(a) Rework Example 31.4 to find the apparent distance to a...

(a) Rework Example $31.4$ to find the apparent distance to a fish that's located $12.0 \mathrm{~cm}$ from the far wall of the aquarium. Assume the cat is looking in on a line that includes the fish and the center of the cylindrical aquarium. (b) In Example $31.4$, the fish appeared closer than its actual distance. Is that the case here?

Key Concepts

Snell's Law

Snell's Law is a fundamental principle in optics that governs how light rays bend when passing from one medium to another with a different refractive index. It quantitatively relates the angles of incidence and refraction to the indices of refraction of the two media, ensuring that the path of the light ray is correctly predicted as it crosses an interface.

Index of Refraction

The index of refraction is a measure of how much light slows down in a medium compared to its speed in a vacuum. It is crucial for determining the degree of bending of light rays at an interface and is central to computing the apparent position of objects when they are viewed through different media, such as air and water.

Apparent Depth

The phenomenon of apparent depth occurs when an object submerged in a medium like water appears to be located at a different position than its true physical location due to the refraction of light. This effect is explained through geometric relationships and Snell's Law, and it is particularly noticeable when viewing underwater objects across a plane or curved interface.

Geometrical Optics in Curved Interfaces

When light rays pass through curved surfaces, such as those of a cylindrical aquarium, the analysis involves additional geometric considerations compared to flat interfaces. However, if the observer’s line of sight goes through the center of curvature, the curved interface can behave similarly to a flat one, making it possible to use simplified geometric optics methods—such as similar triangles—to determine apparent distances.

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