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Chapter 40 Problems 1, 2, 3 = straightforward, intermediate, challenging Section 40.1 Blackbody Radiation and Planck’s Hypothesis 1. The human eye is most sensitive to 560-nm light. What is the temperature of a black body that would radiate most intensely at this wavelength? 2. (a) Lightning produces a maximum air temperature on the order of 104 K, whereas (b) a nuclear explosion produces a temperature on the order of 107 K. Use Wien’s displacement law to find the order of magnitude of the wavelength of the thermally produced photons radiated with greatest intensity by each of these sources. Name the part of the electromagnetic spectrum where you would expect each to radiate most strongly. 3. A black body at 7 500 K consists of an opening of diameter 0.050 0 mm, looking into an oven. Find the number of photons per second escaping the hole and having wavelengths between 500 nm and 501 nm. 4. Consider a black body of surface area 20.0 cm2 and temperature 5 000 K. (a) How much power does it radiate? (b) At what wavelength does it radiate most intensely? Find the spectral power per wavelength at (c) this wavelength and at wavelengths of (d) 1.00 nm (an x- or γ ray), (e) 5.00 nm (ultraviolet light or an x-ray), (f) 400 nm (at the boundary between UV and visible light), (g) 700 nm (at the boundary between visible and infrared light), (h) 1.00 mm (infrared light or a microwave) and (i) 10.0 cm (a microwave or radio wave). ( j) About how much power does the object radiate as visible light? 5. The radius of our Sun is 6.96 × 108 m, and its total power output is 3.77 × 1026 W. (a) Assuming that the Sun’s surface emits as a black body, calculate its surface temperature. (b) Using the result of part (a), find λmax for the Sun. 6. A sodium-vapor lamp has a power output of 10.0 W. Using 589.3 nm as the average wavelength of this source, calculate the number of photons emitted per second. 7. Calculate the energy, in electron volts, of a photon whose frequency is (a) 620 THz, (b) 3.10 GHz, (c) 46.0 MHz. (d) Determine the corresponding wavelengths for these photons and state the classification of each on the electromagnetic spectrum. 8. The average threshold of darkadapted (scotopic) vision is 4.00 × 10–11 W/m2 at a central wavelength of 500 nm. If light having this intensity and wavelength enters the eye and the pupil is open to its maximum diameter of 8.50 mm, how many photons per second enter the eye? 9. An FM radio transmitter has a power output of 150 kW and operates at a frequency of 99.7 MHz. How many photons per second does the transmitter emit? 10. A simple pendulum has a length of 1.00 m and a mass of 1.00 kg. The amplitude of oscillations of the pendulum is 3.00 cm. Estimate the quantum number for the pendulum. 11. Review problem. A star moving away from the Earth at 0.280c emits radiation that we measure to be most intense at the wavelength 500 nm. Determine the surface temperature of this star. 12. Show that at long wavelengths, Planck’s radiation law (Equation 40.6) reduces to the Rayleigh–Jeans law (Equation 40.3). Section 40.2 The Photoelectric Effect 13. Molybdenum has a work function of 4.20 eV. (a) Find the cutoff wavelength and cutoff frequency for the photoelectric effect. (b) What is the stopping potential if the incident light has a wavelength of 180 nm? 14. Electrons are ejected from a metallic surface with speeds ranging up to 4.60 × 105 m/s when light with a wavelength of 625 nm is used. (a) What is the work function of the surface? (b) What is the cutoff frequency for this surface? 15. Lithium, beryllium, and mercury have work functions of 2.30 eV, 3.90 eV, and 4.50 eV, respectively. Light with a wavelength of 400 nm is incident on each of these metals. Determine (a) which metals exhibit the photoelectric effect and (b) the maximum kinetic energy for the photoelectrons in each case. 16. A student studying the photoelectric effect from two different metals records the following information: (i) the stopping potential for photoelectrons released from metal 1 is 1.48 V larger than that for metal 2, and (ii) the threshold frequency for metal 1 is 40.0% smaller than that for metal 2. Determine the work function for each metal. 17. Two light sources are used in a photoelectric experiment to determine the work function for a particular metal surface. When green light from a mercury lamp (λ = 546.1 nm) is used, a stopping potential of 0.376 V reduces the photocurrent to zero. (a) Based on this measurement, what is the work function for this metal? (b) What stopping potential would be observed when using the yellow light from a helium discharge tube (λ = 587.5 nm)? 18. From the scattering of sunlight, Thomson calculated the classical radius of the electron as having a value of 2.82 × 10–15 m. Sunlight with an intensity of 500 W/m2 falls on a disk with this radius. Calculate the time interval required to accumulate 1.00 eV of energy. Assume that light is a classical wave and that the light striking the disk is completely absorbed. How does your result compare with the observation that photoelectrons are emitted promptly (within 10–9 s)? 