A light ray enters the atmosphere of the Earth and descends...

A light ray enters the atmosphere of the Earth and descends vertically to the surface a distance $h=100 \mathrm{km}$ below. The index of refraction where the light enters the atmosphere is 1.00 , and it increases linearly with distance. to have the value $n=1.000293$ at the Earth's surface. (a) Over what time interval does the light traverse this path? (b) By what percentage is the time interval larger than that required in the absence of the Earth's atmosphere?

Key Concepts

Time Delay

Time delay refers to the additional time required for light to traverse a distance in a medium compared to the time it would take in a vacuum. This concept is essential when comparing the actual travel time of light in a refracting medium with its theoretical vacuum travel time.

Integration in Variable Media

When the refractive index varies with position, as in a graded atmosphere, the travel time and optical path length must be obtained by integrating over the path. This method accounts for the gradual changes in the medium's properties and yields a more accurate analysis of the light's journey.

Optical Path Length

Optical path length is the product of the physical distance traversed by light and the local refractive index of the medium. It provides a measure of the effective distance that light 'feels' as it moves through a material, incorporating the effects of slowing down due to the medium.

Speed of Light in a Medium

This concept states that light travels slower in a medium than in a vacuum, with the speed given by the vacuum speed divided by the refractive index. This reduction in speed is critical in analyzing light propagation across media with different refractive indices.

Refractive Index

The refractive index quantifies how much the speed of light is reduced inside a material compared to its speed in vacuum. It is a fundamental property of materials that determines the bending of light at interfaces and influences the propagation speed within the medium.

Transcript

00:01 Well, the index of refraction n is a function of the distance x.

00:03 So index of refraction n is a function of x traveled in the atmosphere and it increases linearly.

00:13 So we can express the index of refraction function using the function of a straight line formula, n, which is a function of x, distance is equal to mx plus n -not.

00:30 Right so before entering the atmosphere at the index of refraction n is equal to 1 .00 and so refraction is it is it is equal to 1 .00 so it n is equal to 1 .00 and it x is equal to 0 we have n x is equal to 0 is equal to 1 .00 is equal to 0 plus n0 so n .0 is equal to 1 .00 right and further nx is equal to mx plus 1 .00 right so when the light reaches the earth surface after distance h is equal to 100 meter the index of refraction is n is equal to 1 .00293 thus we have 1 .00293 is equal to m into 100 multiply by 10 to the power 3 meter plus 1 .00 so it's not this height is hundred multiply with 10 to the power 3 meter plus 1 .00 here, right? and further simplifying this we have m is equal to 2 .93 multiply by 10 to the power minus 8 per meter.

02:32 So the dimensions of the unit of here, m is per meter.

02:40 The general form of the index of refraction is a function of the distance travel is and which is a function of x equals m which is 2 .93 multiplied by 10 to the power minus 8 per meter into x plus 1 .0 .0.

03:02 Now we do part a now let's do part a well velocity is dx divided by d t right and from here d t can be written is dx divided by v right so we can get the time interval it takes for the light travers the distance h to the earth surface for date integrating over the distance from the entering the atmosphere to the earth's surface...

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