A penny is at the bottom of a bowl full of water. When you...

A penny is at the bottom of a bowl full of water. When you look at the water surface from the side, with your eyes at the water level, the penny appears to be just barely under the surface and a horizontal distance of $3.0 \mathrm{~cm}$ from the edge of the bowl. If the penny is actually $8.0 \mathrm{~cm}$ below the water surface, what is the horizontal distance between the penny and the edge of the bowl? [Hint: The rays you see pass from water to air with refraction angles close to $90^{\circ} .$ ]

Key Concepts

Geometric Optics and Trigonometry

Geometric optics uses principles of geometry and trigonometry to trace the paths of light rays as they travel through different media. In problems like this, trigonometric relationships (for example, relating the depth of an object to its apparent horizontal displacement via the tangent of the angle of incidence) are used to derive unknown distances or positions from known physical parameters.

Critical Angle

The critical angle is the angle of incidence in the denser medium (water, in this case) above which light is completely reflected back into that medium rather than refracting out into the less dense medium (air). When observed rays approach this angle, the emergent ray in air travels nearly parallel to the surface, which is key to determining apparent positions in problems involving refraction.

Snell's Law

This law describes how light bends (or refracts) when entering a medium with a different refractive index. It is expressed by the equation n? sin ?? = n? sin ??, where n represents the refractive index and ? the angle of the light ray relative to the normal at the interface. This principle is the foundation for understanding how light behaves at the boundary between media such as water and air.

Refractive Index

The refractive index of a medium quantifies how much the speed of light is reduced inside that medium compared to vacuum. It plays a crucial role in determining the degree of bending that occurs when light passes from one substance to another. Differences in refractive indices between materials like water and air account for the refraction effects seen in optical phenomena.

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