PROBLEMS 1, 2, 3 = straightforward, intermediate, challenging = full solution available in Student Solutions Manual/Study Guide = biomedical application Section 27.1 Blackbody Radiation and wavelength of this source, calculate the number Planck’s Hypothesis of photons emitted per second. Section 27.2 The Photoelectric Effect and the 7. An FM radio transmitter has a power output Particle Theory of Light of 150 kW and operates at a frequency of 99.7 MHz. How many photons per second does the 1. (a) What is the surface temperature of transmitter emit? Betelgeuse, a red giant star in the constellation of Orion, which radiates with a peak wavelength 8. The threshold of dark-adapted (scotopic) of about 970 nm? (b) Rigel, a bluish-white star vision is 4.0 × 10–11 W/m2 at a central in Orion, radiates with a peak wavelength of 145 wavelength of 500 nm. If light with this nm. Find the temperature of Rigel’s surface. intensity and wavelength enters the eye when the pupil is open to its maximum diameter of 8.5 2. (a) Lightning produces a maximum air mm, how many photons per second enter the temperature on the order of 104 K, while (b) a eye? nuclear explosion produces a temperature on the order of 107 K. Use Wien’s displacement law to 9. A 1.5-kg mass vibrates at an amplitude of 3.0 find the order of magnitude of the wavelength of cm on the end of a spring of spring constant 20 the thermally produced photons radiated with N/m. (a) If the energy of the spring is quantized, greatest intensity by each of these sources. find its quantum number. (b) If n changes by 1, Name the part of the electromagnetic spectrum find the fractional change in energy of the where you would expect each to radiate most spring. strongly. 10. A 0.50-kg mass falls from a height of 3.0 m. 3. (a) Assuming that the tungsten filament of a If all of the energy of this mass could be lightbulb is a blackbody, determine its peak converted to visible light of wavelength 5.0 × wavelength if its temperature is 2 900 K. (b) 10–7 m, how many photons would be produced? Why does your answer to part (a) suggest that more energy from a lightbulb goes into heat than 11. When light of wavelength 350 nm falls on into light? a potassium surface, electrons are emitted that have a maximum kinetic energy of 1.31 eV. 4. Calculate the energy, in electron volts, of a Find (a) the work function of potassium, (b) the photon whose frequency is (a) 620 THz, (b) 3.10 cutoff wavelength, and (c) the frequency GHz, (c) 46.0 MHz. (d) Determine the corresponding to the cutoff wavelength. corresponding wavelengths for these photons and state the classification of each on the 12. Electrons are ejected from a metallic surface electromagnetic spectrum. with speeds ranging up to 4.6 × 105 m/s when light with a wavelength of λ = 625 nm is used. 5. Calculate the energy in electron volts of a (a) What is the work function of the surface? (b) photon having a wavelength in (a) the What is the cutoff frequency for this surface? microwave range, 5.00 cm; (b) the visible light range, 500 nm; and (c) the x-ray range, 5.00 nm. 13. Molybdenum has a work function of 4.20 eV. (a) Find the cutoff wavelength and threshold 6. A sodium-vapor lamp has a power output of 1 frequency for the photoelectric effect. (b) 000 W. Using 589.3 nm as the average Calculate the stopping potential if the incident light has a wavelength of 180 nm. approximately 1.0 × 10–8 m to 1.0 × 10–13 m. 14. Lithium, beryllium, and mercury have work Find the minimum accelerating voltages functions of 2.30 eV, 3.90 eV, and 4.50 eV, required to produce wavelengths at these two respectively. If 400-nm light is incident on each extremes. of these metals, determine (a) which metals exhibit the photoelectric effect and (b) the 20. Calculate the minimum wavelength x-ray maximum kinetic energy for the photoelectrons that can be produced when a target is struck by in each case. an electron that has been accelerated through a potential difference of (a) 15.0 kV, (b) 100 kV. 15. From the scattering of sunlight, Thomson calculated that the classical radius of the electron 21. What minimum accelerating voltage would has a value of 2.82 × 10–15 m. If sunlight having be required to produce an x-ray with a an intensity of 500 W/m2 falls on a disk with this wavelength of 0.0300 nm? radius, find the time required to accumulate 1.00 eV of energy. Assume that light is a classical Section 27.4 Diffraction of X-Rays by wave and that the light striking the disk is Crystals completely absorbed. How does your value compare with the observation that 22. A monochromatic x-ray beam is incident on photoelectrons are promptly (within 10–9 s) a NaCl crystal surface where d = 0.353 nm. The emitted? second-order maximum in the reflected beam is found when the angle between the incident beam 16. An isolated copper sphere of radius 5.00 cm, and the surface is 20.5°. Determine the initially uncharged, is illuminated by ultraviolet wavelength of the x-rays. light of wavelength 200 nm. What charge will the photoelectric effect induce on the sphere? 