A fish is at a depth d under water. Take the index of...

A fish is at a depth $d$ under water. Take the index of refraction of water as $4 / 3$. Show that when the fish is viewed at an angle of refraction $\theta_{1}$, the apparent depth $z$ of the fish is
$$z=\frac{3 d \cos \theta_{1}}{\sqrt{7+9 \cos ^{2} \theta_{1}}}$$

Key Concepts

Geometric Optics and Trigonometric Relationships

This concept involves applying the principles of ray optics and using geometric and trigonometric methods to describe the path of light rays. It includes using cosine and square root relationships to map out how light rays traverse the interface between media and how their directional components relate to perceived positions. These tools are essential when deriving the mathematical expression for apparent depth in terms of angles and actual depth.

Snell's Law

This fundamental law of refraction relates the angles of incidence and refraction to the indices of refraction of the two media. It is essential in determining how light rays bend when transitioning between media with different optical densities, and its application is crucial in deriving relationships between the real and apparent positions of objects observed through a refractive medium.

Refractive Index

The refractive index quantifies how much light slows down in a medium compared to a vacuum. It is a key parameter in optics since differences in refractive indices between media determine the bending of light rays under Snell's Law. In problems involving water and air, knowing the refractive index of water is crucial to ascertain how light escapes and reaches an observer.

Apparent Depth

Apparent depth refers to the perceived position of an object submerged in a refractive medium, as seen by an observer. Due to refraction, the object appears to be at a shallower depth than its actual position. The calculation of apparent depth involves combining geometric relationships with refractive principles, which is central to analyzing the problem at hand.

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