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, the bubble cannot be seen by looking down into the cube at any angle. However, when a dime (diameter $1.75 \mathrm{cm}$ ) is placed directly over it, the bubble can be seen by looking down into the cube. What is the range of the possible depths of the air bubble beneath the surface?

Key Concepts

Critical Angle and Total Internal Reflection

When light travels from a medium with a higher refractive index to one with a lower refractive index, there is a specific angle—the critical angle—beyond which all the light is reflected back into the medium rather than refracted. The critical angle is determined by the refractive indices of the two media and is given by sin ?_c = n? / n?. The concept of total internal reflection, which occurs at this angle or greater, is crucial for understanding when light originating from objects beneath a surface will or will not emerge, affecting their visibility.

Geometrical Optics and Ray Tracing

Geometrical optics involves tracing the paths of light rays as they interact with surfaces, which allows for the prediction of image positions and the behavior of light in complex setups. In this context, ray tracing is used to analyze how light emanating from a subsurface bubble is refracted at the interface, leading to a lateral shift that determines whether the bubble’s image falls within or outside the area covered by an overlying object.

Snell's Law

Snell’s law is a fundamental principle in optics that relates the angles of incidence and refraction when light passes between two media with different refractive indices. It is expressed as n? sin ?? = n? sin ?? and is instrumental in determining how much light is bent at an interface. In problems involving refraction at a flat surface, such as light emerging from a Lucite cube into air, Snell’s law provides the basis for calculating the angles at which light exits the material.

Lateral Displacement in Refraction

Lateral displacement refers to the sideways shift of a light ray as it refracts when passing from one medium to another. In optical systems with flat interfaces, the displacement is determined by the depth of the object and the angle at which the ray emerges, often involving trigonometric relations such as the tangent of the critical angle. This concept links the geometry of the setup—such as the size of an overlaying coin and the depth of a defect—with the visibility of the subsurface feature.

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