Phy 133 Section 1: f. Geometric Optics: Assume the rays follow straight lines. (No diffraction). v 1 λ 1. = v 2. λ 2. = c λ 2. c λ 1.

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1 Phy 133 Section 1: f Geometric Optics: Assume the rays follow straight lines. (No diffraction). Law of Reflection: θ 1 = θ 1 ' (angle of incidence = angle of reflection) Refraction = bending of a wave due to a change in its medium. Cause: c = 3 x 10 8 m/s is the speed of light in a vacuum. In any material, it slows to some speed, v. 1 (Sec. 10: c =.) μ 0 ε 0 definition: Index of Refraction: n = c v When light hits the boundary between two materials at an angle, there is a short time when each end of a wave is in a different material. Having each end going at a different speed makes it swing around, so ray changes direction where it crosses. f must be the same in both media. f 1 = f 2 v 1 λ 1 = v 2 λ 2 Flip it upside down, multiply by c: c λ 1 v 1 = c λ 2 v 2 Substitute n = c v : n 1 λ 1 = n 2 λ 2 From, λ 1 = H sinθ. From, λ 2 = H sinθ n 1 (H sin θ 1 ) = n 2 (H sin θ 2 ) Snell's law: n 1 sin θ 1 = n 2 sin θ 2

2 -2- Example 1-1. Find θg: Say you're underwater, shining light toward the surface. As you increase θ 1 in the water, θ 2 in the air always has to be larger, so it gets to 90 first. Beyond this "critical angle," there is no refracted beam. All of the light goes into the reflected one. - Formula for critical angle Total Internal Reflection: Example: find the critical angle for diamond in air. ans: sin θ c = n 2 n 1 = θ c = 24.4 Fiber optics: Due to total internal reflection, light does not escape from inside a glass fiber. (I will demonstrate by shining a laser in the end of a bent plastic rod.) Plane mirror:

3 -3- VIRTUAL (or imaginary) IMAGE - one which light rays do not pass through. (Can't be projected on a screen.) Example: plane mirror. REAL IMAGE - Rays do pass through image. (So, can be projected on a screen.) Example: movie theater. Spherical Concave Mirror (shown in cross section.) C is the center of curvature. Every point on the mirror is the same distance from point C. F is the focal point, where parallel rays cross after reflection. Converging lens: Ray diagrams: 3 rays easiest to draw accurately:

4 -4-1. In parallel to axis (the line through C and F), out through F. 2. In through F, out parallel to axis. 3. Mirror: Ray through C reflects back on itself. Lens: Ray through center of lens goes straight. s o = object distance s i = image distance h = height If object was inside focal point, image would be virtual & erect instead. Example 1-2. An object 10 cm tall is placed 40 cm from a mirror with a 140 cm radius of curvature. Use a ray diagram to find the image s (a) position, (b) size and (c) whether it s erect or inverted. To calculate an answer instead of measuring a diagram: Similar triangles h i h o = s i s o

5 -5- These similar triangles s i f f = h i h o s i f f = s i s o (using h i h o = s i s o ) s i f 1 = s i s o Dividing by s i, 1 f 1 s i = 1 s o 1 f = 1 s i + 1 s o definition: Magnification: M = h i h o h i h o = s i s o Insert a minus because we will call h i negative when inverted. s i is negative for a virtual image. (The side the rays go to in the end is where s i is poisitive.) Example 1-3. Calculate the answer to 1 2. Diverging (convex) mirror, diverging (concave) lens. The image is always virtual. Same equations and diagrams except f and R are negative, meaning the opposite side from how we did it before. Example 1-4: Find the position, size and character of the final image.

6 Brief review of waves from PHY 131, section Section 2: Superposition Principle: Displacement of medium due to several waves at once is the vector sum of the individual displacements. Adding the wave functions of two identical waves going through each other in opposite directions gives the wave function for a standing wave. Using pictures instead of functions: At each of these four instants, the top picture shows what the string would look like with only wave 1 on it, the middle picture shows what the string would look like with only wave 2 on it, and the bottom picture, which comes from adding the two above it, is what the string looks like with both at once. So, the string moves like so: (a standing wave) Distance between nodes, between antinodes, etc. Resonance: When a driving force creates a large amplitude vibration. (Due to driving system at a resonant frequency.)

