Question

A bird in air is at a height y from the surface of water. A fish is at a depth x below the surface of water. The refractive index of water is μ. The apparent distance of fish from the bird is :
A. \[x+\dfrac{y}{\mu }\]
B. \[\mu x+y\]
C. \[\dfrac{x}{\mu }+y\]
D. \[\dfrac{x}{\mu }-y\]

Answer 1

Hint: Using the apparent depth and real depth formula where the depth of the object is raised when looking from a rarer medium when the object is kept at a denser medium. The formula for the apparent depth calculation is:
\[\text{Apparent depth}=\dfrac{\text{real depth}}{{{\mu }_{m}}}\]
where \[{{\mu }_{m}}\] is the refractive index of the material where the substance is kept with m denoting the material.

Complete step by step answer:
We need to find the apparent depth of the bird and fish from the water surface, the height of the bird from the water surface is given as y and the height from the water surface to the fish is given as x. Hence, the real depth of the fish and the bird is given as:
Real depth is x for fish
Real depth is y for bird
Now for the apparent depth we use the formula as:
\[\text{Apparent depth}=\dfrac{\text{real depth}}{{{\mu }_{m}}}\]
The apparent depth of the bird when looked by the fish from below the surface of the water level is:
\[\text{Apparent depth}=\dfrac{\text{x}}{{{\mu }_{water}}}\]
The apparent depth of the fish when looked by the bird from above the surface of the water level is:
\[\text{Apparent depth}=\dfrac{\text{y}}{{{\mu }_{air}}}\]
After getting the apparent distance from the bird to the water surface and from water surface to the fish, we will add the entire distance from fish to bird as the total apparent distance between them by putting the values into the sum as:
\[\text{Apparent depth}=\dfrac{\text{x}}{{{\mu }_{water}}}+\dfrac{y}{{{\mu }_{air}}}\]
The refractive index of air is 1 and placing the value of ${{\mu }_{air}}=1$ in the above formula, we get:
\[\text{Apparent depth}=\dfrac{\text{x}}{{{\mu }_{water}}}+\dfrac{y}{1}\]
\[\text{Apparent depth}=y+\dfrac{\text{x}}{{{\mu }_{water}}}\]
Therefore, the apparent distance from fish to bird \[=y+\dfrac{\text{x}}{{{\mu }_{water}}}\]

Note:
Although the refractive index of water is near 1 but the most close to 1 is the refractive index of air hence, we use the refractive index of air as 1. In this case the apparent depth and real depth of the eagle is the same as the eagle is viewing the fish directly downwards.