A large Lucite cube (n=1.59) has a small air bubble (a...

A large Lucite cube $(n=1.59)$ has a small air bubble (a defect in the casting process) below one surface. When a penny (diameter, $1.90 \mathrm{~cm}$ ) is placed directly over the bubble on the outside of the cube, one cannot see the bubble by looking down into the cube at any angle. However, when a dime (diameter, $1.75 \mathrm{~cm}$ ) is placed directly over it, one can see the bubble by looking down into the cube. What is the range of the possible depths of the air bubble beneath the surface?

Key Concepts

Snell's Law

Snell's Law relates the angles of incidence and refraction when light passes between two media with different indices of refraction. It is essential in determining how light bends at the interface between a dense medium (Lucite) and a less dense medium (air), which in turn affects how objects inside the medium are seen from outside.

Critical Angle

The critical angle is defined as the incidence angle above which light traveling within a denser medium (like Lucite) will be entirely reflected at the interface with a less dense medium (air) rather than being refracted out. This concept is crucial for understanding the conditions under which internal features, such as an air bubble, become hidden due to total internal reflection.

Total Internal Reflection

Total internal reflection occurs when light traveling in a medium with a higher index of refraction strikes the boundary with a lower index medium at an angle greater than the critical angle, resulting in all the light being reflected back into the original medium. This phenomenon determines whether light from the bubble can exit the Lucite and be seen by an observer.

Geometrical Optics and Ray Diagrams

Geometrical optics uses ray diagrams to map out the paths of light as it travels through different media and interfaces. This approach helps visualize and analyze how light rays from the bubble are refracted or reflected, which is key in determining the conditions under which the bubble is visible or obscured when covered by objects of different sizes.

Similar Triangles in Optical Geometry

The use of similar triangles in optical geometry allows for the determination of relationships between physical dimensions and distances, such as the diameter of the covering object and the depth of the bubble. Through these relationships, one can infer the range of depths where the bubble’s image will be either revealed or hidden based on the geometry of the light paths.

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