7 There are several examples (Oscillating circuits, PHY 132). For a standing wave on a string: -7- Must be a node where the string is fixed. ( boundary condition ) Only certain wavelengths fit on the string. (Since nodes are.5λ apart, the string must hold a whole number of half-wavelengths.) It can only vibrate at the frequencies corresponding to those wavelengths. This is why there are resonant frequencies. (At these frequencies, as waves reflect over and over, all those going one way line up and all those going the other way line up.) The vibration patterns where this happens are called normal modes. Example 2-1: Waves on a clothesline 3.0 m long travel at 30 m/s. Find its three lowest resonant frequencies. Harmonic sequence: All f's are integer multiples of f 1. Complex Waves: Pluck or striking a string doesn't single out any particular resonant frequency. Rather, the string goes into a superposition of many of these modes of vibration. Fourier's Theorem: Any periodic function can be thought of as a superposition of sine and cosine waves: y(t)= n=1 (A n sin ω n t + B n cos ω n t) where f 1 = 1/T and f n = nf 1 To find A n and B n : more advanced course. Each term in the series represents a harmonic in the sound. (Its intensity depends on A n and B n.) Terminology: harmonics, overtones, and so on. What determines pitch and quality of a sound. (Loudness amplitude (or intensity), from PHY 131.)

8 -8- Example 2-2: A guitar string 56.0 cm long produces an A (440 Hz) without fingering. Where should you put your finger to get a C (523 Hz)? Wind instruments: Standing sound waves in a pipe. Boundary conditions. Example 2-3: A clarinet is closed at one end (by the player's mouth) and open at the other. Find the two lowest resonant frequencies of a clarinet 60 cm long. Beats: Interference between waves of slightly different frequencies.

9 -9- Section 3: Interference. For example, Coherent waves (identical except possibly for amplitude) Or, if you move the top speaker another wavelength, it's the same picture with one more wave drawn onto its left edge. In general, Constructive Interference if path difference = mλ where m = 0 or 1 or 2 or... On the other hand, The positive parts of one wave arrive at the same time as the negative parts of the other, so they cancel. The same thing happens if the path difference is 1½λ or 2½λ or... Destructive Interference if path difference = (m + ½)λ Example 3-1. λ = 30.0 cm for the sound coming from the speakers. What values of x make the loudness at the microphone a maximum?

10 -10- Diffraction = the bending of waves around an obstacle. (Or through an opening.) If you move the surface of some water up and down with your finger, circular ripples move away from that point. If the water in this opening gets moved up and down by the wave instead of your finger, the same thing happens. Double slit interference: Light falls on a barrier with two narrow, parallel slits. A cross-sectional view is on the left. Diffraction makes the slits act as coherent sources of interfering waves. paths of different lengths Waves spread due to diffraction, and overlap. Example 3-2: Monochromatic (only one λ) 380 nm light passes through two slits 50 microns apart. Find θ for the first order maximum. Example 3-3: This pattern is projected on to a wall 2.50 m from the slits. How far apart are the 0 th and 1 st order maxima? A variation on double slit interference: Phase change from reflection:

11 -11- You will observe a dark fringe at the center of the pattern, not bright. Reason: Phase changes 180 when light reflects from a higher n medium. - Wavelength in a material. Thin film interference: Light rays reflect from opposite sides of a film, like a soap bubble. (Film s index is higher or lower than n on both sides.): 2t = path difference. If 2t 0, destructive interference (due to phase shift.) If 2t = λ n /2, constructive interference. If 2t = λ n, destructive interference. And so on. λ n = wavelength in material of index n λ v = wavelength in a vacuum Thin film whose index is higher than n on one side and lower than the other. (Example: Nonreflective lenses have a coating whose thickness gives destructive interference. This makes all the light go through it, instead of some being reflected.)

12 -12- If 2t 0, constructive interference. Formulas for thin film interference. With white light, some λ's match the thickness of the film to interfere constructively, while others match it destructively. This is why you see colors in a bubble, or oil floating on a puddle: The thickness of the film varies between different places, and the reflected λ varies with it. Example 3-4: How thick should the coating be to make air the lens completely nonreflecting to 550 nm light? n = 1.38 _ n = 1.55 glass Example 3-5: A lens on a flat piece of glass is illuminated with 500 nm light. How thick is the air gap at point P?

