Light Basics and Geometrical Optics
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This text is meant to accompany class discussions. It is not everything there is to know about uniform circular motion. It is meant as a  prep for class. More detailed notes and examples are given in the class notes, presentations, and demonstrations. See the links below.
 
Index of refraction

 

Light travels at 3.00 x 108 m/s. This is not a law but a postulate. It is a supposition we treat like a law but it is not a law. Many concepts around time dilation, mass dilation, gravity dilation, general relativity, and special relativity depend on this postulate.

If I shine a beam of light from the Earth to the Moon and reflect it to the Earth. I will find, with a precise enough measurement tool, that the speed of light is not traveling at 3.00 x 108 m/s . It's slower. In fact if I pass a beam of light through a very, very, very, dense vapor of sodium atoms in a 17 m long tank, it will take about 1 second to travel from end to end. How can this happen if the speed of light is always 3.00 x 108 m/s? (This speed of light in the dense sodium vapor is nearly 17 m/s.)

 

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  • Answer
 

The speed of light is always 3.00 x 108 m/s. But the average speed of light can be less. When light travels through an optically dense material, the photons do not travel straight across and every time the photons impact an atom, the atom excites and de-excites before releasing a photon. This takes time and slows down the photon.

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See in the animation how the light is still moving at 3 x 108 m/s between atoms. But the average speed is less than 3.00 x 108 m/s.

 


One of the properties we give materials that allow for the transmission of light is an "index of refraction." Its variable is "n." It is the ratio of the speed of light through a reference substance over the speed of light through another material. Unless stated, the index of refraction is alway in reference to the speed of light in a vacuum, 3.00 x 108 m/s.

Because the average speed of light in a substance is always slower than the speed of light in a vacuum, the index of refraction is always positive and greater than 1, (for all natural materials.) Some metamaterials can have negative indexes of refraction and some plasmas can be less than 1. But they are not the "natural" materials we typically have contact with.

 

Refraction

 

Refraction occurs when a wave is "bent." as it travels from one material to another.

In the example image, the incident light ray is the source, If ithits a surface with a higher index of refraction. Most of the ray is transmitted and is bent either towards to away from the normal line. A very small piece of the ray is reflected following the law of reflection.

The refracted ray is bent towards the normal if the substance it is traveling in has a higher index of refraction that the substance if comes from. The light ray is reversible, this means that the light ray bends away from the normal line if it enters a substance with a lower refraction index. (This concept about how it bends is important.)

The greater the difference be indexes of refractions of the substances, greater the difference in the incident and refracted light rays' angles.

This is an image of a fork in water. Notice how the image under water does not line up with the image above the water. This is due to the refraction of the light in the water.

Example
  • Question
  • Solution

A ray of light with a wavelength of blue light (475 nm long) travels between two pieces of plastic as shown below. (a) Draw what happens after the light hits the surface. Label all rays. (b) Calculate the angle of refraction for the incident ray.

 

Note: All frequencies, i.e. colors, of light do not refract at the same angle. Because of this, when white light, which contains all of the frequencies of visible light, hits a prism the different colors appear to spread out. Long wavelengths bend the least and shorter wavelengths bend more. In the visible light spectrum, red bends the least and violet bends the most.

By Prism-rainbow-black.svg: *Prism-rainbow.svg: Suidrootderivative work: Sceptre (talk)Prism-rainbow.svg: Suidrootderivative work: Sceptre (talk) - Prism-rainbow-black.svgPrism-rainbow.svg, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=6898196 accessed 02/24/2016

 

Critical Angle

 

When light travels from a higher index of refraction to a lower index of refraction is bends away from the normal. At just the right angle the refracted ray bends 90 degrees from the normal. At this angle, the incident angle is called the critical angle. You can only get a "critical angle" when the light travels from high index to a lower index of refraction. When the incident light ray is LESS THAN or equal to the incident angle, the light refracts according to Snell's law. At the critical angle, the light ray refracts 90 degrees. When the incident light ray is GREATER than the critical angle, the light ray reflects of the surface.

The incident light ray is at the critical angle because the refracted angle is 90 degrees.

Critical angle summary and example

This video can be found on YouTube at https://youtu.be/Ul-lRCiEI9A

 

This image is a great example of the critical angle. The picture is taken under water while looking up to the surface. The refraction index of water, (n=1.33,) is greater than that of air (n=1.00). When the incoming light is less than the critical angle you can see above the surface. When the angle is greater than the critical angle, you see a refection of the dark ocean bottom.

 

The lower picture is a side view of the underwater picture that is showing where the light and dark area come from. Incident angles less than or equal to the critical angle, θc, are refracted. Incident angles greater than the critical angle are reflected. Outside of the cirlce of light is dark because the reflected rays are picking up the dark color of the ocean floor. The image above, with the light circle inside of the dark surroundings, is refered to as "Snell's Window."

 

Example 2
  • Question
  • Hint
  • Solution
A beam of light travels from a plastic with an index of refraction of 2.44 to another material with an index of refraction of 1.17. What happens to the light ray after hitting the boundary betweenthe tow materials?
  • You need to figure out if the light ray is ging to reflect or refract since it is traveling from a high index of refraction to a lower index of refraction.
  • Do this by calculating the critical angle, θc, & checking if the incident angle with the normal line is greater than or less than θc. If it is greater, then the incident ray will be reflected at the incident angle and it will never travel to the other material. If is if less than the θc, then use Snell's law to determine the angle of refraction.
  • Note: to calculate θc the refracted angle is 90 degreees ...by definition.

 

 
 

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by Tony Wayne ...(If you are a teacher, please feel free to use these resources in your teaching.)