Chapter 3-5 Key Homework Problems These problems are to be worked on in discussion. They contain most of the physics concepts needed to solve other problems. The other problems have solutions that can be used to help you with these more basic problems. The solutions to these problems are intentionally not given so you can learn to really solve things yourself. For each problem, state the physics principle (in words) that you will use to solve the problem and choose the best formulas from the formula sheet to answer the problem. Then write an algebraic expression for the answer based on the formulas and finally compute a numerical result if that is possible. 1. A Red Dwarf star system is being desperately considered for colonization. It is measured that the star’s surface temperature is 3100 K and that its radius is 1.3X108 m. (see 3:2, 3:4) The Wien displacement law says how the peak wavelength depends on temperature. Stephan’s law gives the power output per square meter. We can get the total power output for the star by multiplying by 4πR2. WE can get the intensity at a distance r from the star by dividing the total power by 4πr2. a. At what wavelength will the EM radiation from the star peak? b. What color is that? c. What frequency does this wavelength correspond to? d. What is the EM radiation power per square meter at the surface of the star? e. The peak power per unit frequency is not at the same frequency you calculated in part c. Why. (see ) f. The power per unit area from the sun just outside the earth’s atmosphere is about 1000 W/m2. At what distance from the Red Dwarf, will the total intensity of EM radiation be 1000 W/m2? g. Is this star a possible new home for humanity? 2. Assume that the sun behaves as a black body with a temperature of 5800 K and that the 2 solar power incident upon a blacktop parking lot is 600 W/m . Assume that the blacktop radiates as a black body but that because of the green house effect only 40% of the energy emitted by the parking lot escapes with the rest reabsorbed by the blacktop. We need to balance the heating of the blacktop due to BB radiation from the sun, with the cooling of the blacktop due to its own blackbody radiation. The greenhouse effect will make a difference in the temperature of the blacktop at which energy flow is balanced. a. What would be the equilibrium temperature of the blacktop with zero greenhouse effect? b. What would be the equilibrium temperature of the blacktop with this 40% escape greenhouse effect? 3. Challenge: The Cosmic Microwave Background has a black-body spectrum for a Temperature of 2.76 degrees Kelvin. This is the temperature of the night sky. (see 3:2, 3:4) a. At what wavelength will the maximum amount of radiation be emitted? b. What power per square meter flows into the earth’s atmosphere from the CMB? c. Night on the moon lasts for about 14 days. Since the moon has no atmosphere, there is no greenhouse effect. Estimate the moon’s temperature during the day. d. Estimate the moon’s temperature after 10 earth days with no sunlight; assuming the surface is well insulated from the moon’s interior. 4. Photoelectric Effect: Monochromatic light is incident upon a material with a work function of 2.3 eV. (see 3:13, 3:15, 3:20) a. What is the minimum frequency of the light needed to knock electrons out of the material. b. What is the maximum wavelength for the light to knock out electrons? c. If a retarding voltage of 1.1 volts is applied to this photocell, what would the maximum wavelength be to observe a photocurrent? d. The photocurrent of a photocell is cut off by a retarding potential of 1.32 V for radiation of wavelength 450 nm. Find the work function for the material in eV. 5. A spectral line of sodium is observed using a diffraction grating with slit spacing of 3 microns. Aside from the very bright line at zero deflection angle, the lowest angle at which the spectral line is observed is 7.4 degrees. a. What is the wavelength of photons in that line? b. At what angle would we next observe this line? 6. Two narrow slits 0.7 mm apart, are illuminated with monochromatic light and the usual interference between the two slits is observed. The top picture shows the diffraction pattern if the right hand slit is covered. The bottom picture shows the 2-slit interference pattern. a. Explain the double-slit pattern including the tails. b. If we measure the peak intensity (at the center) in the top and bottom distributions, what would the ratio of bottom to top be? c. If the intensity is turned down so that there is only one photon in the detector at a time, what will the pattern of a single photon look like? d. What will the observed distribution of photons be after enough time to accumulate measurements of single photons? 7. X-rays with an energy of 0.6 MeV undergo Compton scattering from a target. If the scattered x-rays are detected at 40 degrees relative to the incident rays, a. What are the x-rays and off what are they scattering? b. What is the wavelength of the incoming x-rays? c. How was the Compton scattering formula you will use for part d derived? d. What is the wavelength of the scattered x-rays? 8. A single photon from an unpolarized wave travelling in the z direction is directed through a series of three polarizers: first a polarizer with its axis in the x direction, then a polarizer at 30 degrees to the x axis, then finally a y polarizer. a. What is the probability that the photon makes it through the first (x) polarizer? b. If it comes through the first polarizer, what is its polarization state in between the first two polarizers? c. What is the probability that the initial photon makes it through the first two polarizers? d. If it comes through the first two polarizers, what is its polarization state in between the second and third polarizers? e. What is the probability that the initial photon makes it through all three polarizers? f. If it comes through all three polarizers, what is its polarization state? g. If it comes through all three polarizers, what was its polarization state in between the first and second polarizers? h. Do the probabilities for all the possible outcomes add to 1? 