13 -13- Section 4: Michelson Interferometer: Used for precise position measurements: If movable mirror moves just a fraction of a wavelength, the interference pattern changes. Example 4-1: The air, originally at one atmosphere, is pumped out of the chamber. How many bright fringes pass a certain point on the screen? Diffraction grating: Many equally spaced parallel slits (or scratches). Constructive interference at the same angles as with two slits. But, dark if θ is even a little off. With white light, constructive interference of each λ is at a slightly different angle, separating the light into a spectrum. Example 4-2: Light from a mercury vapor lamp falls on a grating with 8000 lines per cm. find the angle where the 546 nm line appears in the 2 nd order spectrum. Does a third order exist?

14 -14- X - ray diffraction: Single slit interference. Different parts of slit are different distances from this point, so some rays cancel others. Spreading and Resolution: Passing through an opening makes a wave spread out. (Single slit interference.) So, light from a pointlike source spreads into a nonpointlike blob when entering an optical instrument. This spreading limits the resolution of optical instruments. (The minimum separation where you can distinguish two objects.) Images (diffraction patterns) are considered barely resolved when center of one is at first minimum of other. Example 4-3: How far from your eye to a pair of barely resolvable headlights 5 feet apart? Assume: 1. Circular pupils 2 mm wide, 2. λ = 500 nm, 3. Only diffraction limits resolution. Polarization: Linear polarization: E stays parallel to y axis, making a cosine curve:

15 -15- (Circular polarization: E rotates in yz plane, making a corkscrew around x axis. Elliptical polarization: E changes both magnitude and direction.) Ordinary light is usually unpolarized: a superposition of all these. E varies at random in yz plane. To polarize light: 1. By selective absorption. Use a material which absorbs light along one axis but not the other. 2. By reflection. E is reflected more strongly than E, partially polarizing the reflected ray. Complete polarization: n = tan θ p Brewster's law 3. By double refraction. In certain crystals, rays of different polarizations refract through different angles. 4. By scattering. Scattering is absorption and re-radiation by a system of particles. Different polarizations make electrons in the particles vibrate in different directions, reemitting the light in different directions. For example, scattering by air molecules partially polarizes light from the sky.

16 -16- Review of Sec. 1 4: The format of my tests is the same as the same as that for this review sheet: The test has five parts, each worth 25 points. The best four you do will be counted. The actual test will not necessarily be similar to the questions below; including something similar to every possible test question would make this review vastly longer. Rather, these are "rough spots," where many people seem to need more work. At least part of your test preparation should be to carefully go back through the notes. Be sure you know the terminology, that you understand the basic ideas and relationships, and that you follow how I've applied them in the sample problems. 1. A certain string on a cello is 88 cm long, and produces an "A" (220 Hz) when bowed or plucked. a. What is the speed of waves on this string? b. If the effective length is reduced to 79 cm by fingering, what is the frequency then? ans: 387 m/s, 245 Hz 2. An object and two lenses are arranged along the x axis as shown. At what value of x is the final image located? Is it real or virtual? ans: 5.0 cm, virtual 3. A ruby laser beam (f = 4.32 x Hz) is sent out from a 2.7 m diameter telescope to the moon, km away. What is the radius of the red spot it makes on the moon? ans: 120 m 4. Light falling on a pair of glass plates reflects from the top and bottom of the space between them. At one particular point, there is a dark interference fringe when 420 nm light is used. At this same point, there is a bright interference fringe when 504 nm light is used. What is the narrowest possible value for the gap between the plates at this point? ans: 630 nm 5. Short answer, 5 points each: a. When a diffraction grating is illuminated with white light, which color in the first order spectrum appears closest to the central maximum?

17 -17- b. When light crosses from air into glass, what happens to its i. speed? ii. frequency? iii. wavelength? c. The light used in a double slit experiment has a wavelength of.50 μm. Is interference constructive or destructive at a point which is 20.0 μm from one slit and 20.5 μm from the other? d. When a certain pendulum is driven at 1.0 Hz or 2.0 Hz or 1.7 Hz or a lot of other frequencies, it just quivers around a bit. But, at 1.5 Hz, it is soon making large swings back and forth. What do we call this phenomenon which occurs at 1.5 Hz? e. A standing sound wave in a tube has an antinode of vibration at an open end, as stated on your formula sheet. Is the variation in pressure also a maximum there, or is it a minimum?