Apparent Depth
One outcome of refraction is the illusion that things appear closer to the surface than they actually are; that is their apparent depth.
In the diagram above, the fish appears to be much closer to the surface than it actually is. In general, when light emerges perpendicularly to the surface of a liquid, the submerged object's apparent depth equals its true depth divided by the liquid's index of refraction. |
Refer to the following information for the next question.
When trying to pick up a coin in a fountain or a pool many of us have "thought" that the water was shallower than it truly was. We were surprised when we reached down and the water rose much higher than we expected.
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Mirage
A second outcome of refraction is a mirage. When light waves pass over hot surfaces the heat gradient gradually alters the air's index of refraction, causing the light wavefronts to bend away from the hot surface. When those refracted rays strike our eyes, we "dot them back" to form a virtual image. The image appears as a reflection of the object, giving the impression of a wet surface. In the following diagram, the refracted rays from the real rose result in our eye "seeing" a mirage of rose.
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Total Internal Reflection
When the light source is in the denser medium as the angle of incidence increases so does the refracted angle in the less dense medium into which it escapes. Eventually, the refracted angle reaches 90º and the light is totally internally reflected BACK into the more optically dense medium. The unique angle when this phenomena first occurs is called the critical angle, θc.
In the diagram shown below, you can see that rays #1, #2, and #3 are refracted from the prism along its hypotenuse; while rays #4 and #5, after being totally internally reflected off the hypotenuse, are only refracted out of base of the prism. Also notice the relative intensity of the five rays emerging from the base: #4 and #5 are equally bright while #1, #2, and #3 show less intensity since they were only partially internal reflected off the hypotenuse.
The following formula allows you to calculate the critical angle at which all the light is totally internally reflected and none is refracted
n1 sin(θ1) = n2 sin(θ2)
n1 sin(θc) = n2 sin(90º)
n1 sin(θc) = n2
sin(θc) = n2/n1
Notice that this is merely a rearrangement of Snell's Law in which θ2 = 90º.
Remember that total internal reflection can ONLY occur when the light BEGINS in the denser medium - for example, the light starts in water and is bounced back into the water at the water-air interface. Notice that θreflected = θcritical according to the Law of Reflection.
Let's practice.
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Internal reflection is what causes diamonds to glitter and sparkle. Dispersion results in the different colors escaping at unique angles.
ideal cut
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too deep
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If the diamond is cut too deeply the light is internally reflected only once before it escapes (refracts) from the opposing side. If it is cut too shallowly, the light escapes before it is ever internally reflected even the first time.
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too shallow
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Total internal reflection is also the principle behind optical fibers. The index of the fiber is calculated so that the light is "bounced back and forth" within the fiber instead of escaping out of the fiber. Note in the diagram below that the light travels in straight paths between bounces.
The greater the fiber's index of refraction, the smaller its critical angle. Consequently, the more likely a ray of light will strike the fiber's inner surface at an angle greater than θc and being totally internally reflected.
Rainbows
A combination of total internal reflection and
dispersion is responsible for the formation of
rainbows. The critical ingredients are that the sun be behind your back and the rain falling in front of you. The raindrops act as tiny prisms to reflect and disperse the sunlight. Every raindrop disperses an entire spectra of colors; however, you only "see" one color from each raindrop - so it takes a multitude of raindrops to form a rainbow. The top raindrops will produce the "red bow" while the bottom raindrops produce the "violet bow."