www.lsmsa.edu teacher Robert Dalling's Physics Lectures (see also www.ushumans.net)
http://micro.magnet.fsu.edu/primer/lightandcolor/java.html for lenses and mirrors and all optics
What is light? Light interacts with the material of your eyeball as if light is something real, but, since you can not hold light in your hand, can light be real. How about sound? Can you hold sound in your hand? Waves are different “creatures” in that they exist only while wiggling. (Click to hear musical light.)
Produce a water wave by standing waist deep in a water filled tub and pressing to oscillate a plate, making waves that move outward. The frequency of the emitted wave matches the frequency of your plate pushing. If two people stand one wavelength apart and wiggle the plate at the same frequency, then their two waves will constructively sum together. They can also destructively interfere. Place them half a wavelength apart. Place three persons in a triangle and the waves will add in some directions and subtract in others. This means waves will move in certain directions and not in others. This is how groups of radio station antennae towers direct the signals toward the nearby city and not where there are less people.
Produce a gravitational wave by wiggling a mass. Lines of force point back to the location of the source. Suddenly move the mass below off to the right, then slow it, stop it, and move it back. The lines of force point to the "last known" location of the mass. Regions farther out point further back in time to the old location of the mass.
. .
. .
. . .
. .
. . . .
. . .
. . ■ - - - - > ■
The kinks wiggle back and forth in response, break off, and travel away. A wave occurs at a distance. Frequency of wave there due to frequency of wiggling of the source. That's why wave frequency doesn't change when waves moves into another medium. Gravitational wave moves outward at the speed of light.
Remember c=3x108 m/s and then lambda = v / frequency. The four largest moons of Jupiter were easily seen by Galileo in a 40 power telescope he made. The period squared and radius cubed equation weighed Jupiter and showed there are objects not orbiting the Earth or Sun. First good estimate of c obtained from delays in emergence of these moons from behind Jupiter. The delay occurs when light from one of its moons has to travel cross the added path when the Earth is on the side opposite the sun rather than on the same side of the sun. Knowing the size of the Earth-sun distance, d=2r, the speed of light is d=ct, where t = the 22 minute delay. Roemer measured c=3/4 today's value because the orbital radius of the Earth about the Sun was thought to be 3/4 of what we more-accurately measure today.
In a few chapters we'll learn about electrical charge, such as the static jolt you get sometimes when touching metal and that F=kqq/r2. Of interest in this chapter is that a wiggling electrical charge produces light waves in a similar manner to a wiggling mass producing gravitational waves. Sometimes a charge--electron--is forced to wiggle up and down in the antennae of a radio station at the desired frequency (the height of the tower is either one-half or one-quarter wavelength). Light is an aspect of electromagnetism. The frequency of produced light is equal to the frequency with which the charge is wiggled.
The electrical charges that make up matter are caused to wiggle whenever a light wave passes by, and this in turn causes them to emit light waves of their own. Incoming light makes a surface atom wiggle and in turn this wiggling emits light after a short delay. This is why the speed of light differs in various materials. Speed of light differs in different materials. The index of refraction n = c/v.
A diatomic atom can also rotate as a dumbbell and wiggle like an oscillating spring. Each frequency of motion corresponds to a color of light, most colors are invisible to us.
In a block of metal, there are many "free" electrons that are not bound to one particular atom and roam around the entire structure. It takes a bit of work to pull an electron out of the structure. These free electrons catch and throw back incoming light, making metal bright. They also readily transmit heat.
See em-spectrum wall chart. See all frequencies on page Z-374 and Cameron page 313.
When light hits the surface of a material, that material will absorb, reflect, and transmit portions of the light. Each material absorbs, reflects, transmits its unique set of frequencies. Glass absorbs no visible light, so we can see through it. It is transparent. The wooden desk, and nearly every other object, absorbs visible light, making them opaque. Glass absorbs infrared and ultraviolet light. We know that our tissue does not absorb x-rays but our bones do.
Glass was first made in ??th century Egypt from desert sand. We've all seen how glass bends light. Lenses were made and studied in the medieval Islamic equator. Passed on to Europeans in 14th century who made singly-held eye glasses, which were rumored to allow one to see into the future and to make the illiterate read. In the U.S., we learn that Ben Franklin arranged a wire holder to make them fall off our nose and ears every three minutes. I use velcro.
Right after Roentgen took the first x-ray--of his wife's hand (Cameron page 387) rather than risking his own, as he peeked around a distant corner--shoe stores began using x-ray machines to check the fit of shoes over our feet. Customers would stand in their shoes over the x-ray machine and get blasted. There were hopeful but incorrect reports of various rays having magical properties. The n-ray was reported to allow ...
│ ┌┐
│ └┘
│
──────────┘
>°
Here is Roentgen hiding behind a wall while x-raying his wife. Whenever people came to his lab they saw him hiding behind the corner and figured the rays wouldn't work otherwise. They still follow this practice at the doctor's office today.
Fluorescence is absorption at higher and reradiation at lowered frequencies. Black light poster absorb UV, which is of a higher frequency than visible, and re-radiate at visible frequencies.
UV below 290 nm kills germs.Which wavelength do germs use to kill us
Vitamin D is produced when UV strikes our skin. As we tan (melanin), we reflect more light and become cooler. Skin cancer most often caused by UV light. See there are more sunspots on the sun side of your skin. UV might cause our eye lenses to eventually become cloudy: this is a cataract.
Cloud tops are bright because we see light that encountered a single water drop bounced into our eye. Clouds are always bright from above. From below, they are darker because the cloud also absorbs light. The intensity of the light decays exponentially. The taller the cloud, the more absorption that has occurred, and the darker will be the cloud. Really dark clouds are really tall and block the sun. Clouds full of falling raindrops are darker yet because the raindrops are larger than cloud drops and absorb light. The size of the drops--or hailstone--increase with the number of complete circles taken by the air up and down within the thunder cloud.
We see the exponential decay of light intensity when advertising lights suddenly end in the sky, and in a fog. You can see 10 to 1000 meters through fog, depending on the amount of suspended water vapor per unit volume being 0.01 to 10 gm/m3 (Lynch p148).
When light hits your shirt, its like the cloud top in that the light bouncing toward your eye has done so after a single collision with a shirt molecule. Place a drop of water on your shirt and that spot will appear darker because light is bouncing around within the water, which is more massive than the nearby shirt. With each reflection some light is being absorbed. When a piece of light finally bounces outward in your direction, its intensity is less than the light hitting your eye after a single bounce off your shirt.
Remember the Maxwellian velocity distribution from the heat and temperature chapter. The range in speeds or temperatures of charges on the surface of the sun, at temperature 5700 °C, produces light:
.
. .
. .
% . .
^ .
│ . .
│ .
│ . .
│ . .
│ . .
└───────────────────────────────────────────────────────> lambda(nm)
0 400 700 1000 2000 3000
ΓX UV V B G R Infra-red > micro TV/FM
f=c/l
f(1012 Hz)
750 400 300 150 100
The sun emits most light in the range red, green, blue having wavelengths 700, 550, 400 nm = 10-9 meter or frequencies f=v/l=4, 5.5, 7.5 x 1014 /s.
We see those colors available, which are red, green, and blue. We see best at 550 nm = yellow-green really, So some fire trucks, golf balls, and tennis balls are painted yellow-green.
It would do us no good to see x-ray, TV, FM, microwave frequencies because the sun emits few. We feel infra-red as heat.
Animals on planets orbiting other stars will see in the range of light most emitted by their star.
Snakes see a bit in the infra-red to locate prey in the dark (This means that astronomers could train them to use infra-red telescopes.)
Most every mammal is color blind but sees much better in the dark than we do. For example, cats see night to be 10x brighter than we do: they wonder why we stumble around in what to them is just dim light. The reflective coating at the back of the cat's eye sends light past its rods a second time, so they can catch more.
$$ Light is an electromagnetic phenomena
Light waves come in many colors, most of which are invisible to us. Some of these are called radio, microwave, infrared, ultraviolet, radar, x-ray, and gamma-ray. This means that the operation of lenses, mirrors, cameras, optical fibers, infrared remote control devices and night-time viewers, radar detectors for the weather and the speed of automobiles, microwave ovens, medical x-rays, and other such devices are also described by Maxwell's equations because they involve electromagnetic waves.
Light occurs when the electrical charges of an atom are wiggled. The outer electrons often make light in the visible range. Nuclear charges (protons) make gamma rays. There are numerous arrangements of wiggling electrons. We'll get back to this in a few weeks.
Anything having to do with light, is described by F=ma using the electric force F=kqq/r2 (Maxwell's equations). Some examples of light phenomena include the rainbow patterns seen on a soap bubble or on spilled oil sheets, polarized sunglasses, the highly reflective paint used on highway signs, telescopes, binoculars, a mirage on a hot day, the colors separated by water drops and the resulting rainbow, the halo that is sometimes seen around the Sun or the Moon, the blue sky, the red sunset, the color of every object, and the streaks of light you see when you squint your eyes while looking at a lightbulb.
