A light ray enters the atmosphere of a planet where it...

A light ray enters the atmosphere of a planet where it descends vertically to the surface a distance $h$ below. The index of refraction where the light enters the atmosphere is $1.000,$ and it increases linearly to the surface where it has the value $n .$ (a) How long does it take the ray to traverse this path? (b) Compare this to the time interval required in the absence of an atmosphere.

Key Concepts

Index of Refraction

The index of refraction is a fundamental optical property that quantifies how much a medium slows down light relative to its speed in a vacuum. It defines the relationship between the speed of light in vacuum and in a medium through the simple relation v = c/n. This concept is central to understanding how light propagates through different media, especially when the optical density changes continuously.

Variable Speed of Light in a Medium

Because the speed of light in any medium depends on the local index of refraction, a spatially varying index causes the speed of light to change along its path. When the index increases or decreases gradually, the light’s velocity is not constant, and its travel time must be calculated by considering these variations locally.

Calculus Integration in Variable Media

When the refractive index is not uniform, the travel time is computed using integration. By expressing the differential time d? in terms of the local speed (or equivalently the local index of refraction), one can integrate along the path of the light to determine the total transit time. This approach is necessary in scenarios where the medium’s properties vary continuously.

Time Delay Comparison

Comparing the time interval of light propagation in a medium with a variable index of refraction to that in vacuum involves understanding the concept of optical path length. Even if the geometric distance is the same, the presence of an atmosphere or any medium where the speed of light is reduced leads to a delay. This comparison highlights how material properties influence the effective travel time of light.

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