An under water swimmer looks upward at an unobstructed overcast sky. The vertex angle does the sky appear to subtend at the eye of swimmer is (critical angle for water air interface is C)

Options

A. C
B. C/2
C. 2C
D. 3C

Explanation

To solve the problem of determining the vertex angle that the sky appears to subtend at the eye of an underwater swimmer, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Setup**:
- The swimmer is underwater, looking up towards the sky.
- We need to consider the critical angle (C) for the water-air interface, which is the angle of incidence above which total internal reflection occurs.

2. **Identifying the Critical Ray**:
- When the swimmer looks up, light rays from the sky enter the water.
- The critical ray is the ray that makes an angle of C with the normal at the water-air interface. Beyond this angle, light will not exit the water but will be totally internally reflected.

3. **Drawing the Diagram**:
- Imagine a vertical line representing the water surface.
- Draw the normal line (perpendicular) to the surface at the point where the swimmer is looking up.
- The critical angle (C) is measured from the normal to the critical ray.

4. **Calculating the Vertex Angle**:
- The critical ray makes an angle C with the normal on both sides (left and right) of the vertical line.
- Therefore, the angle between the two critical rays (the vertex angle θ) is given by:
\[ \theta = C + C = 2C \]

5. **Conclusion**:
- The vertex angle that the sky appears to subtend at the eye of the swimmer is \(2C\).

### Final Answer:
The vertex angle that the sky appears to subtend at the eye of the swimmer is \(2C\).

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