Document Sample
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.
                                                          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.
                                                        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–
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
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                                2m                
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
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 
                                   
                                   
                         me c  

where me is the mass of the electron. (Hint:
Begin with conservation of energy,
E  me c 2        pc2  me c 2 2 . (b) If one
of these photons strikes platinum (φ = 6.35 eV),
determine the resulting maximum speed of the