23. Potassium iodide has an interplanar spacing The work function for copper is 4.70 eV. of d = 0.296 nm. A monochromatic x-ray beam shows a firstorder diffraction maximum when 17. When light of wavelength 254 nm falls on the grazing angle is 7.6°. Calculate the x-ray cesium, the required stopping potential is 3.00 wavelength. V. If light of wavelength 436 nm is used, the stopping potential is 0.900 V. Use this 24. The spacing between certain planes in a information to plot a graph like that shown in crystal is known to be 0.30 nm. Find the Figure 27.6, and from the graph determine the smallest grazing angle at which constructive cutoff frequency for cesium and its work interference will occur for wavelength 0.070 nm. function. 25. X-rays of wavelength 0.140 nm are 18. Ultraviolet light is incident normally on the reflected from a certain crystal, and the first- surface of a certain substance. The binding order maximum occurs at an angle of 14.4°. energy of the electrons in this substance is 3.44 What value does this give for the interplanar eV. The incident light has an intensity of 0.055 spacing of this crystal? W/m2. The electrons are photoelectrically emitted with a maximum speed of 4.2 × 105 m/s. Section 27.5 The Compton Effect How many electrons are emitted from a square centimeter of the surface each second? Assume 26. X-rays are scattered from electrons in a that the absorption of every photon ejects an carbon target. The measured wavelength shift is electron. 1.50 × 10–3 nm. Calculate the scattering angle. Section 27.3 X-Rays 27. Calculate the energy and momentum of a photon of wavelength 700 nm. 19. The extremes of the x-ray portion of the electromagnetic spectrum range from 28. A beam of 0.68-nm photons undergoes 37. The nucleus of an atom is on the order of 10– 14 Compton scattering from free electrons. What m in diameter. For an electron to be confined are the energy and momentum of the photons to a nucleus, its de Broglie wavelength would that emerge at a 45° angle with respect to the have to be of this order of magnitude or smaller. incident beam? (a) What would be the kinetic energy of an electron confined to this region? (b) On the basis 29. A 0.001 6-nm photon scatters from a free of this result, would you expect to find an electron. For what (photon) scattering angle will electron in a nucleus? Explain. the recoiling electron and scattered photon have the same kinetic energy? 38. After learning about de Broglie’s hypothesis that particles of momentum p have wave 30. X-rays with an energy of 300 keV undergo characteristics with wavelength λ = h/p, an 80- Compton scattering from a target. If the kg student has grown concerned about being scattered rays are deflected at 37.0° relative to diffracted when passing through a 75-cm-wide the direction of the incident rays, find (a) the doorway. Assume that significant diffraction Compton shift at this angle, (b) the energy of the occurs when the width of the diffraction aperture scattered x-ray, and (c) the kinetic energy of the is less than 10 times the wavelength of the wave recoiling electron. being diffracted. (a) Determine the order of magnitude of the maximum speed at which the 31. A 0.110-nm photon collides with a student can pass through the doorway in order to stationary electron. After the collision, the be significantly diffracted. (b) With that speed, electron moves forward and the photon recoils how long will it take the student to pass through backward. Find the momentum and kinetic a doorway in a wall 15 cm thick? Compare your energy of the electron. order-of-magnitude result to the currently accepted age of the Universe, which is 4 × 1017 32. After a 0.800 nm x-ray photon scatters from s. (c) Should this student worry about being a free electron, the electron recoils with a speed diffracted? equal to 1.40 × 106 m/s. (a) What was the Compton shift in the photon’s wavelength? (b) 39. De Broglie postulated that the relationship Through what angle was the photon scattered? λ = h/p is valid for relativistic particles. What is the de Broglie wavelength for a (relativistic) 33. A 0.45-nm x-ray photon is deflected through electron whose kinetic energy is 3.00 MeV? a 23° angle after scattering from a free electron. (a) What is the kinetic energy of the recoiling 40. At what speed must an electron move so that electron? (b) What is its speed? its de Broglie wavelength equals its Compton wavelength? (Hint: This electron is relativistic.) Section 27.7 The Wave Properties of Particles 41. The resolving power of a microscope is 34. Calculate the de Broglie wavelength for an proportional to the wavelength used. A electron that has kinetic energy (a) 50.0 eV and resolution of approximately 1.0 × 10–11 m (0.010 (b) 50.0 keV (ignore relativistic effects). nm) would be required in order to “see” an atom. (a) If electrons were used (electron 35. (a) If the wavelength of an electron is equal microscope), what minimum kinetic energy to 5.00 × 10–7 m, how fast is it moving? (b) If would be required for the electrons? (b) If the electron has a speed of 1.00 × 107 m/s, what photons were used, what minimum photon is its wavelength? energy would be needed to obtain 1.0 × 10–11 m resolution? 