# Notes on Homework Please tear the "ruffle edges" off of homework done in a notebook and torn out If the homework is two pages long, staple the pages together.

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Notes on Homework Please tear the "ruffle edges" off of homework done in a notebook and torn out If the homework is two pages long, staple the pages together If you have a problem with the grading, you think something was graded incorrectly, points were added incorrectly, or something was entered wrong into D2L, put your homework and a written explanation in Molly’s mailbox.

Chapter 2: Geometrical Optics Reflection –Half-silvered mirrors –Locating an image –Multiple reflections Refraction –Total internal reflection –Fiber Optics –Mirages Dispersion –Prisms –Rainbows –Diamonds

Transmission and Reflection Incident ray Transmitted ray Reflected ray The incident ray causes electrons in the material to oscillate, which generates a new wave. Some of this new wave travels backwards as the reflected wave, some travels forward, and combines with the incident wave as the transmitted wave.

Half-silvered mirrors Silver is a good mirror because of all the oscillating charges. But what if the layer is too thin to allow the electrons to move enough to fully cancel the forward going wave? This is called a “half-silvered” mirror, because half the light is transmitted and half is reflected. Often called “one way mirrors”, this is not technically correct. Light passes both ways through the mirror, but typical use leads them to appear to be only one way mirrors.

Half-Silvered Mirrors: Application

Clicker Question If someone turns the light on in the policeman’s room, what does the suspect see? A.Only his reflection B.Only the policeman C.Both his own reflection and the policeman ??

Law of Reflection Ray tracing: 1.Draw the incident ray 2.Draw the “normal”, a line at right angles to the surface, at the point where the incident ray hits 3.The reflected ray will be at the same angle to the normal as the incident ray incident ray normal to surface reflected ray reflective surface (mirror) This law is independent of the reflective surface material. It only matters that the surface is reflective.

Notes on Ray Tracing There are an infinite number of rays that you can draw, but some of them are more useful than others: Pinhole camera Image Object

Notes on Ray Tracing There are an infinite number of rays that you can draw, but some of them are more useful than others: Pinhole camera Image Object We only NEED these two rays to find the position and size of the image

Specular and Diffuse Reflections From a mirror or mirror-like surface From a rough or highly textured surface, like a sheet of paper, cloth, or fur

Clicker Question Where are you most likely to see a good reflected image of yourself? A.A pool of water B.A polished metal sheet C.A piece of window glass

Reflections from the surface of water

Reflected Image Formation

Virtual Images The virtual image is in the same place regardless of the location of the viewer The image is called virtual because no real rays reach the image, and it cannot be seen by putting a screen at its position

Image Formation in Flat Mirrors

Multiple Mirrors B A virtual image can act as an intermediate image and generate its own virtual image A C

Multiple Mirrors The mirrors do not have to be at right angles To avoid ray-tracing at weird angles, you can extend one of the mirrors and position the image in the appropriate spot

Retroreflectors Retroreflectors are a special set of multiple mirrors that reflect incoming rays back the way they came, parallel to the incoming ray They are used in bike reflectors and other safety equipment

Refraction When going from a fast to a slow medium, light bends towards the normal to the surface When going from a slow to a fast medium, light bends away from the normal to the surface The speed of the medium is related to its index of refraction, n –n = 1.3 for glass –n = 1.5 for water Remember there is still a reflection as well Air (fast medium) Glass or water (slow) Normal Glass or water (slow) Normal Air (fast medium)

Refraction Analogy Incident ray Refracted ray in slower medium Rows of soldiers before they hit rough (muddy) slow terrain Rows of soldiers after they hit slow terrain Normal to front edge of slower terrain Muddy terrain The rows of marching soldiers are analogous to the wave fronts of light When the soldiers hit muddy terrain they slow down This causes the rows or parts of rows in the mud to move less far in a given time. Another way of saying this is that the ray perpendicular to the rows of soldiers outside the mud bends towards the normal to the muddy region for rows in the mud

Refraction Analogy Incident ray Refracted ray in slower medium Rows of soldiers before they hit rough (muddy) slow terrain Rows of soldiers after they hit slow terrain Normal to front edge of slower terrain Muddy terrain You can imagine that the muddier the terrain, the more the soldiers will slow down, and the more the “ray” will bend towards the normal. Different optical materials are differently “muddy”, which we identify using the index of refraction. Higher index materials are “muddier” and bend the ray more than lower index materials n = 1.3 for glass n = 1.5 for water n = 2.4 for diamond

Locating an Image Underwater The ray from the fish’s fin is bent away from the normal as it exits the water We interpret the ray as having arrived in a straight line. Adding one more ray from the fish’s fin (the ray normal to the surface is chosen because it is not bent) allows us to locate the image of the fish There is still a reflection here as well

Clicker Question Where does the fish see the fisherman? A.Where he is B.Not at all C.Higher than where he is D.Lower than where he is

Clicker questions Ch. 2 10.The light and fish are both underwater in the lake. Where does the fish see the light and/or the image of the light? a)Higher b)Lower c)Where the light actually is d)(c) and (a) e)(c) and (b) Top of lake

Total Internal Reflection Recall that rays moving from glass (slow) to air (fast) bend away from the normal The amount by which they bend away depends on the material For any material, we can find an angle where there is no transmitted ray, called the critical angle Glass

Fiber Optics

Mirages Occur when the index of refraction increases as you travel into the medium In our analogy with the marching soldiers this means they go slower and slower as they advance. Imagine the mud getting deeper and deeper as they advance. The ray keeps bending more and more towards the normal as the mud deepens In the limit as the muddier transitions become more gradual the refracted ray looks curved Incident ray Muddy Muddier Muddiest

Mirages Increasing temperature Hot air is less dense, and is thus a “faster” medium than cold air. This leads to the ray bending away from the normal. Because we interpret rays as traveling in straight lines, we see a “reflection” in the road, and assume it is wet. (color indicates temperature, not a sunset or sunrise)

“Flying Islands” Mirage

Decreasing temperature Cold Ocean The temperature near the surface of the ocean is often cooler than the air above it, we see the upside down version of the road mirage, as the ray travels from cool to warm air, and is bent away from normal. Again, we interpret rays as traveling in straight lines, and thus the island appears to float above the ocean (color indicates temperature, not a sunset or sunrise)

Dispersion So far, what we’ve learned about refraction means that in a prism would refract like this: But we’ve seen that what really happens is this:

Dispersion Dispersion means that different colors (frequencies/wavelengths) of light have different speeds through a material (different index of refraction), and thus refract at different angles. In most transparent optical materials, blue (high frequency) light has a higher index than red (low frequency), and thus blue light is bent more than red.

Clicker Question Where does blue come out of prism, compared to red? red blue red blue A) B) C) Somewhere in the middle

What is the normal to a curved surface and how is it used to find rays? To find the normal to a curved surface at a point where a ray hits that surface (and will be reflected or refracted) –First draw a tangent line to the curve (or tangent plane to the surface) –The normal is perpendicular to that line or plane and going through the point –Once you have drawn the normal you can draw the reflected or refracted ray

Rainbows

Diamonds Diamond has a very high index of refraction leading to a small critical angle for total internal reflection Diamonds are cut such that most light entering will hit a back facet at more than the critical angle, reflecting many times before returning out the front This long path through the diamond leads to a lot of dispersion, spreading the colors out 24.5° Diamond: n=2.4

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