19. Review problem. An isolated copper sphere of radius 5.00 cm, initially uncharged, is illuminated by ultraviolet light of wavelength 200 nm. What charge will the photoelectric effect induce on the sphere? The work function for copper is 4.70 eV. 20. Review problem. A light source emitting radiation at 7.00 × 1014 Hz is incapable of ejecting photoelectrons from a certain metal. In an attempt to use this source to eject photoelectrons from the metal, the source is given a velocity toward the metal. (a) Explain how this procedure produces photoelectrons. (b) When the speed of the light source is equal to 0.280c, photoelectrons just begin to be ejected from the metal. What is the work function of the metal? (c) When the speed of the light source is increased to 0.900c, determine the maximum kinetic energy of the photoelectrons. Section 40.3 The Compton Effect 21. Calculate the energy and momentum of a photon of wavelength 700 nm. 22. X-rays having an energy of 300 keV undergo Compton scattering from a target. The scattered rays are detected at 37.0° relative to the incident rays. Find (a) the Compton shift at this angle, (b) the energy of the scattered x-ray, and (c) the energy of the recoiling electron. 23. A 0.001 60-nm photon scatters from a free electron. For what (photon) scattering angle does the recoiling electron have kinetic energy equal to the energy of the scattered photon? 24. A 0.110-nm photon collides with a stationary electron. After the collision, the electron moves forward and the photon recoils backward. Find the momentum and the kinetic energy of the electron. 25. A 0.880-MeV photon is scattered by a free electron initially at rest such that the scattering angle of the scattered electron is equal to that of the scattered photon (θ = φ in Fig. 40.13b). (a) Determine the angles θ and φ. (b) Determine the energy and momentum of the scattered photon. (c) Determine the kinetic energy and momentum of the scattered electron. 26. A photon having energy E0 is scattered by a free electron initially at rest such that the scattering angle of the scattered electron is equal to that of the scattered photon (θ = φ in Fig. 40.13b). (a) Determine the angles θ and φ. (b) Determine the energy and momentum of the scattered photon. (c) Determine the kinetic energy and momentum of the scattered electron. 27. In a Compton scattering experiment, an x-ray photon scatters through an angle of 17.4° from a free electron that is initially at rest. The electron recoils with a speed of 2 180 km/s. Calculate (a) the wavelength of the incident photon and (b) the angle through which the electron scatters. 28. A 0.700-MeV photon scatters off a free electron such that the scattering angle of the photon is twice the scattering angle of the electron (Fig. P40.28). Determine (a) the scattering angle for the electron and (b) the final speed of the electron. Figure P40.28 29. A photon having wavelength λ scatters off a free electron at A (Fig. P40.29) producing a second photon having wavelength λ’. This photon then scatters off another free electron at B, producing a third photon having wavelength λ’’ and moving in a direction directly opposite the original photon as shown in Figure P40.29. Determine the numerical value of Δλ = λ’’ – λ. 30. Find the maximum fractional energy loss for a 0.511-MeV gamma ray that is Compton scattered from a free (a) electron (b) proton. Section 40.4 Photons and Electromagnetic Waves 31. An electromagnetic wave is called ionizing radiation if its photon energy is larger than say 10.0 eV, so that a single photon has enough energy to break apart an atom. With reference to Figure 34.12, identify what regions of the electromagnetic spectrum fit this definition of ionizing radiation and what does not. 32. Review problem. A helium–neon laser delivers 2.00 × 1018 photons/s in a beam of diameter 1.75 mm. Each photon has a wavelength of 633 nm. (a) Calculate the amplitudes of the electric and magnetic fields inside the beam. (b) If the beam shines perpendicularly onto a perfectly reflecting surface, what force does it exert on the surface? (c) If the beam is absorbed by a block of ice at 0°C for 1.50 h, what mass of ice is melted? Section 40.5 The Wave Properties of Particles 33. Calculate the de Broglie wavelength for a proton moving with a speed of 1.00 × 106 m/s. Figure P40.29 34. Calculate the de Broglie wavelength for an electron that has kinetic energy (a) 50.0 eV and (b) 50.0 keV. 35. (a) An electron has kinetic energy 3.00 eV. Find its wavelength. (b) What If? A photon has energy 3.00 eV. Find its wavelength. 36. (a) Show that the wavelength of a nonrelativistic neutron is λ 2.86 10 11 m Kn where Kn is the kinetic energy of the neutron in electron volts. (b) What is the wavelength of a 1.00-keV neutron? 