition that the light ray exits from the opposite face without undergoing total internal reflection (TIR).

### Step-by-Step Solution:

1. **Understanding the Prism and Angles**:
- A prism has an apex angle denoted as A.
- When light enters the prism, it refracts at the first face and exits at the second face.
- The angles of refraction at the two faces are denoted as r1 and r2.

2. **Conditions for No Total Internal Reflection**:
- Total internal reflection occurs when the angle of incidence exceeds the critical angle (c) for the material of the prism.
- For light to exit the prism without TIR, the angle of refraction (r2) must be less than the critical angle (c).

3. **Relation Between Angles**:
- The relationship between the prism angle (A) and the angles of refraction can be expressed as:
\[ A = r1 + r2 \]
- For light to emerge from the prism, we must ensure that both r1 and r2 are less than the critical angle (c).

4. **Analyzing Different Ranges of A**:
- **Case 1**: When A < c
- In this case, both r1 and r2 can be adjusted such that they remain below the critical angle. Therefore, i_min can be zero, meaning that light can enter the prism at any angle.
- **Case 2**: When c < A < 2c
- Here, r1 can still be less than c, but r2 may approach or exceed c, leading to the possibility of TIR. However, there is still a range of i_min that allows for emergence.
- **Case 3**: When A > 2c
- In this scenario, r2 will definitely exceed the critical angle, leading to TIR. Thus, i_min will again be zero since no light can emerge.

5. **Conclusion**:
- From the analysis, we find that the minimum angle of incidence (i_min) is zero when A is less than c and also when A is greater than 2c.
- Between these ranges, there exists a non-zero minimum angle of incidence. Therefore, the graph that represents this relationship will show that i_min is zero for A < c and A > 2c, and it will rise as A approaches the critical angle.

### Final Answer:
The correct graph representing the relationship between `i_(min)` and `A` is **Option B**, which shows that i_min is zero for A < c and A > 2c, and increases in between these ranges.