It was claimed above that many seemingly unrelated phenomena are simply different aspects of a more fundamental law of nature. The list of light-phenomena illustrates this point. Each phenomenon just described is simply a different aspect of electromagnetism. Each is described by Maxwell's equations, and results from pushing on an object with the electromagnetic force.
$$ illumination
The sun and lightbulbs emit light from their wiggling electrons. The everyday objects we see on the Earth do not make light from nuclear processes, they only reflect light coming from the sun or bulbs heated to 1000°F.
An object is seen when it is brighter than its background. The army is interested in this when targeting tanks and other things to kill.
Emitted light is measured in luminous flux = P with units of lumens = lm. A 100-watt lightbulb emits 1750 lm.
Luminous intensity also measured in candle power = candela=cd=P/4π.
This energy spreads out in 3D sphere having surface area 4πr2. The amount of illumination on an object placed in this light is measured in illuminance = E = lm/m2 = lux = lx. E = P/area = P/4πr2 when spreading in 3d sphere. Illuminance goes down as 1/r2.
Problem 42 page 391.
Illumination E = P/4πr2. If P is halved then how must d be changed so that E remains unchanged? E = P1/4πr12 = P2/4πr22 with P2=P1/2 this is P1/4πr12 = P1/2 / 4πr22, cancel P1/4π to get 1/r12 = 1/2r22, invert to get r12 = 2r22 or r2=r1/sqrt(2)=(3.3 meter)/1.414=2.3m.
$$ demo color combinations of RGB by overlapping colored light
White light contains an equal mixture of the primary colors red, blue, and green (RGB monitor).
Mix pairs of primary colors to get secondary colors (see Z page 383)
R+G=yellow, B+G=cyan, R+B=magenta
Page 390 problem 27: r+g=yellow. And problem 28=none
Note: R+G=yellow, B+G=cyan, R+B=magenta
r+b+g=white
white - y = b+g =cyan
w - b = r+g = yellow
w - g = r+b = magenta
A primary color added to its complimentary or opposite gives white = r+b+g. So x+green=white or x = white - green = (r+b+g)-g = r+b = magenta and the complimentary or opposite of green is magenta.
That is, magenta and green are complimentary because
m+g=r+b+g=w
yellow + blue = ?
since yellow = r+g we have y+b=(r+g)+b = white
so yellow and blue are so-called "complimentary" colors
Page 390 problem 27: r+g=yellow.
Also, x+b=white so x=w-b=(r+g+b)-b=r+g=yellow. So the complimentary or opposite of blue is yellow.
cyan and red are complimentary or opposite because
c+r=b+g+r=w
Some materials remove one color, or transmit one, or reflect one. Other materials affect 2 colors.
Red ink absorbs blue and green from white, w=r+b+g so w-b-g=r
Page 390 problem 25:
a) We said Red ink absorbs blue and green from white. The cellophane either transmits or absorbs the blue and green
b) w=r+b+g, The blue and green are being absorbed, leaving red. cellophane transmits red. We now know that in part a) it was absorbed not transmitted.
A primary pigment absorbs just one of r, g, or b:
yellow absorbs blue
RGB are also secondary pigments in that they absorb two of rgb and reflects the other:
primary pigment colors YCM=secondary light color (Yel cyan magenta)
secondary pigment colors (RBG) = primary light colors, which are RGB
Color TV produces a desired color by lighting combinations of g+r+b. It is a color additive process.
Inkjet printers produce colors by mixing the subtracting colors magenta, cyan, and yellow. Each colored line is absorbing its complimentary color and reflecting the color you see.
An example of limited human imagination is that we can not invent a new color offscale beyond red and blue.
Page 390 problem 26:
Here, no absorption occurs. If you are inside the soap bubble, blue is reflected at the surface and w-b=r+g transmitted.
Hewitt problems, pages 484
If sunlight was pure green, which color should you were on hot days? Ans: green or white would reflect the sun's color. Wearinh white clothes always reflects the sun and so aare cooler. Which on cold days? Ans: the complimentary opposite of green, which is magenta because white - green = (r+b+g) - g = r+b = magenta.
If a material absorbs or subtracts yellow from white light, which color remains? white - yellow = r+b+g - yellow = r+b+g - (r+g) = b. So blue remains.
If a material absorbs a tiny bit of blue. Which color remains? w-b=(r+b+g)-b = r+g = yellow.
If a material absorbs a tiny bit of red. Which color remains? w-r=(r+b+g)-r = b+g = cyan.
Hewitt Problems page 484.
10) White paper appears white in white light, blue in blue light, red in red light because it reflects all colors.
9) What color does red cloth appear to have when illuminated by red light? Ans: Red because it reflects red and absorbs the other two, which are green and blue. These add up to r+b=cyan. Cyan is absorbed. It appears black if ilimunated with cyan light.
12) Which spotlight color should be used to make an actor's yellow clothes suddenly appear black? Ans: The cloth reflects yellow = r+g and absorbs the other primary=blue. Shine blue light, which it absorbs, and it will appear black. Similarly, a yellow deep-sea fish illimuninated by the scant sunlight, of which only blue remains, reaching such depths, will appear black and be invisible. Red fish will be almost as dark but blue fish will be the brightest of all.
Some deep sea fish live where only a tad of sunlight reaches. Most of the remaining sunlight is blue. To be invisible, this fish should absorb blue and reflect red and green. Since w - b = (r+b+g) - b = r+g = yellow, this hardest-to-see fish should be yellow.
13) Shine separate blue and yellow =r+g light from two lamps on a white screen. What color is seen where the two beams overlap? Answer. White light is seen where the two lights overlap. What if white light is sent through a blue filter, which removes r+g, and then through a yellow filter, which removes r+g? Answer: black.
14) TVs light combinations of r, b, and g phosphorescent (glow for a 1/60 sec when hit with an electron) dots to produce other colors. Which colored dots are lit to give yellow? Ans r+g. To give white? Ans: r+g+b. To give magenta? Ans: R+B.
17) If a photo shows a blue sweater, what color is the sweater in the phot's negative? Ans: The complimentary color of blue is yellow because blue + yellow = b + (r+g) = white.
18) What color is transmitted through overlapping cyan and magenta? cyan passes b+g (to become added to make that cyan) but absorbs red. Magenta passes r+b to become magenta but absorbs the other primary, which is g. Ans: blue gets through both filters.
21. Magenta + yellow + cyan =?= (r+b) + (r+g) + (b+g) = white
22. g+b=cyan = white - ?= white - red.
23. Bananas reflect yellow =r+g but absorb the other primary=b. Illimuniate it with which color to make it appear black? Ans: Blue.
Note: R+G=yellow, B+G=cyan, R+B=magenta
r+b+g=white
white - y = b+g =cyan
w - b = r+g = yellow
w - g = r+b = magenta
$$ Hewitt shadows
Distinct shadows when a small light source illuminates a large object: H page 460. The sun is a large object illuminating a smaller moon. The distinct moon-shadow is a total solar eclipse seen in narrow bands on the surface of the Earth. By the way, at each spot on the earth, the lunar eclipse pattern repeats every 19.2 years: H page 461. The shadow of the Earth covers the entire moon so that lunar eclipses are seen throughout the daylit world. The sky is red during sunset because red is last to refract or bend around the curved earth atmosphere. Refraction of sunlight by the Earth's atmosphere makes the moon red while in the penumbra.
$$ human vision
Demo:
Hewitt page 462-3 says we have a 6.5° blind spot, which is 11 moon diameters but our brains fill in the missing info. Draw an x and o on your paper or use Cameron's five coins placed 10 cm apart (page 349). Look at the central coin and vary your viewing distance until a coin disappears. The missing info brain work is hinted at in Cameron page 362. He says to hold a six-inch paper towel tube in the middle and then look through it to see a hole in your hand.
Hewitt page 462-3 also says we see moving periphery objects but not their color. We need to see peripheral predators. Objects also seem larger when poking out from behind hiding places, like tress, rocks, and rooftops. I guess this makes the rising sun and moon seem larger.
Our eyes have cones and rods. Rods are sensitive to light and dark and allow night-time black and white vision, which we saw above is better in nocturnal mammals than humans and the small number of daylight mammals. Cones see colors. Rods see black and white light having 0.1% the intensity of light needed for color vision (Cameron p337). The three types of cones in our eyes are sensitive to low, middle, and high frequencies of light.
Primates and squirrels are the only mammals having three types of cones; the others have 0, 1 or 2. It may be that fruit eating primates developed color vision to gauge the ripeness of fruit. Female color vision is better than that of males. Does this mean our female ancestors were ...
H page 464: pupil size increases when we see something we like.
Some signal processing occurs right in the eye, not in the brain.
H page 465-6 shows brightness illusion and several others.
(Cameron cylindrical lens focuses a line: p366) Discuss that while spherical lenses focus light onto a tiny circular spot, a cylindrical lens instead focuses into a line along its central axes. Some colonial people placed a rotatable cylindrical lense in their eyeglass frame. They would rotate the lens to the right angle to see more clearly.
Demo:
optical illusions
$$ pinhole camera
demo pinhole camera,
if you placed film out in the open, you'd get a total white picture as light came in from every angle and object. The pinhole means you get light onto the film only from those objects in front of the hole and not from objects off to the side. Trace rays from objects, through the hole and onto the film. See the image is flipped. Larger hole allows more light and shorter exposure times.