37. The nucleus of an atom is on the order of 10–14 m in diameter. For an electron to be confined to a nucleus, its de Broglie wavelength would have to be on this order of magnitude or smaller. (a) What would be the kinetic energy of an electron confined to this region? (b) Given that typical binding energies of electrons in atoms are measured to be on the order of a few eV, would you expect to find an electron in a nucleus? Explain. 38. In the Davisson–Germer experiment, 54.0-eV electrons were diffracted from a nickel lattice. If the first maximum in the diffraction pattern was observed at φ = 50.0° (Fig. P40.38), what was the lattice spacing a between the vertical rows of atoms in the figure? (It is not the same as the spacing between the horizontal rows of atoms.) Figure P40.38 39. (a) Show that the frequency f and wavelength λ of a freely moving particle are related by the expression 2 1 1 f 2 2 λ λC c where λC = h/mc is the Compton wavelength of the particle. (b) Is it ever possible for a particle having nonzero mass to have the same wavelength and frequency as a photon? Explain. 40. A photon has an energy equal to the kinetic energy of a particle moving with a speed of 0.900c. (a) Calculate the ratio of the wavelength of the photon to the wavelength of the particle. (b) What would this ratio be for a particle having a speed of 0.001 00c ? (c) What If? What value does the ratio of the two wavelengths approach at high particle speeds?(d) At low particle speeds? 41. The resolving power of a microscope depends on the wavelength used. If one wished to “see” an atom, a resolution of approximately 1.00 × 10–11 m would be required. (a) If electrons are used (in an electron microscope), what minimum kinetic energy is required for the electrons? (b) What If? If photons are used, what minimum photon energy is needed to obtain the required resolution? 42. After learning about de Broglie’s hypothesis that particles of momentum p have wave characteristics with wavelength λ = h/p, an 80.0-kg student has grown concerned about being diffracted when passing through a 75.0-cm-wide doorway. Assume that significant diffraction occurs when the width of the diffraction aperture is less that 10.0 times the wavelength of the wave being diffracted. (a) Determine the maximum speed at which the student can pass through the doorway in order to be significantly diffracted. (b) With that speed, how long will it take the student to pass through the doorway if it is in a wall 15.0 cm thick? Compare your result to the currently accepted age of the Universe, which is 4 × 1017 s. (c) Should this student worry about being diffracted? Section 40.6 The Quantum Particle 43. Consider a freely moving quantum particle with mass m and speed u. Its energy is E = K = ½ mu2. Determine the phase speed of the quantum wave representing the particle and show that it is different from the speed at which the particle transports mass and energy. 44. For a free relativistic quantum particle moving with speed v, the total energy is E = hf = ħω = p 2 c 2 m 2 c 4 and the momentum is p = h/λ = ħk = γmv. For the quantum wave representing the particle, the group speed is vg = dω/dk. Prove that the group speed of the wave is the same as the speed of the particle. Section 40.7 The Double-Slit Experiment Revisited 45. Neutrons traveling at 0.400 m/s are directed through a pair of slits having a 1.00-mm separation. An array of detectors is placed 10.0 m from the slits. (a) What is the de Broglie wavelength of the neutrons? (b) How far off axis is the first zerointensity point on the detector array? (c) When a neutron reaches a detector, can we say which slit the neutron passed through? Explain. 46. A modified oscilloscope is used to perform an electron interference experiment. Electrons are incident on a pair of narrow slits 0.060 0 μm apart. The bright bands in the interference pattern are separated by 0.400 mm on a screen 20.0 cm from the slits. Determine the potential difference through which the electrons were accelerated to give this pattern. 47. In a certain vacuum tube, electrons evaporate from a hot cathode at a slow, steady rate and accelerate from rest through a potential difference of 45.0 V. Then they travel 28.0 cm as they pass through an array of slits and fall on a screen to produce an interference pattern. If the beam current is below a certain value, only one electron at a time will be in flight in the tube. What is this value? In this situation, the interference pattern still appears, showing that each individual electron can interfere with itself. Section 40.8 The Uncertainty Principle 48. Suppose Fuzzy, a quantum– mechanical duck, lives in a world in which h = 2πJ · s. Fuzzy has a mass of 2.00 kg and is initially known to be within a pond 1.00 m wide. (a) What is the minimum uncertainty in the component of his velocity parallel to the width of the pond? (b) Assuming that this uncertainty in speed prevails for 5.00 s, determine the uncertainty in his position after this time interval. 