PHY 122 Section 9: Vibration & Waves - Genesee Community College

- 44 - (This is the experiment that first proved light is a wave.) Problem 12-1: Light with a wavelength of 500 nm is used in this double slit experiment. How many nm is x, the amount by which the path on the bottom is longer than the path on the top? With the screen far from the slits, some trig shows that the path difference equals d sin θ. Constructive interference: d sin θ mλ Destructive interference: d sin θ (m ½)λ m 0, 1, 2, , screen far from the slits. Problem 12-2: Monochromatic (only one λ) 380 nm light passes through two slits 50 microns apart. Find θ for the first order maximum. Ans: .435 Problem 12-3: This pattern is projected on to a wall 2.50 m from the slits. th st How far apart are the 0 and 1 order maxima? (Hint: Consider the triangle shown.) Ans: .0190 m

- 45 - 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. d sin θ mλ Maxima from a grating With white light, constructive interference of each λ is at a slightly different angle, separating the light into a spectrum. Problem 12-4: Light from a mercury vapor lamp falls on a grating with 8000 lines per cm. Find the nd angle for the 546 nm line in the 2 order spectrum. Does a third order exist? Ans: 60.9 , no Thin Film Interference: Example: Coating on nonreflective lenses makes all the light go through it, instead of some being reflected: 2t path difference. Reflected rays cancel each other. If n of film is between ns on either side: Constructive interference: 2nt mλv Destructive interference: 2nt (m ½)λv m 0, 1, 2 λv wavelength in a vacuum

- 46 - If n of film is higher or lower than n on both sides: Constructive interference: 2nt (m ½)λv Destructive interference: 2nt mλv The formulas contain n (film’s index of refraction) because slowing the light shortens λ. How the wavelength in the material (λn λv/n) compares to the path difference (2t) determines what kind of interference there is. The reason for two sets of formulas is that there can be a change due to reflection: phase If both rays or neither are phase shifted, the top box applies. (Notice constructive interference when t 0.) With one shifted, the bottom applies. (Destructive if t 0.) With white light, some λs match the thickness of the film to interfere constructively, while others match it destructively. This is why you see colors in bubbles, or a layer of oil floating on water: As the thickness of the film varies between places, the reflected λ varies with it. Problem 12-5: How thick should the coating be to make the lens completely nonreflecting to 550 nm light? Ans: 99.6 nm air n 1.38 n 1.55 Single slit interference. Central maximum: θ 0 Minima: a sin θ mλ Increasingly faint max.: a sin θ (m ½)λ m 1, 2, 3, (no 0), screen far from the slit. . glass

- 47 - Spreading and Resolution: Passing through the opening into an optical instrument spreads light out due to single slit interference: a sin θ (1)λ The image of a pointlike source, like a star, is a blob this big (with fainter light around it) instead of a pointlike image. Two images are thought of as barely resolved if the blobs overlap halfway: Rectangular opening: θmin λ a Circular opening: θmin 1.22 λ D D diameter A shortcut useful in the next problem: Example: The Washington Monument is 555 feet tall. To an observer level with its midpoint, it looks 1.00º tall. How far away is it? Ans: Problem 12-6: 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. Ans: 1.64 x 104 ft

- 48 Review of Sections 9-12: 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 2.0 cm tall is placed 4.5 cm from a converging lens with a focal length of 10.0 cm. By constructing a ray diagram, find the image’s position, size and character. Ans: -8.2 cm, 3.6 cm, virtual & erect 14 3. A ruby laser beam (f 4.32 x 10 Hz) is sent out from a 2.7 m diameter telescope to the moon, 384 000 km away. What is the radius of the red spot it makes on the moon? Ans: 120 m 4. A certain sound has an amplitude of .00001 cm and a frequency of 171.5 Hz. Sketch a graph showing the displacement of the air as a function of position for this wave. Label both axes with appropriate units. 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? b. The speed of light in glass is (greater than? less than? equal to?) the speed of light in diamond. 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. If you triple your distance from a light source, what happens to the intensity of the light? e. What property of a light wave is perceived as its i. color? ii. brightness?