9. A narrow beam of unpolarized light with an intensity I0 is incident upon two polarizers. The angle of the second polarizer is 57 degrees different from the first. a. What will be intensity of light after the two polarizers? b. What will be the polarization of the light after the two polarizers? 10. Rutherford Scattering: A parallel beam of α particles with fixed kinetic energy is normally incident on a piece of lead foil. (a) If 1000 particles per second are detected in a 0.01 steradian detector at 10°, (see 4:8, 4:9) a. how many per second will be counted at 120°? b. how many per second will be counted at 0.5°? c. how many per second will be counted at 180°? d. Recall that the Rutherford scattering cross section was derived based on Coulomb scattering from a point nucleus. e. if the kinetic energy of the incident α particles is reduced by a factor of 3, how many per second will be detected at 90°? f. The radius of a nucleus is approximately given by 1.2X10-15 A1/3 where A is the number of nucleons (208 for lead). Calculate the alpha particle energy needed to probe inside the nucleus and thus get a deviation from the Rutherford formula for scattering angles near 180°. g. Similarly, calculate the alpha energy needed to see deviations for a C12 target. 11. Light from excited Hydrogen atoms is observed with a wavelength of 4.05 microns. (see 4:11, 4:12, 4:13, 4:18) a. What atomic transition could this light come from? b. What is the wavelength of Lyman alpha (n=2 goes to n=1)? 12. Even though all particles and forces are quantized, quantum particles all have wave functions and the wavelength of these wave functions determine much about the physical world. De Broglie stated the simple relationship between m om entum and wavelength λ=h/p . This may seem different from the wavelength for photons but given that E=pc for massless particles like the photon, we get λ=h/p= hc/E, as usual for photons. The place where many mistakes can be made is in computing the momentum from the Kinetic energy for massive particles. In the problems below we give the Kinetic energy of particles and ask you for the momentum or wavelength. (see 5:2, 5:6, 5:7, 5:9, 5:10) a. Derive the expression for p in terms of the Kinetic energy K using the correct relativistic expression. This is always correct but could suffer from numerical inaccuracy. b. Derive the expression for p in the nonrelativistic limit (pcmc2). d. In each of the three above cases write an expression for the wavelength taking advantage of our units like MeV/c2 for mass. e. What is the wavelength of a 2 eV photon? f. What is the wavelength of a 2 eV electron? g. What is the wavelength of a 2 eV proton? h. What is the wavelength of a 2 MeV photon? i. What is the wavelength of a 2 MeV electron? j. What is the wavelength of a proton that is accelerated through a potential difference of 2 million Volts. 13. The “seeing” ability, or resolution, of radiation is determined by its wavelength. The wavelength needs to be at least as small as the feature we wish to “see”. (see 5:4, 5:5,5:11) a. If the size of an atom is of the order of 0.1 nm, what Kinetic energy electron is needed to have a wavelength small enough to “see” atoms? b. What Energy photon is needed to have a wavelength small enough to “see” atoms. c. If the size of a proton is of the order of 1 fm, what Kinetic energy electron is needed to have a wavelength small enough to “see” protons inside a nucleus? d. What is the smallest size we can see with visible light? 14. The Heisenberg uncertainty principle , ΔpΔx ≥ ! 2 , tells us that if we confine a particle in a small space, its momentum becomes uncertain and hence its Kinetic energy becomes larger. For NR particles, K= p 2 Δp 2 !2 ≥ ≥ , a lower limit on the kinetic energy can 2m 2m 8mΔx 2 be estimated. (see 5:19, 5:20) a. If a proton is confined within a nucleus so that Δr=12 fm, estimate the uncertainty on the momentum of the proton? Estimate its Kinetic energy. -14 b. For an electron to be confined to a nucleus, its Δx would have to be less than 10 m. Use the uncertainty principle to estimate the kinetic energy of an electron so confined? c. Estimate the size of a Hydrogen atom based on the uncertainty principle. 15. For a given potential, we can estimate the energy of the ground state using the uncertainty principle. We can take the uncertainty principle and write Δp in terms of Δx, then estimate that p~Δp and x~Δx. Now we can write the total energy in terns of x and minimize it. The minimum value is an estimate for the ground state energy. (see 5:24) a. Use the uncertainty principle ΔpΔx ≅ ! to estimate the ground state energy of Hydrogen. (We do not use the minimum uncertainty because the Hydrogen wave function will not be the minimum uncertainty wave packet and we are ignoring the 3 dimensions. What is Δx? What are the potential and kinetic energies? b. Use the uncertainty principle, ΔpΔx ≅ ! 2 , to estimate the potential energy in the ground state of the 1D harmonic oscillator. What are the kinetic and potential energies in this estimate?
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