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```Chapter 40 Problems
1, 2, 3 = straightforward, intermediate,
challenging
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
greatest intensity by each of these sources.
Name the part of the electromagnetic
spectrum where you would expect each to
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
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,
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
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
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
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 ?
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
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
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
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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