A light ray enters the atmosphere of a planet and descends...

A light ray enters the atmosphere of a planet and descends vertically $20.0 \mathrm{~km}$ to the surface. The index of refraction where the light enters the atmosphere is $1.000$, and it increases linearly to the surface where it has a value of $1.005 .$ (a) How long does it take the ray to traverse this path? (b) Compare this to the time it takes in the absence of an atmosphere.

Key Concepts

Comparison to Vacuum Propagation

Comparing the time taken by light to traverse a medium versus a vacuum highlights the effect of the medium’s refractive properties on light speed. In a vacuum, light travels at its maximum speed, c, and the travel time is simply the distance divided by c. When light passes through a medium with a refractive index greater than one, even if only slightly so, the effective speed is reduced, resulting in a longer travel time—a concept particularly useful for understanding delays in optical and atmospheric physics.

Time of Flight Calculation in a Varying Medium

Calculating the time it takes for a light ray to traverse a medium with a varying index of refraction involves integrating the differential contributions to the time delay over the path of the ray. Since the local speed of light varies as v = c/n(z), where n(z) is a function of position, the total time is found by integrating dt = n(z) dz/c. This method is essential for accurately assessing the overall delay in light propagation compared to a uniform medium.

Index of Refraction

The index of refraction is a dimensionless number that quantifies how much the speed of light is reduced in a medium compared to its speed in a vacuum. It is given by n = c/v, where c is the speed of light in vacuum and v is the speed of light in the medium. This concept is fundamental in understanding light propagation through various substances and media, as differences in the index lead to phenomena such as refraction and bending of light rays.

Speed of Light in a Medium

The speed at which light travels in a medium is determined by the medium’s index of refraction. Specifically, light travels as v = c/n. This relationship is crucial when analyzing how light moves through materials where the refractive index can vary, such as atmospheric layers, since changes in n directly affect the transit time and path of the light ray.

Graded-Index (Variable Index) Media

A graded-index or variable-index medium is one in which the index of refraction changes continuously throughout the medium. In such cases, light may follow curved trajectories due to the continuously varying speed, and its transit time must be computed by integrating the effects of the local refractive index along its path. This concept is important for understanding real-world scenarios like the Earth’s atmosphere, where conditions such as temperature or density variations cause changes in the refractive index.

Transcript

Please give Ace some feedback