Flipped image as in our own eye, which is nothing but a pinhole and a focusing lense. Newborns take a few weeks until their brain does an extra flip. One scientist wore lenses over his eyes so that everything was upside down. After six months his brain produces an extra flip. Then he removed the lenses and waited six months to again be reflipped. He dropped many hamburgers and children and likely broke all seven arms during this year.
$$ polarization
see page 387 showing a vertically oriented wave passing through a vertical slit but not a horizontal slit.
demo polarization of rope wave.
Level 3:
The electric force is a vector. A traveling light wave is a traveling electric-force wave. It is a vector having components perpendicular and tangential to any surface which it encounters. The reflected wave has differing components. There are complicated equations involving n giving outgoing components Ex and Ey in terms of incoming components. At a certain incoming angle, the reflected wave will have one zero component and one non-zero component. It will be polarized.
This is Brewster's angle for 100% polarization: tanΘp=nr/ni.
example:
what is the Brewster's angle for incoming light in air striking water having index n=1.33? Θp= tan-1(n) = tan-1(1.33)=53°.
Demo Θp for water with polarizing filter.
example:
what is the Brewster's angle for light in air hitting glass having index n=1.52? Θp= tan-1(n) = tan-1(1.52)=56.7°.
As sunlight scatters from air it is partially polarized. The percent polarization differs through ring shaped regions centered around the sun. This is seen when you look around the sky with your polaroid sunglasses. Bees and homing pigeons make use of this effect to navigate. Recently, dung beetles were found to use moonlight polarization to aid navigation.
Some organic molecules rotate the polarization angle of passing light. The amount of rotation is proportional to the concentration of that material in solution.
$$ interference
interference when path length = (2n+1) 1/4 wavelengths, n=0,1,2..
demo soap bubble interference
soap bubble
two radio towers separated by 1/2 wavelength
$$ causes of color article
$$ light devices
cd and dvd
blue light laser has smaller wavelength so more densely packed digital info.
$$ scattering
for wavelengths much larger than the scattering object, Rayliegh scattering (1871) is proportional to the fourth power of frequency (Jackson page 423). Atmospheric molecules are 10-12 meter in diameter and scatter visible light having wavelength 550 nm = 5.5x10-7 meter. Blue light is scattered 10 times as much as is red, so the sky is blue. Violet is more strongly scattered but our eyes are less sensitive to violet then blue, so we call it blue. We perceive it to be predominantly blue. Might squirrels or eagles see violet? Can we measure this?
Haze = light scattering
http://www2.nature.nps.gov/air/edu/someair/bigpicture/IIIA5a.html
The Moon has no atmosphere, so the stars are visible during daylight hours.
A good portion of us are color blind, or do not see one color, e.g. green.
The overhead sun traverses 30 km = 19 miles of air. Draw two concentric spheres. One of radius 6.4x106 meter. The other having radius 6.37x106 m + 30000 m =6.4 x106 m. How far is the horizon in the direction of the setting sun?
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. . .│h .
r+h . . . . .
. .. . .
. . . . .
r .
.
The Earth has radius r, the atmosphere height h, the sunset line of length d goes from the outer circle, radius r+h, then tangent to Earth. The triangle says r2+d2=(r+h)2 so d=sqrt[(r+h)2-r2]=620 km, compared to 30km straight up. 620/30=21. But the air density decays exponentially upward. So the air near the ground counts much more. This means setting sunlight traverses 38 times as much air as noon-time sunlight. Sunlight traverse enough air for all the blue to have been removed, leaving red. The sunset is red.
Scattering from water vapor makes sky less blue.
Lynch plot on page 149 shows the scattering efficiency of drops of various radii. The relative proportions of scattering, absorption, or refraction of various atoms, dust, and other particles
Dust and larger particles scatter lower frequencies, and help color the sunset but also make the daytime sky whitish because a range in the size of the scattering particles would scatter all frequencies and add up to white. The largest particles absorb rather than scatter and make that brown haze of smog. (Easy but more expensive to arrange it such that homes, autos, and factories collect all by-products and so emit nothing into the environment.)
$$ transmission, reflection, and absorption.
When a wave moves from one medium into another, portions are transmitted, reflected, and absorbed.
Each material absorbs and transmits differing frequencies. Glass transmits UV to the car seats. Which try to re-radiate at lower, infra-red frequencies which the glass does not transmit. The interior of the car heats up. Place a gallon of water in there and it will also heat up, too.
The amount absorbed is that exponential intensity decay we met before I=Ioekx
level 3: (Westgard page 273)
reflected and transmitted fractions: when Θi = 0° (and E is normal)
T = 2/(1+nr/ni) and R = │ ( 1 - nr/ni ) / (1 + nr/ni) │
which gives R=0 when nr=ni (impedance matching)
example:
going from air ni=1 to glass nr=1.5 we get
T=2/(1+1.5/1)=0.8 and R=(1-1.5) / (1+1.5) = 0.2
and going from air ni=1 to diamond nr=2.5 we get
T=2/(1+2.5/1)=.57 and R = │ (1-2.5) / (1+2.5) │= 0.43
Diamond reflects twice the light incident normally plus has low critical angle below. (Ask the internet for Diamond mining in Sierra Leon and Liberia.) Does anyone have a diamond on their shoe to demo? Hewitt page 503 discusses diamonds.
level three homework:
If nr>ni then nr/ni>1 and R=( nr/ni - 1 ) / (1 + nr/ni). To get R=T/2, what must be nr/ni? Answer: 1=nr/ni-1 or 2=nr/ni
level three homework:
For nr>ni we have nr/ni > 1 so 1 - nr/ni < 0 and
│ 1 - nr/ni │ = nr/ni - 1, then solve for nr/ni such that R=T.
Answer: T = 2/(1+nr/ni) = R = ( nr/ni - 1 ) / (1 + nr/ni) denominators cancel to get 2=nr/ni+1 or 3=nr/ni or nr=3ni. Get the same result for nr<ni.
intensity of water on shirt absorbs more light, hence less is reflected and the wet area of the shirt seems darker,
see no scattered blue sky from bottom of deep mine shaft,
All light motions are identical in the reverse direction. When light travels from either direction, left to right, or right to left, the same results are obtained.
During the day, you can see the street outside your window but a pedestrian can barely see in. The inside might be 90% darker. This situation reverses at night.
"One-way" mirror: reflects 95% and transmits 5% into a dim room. Still two-way, about 0.95% one-way. Light a cigarette in the dim room and the people in the lighted room will still see it. Cup your eyes against the mirror and you might see into the darkened room. If the darker side is slightly lit, and you're in the lighter side and shut off your lights then you will suddenly be on the darker side. you would then see the dim other side.
$$ reflection
mirror: angle of incidence = angle of reflection.
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Why does a mirror reverse left and right but not up and down? Trace the rays. The mirror actually reverses front and back, not left and right, or top and bottom. Your front is flipped, you are rotated 180°.
To see your whole body, from head to toe, how long must be a mirror? Just half your height. A ray from your toe hits your eye a distance above the ground equal to half your height--not matter the distance between you and the mirror. for the same reason, a handheld mirror shows an area of your face equal to twice the area of the mirror--no matter how far away the mirror is.
│
■ . │ or place another person over her
/ \ . and replace the mirror with a holy wall
|. │
│
To see this you might replace the mirror with a hole cut in the wall to allow a person on the left to see the whole person placed on the right.
Place a mirror under a table. You don't see the top of the table, yo see its underside. Place a tall table holding a plate into a shallow pond, then look at it from beneath the water. You will see the bottom of the table but not the plate. Place the table on the ground and plavce a mirror halfway between you and the mirror, you will see the same view as when you were underwater. You will not see an inverted image of the table. Look for this in paintings to see if the artist is correct or wrong.
example
Take two mirrors placed at right angles to each other
outgoing
│ .
│ .
│ ß .
│ . incoming
b.─────── line 2 .
│ . .
│ß .L │ .
│ . Θ│Θ .
│90 α . │ .
└─────────.───── line 1
c │a
│
│
Incoming ray strikes point a and reflects to point b. Line L goes from point a to b.
α + Θ = 90 so α = 90 - Θ
Triangle abc gives 180°=α+ß+90 so that ß=90-α, which is Θ. This means that the outgoing angle matches the incoming angle, for any Θ.
.
│ . . │ .
│ . . │ .
│ . . │ .
│ . . │ . .
│. . . .
. . │ . .
└.─────────────── └───────────.──
Same thing happens in three dimensions.
The double mirrors reverse left and right twice. Wink your left eye and you'll see your sides as others do. If your car's rearview mirror was a double mirror then images wouldn't be reversed. Too late to switch now, we'll all crash.
demo 2d and 3d reversing mirrors. Want to see something really really hideously ugly? Can anyone see themselves in this mirror?
One such mirror left on the moon to reflect laser light and enable a measurement of the earth-moon distance.
Some shop windows are tilted to avoid having sunlight reflect back at the viewer who then could not see the products.
The rearview mirror of our cars are similarly tiltable. The daylight setting reflects light off back side of mirror's glass. The nighttime setting reflects light off the front of the glass.
example
serway problem 22.11 page 725
.
normal . │
. . │ ß .
. . a
. . . │
. . │
c■ │ .
. σ . α │ α .