- 49 Section 13: Atoms Wave-particle duality: Experiments in the early 1900’s showed that light is, in some ways, more like a stream of particles than a wave. A good picture (although not perfect) might be to think of light as a lot of little flashes rather than a continuous wave like you see on a pond. Each particle, or flash of light, is called a photon, and has an energy of E hf -34 h Planck’s constant 6.63 x 10 J·s Problem 13-1: What is the energy of a photon with a 500 nm wavelength? Ans: 3.98 x 10-19 J de Broglie wavelength: What’s so special about photons? Maybe all particles have this dual nature, with a wavelength of h mv This was confirmed when electrons, neutrons, and even whole atoms were diffracted from crystals, just like x-rays. Problem 13-2: Calculate the wavelength of a 75 kg person running at 5.0 m/s. Ans: 1.77 x 10-36 m (Because λ is extremely small, the amount you diffract when running through a doorway is too small to measure. So, you don’t notice the wave nature of objects in everyday life.) Demonstration: How can something be both a wave and a particle, when they are as different as black and white? A blob of correction fluid on clear plastic looks white when front lit and black when back lit. It's black or white, depending on how you look at it. Similarly, light is a wave or a particle, depending on whether you look at it in a diffraction experiment, or a photoelectric experiment. (Either model is oversimplified.) The Bohr model of hydrogen (as explained by de Broglie): Think of an electron as a wave on a circular path around the nucleus. To get constructive interference, the circumference of the orbit must be a whole number of wavelengths. 2πr n n 1 or 2 or 3 or So, the electron can only exist at certain distances from the nucleus. From Newtonian mechanics and Coulomb’s law, the energies corresponding to these allowed orbits are:

- 50 - En – 13.6 eV n2 -19 1 eV 1 electron volt 1.60 x 10 J (energy an electron gains going through one volt.) (The lowest energy level is called the “ground state.” Higher ones are called “excited states.”) Example 13-3: When hydrogen’s electron is in the n 2 level, the radius of its orbit is .212 nm. (a) What is the wavelength of the electron? (b) What is the electron’s energy? (c) How much energy is lost when it drops to n 1? (d) What wavelength of light is given off dropping to n 1? Ans: .666 nm, –3.40 eV, 10.2 eV, 122 nm Problem 13-4: When the electron in hydrogen is in the n 3 level, the radius of its orbit is .476 nm. (a)What is the wavelength of the electron? (b)What wavelength of light is given off dropping to n 2? Ans: .997 nm, 659 nm Spectra: Look at any light source through a grating; you’ll see one of three things: Continuous spectrum: All λs present. Emission spectrum: Only certain λs present. (A pattern of bright colored lines on a dark background.) Absorption spectrum: Only certain λs absent. (Dark lines.) (Pass a continuous spectrum through a gas, and it will absorb the same λs it would emit if hot.) Pattern of lines acts as a "fingerprint" to identify what made the spectrum. For example, this is how we know what stars are made of. No one could explain line spectra, until Niels Bohr suggested that the electrons in atoms are restricted to energy levels. A photon is absorbed or emitted “all or nothing” when an electron jumps or drops from one energy level to another. From what we just went over, he derived: 1 1 1 (1.097 x 107 ) 2 – 2 λ ni nf (λ in meters) This exactly matches hydrogen’s observed spectrum. Problem 13-5: Calculate the wavelength given off when hydrogen goes from n 3 to n 2, the short way. Ans: 656 nm (A little different due to how numbers were rounded.)

- 51 Laser: Stimulated emission: Electron is knocked down to a lower energy, instead of falling spontaneously. Example, Helium - neon laser: Neon atoms in tube undergo stimulated emission: Beyond the Bohr Model: A more detailed analysis shows the main energy levels contain sublevels. A complete description needs 4 quantum numbers, not just n: n, l, ml and ms In the n 1 level, there are 2 possible combinations of these numbers. In n 2 there are 8. (Take chemistry for details.) Pauli Exclusion Principle: No two electrons can have all four quantum numbers the same. So, in multi-electron atoms, the electrons form “shells” because there isn’t room for them all in the lowest level. (2 in the inner shell, 8 in the next, etc.) Solids: (Many atoms forming a crystal lattice; a regular array) Energy bands: When close, atoms disturb each other's energy levels. The atoms’ slightly different levels, taken together, form bands. (Each band is actually many very close levels.) example, sodium at 0 K: Electrons in the band which is partly filled are the ones which flow when there is an electric current.