36. Through what potential difference would an electron have to be accelerated from rest to give Section 27.9 The Uncertainty Principle it a de Broglie wavelength of 1.0 × 10–10 m? 42. A 50.0-g ball moves at 30.0 m/s. If its speed is measured to an accuracy of 0.10%, what is the minimum uncertainty in its position? 43. A 0.50-kg block rests on the icy surface of a frozen pond, which we can assume to be frictionless. If the location of the block is measured to a precision of 0.50 cm, what speed must the block acquire because of the measurement process? Figure 27.47 44. Suppose Fuzzy, a quantum-mechanical duck, lives in a world in which h = 2π J · s. Fuzzy has 48. An x-ray tube is operated at 50 000 V. (a) a mass of 2.00 kg and is initially known to be Find the minimum wavelength of the radiation within a pond 1.00 m wide. (a) What is the emitted by this tube. (b) If this radiation is minimum uncertainty in his speed? (b) directed at a crystal, the first-order maximum in Assuming this uncertainty in speed to prevail for the reflected radiation occurs when the grazing 5.00 s, determine the uncertainty in position angle is 2.50°. What is the spacing between after this time. reflecting planes in the crystal? 45. Suppose optical radiation (λ = 5.00 × 10–7 49. The spacing between planes of nickel m) is used to determine the position of an atoms in a nickel crystal is 0.352 nm. At what electron to within the wavelength of the light. angle does a second-order Bragg reflection What will be the resulting uncertainty in the occur in nickel for 11.3-keV x-rays? electron’s velocity? 50. Johnny Jumper’s favorite trick is to step out 46. (a) Show that the kinetic energy of a of his 16th story window and fall 50.0 m into a nonrelativistic particle can be written in terms of pool. A news reporter takes a picture of 75.0-kg its momentum as KE = p2/2m. (b) Use the results Johnny just before he makes a splash, using an of (a) to find the minimum kinetic energy of a exposure time of 5.00 ms. Find (a) Johnny’s de proton confined within a nucleus having a Broglie wavelength at this moment, (b) the diameter of 1.0 × 10–15 m. uncertainty of his kinetic energy measurement during such a period of time, and (c) the percent ADDITIONAL PROBLEMS error caused by such an uncertainty. 47. Figure P27.47 shows the spectrum of 51. Photons of wavelength 450 nm are incident light emitted by a firefly. Determine the on a metal. The most energetic electrons ejected temperature of a blackbody that would emit from the metal are bent into a circular arc of radiation peaked at the same frequency. Based radius 20.0 cm by a magnetic field with a on your result, would you say firefly radiation is magnitude of 2.00 × 10–5 T. What is the work blackbody radiation? function of the metal? 52. A 200-MeV photon is scattered at 40.0° by a free proton initially at rest. Find the energy (in MeV) of the scattered photon. 53. A light source of wavelength λ illuminates a metal and ejects photoelectrons with a maximum kinetic energy of 1.00 eV. A second light source of wavelength λ/2 ejects photoelectrons with a maximum kinetic energy of 4.00 eV. What is the work function of the 58. In a Compton scattering event, the metal? scattered photon has an energy of 120.0 keV and the recoiling electron has a kinetic energy of 54. Red light of wavelength 670 nm produces 40.0 keV. Find (a) the wavelength of the photoelectrons from a certain photoemissive incident photon, (b) the angle θ at which the material. Green light of wavelength 520 nm photon is scattered, and (c) the recoil angle of produces photoelectrons from the same material the electron. (Hint: Conserve both mass-energy with 1.50 times the maximum kinetic energy. and relativistic momentum.) What is the material’s work function? 59. A woman on a ladder drops small pellets 55. How fast must an electron be moving if all toward a spot on the floor. (a) Show that its kinetic energy is lost to a single x-ray photon according to the uncertainty principle, the (a) at the longwavelength end of the x-ray average miss distance must be at least electromagnetic spectrum with a wavelength of 1 1 1.00 × 10–8 m; (b) at the shortwavelength end of h 2 H 4 x 2g the x-ray electromagnetic spectrum with a 2m wavelength of 1.00 × 10–13 m? where H is the initial height of each pellet above the floor and m is the mass of each pellet. (b) If 56. Show that if an electron were confined inside H = 2.00 m and m = 0.500 g, what is Δx? an atomic nucleus of diameter 2.0 × 10–15 m, it would have to be moving relativistically, while a 60. Show that the speed of a particle having de proton confined to the same nucleus can be Broglie wavelength λ and Compton wavelength moving at less than one-tenth the speed of light. λC = h/(mc) is c 57. A photon strikes a metal with a work v function of φ and produces a photoelectron with 2 a de Broglie wavelength equal to the wavelength 1 C of the original photon. (a) Show that the energy of this photon must have been given by me c 2 2 E me c 2 where me is the mass of the electron. (Hint: Begin with conservation of energy, E me c 2 pc2 me c 2 2 . (b) If one of these photons strikes platinum (φ = 6.35 eV), determine the resulting maximum speed of the photoelectron.