. Θ .n ε . │ .
d. . . . . . . . ■ . .
b
ε = 90 - α
Triangle bcd gives 180 = Θ + σ + ε = Θ + σ + 90 - α
so σ = 90 - Θ + α
The angle between ray bc and the normal to ray dc is δ
δ = 90 - σ = 90 - ( 90 - Θ + α ) = Θ - α
Triangle abc gives 180 = 2α + 2δ + ß so
ß = 180-2α-2δ
ß = 180-2α-2( Θ - α ) = 180° - 2Θ
$$ Refraction
Refraction occurs at a boundary between two media having different sppeds of light. The change in speeds causes the waves to rotate toward the slower end. When light travels from one medium to another, its frequency is unchanged in the new material because the wiggling source hasn't changed--but wavelength in the new material changes to v-new / f.
transmission (bending is refraction)
Speed of light differs in various materials, v=c/n, where n is the index of refraction. example: For water, n=1.33 and for glass n=1.5 (see table on Z page 397 plus Cameron page 341 shows n for 5 parts of the eye, n= 1.33 - 1.41 except the lens which has layers with n ranging from 1.07 to 1.7. Incoming light hits the cornea first, having n=1.37). The speed v changes so lambda does too but f never changes as the wave moves from one material into another.
We saw that wave fronts pivot or bend toward the slower-moving side. see page 400 for refraction of light entering a cubical glass:
\ | /
\ fast | slow /
\ | /
\ | /
\ | /
\ | /
There is a science fiction story about life on a planet where R, n, and vertical temperature profile result in light bending clear around the planet. Picture concentric spheres of decreasing ns. A ray headed 45° off the ground's normal, or tangential to the surface, will bend downward, pivoting towards its slower moving end.
In a special case, you get a similar thing for shallow angled sound or light waves reflecting all along the inner wall of a 360° cylindrically curved wall.
v = lambda f so lambda1 = v1 / f and in another material lambda2 = v2 / f. The ratio lambda1 / lambda2 = v1/f / v2/f = v1/v2
= c/n1 / c/n2 = n2 / n1 and also lambda1 n1 = lambda2 n2
compared to the wavelength in a vacuum = lambda0, n=lambda0/lambda-n
Refraction equations are
sinΘr vr ni lambdar
----- = -- = -- = -------
sinΘi vi nr lambdai
Snell's law: nisinΘi = nrsinΘr
Notice that for Θi=90° there is no refraction-bending.
Most homework problems want to know Θr=? so we solve once and for all for
Θr = sin-1( ni/nr sinΘi ).
This makes light bend toward the normal when moving from smaller to higher n.
n
───── . │
. Θi│
───── lambdai . │
. │
───── ni < nr .│
...............................│....
───── nr │.
───── lambdar │
───── │ .
│Θr
│ .
reverse process: light is bending away from the normal when moving into lower n
Want to reach a spot in a river in the least time? Should you run along the bank until directly opposite spot then jump in, or jump in erlier and swim, or jump in right away and swim the whole way? The solution to this "least time" question is the path followed by a refracting light ray. The light traverses the distance on the path that minimizes the journey's time.
Makes the pencil look bent when placed in a glass of water. Why? Because light waves bend toward the region where they move more slowly.
example S 22.4
n=1.52 for crown glass, Θi=30° then
Θr = sin-1( ni/nr sinΘi ) = sin-1( 1/1.52 sin30 ) = 19.2°
example S 22.5
What is the speed of light in fused quartz, having n=1.458?
vg=c/n= 3 x 108 m/s / 1.458 = 2.06 x 108 m/s.
For a wavelength of lambda = 589 nm in air, what does it become in fused quartz?
lambda = lambdao/n = 589 nm / 1.458 = 404 nm (nm = 10-9 m)
And what is f?
f=c/lambda=vg/lambdag = 2.06 x 108 m/s / 404 x 10-9 m= 5.14 x 1014/s
example
S problem 22.12 page 725
Light of wavelength 436 nm in air enters a fishbowl filled with water. then exits through a crown-glass wall. Find its wavelength in water and glass.
lambda = lambda-vacuum/n
lambda-water = lambda-vacuum/n-water = 436 nm / 1.33 = 328 nm
lambda-glass = lambda-vacuum/n-glass = 436 nm / 1.5 = 291 nm
example
Find Θr for light moving from air to lens having nr=1.41. Repeat for light moving from water into that lens. To focus their eyes, divers need goggles to supply a thin layer of air. This is the reason we can't focus under water. (Recent research found newborns kept swimming maintain their stronger eye muscles and so can focus underwater.) Fish can't focus on land--they would need water-filled goggles. We focus by changing the radius of curvature of our lens; Fish focus by moving the lens forward and backward.
example
a ray passing through a slab emerges parallel to the incoming light but is offset. (Z problem 19 page 411, S prob 19 and 20 page 726).
. Θi
. │
.│
┌──┴────────┐
│ . │
│ Θr │
│ . │
└──────┬────┘
│ .
│ .
│Θ3 .
sinΘr= ni/nr sinΘi
sinΘ3= nr/ni sinΘr = nr/ni ni/nr sinΘi = sinΘi
so Θ3=Θi
Get the same result for several layers.
Example (Z 17.47 and S 22.20)
By how much, w, is the outgoing ray offset from the incoming ray?
. .
. .
. . .
normal │ Θi . w . .
^ │ . .
b│ c│ . .
─┬─ ┌──────────────────────┼────.──────────────.────┐
│ │ │-x- │ . │
│ │ │ . .│ . │
│ │ .│ α │ . │
│ │ . │ . │ . │
y │ . │Θr │ . │
│ │ . │ .Θi│ . │
│ │ .90° │ L . │
│ │ . . │. . │
│ │ . d . │ . │
─┴─ └──────────────────────.────────────────────────┘
. │ a
. │
. Θi│
.
.
Draw Θi and Θr then a, b, c, d
Label incoming, refracted, outgoing, and their extensions
label ΘI=Θi, and show two parallel lines cut by a transversal, a different set of transversal lines gives Θi = α + Θr so α = Θi-Θr
Triangle abc:
L is the hypotenuse of this triangle:
L cosΘr = y so L = y/cosΘr
From triangle adc: L sinα = d. With L and α from above
d = ( y / cosΘr ) sin( Θi - Θr )
= ( 25 mm / cos28 ) sin( 45 - 28 ) = 8.3 mm
The offset distance depends on n, Θi, and the thickness of the lens.
example
Z problem 17.46
normal b
. .
. .60 .
. a . .
Θi ■ . A B .
. . .C Θr . .c
. . 60 d.n 60 ■
. . . . . . . . . .
Θi=45 so then
Θr = sin-1( ni/nr sinΘi ) = sin-1( 1/1.52 sin45 ) = 28.1°
What are angles A, B, and C?
A = 90 - Θr = 90 - 28.1 = 61.9°
Triangle abc gives 180° = A + 60 + B so
B=180-60-61.9=58.1°
Angle D = 90 - B = 90 - 58.1 = 31.9
Triangle adc gives 180 = Θr + D + C so
C = 180 - 28.1 - 31.9 = 120°
For the refraction at the exit from incident angle D:
Θr' = sin-1( ni/nr sinΘi ) = sin-1( 1.52/1 sin31.9 ) = 53.4°
example
S 22.14 page 725
n=c/v=3/2.17=1.38
Θr = sin-1( ni/nr sinΘi ) = sin-1( 1/1.38 sin23.1 ) = 16.5°
example
S 22.17 page 725
ni/nr=vr/vi=0.9 where i is the material surrounding the liver
Θr = sin-1( ni/nr sinΘi ) = sin-1( 0.9 sin50 ) = 43.6°
. Θi
│ d d │
──⋅────┼────┼──────
h │. │
│Θr. │
──┴────⋅────────────
│
tanΘr = d/h so h = d/tanΘr = (6 cm)/tan43.6 = 6.3 cm
example
S 22.21 page 726
R=10 cm
L = 5 cm
ni=1
nr=2
. .
. │
─┬─ ──────■. │
L . │α . R │
│ │ . │
─┴─ .────┴──────────┘
R cosα = L so α = cos-1( L/R ) = cos-1( 5/10 ) = 60°
n
.
.
.
. . .
Θi .a . │
──────────────┬──────────────────── │
. │ .. │
│ . . │
. │ . . │
│ . Θr . │
. │ . . │
│ . Θi' . │b
. │ R . ────ε ■────────── n
. │
. │
. δ │
Θi . │c
─────────────────┘
Θi = 90 - α = 30°
and
Θr = sin-1( ni/nr sinΘi ) = sin-1( 1/2 sin30 ) = 14.5°
δ = 90 - Θi = 90 - 30 = 60°
triangle abc: 180 = Θr+δ+ε = 14.5 + 60 + ε so
ε = 180 - 14.5 - 60 = 105.5
ε swings from ray ab to ray bc
Θi' = ε - 90 = 105.5 - 90 = 15.5
then, exiting is
Θr' = sin-1( ni/nr sinΘi ) = sin-1( 2/1 sin15.5 ) = 32.3°
$$ total internal reflection
when moving into a higher n2, Θr=90° occurs for the critical angle sinΘc=n2/n1.
