Question

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 $ z $ is the depth of the fish from the surface, then the distance of the bird as estimated by the fish is

• A. \( x + y \left(1 - \frac{1}{n}\right) \)
• B. \( x + ny \)
• C. \( x + y \left(1 + \frac{1}{n}\right) \)
• D. \( y + z \left(1 - \frac{1}{n}\right) \)

Hint:

When considering refraction, remember that the apparent distance of an object in a medium with refractive index \( n \) is given by \( n \times \text{real distance} \). This formula is crucial for solving problems involving refraction at boundaries.

Answer 1

The Correct Option is B

To solve the problem of how a fish perceives the distance of a bird above it, we need to consider the effects of refraction at the water-air interface. When light travels from a medium with a higher refractive index (water, in this case) to a medium with a lower refractive index (air), it bends away from the normal. This makes objects appear closer than they actually are.

Let's denote:

• \( n \) as the refractive index of water.
• \( y \) as the actual height of the bird above the water surface.
• \( z \) as the depth of the fish below the water surface.

The perceived distance by the fish is due to the apparent shift caused by the refractive index. The formula that relates the actual height of the bird with its perceived height involves multiplying the actual height by the refractive index:

\[ \text{Apparent height} = n \times y \]

Hence, the distance of the bird as estimated by the fish includes this perceived height:

\[ \text{Estimated Distance} = \text{Apparent height} = n \times y \]

Thus, the correct expression for the distance of the bird as estimated by the fish is: \( x + ny \)

Answer 2

To determine the distance of the bird as estimated by the fish in water with a refractive index \( n \), we first need to understand the concept of apparent depth. The apparent depth when viewed from a medium with a refractive index different from that of the medium containing the object is calculated using the formula:

Apparent Depth = \(\frac{\text{Real Depth}}{\text{Refractive Index}}\)

In this scenario:

• The real height of the bird in the air is \( y \).
• The refractive index of air is approximately 1, while the refractive index of water is \( n \).
• Since light is going from water to air, the apparent height as seen by the fish is scaled by the refractive index of the water.

Therefore, the apparent distance of the bird from the fish's perspective in water is:

\( \text{Apparent Distance} = n \times y \)

This means the fish perceives the bird to be further away because of the denser medium it is in.

Thus, the distance of the bird as estimated by the fish is given by:

\( x + ny \),

where \( x \) is another factor or initial distance often considered as a reference point.

Therefore, the correct answer is:

Option: \( x + ny \)