- 52 - Insulators have no partially filled band: Semiconductors (ex: silicon) Doping (adding impurity atoms): Donor: An atom with too many electrons to fit into the lattice gives one off, adding to conduction electrons. n-type semiconductor: Current is mostly electrons. Acceptor: an atom that gains an electron, making a hole. p-type semiconductor: Current is mostly holes. p-n junction: Free charges leave junction. Very small I because few free charges are left to flow. Lots of charges flow in, so I is big. The junction is a diode, conducting in only one direction. (It rectifies: converts AC into pulses of DC.)

- 53 - Some holes get stuck on donor atoms in the thin center layer. They repel additional current. The more ib drains off, the larger ie and ic. So small base current controls large e to c current. Used as amplifiers, and as "switches" for digital circuitry. Problem 13-6: a. The energy bands of a certain solid are as shown. Is this material an insulator, a semiconductor or a conductor? b. Does having electrons flow into the middle layer of this transistor increase, decrease, or have no effect on the main current flowing horizontally? c. i. Draw a few holes and electrons in each picture. Draw arrows showing which way they are moving. ii. In which case is the current large? Light Emitting Diode (LED): Electrons at the bottom of the conduction band encounter holes at the top of the valence band at a p-n junction. When they combine, they release an amount of energy equal to the gap between these bands. If the energy is released as light rather than heat it is an LED. Problem 13-7: If the LED is made from a semiconductor with an energy gap of 1.88 eV, what is the color of the light it gives off? (Violet 380 – 440 nm, Blue 440 – 490 nm, Green 490 – 560 nm, Yellow 560 – 590 nm, Orange 590 – 630 nm, Red 630 – 750 nm) Ans: Red. (660 nm)

- 54 Section 14: The Nucleus: A nucleus is held together against the repulsion between protons by the Strong force: Protons and neutrons all attract each other. (Very short range: Does not operate outside the nucleus.) Number of protons in nucleus determines which element it is. Isotopes: The different nuclei of an element: Differ in number of neutrons. Example: the 3 isotopes of hydrogen: mass number total number of nucleons (ps & ns) 1 1H 3 1H 2 1H atomic number number of protons Example: how many neutrons in a Ans: 208 - 82 126 208 82Pb nucleus? 2 Binding energy (c )(difference between mass of nucleus and particles making it up) Z atomic number N no of neutrons Eb (in MeV) (ZmH Nmn - matom)(931.5) 12 (masses in atomic mass units: 1 u 1/12 mass of a C atom) mH mass of 11H 1.007825 u mn mass of neutron 1.008665 u Problem 14-1: Find the average binding energy per nucleon of 15.994915 u. Ans: 7.98 MeV Radioactivity: -Alpha "rays” (least penetrating): example: 238 92U 234 4 90Th 2He alpha particle (helium nucleus, 2p's & 2n's) Mass numbers add up to same thing on both sides. Atomic numbers add up to same thing on both sides. 16 8O if the mass of one atom is

- 55 -Beta "rays” (intermediate penetration): example: 14 6C 14 0 7N -1e beta particle (electron) antineutrino ( : Greek "nu") Neutrino: Does not feel strong force, electromagnetism (no charge), or gravity (little or no rest mass). Therefore, it interacts very weakly with matter. -Gamma rays (most penetrating) are photons. Emitted when an excited nucleus drops to a lower energy level. example: 123 53I 123 53I γ gamma ray (photon) Problem 14-2: Complete the equations: Half-life: Time for half the nuclei of a substance to decay. example: 14C has a half-life of 5700 years, meaning that an object containing 1.0 gram of 14C 5700 years ago would contain .5 gram today. (The rest decayed into 14N.) 5700 years from now, it would be halved again, to .25 g. N N0 (.5)t T1/2 R R 0 (.5)t T1/2 N number of nuclei present R “decay rate” or "activity" (R ΔN/Δt) units: SI: 1 becquerel 1 Bq 1 decay/sec 10 also popular: 1 curie 1 Ci 3.7 x 10 Bq