(see Z fig 17.2 page 403.)
example:
What are the critical angles for light emerging from water, glass and diamond into air?
water: Θc=sin-1(1/1.33) = 49°.
glass: Θc=sin-1(1/1.5) = 42°.
diamond: Θc=sin-1(1/2.42) = 24.4°.
cubic zirconia has n = 2.2, which is close to diamond.
example
Z 17.52 page 412.
What is Θr when moving from ni=1.33 into nr=1?
Θr=sin-1[ ni/nr sin(Θi) ] = sin-1[ 1.33/1 sin(55) ] = sin-1[ 1.09 ] which is undefined. This tells us that total internal reflection occurred instead of refraction out of the bucket.
When a submerged diver looks up toward the sun, the 180° horizon is compressed into 2x49°=98° with waves making fuzzy edges (see Lynch page 79). This is the same thing a fish sees.
optical fiber
Light bounces repeatedly off the walls but also decays in amplitude and so requires repeated boosting.
. .
─┐ . . . .
. └────┐. . ┌──
. └─┐. . ┌────┘ . .
. └───┐. . ┌────┘.
. └───────┘ .
. . .
Doctors use optical fibers to bend light around corners. A car dashboard needs only one lightbulb if an optical fiber directs the light to each gauge. Much communication is done with fiber optics. Copper cables using electricity are being replaced with fiber optic cables using light. Much faster and more info packed in.
The hair of a polar bear is a hollow optical fiber. The hair traps ultraviolet light and conducts it down to the bear's black, light absorbing skin. This maximizes the amount of absorbed heat of sunlight.
Diamond's small critical angle means for angles 24 to 180 all light is internally reflected. Light likely bounces around a few times and emerges closer to normal than does light escaping from glass. This makes the diamond seem brighter because more light emerges normal to its planes and so toward you than off to the side.
Much internal reflection for diamond would make it a good fiber optic cable.
Binoculars (periscopes can, too) contain prisms using total internal reflection. see hewitt page 503.
. outgoing
. . ^
. . │
. ┌─────────┐ . │
│ │ │
│ │ │
│ │ │
│ . │ │ .
│ . │ │ .
│ .└───────┘.
│ . .
ingoing .
$$ dispersion
In addition, diamond's high dispersion makes red hit your eye causing "fire." Plus, diamond's total internal reflection angle for blue light is less than for red light. Is cubic zirconia as dispersive? Are diamonds the meaning of life?
demo prism
measure wavelength with diffraction or refraction spectrometer
demo spectrometer
Each chemical atom emits its own frequencies (we'll see its Bohr's orbiting electrons changing levels). Geographic source of archaeological chemical, we did beryllium in copper mix, can determine any chemical from a table of known wavelengths. Show CRC table of wavelengths.
Index of refraction differs slightly for differing wavelengths. This is the reason prisms separate white light into that band of colors.
show n(f) plot for water in Jackson page 291
example:
(S page 727 problem 32)
Compare refraction angles for blue nb=1.650 and red nr=1.615 light moving from air ni=1 into glass when Θi=30°.
Θb = sin-1[ ni/nb sin( Θi ) ] = sin-1[ 1/1.650 sin( 30 ) ] = 17.6°
Θr = sin-1[ ni/nr sin( Θi ) ] = sin-1[ 1/1.615 sin( 30 ) ] = 18.0°
difference in angle is 0.4°.
Dispersion causes chromatic abberation
demo
blue ring on 3x airplane scope
green ring on sun
Sunlight refracts through the atmosphere of Earth and lands on the eclipsed moon, which appears red because it refracts (disperses) last, blue is first.
The setting sun and moon are actually a few diameters below the visible horizon. That's why they appear to be squished (see Lynch page 48).
$$ rainbow
see Z page 409, better is Hewitt page 499.
Show incoming light hitting a sphere along its center line. Then incoming rays just above and just below. As we work toward the upper edge, see Bohren page 172, the light escapes increasingly toward the bottom but reaches a lower limit before once again rising higher. This is the primary rainbow angle.
The light forming the rainbow is coming from zillions of water drops, some are a few meters away from you and others are hundreds. Each drop uses all its might to contribute its portion. A drop that is shaded from the sun can not contribute.
A rainbow is circularly shaped because the drops are. The circle is centered around the sun, which is at an angle above your back. This slant makes the rainbow appear parabolic to you. The angle between the sun and the exiting red light is 42°; blue light exits at 40°. If the sun is more than 42° above the horizon, the primary bow will not point at you--at 51°, the secondary bow can not be seen. If the sun is on the horizon then you can see 180° worth of rainbow. From an airplane, one can see the entire circle. Someone standing nearby, sees a rainbow from a different collection of drops. The red light you see is coming from drops lower in the air than those drops sending blue light your way.
We sometimes see two rainbows. The first rainbow is the brightest. Since some is reflected and some transmitted, the second rainbow is the result of an extra reflection, which makes its colors occur in reverse order as the first. There are several rainbows but each is about 10% as bright as the previous. About 15 rainbows can be detected using sensitive light meters in the lab (Bohren p177 shows many):
Each and every culture has a unique explanation of the rainbow
(Bohren p25):
1) the Mojave of Arizona say God needs to display all of the colors of light to end a thunderstorm
2) Homer and one African people say the rainbow is a giant snake out to graze after the storm
3) An Eastern European people said it sucked water from the seas to sprinkle it as rain
4) It is a stream that souls in heaven drink
Some say to point at the rainbow is bad luck causing an ulcer, to be struck by lightning, or the loss of that finger. Those who pass beneath the rainbow will change sex. That's what happened to Arnold Shwartzenegger. Finding the end of the rainbow brings health, beauty, pearls, gold.
$$ concave and convex mirrors
Draw light rays from a nearby object, they form a steep triangle. But light rays from a distant object form a barely slanted triangle. These rays coming from infinity are parallel.
A spherical (radius r), concave mirror focuses light from infinity through its focal point, which is located at half its radius, f=r/2. It does this for incoming light near the center line of the mirror. Light from an object moves toward the mirror. A ray striking the center of the mirror surface always reflects back at the same angle.
Demo that a 10-cm concave mirror (or a convex lens) focuses sunlight, which is coming through the window, onto your palm held 10-cm from the mirror or lens. Often in physics, the number 20 is 90% of infinity. For the mirror, infinity is about 20 f. Let each student hold the mirror and lens and see when their eye is farther away than a focal length then the mirror image is inverted and smaller. Place eye at f and see a big blur because the image is at infinity. Place eye closer than f and the image is right side up and enlarged. Place eye at r and image is at r. If you were in the center of a mirror-lined sphere, anywhere you look you see right back to yourself. Estimate magnification by simultaneously looking through and around the lens.
Show 12-inch concave and convex mirror. Convex mirrors always have upright, reduced image. Car side mirrors are convex. Blind traffic intersections sometimes have convex mirrors.
Concave mirror
f = r / 2 is positive f because its on the real image side,
light from infinity (=20f) comes in as parallel rays and is focused through f
Case I:
<--------- object beyond focal point ---->
forms real, inverted images
1 .
^ .......................................
| 2. . .
. r f .
. .
. . .
. .
. .
Draw four rays.
Ray 1 comes in from infinity, parallel to the axis of the mirror and bounces through focal point f.
Ray 2 passes first through f, bounces off the mirror and then moves straight out to infinity moving parallel to the axis of the mirror. This is the reverse of ray 1.
Ray 3 passes first through the center of curvature point r, bounces off the mirror and returns straight through that center of curvature point r, once again.
Ray 4 bounces off center of mirror surface and reflects with θi = θf. (Can’t draw accurately.)
Two lines make a point. The third and fourth lines check and double check that you are right.
. /
.
/ . ■) eye here looking toward mirror
\ .
. \
Draw a convex mirror and show that the rays dont cross but appear to emenate from a point behind th mirror.
Algebraically (the lens/mirror equation)
1/f = 1/di + 1/do
(notice if do=∞ then 1/do=0 and di=f)
(notice if do=2f=r then 1/f = 1/2f + 1/di, so di=2f=r)
= ( di+do ) / dido
so f = di do / (di + do). Some homework problems give di, do and ask for f. Others ask for di:
1/di = 1/f - 1/do = ( do - f ) / fdo
di = fdo / ( do - f )
(notice (do-f)>0 when do>f)
do or di, no difference, so interchange o and i subscripts to avoid repeating the algebra
do = fdi / ( di - f )
magnification
An object of height ho, its image appears to have height hi, the magnification is
m= hi/ho = -di/do
A negative m means the image is inverted.
Brightness of image depends on area of mirror or lens. Their magnification does not. I sat my 6" telescope mirror on my car seat and went to work. Its 48" focal length was almost the distance between the seat and the headliner. Through the day the sun passed overhead and burned a 3" swath across the headliner. Whitehead's mirror caught her basement on fire when the sun went passed the window later in the year.
Concave mirror--first surface is caved inward, f on real side and positive, object and image positive on real or first side otherwise negative:
Case II:
object between focal point and mirror <--->
forms virtual, upright images
.
.
. │
r f │ . │
o . i
.
.
.
