A fish in water (refractive index n ) looks at a bird vertically above in the air. If y is the height of the bird and x is the depth of the fish from the surface, then the distance of the bird as estimated by the fish is

Options

A. `x + y (1+(1)/(n))`
B. `y+x(1-(1)/(n))`
C. `y + x (1 - (1)/(n))`
D. `x + y (1 - (1)/(n))`

Answer

To solve the problem, we need to determine the distance of the bird as estimated by the fish, taking into account the refraction of light at the water-air interface. Here’s a step-by-step solution:

### Step 1: Understand the Setup
We have a fish located at a depth \( x \) below the water surface and a bird at a height \( y \) above the water surface. The refractive index of water is \( n \).

### Step 2: Identify the Refraction Effect
When light travels from a denser medium (water) to a rarer medium (air), it bends away from the normal. The apparent position of the bird as seen by the fish will be shifted due to this refraction.

### Step 3: Calculate the Normal Shift
The normal shift can be calculated using the formula:
\[ \text{Normal Shift} = y \cdot (n - 1) \]
This shift accounts for how much the light bends as it exits the water.

### Step 4: Determine the Apparent Height of the Bird
The apparent height of the bird as perceived by the fish will be the actual height \( y \) plus the normal shift:
\[ \text{Apparent Height} = y + y \cdot (n - 1) = y + (n - 1)y = ny \]

### Step 5: Calculate the Total Distance Perceived by the Fish
The total distance from the fish to the bird, as estimated by the fish, will be the sum of the depth of the fish \( x \) and the apparent height of the bird:
\[ \text{Distance} = x + ny \]

### Final Result
Thus, the distance of the bird as estimated by the fish is:
\[ \text{Distance} = x + ny \]