- 56 R Re-arranging that, t T1/2 ln(R ) 0 ln(.5) Problem 14-3: A certain substance has a half-life of five years. If its activity today is 1000 decays/sec, what will its activity ten years from now be? 14 12 Problem 14-4: Radioactive dating: All living things contain the same ratio of C to C, giving each 14 gram of carbon an activity of .25 Bq. The half life of C is 5730 years. If a 100 gram piece of charcoal has an activity of 17 Bq, about how old is it? Ans: 3190 yr Problem 14-5: Iodine 123, half-life 13.2 hours, is used as a medical radioactive tracer. If a dose has an initial activity of 300 μCi, what will the activity be two days later? (In practice you would be exposed to even less, because your body excretes some of it.) Ans: 24.1 μCi Disintegration energy/ reaction energy: Disintegration: Radioactive decay, as above. Reaction: Shooting some particle, a, at a nucleus can transmute it into another element: a X Y b . 2 Q (total m before - total m after)c (c2 931.5 MeV/u) (Q KE of Y & other particles, energy of γ rays, etc.) Problem 14-6: Find the Q value for the α decay of 4 222.017574 u, He: 4.002603 u. Ans: 4.87 MeV 226 Ra. Mass of If Q is positive, the process can happen spontaneously. If Q is negative, the process can’t happen spontaneously. Nuclear reactions: Fission: Split a nucleus into smaller ones. examples: original "A" bomb, nuclear reactors. Fusion: Combine nuclei. examples: H-bomb, the Sun. 226 Ra: 226.025406 u, 222 Rn:

- 57 Fission: 1 example: 0n 235 92U 141 56Ba 92 36Kr 1 3( 0n) energy (many other possibilities) (Mass numbers and atomic numbers must add up to same thing on both sides.) 235 140 Problem 14-7: a. Write the equation for 92U splitting into 54Xe and 235 132 101 b. Write the equation for 92U splitting into 50Sn and 42Mo. 94 38Sr. Chain reaction: The neutrons released go on to split other 235U's, releasing even more neutrons, etc. Fusion: Example: Requires great temperature and/or pressure. Effects of radiation: Ionizing radiation damages molecules in cells. If the cell can't repair the damage: - Radiation sickness: Many dead/damaged cells, causing blood abnormalities at a certain dosage; nausea, hair loss, etc at a higher dosage; death at a still higher dose. - Cancer: If there is damage to a cell's genetic material, it can start dividing out of control. -Birth defects: Damage to genes in reproductive cells can cause mutations, passed down to all future generations.

- 58 Review of Sec. 1 - 14: 1. How many amps flow through the 5.00 Ω resistor? Ans: 1.50 A 2. Assume that a clarinet is a cylindrical tube full of air at room temperature (speed of sound 343 m/s), 60 cm long, open at one end and closed at the other. a. Find the frequency and wavelength of the lowest note which can be played on it with all the side holes shut. b. Find the frequency and wavelength of the next lowest note. (It’s not just twice the frequency from a.) Ans: 143 Hz, 2.40 m, 429 Hz, .800m 3. Two speakers, located 4.00 m apart on a vertical pole, are driven by the same electronic signal. As a man whose ears are level with the lower speaker walks toward it, the loudness rises and falls due to constructive and destructive interference. The most distant minimum is when L 8.90 m. What is the frequency of the sound? Ans: 200 Hz 4. Consider the nuclear reaction 2H 6Li 24He. The masses of these nuclei are: 2H, 2.014 102 u; 6Li, 6.015 122 u; and 4He, 4.002 603 u. Assume the hydrogen and lithium nuclei collide at very low speeds, meaning that their initial energies are entirely in the form of mass. What is the total kinetic energy, in joules, of the helium nuclei after the reaction? Ans: 3.58 x 10-12 J -18 5. In an electrostatic air cleaner, a piece of dust with a charge of 3.68 x 10 C is between charged parallel plates 8.00 mm apart. If there are 120 V between these plates, how large is the force on the piece of dust, in newtons? Ans: 5.52 x 10-14 N 6. One coil is placed around another as shown. The solenoid has a cross sectional area of.028 m2 and has 700

A wave a moving disturbance. (Sound, water waves, etc. make some material substance move as they go by. Electromagnetic waves, such as light or radio waves, make electric and magnetic fields change strength.) Transverse waves: The medium is displaced perpendicular to the direction of propagation. Examples: light, string waves, water waves .