Draw ray 1 from object, parallel to axis, bounce off mirror and then pass through f. Draw ray 2 from f through the top of the object, bounce off the mirror and travel parallel to the axis back out to infinity. Draw ray 3 through center of mirror, bounce with reflection angle equal to incident angle. As we try to draw rays 1-3, we see that the rays are not meeting. So we extend them backwards where they will meet and form a virtual image appearing to emenate from behind the mirror.
Example
Serway page 758, problem 6
r=20 cm and f = 10cm.
find di for for each of do = 40 cm, 20 cm, 10 cm
di = fdo/( do -f ) m=-di/do
for do=40 we get di = 10x40/(40-10)=13.3 cm =-13.3/40=-0.33
image inverted and on object side=real
for do=20 we get di = 10x20/(20-10)=20 cm m=-20/20 = -1
image inverted and on object side=real
for do=10 we get di = 10x10/(10-10)= ∞ no image formed
Convex mirror: always produce virtual, upright, reduced images
Case I Case II match, dont matter if do<f or do>f. But must use a negative value for f because f and r are on the virtual side of the mirror.
.
.
1 . .
. . . . . . . .
2...... │ . . .
│ 2 . . f r
3 . .1
.
.
.
Ray 1 from infinity, along axis, "bounce" toward f.
Ray 2 from f straight toward object until it ray hits the mirror, then instead travel along axis out to infinity
Ray 3 from object straight to mirror center and bounce off mirror at reflection angle equal incident angle. The rays don't cross so virtual image.
Summary of mirrors:
lens/mirror equation: 1/f = 1/di + 1/do
which we solved for f, do, and di.
magnification equation m = hi/ho = -di/do
Concave mirror--first surface is caved inward, f on real side and positive, object and image positive on real or first side otherwise negative:
Case I: object beyond focal point forms real, inverted images
Case II: object between focal point and mirror forms virtual, upright images
Convex mirror: always produce virtual, upright, reduced images, f on virtual side and negative,
sign conventions, see Z page 420:
The real side of the mirror is the side where an actual object can be placed and seen. The image is real if its on the same side of the mirror as the object; its negative when its on the other side of the mirror. Rays dont pass through the point, instead they appear to emanate from a point behind the mirror where the ray's backward extensions meet.
+ -
do real neg if the image from a first mirror is the object for 2nd
di real neg
f concave convex
example Serway 23.8 page 758
What happens to the image as you move from do=6f toward a concave mirror? with r=1 meter and f = 0.5 meter and do = 3m downto 0 meter. (Lay a concave mirror on the ground, hold a ball above its center but at a distance greater than f, say 6f with r=1 meter and f=0.5 m. Then drop a ball and describe the motion of the image.) For d=3m we get di=0.6m, (for do=1 get di=1,) do=0.5 get di=infinity, for do=0 get di=0. When the object moves in from do=6f toward the mirror, the image moves outward from di=0.6m away from the mirror and toward infinity. (As do crosses r di=r.) As do crosses f, the image jumps from +infinity to -infinity.
b) the ball and its image are both located at the same point when do=r=1m. The time to fall from y=3m to y=2 m is y=0.5 gt2 so t=sqrt(2y/g)=0.64 s.
example Serway 23.9 page 758
do=1 cm, di=-10 cm. f=dido/(do+di)=-10x1/(1-10)=1.1=r/2 so r=2.2 cm and m=-di/do=--10/1=+10
example Serway 23.10 page 758
A convex mirror with r=-0.55 meter, and f=-0.275 and do=10 m.
di=dof/(do-f)=-0.275x10 / (10- -0.275) = -0.27 m and the magnification is m= -di/do = --0.27/10 = +0.027 the image is upright
example Serway 23.13 page 758
convex mirror = christmas tree ornament, diameter=6cm so r=-3cm and f=-1.5cm. do=+10cm di=? m=?
di=fdo/(do-f)=-1.5x10/(10--1.5)=+1.3 and m=-di/do=+1.3/10=0.13
$$ lenses Finish lens today, homework due Monday for Ch 18
Convex lens
(Next lab is telescope, then difraction grating, then Flying Circus item of your choice)
Convex lens--first surface is curved outward so its edges are thinner than its center, f positive, forms real images on the other side of the lens opposite the object, do positive (when a virtual image from another lens is used as an object then it might be on the virtual side? if forced to define that side as neg), di is positive on the other side of the lens from the object and that image is real, di is negative if on the object side of the lens and that image would be virtual. When choosing signs for do and di, the object side is positive and the other is negative.
Draw three rays.
Ray 1(.) comes in from infinity, parallel to axis of lens, is focused to point f on oppposite side of lens from the object.
Ray 2(■) passes unrefracted through the center of the lens.
Ray 3(°) passes through f on the same side of the lens as is the object, then moves parallel to the axis of the lens as it travels to infinity on the other side of the lens. This is the reverse of ray 1
Z book uses only rays 1 and 2. Two lines make a point, and a third line checks that you are right.
The following diagram of a convex lens is much the same as for the concave mirror.
Case I:
<--------- f=pos, object beyond twice the focal point, do>2f ---->
forms real, inverted images
^.....................
│ ° ■ . . .
│ ° . . .
│ r f . ■ . f r
° . .
°. ° . ° °
. . ■
.
Case II:
<--------- object between f and 2f, f<do<2f ---->
forms real, inverted images
.
. .
. .
r f . . f r
. .
. .
. .
.
Case III:
<--------- object at f, do=f ---->
forms real, inverted images
.
. .
. .
r f . . f r
. .
. .
. .
.
Case IV:
object between focal point and mirror <--->
forms virtual, enlarged, upright images that will
be on the same side of the lens as is the object, di is negative
. ° °
│ .
│ ■ . .
│ ......... .
│ °│ . .
r│ f │ . ■ . .f r
. .
. . ■
. .
.
Concave lens is thicker on its outer edges, the first surface is caved inward, f is neg, always produce virtual, upright, image sometimes enlarged sometimes not
...............
. .
. .
r f . .
. .
. .
. .
...............
Case I
<--------- object at infinity
forms virtual, inverted images. eye on right looking left,
see image emanating from f
rays don't pass
through f but appear
to come from there
............... .
---------------> . . . .
. . .
---------------> r f . . . . .
. . .
---------------> . . . .
. . .
...............
Case II
<--------- object beyond 2f -------->
object between focal point and mirror <--->
forms virtual, enlarged, upright images that will
be on the same side of the lens as is the object, di=neg.
...............
. .
. .
r f . .
. .
. .
. .
...............
Case III
object at focal point <--->
forms virtual, enlarged, upright images that will
be on the same side of the lens as is the object, di=neg.
...............
. .
. .
r f . .
. .
. .
. .
...............
Case IV
object between focal point and mirror <--->
forms virtual, enlarged, upright images that will
be on the same side of the lens as is the object, di=neg.
...............
. .
. .
r f . .
. .
. .
. .
...............
small lenses occur in a mensicus, and in a hair or water-walking insect leg or needle floating on water, or waves on the top of a pool. We then see light refracted through that little "lens" of material.
Highly reflective coating on highway signs contains many small glass beads which return light back to where they came:
.
. .
>.........................
. .
. .
. .
<..........................
. .
We see refraction in drinking glasses and bottles, too--as Picasso told his only girlfriend "even though you're refracted, you're still you."
Hold this lens in your hand.
Notice that the magnification depends on the object distance do.
sign convention: (the real side is opposite actual object side)
positive negative
f converging diverging
f on real side f on virtual side where teh actual object is
do
di real side virtual side
m upright inverted
Some problems ask for one of these four ray diagrams. Most ask for a number from the lens/mirror equation.
Class Quiz:
A converging lens has a radius of 20 cm. Including the sign, what is f? What is di for do=+40cm? What is m? Describe in words if the image is real or virtual, enlarged or reduced in size, upright or inverted. Explain in words how you draw three rays to locate the image.
f=+10 cm.
di=fdo/(do-f)=1040/(40-10)=400/30=40/3=+13.3 cm
m=-di/do=-13.3/40=-0.33
The image is real, inverted, and reduced.
Example:
An object is placed at do=+40cm from a diverging lens with f=-20cm.
What are di and m?
di=fdo/(do-f)=(-20)(40)/(40-(-20))=-13.3
m=-di/do=-13.3/40=-0.33
|m|<1 so image is reduced, m<0 so image is inverted, di<0 so virtual
Example:
An f=+20cm converging lens is used to form a real image 3 times as large as the object. Where should the object be placed? To have a real image, the object must be placed beyond f. The resulting image will be inverted. We have m=-3.
m=-di/do=-3 so di=3do
di=fdo/(do-f)=3do cancel do in numerator to get f/(do-f)=3 or
f/3=do-f or f/3+f=do or 1.3f=do=1.3⋅20=26
Example.
A convex lens of focal length 25 cm is placed a distance of 10 cm from a printed page. What is the image distance and how much larger is the image of the page than the page itself?
Convex means f=+25cm. Given do=+10cm, which is within f so we expect a virtual image.
di=fdo/(do-f)=(25)(10cm)/(10-25)=-16.7
m=-di/do= - (-16.7)/10 = 1.67, which is upright=pos and enlarged
Example:
For f=-80cm, do=+60cm, what are di and f?
di=fdo/(do-f)=(-80)(60)/(60-(-80))=-34.3
m=-di/do=-(-34.3)/60=+0.57
|m|<1 so image is reduced, m>0 so image is upright, di<0 so virtual
Example.