49. An electron (me = 9.11 × 10–31 kg) and a bullet (m = 0.020 0 kg) each have a velocity of magnitude of 500 m/s, accurate to within 0.010 0%. Within what limits could we determine the position of the objects along the direction of the velocity? 50. An air rifle is used to shoot 1.00-g particles at 100 m/s through a hole of diameter 2.00 mm. How far from the rifle must an observer be in order to see the beam spread by 1.00 cm because of the uncertainty principle? Compare this answer with the diameter of the visible Universe (2 × 1026 m). 51. Use the uncertainty principle to show that if an electron were confined inside an atomic nucleus of diameter 2 × 10–15 m, it would have to be moving relativistically, while a proton confined to the same nucleus can be moving nonrelativistically. 52. (a) Show that the kinetic energy of a nonrelativistic particle can be written in terms of its momentum as K = p2/2m. (b) Use the results of (a) to find the minimum kinetic energy of a proton confined within a nucleus having a diameter of 1.00 × 10–15 m. 53. A woman on a ladder drops small pellets toward a point target on the floor. (a) Show that, according to the uncertainty principle, the average miss distance must be at least 2 Δx f m 1/ 2 2H g 1/ 4 where H is the initial height of each pellet above the floor and m is the mass of each pellet. Assume that the spread in impact points is given by Δxf = Δxi + (Δvx)t. (b) If H = 2.00 m and m = 0.500 g, what is Δxf ? Additional Problems 54. Figure P40.54 shows the stopping potential versus the incident photon frequency for the photoelectric effect for sodium. Use the graph to find (a) the work function, (b) the ratio h/e, and (c) the cutoff wavelength. The data are taken from R. A. Millikan, Phys. Rev. 7:362 (1916). by a magnetic field having a magnitude B. What is the work function of the metal? 57. A 200-MeV photon is scattered at 40.0° by a free proton initially at rest. (a) Find the energy (in MeV) of the scattered photon. (b) What kinetic energy (in MeV) does the proton acquire? Figure P40.54 55. The following table shows data obtained in a photoelectric experiment. (a) Using these data, make a graph similar to Figure 40.11 that plots as a straight line. From the graph, determine (b) an experimental value for Planck’s constant (in joule-seconds) and (c) the work function (in electron volts) for the surface. (Two significant figures for each answer are sufficient.) Wavelength (nm) Maximum Kinetic Energy of Photoelectrons (eV) 588 505 445 399 0.67 0.98 1.35 1.63 56. Review problem. Photons of wavelength λ are incident on a metal. The most energetic electrons ejected from the metal are bent into a circular arc of radius R 58. Derive the equation for the Compton shift (Eq. 40.11) from Equations 40.12, 40.13, and 40.14. 59. Show that a photon cannot transfer all of its energy to a free electron. (Suggestion: Note that system energy and momentum must be conserved.) 60. Show that the speed of a particle having de Broglie wavelength λ and Compton wavelength λC = h/(mc) is v c 1 λ/λ C 2 61. The total power per unit area radiated by a black body at a temperature T is the area under the I(λ, T)-versus-λ curve, as shown in Figure 40.3. (a) Show that this power per unit area is I λ, T dλ T 4 0 where I(λ, T) is given by Planck’s radiation law and σ is a constant independent of T. This result is known as Stefan’s law. (See Section 20.7.) To carry out the integration, you should make the change of variable x = hc/λkT and use the fact that 3 x dx 4 0 e x 1 15 (b) Show that the Stefan–Boltzmann constant σ has the value 2 5 k B 2 15c h 4 3 5.67 10 8 W/m 2 K 4 Figure P40.63 62. Derive Wien’s displacement law from Planck’s law. Proceed as follows. In Figure 40.3 note that the wavelength at which a black body radiates with greatest intensity is the wavelength for which the graph of I(λ, T) versus λ has a horizontal tangent. From Equation 40.6 evaluate the derivative dI/dλ. Set it equal to zero. Solve the resulting transcendental equation numerically to prove hc / λmaxkBT = 4.965 . . ., or λmaxT = hc / 4.965 kB. Evaluate the constant as precisely as possible and compare it with Wien’s experimental value. 63. The spectral distribution function I(λ, T) for an ideal black body at absolute temperature T is shown in Figure P40.63. (a) Show that the percentage of the total power radiated per unit area in the range 0 ≤ λ ≤ λmax is A 15 1 4 A B 4.965 0 x3 dx e x 1 independent of the value of T. (b) Using numerical integration, show that this ratio is approximately 1/4. 64. The neutron has a mass of 1.67 × 10–27 kg. Neutrons emitted in nuclear reactions can be slowed down via collisions with matter. They are referred to as thermal neutrons once they come into thermal equilibrium with their surroundings. The average kinetic energy (3kBT/2) of a thermal neutron is approximately 0.04 eV. Calculate the de Broglie wavelength of a neutron with a kinetic energy of 0.040 0 eV. How does it compare with the characteristic atomic spacing in a crystal? Would you expect thermal neutrons to exhibit diffraction effects when scattered by a crystal? 