A fish looks upward at an unobstructed overcast sky. What total angle does the sky appear to subten?(Take refractive index of water as `sqrt(2).)`

Options

A. `180^(@)`
B. `90^(@)`
C. `75^(@)`
D. `60^(@)`

Explanation

To determine the total angle that the sky appears to subtend for a fish looking upward from the water, we can follow these steps:

### Step 1: Understand the Situation
The fish is submerged in water and is looking up towards the sky. The light rays from the sky will enter the water and refract at the water-air interface.

### Step 2: Identify the Critical Angle
The fish can only see the sky if the light rays can exit the water. The critical angle is the angle of incidence above which total internal reflection occurs. For the fish to see the sky, the angle of incidence must be less than or equal to the critical angle.

### Step 3: Use Snell's Law
Snell's Law relates the angles and refractive indices of the two media:
\[ n_1 \sin(\theta_i) = n_2 \sin(\theta_r) \]
Where:
- \( n_1 \) is the refractive index of water (\( \sqrt{2} \))
- \( n_2 \) is the refractive index of air (approximately 1)
- \( \theta_i \) is the angle of incidence (in water)
- \( \theta_r \) is the angle of refraction (in air)

### Step 4: Determine the Critical Angle
At the critical angle, the angle of refraction is 90 degrees (the light travels along the boundary). Thus, we can set up the equation:
\[ \sqrt{2} \sin(\theta_c) = 1 \cdot \sin(90^\circ) \]
This simplifies to:
\[ \sqrt{2} \sin(\theta_c) = 1 \]

### Step 5: Solve for the Critical Angle
Solving for \( \sin(\theta_c) \):
\[ \sin(\theta_c) = \frac{1}{\sqrt{2}} \]
Thus:
\[ \theta_c = 45^\circ \]

### Step 6: Calculate the Total Angle Subtended by the Sky
The fish can see the sky up to the critical angle. The total angle subtended by the sky will be twice the critical angle (as it extends on both sides of the normal):
\[ \text{Total angle} = 2 \times \theta_c = 2 \times 45^\circ = 90^\circ \]

### Final Answer
The total angle that the sky appears to subtend for the fish is \( 90^\circ \).

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