Using a +60cm focal length converging lens, one wishes to obtain an image that is 4 times as large as the object. Where should the object be placed and where will the image be found? To have a virtual image, di will be on the same side of the mirror as the object, so di will be negative. Having di negative makes
m=-di/do=+4 so di=-4do
di=fdo/(do-f)=-4do cancel do in numerator to get f/(do-f)=-4 or
-f/4=do-f or f-f/4=do or 3f/4=do=3/4 60 = +45, which is inside f.
Example.
At what distance should an object be placed from a concave mirror or a convex lens of radius r be placed if the image is to be located at the same location as the object. We get di=do when do=r=2f.
Example.
A convex camera lens with f=15mm takes a picture of an object 20 cm away. What must be the distance between that lens and the film? What is m?
di=fdo/(do-f)=(1.5)(20)/(20-1.5)=1.62 cm.
m=-di/do= -1.62/20 = 0.08
Example.
An object 1 cm in height is placed 10cm from a diverging lens, f=-5cm. Where is the image? What is m?
We have f=-5cm and ho-1 cm and do=10cm
di=fdo/(do-f)= (-5)(10)/(10 - -5) = -3.3, neg means virtual
m=-di/do= -(-3.3)/10 = +0.33, reduced, pos m means upright.
Example.
^ lens1--10-lens2
│ ■ ■ (> look to left through lens
object f=+30 f=-30cm
A convex lens, f=+30cm, is placed 10cm from a concave or diverging lens having f=-30cm. What is the image distance for an object 20 cm in front of the convex lens? What is the total m? do=+20cm.
For the first lens (f=+30cm),
di=fdo/(do-f)= (30)(20)/(20-30)= -60cm, virtual image on object side
m1=-di/do= -(-60)/20 = +3, upright and enlarged.
This image is the object for the second lens (f=-30cm). The image is 60 cm in front of the first lens which is another 10 cm from lens 2. This means the object distance is 60+10 = 70 cm from lens 2, do=70cm and f=-30cm.
di=fdo/(do-f)=(-30)(70)/(70 - (-30)) = -21cm, virtual left of lens2
m2=-di/do= -(-21)/70 = +0.3, upright, reduced.
The total magnification is m = m1m2 = (3)(0.3) = 0.9.
Example.
For what object distance is the image half as far behind a convex mirror as the object is in front of it? f=-20cm. di will be negative di=-do/2.
di=fdo/(do-f) = -do/2 cancel do in numerator so f/(do-f)=-1/2 or
-2f=do-f or -2f+f=do or -f=do, so do=-f=+20cm gives di=-10cm
m=-di/do= -(-10) / 20 = +0.5, upright, reduced, virtual.
Example.
^ lens1--40-lens2
│ ■ ■
object f=+30 f=-50cm
A convex lens, f=+30cm, is placed 40cm from a concave or diverging lens having f=-30cm. What is the image distance for an object 20 cm in front of the convex lens? What is the total m? do=+20cm.
For the first lens (f=+30cm),
di=fdo/(do-f)= (30)(20)/(20-30)= -60cm, virtual image on object side
m1=-di/do= -(-60)/20 = +3, upright and enlarged.
This image is the object for the second lens (f=-50cm). The image is 60 cm in front of the first lens which is another 40 cm from lens 2. This means the object distance is 60+40 = 100 cm from lens 2, do=100cm and f=-50cm.
di=fdo/(do-f)=(-50)(100)/(100-(-50)) = -33cm, virtual left of lens2
m2=-di/do= -(-33)/100 = +0.33, upright, reduced.
The total magnification is m = m1m2 = (3)(0.33) = 1.0.
$$ telescope
to view large objects at large distances
distance between two converging lenses is d=f1+f2, the sum of the focal lengths because light from infinity focuses at f1 and this serves as a point source for lens 2.
$$ microscope
to view small objects at close distances
Visible light has wavelengths 400 to 700 nm so the smallest object resolvable is about 1000 nm in diameter. Biological cells are 5 to 50 times larger than this and so are eaily seen. Since cells are usually transparent, absorbed dyes are used to make them visible.
The phase-contrast microscope detects changes in the index of refraction of the series of materials through which it passes. A beam passing through material is compared to a reference beam to see interference patterns. Differing cell components have differing n and thus become detectable without dyes.
$$ move pinhole camera here
Camera is magic. Takes a picture. Some traditional peoples wonder if it absorbs part of your soul.
If you place a square of photographic film here on this desk. It will absorb light from every source and from every angle and become totally white by blending images from the ceiling, floor, wall, chair, your head, and my shoe. But if we block light from the left and right, top and bottom by placing metal there, then we will only get light coming from front.
The pinhole means you get light onto the film only from those objects in front of the hole and not from objects off to the side. Trace rays from objects, through the hole and onto the film. See the image is flipped. Larger hole allows more light and shorter exposure times.
demo pinhole camera,
This is what our eyes do.
There is a flipped image as in our own eye, which is nothing but a pinhole and a focusing lense. Newborns take a few weeks until their brain does an extra flip. One scientist wore lenses over his eyes so that everything was upside down. After six months his brain produces an extra flip. Then he removed the lenses and waited six months to again be reflipped. He dropped many hamburgers and children and likely broke all seven arms during this year.
Medical physics (Cameron)
Eye glases, glaucoma, near/far sighted, Near/far point, astigmitism, floaters, cataracts.
The eye is about w=2cm deep. When looking at a f=3mm wide (0.03 meter) fly at a distance of d=3 meters, how high h is the image of the fly? From similar triangles or tanΘ=rise/run:
. . ^
h . . . . Fly size F
. . . . . . . . . . . . . . . . . \/
-w-|------------d = 3 m ---------|
2cm
h/2 f/2
- = ---
w d
so h=fw/d=0.003x0.02/3 = 2 x 10-5 meter. This image falls on but a tiny group of light sensors.
Our eyes can focus an object placed at our near point (=nearest focusable object distance), typically of about 25 cm, and within one second can then focus at infinity, which is the eye's relaxed focus point. We can focus first on one object and then on another located more distantly by just one hair's thickness.
Our brain blends images from both eyes, often supplying info blocked from view of one of those eyes. Our brain deduces three-dimensional depth by blending our steroscopic view. We readily get by with one eye.
Our eyes have an internal pressure of 12-23 mm Hg, which is 20/1000 atmosperic pressure. The aqueous humor fills the space between the lens and the cornea. This fluid is contionously produced and carries nutrients to the eye which has no vision-blocking blood vessels. The fluid drains through the canal of Schlemm. If this tube becomes blocked then pressure increase within the eye. This is glaucoma.
Turn off the lights, "cut off in Southernese," and our pupils dilate. The diameter of the iris or hole changes from 3 to 8 mm. It takes 5 min for full dilation to occur. The sensitivity of our color-cones changes in 5 to 10 min--it takes 30 min for rods. Our eyes see in total daylight and in partial darkness, which is a range in intensity of 10 billion to one. The iris opens and closes to vary the amount of light allowed in to our wyes. This adjustment takes a few minutes. Our iris also dilates when we look at something we want. You know if someone loves you because their pupils dilate when they look at you.
A flashlight emits 1018 photons per second = smallest particles of light. A rod fires when two photons hit it. The photons passing unabsorbed hit the back of our eyes where they get absorbed without detection. The back of a cat's eye has a shiny reflector that instead sends those photons back passed the rods again, where they might be absorbed this time.
The lights you see when your eyes are closed are called phosphenes. If you run your eyes you cause some cones to fire. I see concentric green and black wavy lines, exactly as in a 1980s Patty Smyth Scandal video. I'll bet the video maker also saw this pattern?
Blind spot is 5.5° = 11 moon diamters in width.
Eye defects:
Look at the figure of eye components.
The retina converts light signals into the electrical signals processed by the brain. Our senses are chemical and electrical entities. Their is no person watching a TV screen withi our head. Our brains "see" by processing electrical signals.
The retina covers the back half of our eyes, allowing us to detect motion or general movement through a large angular range. Without noticing, we do not see details except near the center of our field of vision, called the macula centralis.
A slightly detached retina can be re-attached with a Laser beam sent through the pupil. It is absorbed in the retina, which is heated. This causes blood to coagulate within a vessel, which then becomes blocked. For some reason this is a good idea.
The cornea is the bump out front that does 2/3 of the eye's focusing. It has a fixed curvature and focus. The cornea repiars scratches. UV and x-rays and such sometimes make the cornea become cloudy and require transplanting. A nonuniform curvature of the cornea results in astigmatism, in which, say, vertical lines are less focused than horizontal lines. (See figure on page 368.)
The lens is bent with muscles to change its raduius and hence its focal point. The eye lens of near sighted people focus light in front of their rods and cones, far sighted eyes form the image behind them. When lens-bending muscles are relaxed, the eye focuses at the "far point," which is usually infinity but for near sighted persons it might be close to the eye. The closest point to which one can focus is the "near point." Children can focus closer in than can adults. Old people focus little. A clouding of the lens is called a cataract. Lasers are used to correct this, making the world seem bright again.
The optometrist prefers diopters, which is 1/f instead of f, that is, a lens of focal length 0.02 meter has a lens strength of 1/f=1/0.02=50 diopters.
