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 solve the problem of how much total angle the sky appears to subtend when viewed by a fish looking upward through water, we can follow these steps:

### Step 1: Understand the Concept of Critical Angle
The fish can only see the sky if the light rays coming from the sky into the water do not exceed the critical angle. The critical angle (\(I_C\)) is the angle of incidence above which light cannot pass through the boundary and is instead reflected back into the water.

### Step 2: Calculate the Critical Angle
The critical angle can be calculated using the formula:
\[ I_C = \sin^{-1}\left(\frac{1}{\mu}\right) \]
where \(\mu\) is the refractive index of water. Given that \(\mu = \sqrt{2}\), we can substitute this value into the formula.

\[ I_C = \sin^{-1}\left(\frac{1}{\sqrt{2}}\right) \]

### Step 3: Evaluate the Critical Angle
The value of \(\frac{1}{\sqrt{2}}\) corresponds to \(\sin 45^\circ\). Therefore:
\[ I_C = 45^\circ \]

### Step 4: Determine the Total Angle Subtended by the Sky
When the fish looks upward, it can see the sky at an angle of \(45^\circ\) from the vertical in both directions (left and right). This means that the total angle subtended by the sky is:
\[ \text{Total Angle} = 45^\circ + 45^\circ = 90^\circ \]

### Final Answer
Thus, the total angle that the sky appears to subtend when viewed by the fish is:
\[ \text{Total Angle} = 90^\circ \]