65. Show that the ratio of the Compton wavelength λC to the de Broglie wavelength λ = h/p for a relativistic electron is λ C E λ m e c 2 2 1 1/ 2 where E is the total energy of the electron and me is its mass. 66. Johnny Jumper’s favorite trick is to step out of his 16th-story window and fall 50.0 m into a pool. A news reporter takes a picture of 75.0-kg Johnny just before he makes a splash, using an exposure time of 5.00 ms. Find (a) Johnny’s de Broglie wavelength at this moment, (b) the uncertainty of his kinetic energy measurement during such a period of time, and (c) the percent error caused by such an uncertainty. 67. A π0 meson is an unstable particle produced in high-energy particle collisions. Its rest energy is about 135 MeV, and it exists for an average lifetime of only 8.70 × 10–17 s before decaying into two gamma rays. Using the uncertainty principle, estimate the fractional uncertainty Δm/m in its mass determination. 68. A photon of initial energy E0 undergoes Compton scattering at an angle θ from a free electron (mass me) initially at rest. Using relativistic equations for energy and momentum conservation, derive the following relationship for the final energy E’ of the scattered photon: E E ' E 0 1 0 2 me c 1 cos 1 69. Review problem. Consider an extension of Young’s double-slit experiment performed with photons. Think of Figure 40.24 as a top view looking down on the apparatus. The viewing screen can be a large flat array of charge-coupled detectors. Each cell in the array registers individual photons with high efficiency, so we can see where individual photons strike the screen in real time. We cover slit 1 with a polarizer with its transmission axis horizontal, and slit 2 with a polarizer with vertical transmission axis. Any one photon is either absorbed by a polarizing filter or allowed to pass through. The photons that come through a polarizer have their electric field oscillating in the plane defined by their direction of motion and the filter axis. Now we place another large sheet of polarizing material just in front of the screen. For experimental trial 1, we make the transmission axis of this third polarizer horizontal. This choice in effect blocks slit 2. After many photons have been sent through the apparatus, their distribution on the viewing screen is shown by the lower blue curve in the middle of Figure 40.24. For trial 2, we turn the polarizer at the screen to make its transmission axis vertical. Then the screen receives photons only by way of slit 2, and their distribution is shown as the upper blue curve. For trial 3, we temporarily remove the third sheet of polarizing material. Then the interference pattern shown by the red curve on the right in Figure 40.24 appears. (a) Is the light arriving at the screen to form the interference pattern polarized? Explain your answer. (b) Next, in trial 4 we replace the large square of polarizing material in front of the screen and set its transmission axis to 45°, halfway between horizontal and vertical. What appears on the screen? (c) Suppose we repeat all of trials 1 through 4 with very low light intensity, so that only one photon is present in the apparatus at a time. What are the results now? (d) We go back to high light intensity for convenience and in trial 5 make the large square of polarizer turn slowly and steadily about a rotation axis through its center and perpendicular to its area. What appears on the screen? (e) What If? At last, we go back to very low light intensity and replace the large square sheet of polarizing plastic with a flat layer of liquid crystal, to which we can apply an electric field in either a horizontal or a vertical direction. With the applied field we can very rapidly switch the liquid crystal to transmit only photons with horizontal electric field, to act as a polarizer with a vertical transmission axis, or to transmit all photons with high efficiency. We keep track of photons as they are emitted individually by the source. For each photon we wait until it has passed through the pair of slits. Then we quickly choose the setting of the liquid crystal and make that photon encounter a horizontal polarizer, a vertical polarizer, or no polarizer before it arrives at the detector array. We can alternate among the conditions we earlier set up in trials 1, 2, and 3. We keep track of our settings of the liquid crystal and sort out how photons © Copyright 2004 Thomson. All rights reserved. behave under the different conditions, to end up with full sets of data for all three of those trials. What are the results? 70. A photon with wavelength λ0 moves toward a free electron that is moving with speed u in the same direction as the photon (Fig. P40.70a). The photon scatters at an angle θ (Fig. P40.70b). Show that the wavelength of the scattered photon is 1 u / c cos h 1 u / c 1 cos λ' λ 0 1 u / c me c 1 u / c Figure P40.70

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