When combining two lenses, the pair have an effective focal length given by 1/feff = 1/f1 + 1/f2. In terms of diopters: the effective lens strength of th combo is the sum of the diopter strength of lens1 plus the diopter strength of lens 2.
Our eye is an average of 17mm deep from cornea to retina, meaning that the focal length of its lens combo is 17mm. Light coming from infinity is focused at 17 mm. The focal length of this eye is 0.017 meter, or the strength of this eye lens machine is 1/f=58.8 diopters. A far-sighted person focuses this light in front of the retina. A near-sighted person focuses infinitly far light behind the retina, which would be back in the brain.
Example.
A near-sighted person with a far-point (most distant object focusable) of 1 meter has 1/f = 1/do + 1/di = 1/1.0 + 1/0.017 = 59.8 diopters or f=0.0167 meters. This eye lens is 59.8 - 58.8 = 1 diopter too strong. The corrective lens would have a strength of -1 diopter.
A healthy eye focused at infinity has 1/f=1/do+1/di=1/∞+1/0.017 or 1/f=59 diopters or f = 0.017 meters while a far-sighted eye focusable down to only 0.5m has 1/f=1/do+1/di=1/0.5+1/0.017 or 1/f=60.8 diopters or f=0.0165 meters. To be able to focus downto 0.25 m (the average), the eye would have 1/f = 1/do+1/di = 1/0.25+1/0.017 or 1/f=62.8 diopters or f=0.0159 meters. A corrective lens needs to add 62.8-60.8=+2 diopters to the focal length of this eye.
Researchers first tried to make contact lenses in the year 1900 but were not successful until 1950. They were developed in Czechoslovakia.
Examples of lenses:
spherical water drops on a leaf focus light onto that leaf and can burn it.
Curved water under a water-walking insect's foot
Half-inch high waves on the surface of a pool refract light to make those bright wiggling lines on the pool floor.
$$ interference
Two waves add numerically, get constructive interference or destructive interference when two people wiggle oppposite ends of a rope or two waves meet in the middle of a pond or two light waves cross paths or two sound waves cross paths.
take two sine waves of equal phase and then add them contructively. Now shift one by half a wavelength and the two will add destructively. Any shift of (m+1/2) lambda does the same. Shifts of integral wavelengths m lambda gets constructive interference.
Remember, within a material the wavelength = wavelength in vacuum/n, where n is the index of refraction fo that material.
Thin film interference.
light comes straight down into a thin film of oil. Some light reflects off this higher n while some light passes into the oil but reflects off its lower surface.
Rule 1: light bouncing off a higher n gets an extra half wavelength shift
Rule 2: Constructive interference when path difference plus shifts are m lambda. Destructive interference when path difference plus shifts are (m+1/2) lambda
Rule 3: in the material, lambda = lambda-vacuum / n.
The colors of some beetles and butterflies is due to thin-film interference in a thin cuticle membrane
Soap bubbles
Blue-tint of the non-reflective coatings on camera lenses.
.
$$ two source interference . .
. . | -
│ . . |
- . . |
| │ . . | Y
d │Θ . . |
| │ . Θ |
- ...................... D ..........| -
│
Path difference is d sin Θ = dark when (m lambda) with m=0,1,2..
= light bands where (m+1/2) lambda
when Θ<11°, sinΘ = tanΘ = rise / run =
$$ difraction
demo spectroscope
Waves emanating from a point source spread out as 3D concentric shells. Plane waves striking a hole in a wall will spread out from the hole as if it were a point source, when the size of the hole (or obstacle--both have identicle effects) is the same order of magnitude as the wavelength. If the wavelength is much smaller than the hole then the wave fronts just pass through like visible light in a doorway. In this case the light wave is "acting like a particle." If the wavelength is much larger than the hole then the wave fronts bounce back and do not pass through the hole. Satelite TV dishes are like that.
.
. .
. .
. . . .
---------- -----------
Dark interference bands where w sinΘ = m lambda, where w is the width of the slit or obstacle.
Waves bend around abjects having the same diameter as a wavelength. A wave of length 100 meters bends around a 100-meter object.
Sound bends around your head v=lmabda f so f=v/lambda = 340/.34=1000 hz.
Yellow-green light wavelength 5.5x10-7 meter is defracted by objects of that size.
The spacing between atomic layers in opal matches this size so that opal is a natrual diffraction grating for visible light.
The halo around the sun or moon occurs when light diffracts through high-altitude ice crystals of hexagonal shape.
Diffraction occurs between your almost-touching fingers, or from the wires of a screen, the threads of an umbrella, the eyelashes in front of squinting eyes,
Diffraction pattern
Gratings:
maxima where sinΘ = m lambda / distance between adjacent slits
= yn / D where yn is the order number and D is the distance to the screen. N slits make the light N times brighter.
History from ushumans
Faience was made before 4000 bc by Ancient Mesopotamians in an attempt to duplicate natural and expensive lapis lazuli stone. Hodges says the first step toward making true glass occurred when faience was accidentally melted. Today's glass is a similar fusion of quartz, soda, and lime. At first, the Mesopotamians were using abrasives to cut and polish the finished product, just as they worked hard stone, but it was later found that the material was more easily molded while it was hot. For more information about glassmaking, visit www.fsus.fsu.edu/SchoolInformation/SchoolRelations/Partnerships/Exhibitions/trialbyfire/main.asp. See also http://en.wikipedia.org/wiki/Glass. By 2000 bc, glassworking techniques had improved in Mesopotamia and Egypt. Many colors were being created, and after 1500 bc, lead was being added to make the resulting glass more shiny and to keep it from contracting while cooling. Lead had been considered a waste product of copper smelting since 3000 bc because it was too soft to be used in tools, but now it was finding other uses.
Hodges says the techniques of glassblowing could not be developed until iron technology had advanced enough to make blowtubes. This occurred in Syria during the first century ad and was then spread by the Romans. (Hodges also says that the rate of Roman technological innovation decreased when they came to rely on the labor of slaves rather than their own cleverness.) The Gies explain that the secrets of Syrian glassblowing were taken to Venice in 1277 ad and that Muslim artisans taught the techniques to the Venetians who then built a European monopoly in fine glass manufacturing. Stained glass windows began being used in European churches in the seventh century ad.
Colonial glassmaking was largely begun by Jan Smedes, who transplanted a Dutch glassworks to New Amsterdam in 1654, and by the Bavarian Casper Wister, who built a New Jersey plant in 1737 employing experts from Rotterdam. Glassmaking secrets were being guarded, just as every factory today keeps its techniques secret. Wister also made window glass. Tunis says that a colonial home might remove and store these valuable panes before going on a long journey.
Glass "bends" light and so came to be used to make lenses. (We saw in Chapter 2 that light is an electromagnetic phenomenon and that this "bending" is described by Maxwell's equations.) We saw that the Europeans inherited their knowledge of lenses from the Islamic world–for example, through their use of the Arabic textbook on optics The Optical Thesaurus, written in 1038 ad by Alhazen. Around the year 1300 ad, Roger Bacon first suggested using lenses to aid people's vision. Within a century, it was a common practice. A person simply tried several lenses to find which was most helpful. Tunis says that early eye-lenses were rumored to enable one to see into the future, to make the blind see, and to make illiterate persons literate. After the year 1500, people who were sensitive to light began using Venetian green glass to reduce glare. People who had trouble seeing both nearby and distant objects carried two different lenses until Ben Franklin first made "bifocals" in 1786. Lenses were spherically shaped: convex lenses help one see nearby objects while concave lenses help one see distant objects. In 1825, George Airy found that cylindrically shaped lenses helped those with astigmatism, which is a slightly misshaped eye that sees better vertically than horizontally. During the eighteenth century, lenses were not yet hung from ears and noses as we do today.
Galileo first used a telescope to make observations of the moon and planets in the year 1609 (see Patrick Moore's Watchers of the Sky and here.) He reported that when viewed with his telescope, planets were seen to have the shape of a disk but stars remained point-like. He also said that hundreds of stars could be seen that were invisible to the unaided eye. The material of the Milky Way had been debated since the time of the first humans, but Galileo’s lense enabled him to report that it simply consisted of innumerable stars. He described the craters of the moon and estimated the height of one lunar "mountain," which was actually the side of a crater, to be 20,000 to 25,000 feet (7500 meters). Its true height is 18,000 feet (6000 meters). Galileo also reported that Venus cycles through the same phases as does the moon. His report that four moons were revolving around Jupiter came as a blow to those who thought that the Earth was the center of the Universe in that all the stars and planets revolved around a stationary Earth. The rings of Saturn were barely discernible in these early telescopes and there were many imaginative explanations of these blurry objects before they were clearly seen to be rings.
Galileo is important in that he was one of the founders of the scientific method. He fully realized the value of making repeatable measurements, as described in Chapter 2. Newton deduced his general motion equation after considering the results of many experiments, such as those made by Galileo involving the motion of objects on inclined planes (for his original texts, see www.mpiwg-berlin.mpg.de/en/Galileo_Prototype/). Science was really born at that time; before then, we were only intellectualizing–that is, guessing from our arm chairs. Our tool-making arts are much older than our scientific procedure.
**** end of history
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