Chapter 40 Problems

1, 2, 3 = straightforward, intermediate,
challenging
Section 40.1 Blackbody Radiation and
Plancks 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 10
4
K,
whereas (b) a nuclear explosion produces a
temperature on the order of 10
7
K. Use
Wiens displacement law to fnd 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 cm
2
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 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 10
8
m,
and its total power output is 3.77 10
26
W.
(a) Assuming that the Suns surface emits as
a black body, calculate its surface
temperature. (b) Using the result of part (a),
fnd
max
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 classifcation of each
on the electromagnetic spectrum.
8. The average threshold of dark-adapted
(scotopic) vision is 4.00 10
11
W/m
2
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,
Plancks radiation law (Equation 40.6)
reduces to the RayleighJeans law
(Equation 40.3).
Section 40.2 The Photoelectric Efect
13. Molybdenum has a work function of
4.20 eV. (a) Find the cutof wavelength and
cutof frequency for the photoelectric efect.
(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 10
5
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 cutof 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 efect and (b) the maximum
kinetic energy for the photoelectrons in
each case.
16. A student studying the photoelectric
efect from two diferent 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 ( =
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 ( = 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/m
2

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 efect induce on the
sphere? The work function for copper is
4.70 eV.
20. Review problem. A light source
emitting radiation at 7.00 10
14
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 Efect
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 ( =
in Fig. 40.13b). (a) Determine the angles
and . (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 E
0
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 ( =
in Fig. 40.13b). (a) Determine the angles
and . (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 of 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
fnal speed of the electron.
Figure P40.28
29. A photon having wavelength
scatters of a free electron at A (Fig. P40.29)
producing a second photon having
wavelength . This photon then scatters of
another free electron at B, producing a third
photon having wavelength and moving
in a direction directly opposite the original
photon as shown in Figure P40.29.
Determine the numerical value of =
.
Figure P40.29
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 ft this defnition
of ionizing radiation and what does not.
32. Review problem. A heliumneon
laser delivers 2.00 10
18
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
felds inside the beam. (b) If the beam
shines perpendicularly onto a perfectly
refecting surface, what force does it exert
on the surface? (c) If the beam is absorbed
by a block of ice at 0C 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
10
6
m/s.
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
m
10 86 . 2

11
n
K

where K
n
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
confned 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 confned 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 fnd an electron in a nucleus?
Explain.
38. In the DavissonGermer experiment,
54.0-eV electrons were difracted from a
nickel lattice. If the frst maximum in the
difraction pattern was observed at = 50.0
(Fig. P40.38), what was the lattice spacing a
between the vertical rows of atoms in the
fgure? (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 of a freely moving particle are
related by the expression
2 2
2

1
C
c
f
+
,
_

where
C
= 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 Broglies
hypothesis that particles of momentum p
have wave characteristics with wavelength
= h/p, an 80.0-kg student has grown
concerned about being difracted when
passing through a 75.0-cm-wide doorway.
Assume that signifcant difraction occurs
when the width of the difraction aperture
is less that 10.0 times the wavelength of the
wave being difracted. (a) Determine the
maximum speed at which the student can
pass through the doorway in order to be
signifcantly difracted. (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 10
17
s. (c) Should this student
worry about being difracted?
Section 40.6 The Quantum Particle
43. Consider a freely moving quantum
particle with mass m and speed u. Its
energy is E = K = mu
2
. Determine the
phase speed of the quantum wave
representing the particle and show that it is
diferent 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 = =
4 2 2 2
c m c p + and the
momentum is p = h/ = k = mv. For the
quantum wave representing the particle,
the group speed is v
g
= d/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 of axis is the frst zero-intensity
point on the detector array? (c) When a
neutron reaches a detector, can we say
which slit the neutron passed through?
Explain.
46. A modifed oscilloscope is used to
perform an electron interference
experiment. Electrons are incident on a pair
of narrow slits 0.060 0 m 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
diference 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 diference 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 fight 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 = 2J 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 (m
e
= 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 rife is used to shoot 1.00-g
particles at 100 m/s through a hole of
diameter 2.00 mm. How far from the rife
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
10
26
m).
51. Use the uncertainty principle to show
that if an electron were confned inside an
atomic nucleus of diameter 2 10
15
m, it
would have to be moving relativistically,
while a proton confned 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 = p
2
/2m. (b)
Use the results of (a) to fnd the minimum
kinetic energy of a proton confned 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 foor. (a)
Show that, according to the uncertainty
principle, the average miss distance must be
at least
4 / 1
2 / 1
2 2

,
_

,
_

g
H
m
x
f

where H is the initial height of each pellet
above the foor and m is the mass of each
pellet. Assume that the spread in impact
points is given by x
f
= x
i
+ (v
x
)t. (b) If H
= 2.00 m and m = 0.500 g, what is x
f
?
Additional Problems
54. Figure P40.54 shows the stopping
potential versus the incident photon
frequency for the photoelectric efect for
sodium. Use the graph to fnd (a) the work
function, (b) the ratio h/e, and (c) the cutof
wavelength. The data are taken from R. A.
Millikan, Phys. Rev. 7:362 (1916).
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 Plancks constant (in
joule-seconds) and (c) the work function (in
electron volts) for the surface. (Two
signifcant fgures for each answer are
sufcient.)
Wavelength Maximum Kinetic
(nm) Energy of
Photoelectrons (eV)
588 0.67
505 0.98
445 1.35
399 1.63
56. Review problem. Photons of
wavelength are incident on a metal. The
most energetic electrons ejected from the
metal are bent into a circular arc of radius R
by a magnetic feld 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?
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 and
Compton wavelength
C
= h/(mc) is
( )
2
C
/ 1+

c
v
61. The total power per unit area radiated
by a black body at a temperature T is the
area under the I(, T)-versus- curve, as
shown in Figure 40.3. (a) Show that this
power per unit area is
( )
4
0
, T d T I

where I(, T) is given by Plancks radiation

law and is a constant independent of T.
This result is known as Stefans law. (See
Section 20.7.) To carry out the integration,
you should make the change of variable x =
hc/kT and use the fact that

0
4 3
15 1

x
e
dx x
(b) Show that the StefanBoltzmann
constant has the value
4 2 8
3 2
4
B
5
K W/m 10 67 . 5
15
2


h c
k

62. Derive Wiens displacement law from

Plancks 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(, T) versus has a horizontal tangent.
From Equation 40.6 evaluate the derivative
dI/d. Set it equal to zero. Solve the
resulting transcendental equation
numerically to prove hc /
max
k
B
T = 4.965 . . .,
or
max
T = hc / 4.965 k
B
. Evaluate the
constant as precisely as possible and
compare it with Wiens experimental value.
63. The spectral distribution function I(,
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

max
is

+
965 . 4
0
3
4
1
15
1 dx
e
x
B A
A
x

independent of the value of T. (b) Using

numerical integration, show that this ratio
is approximately 1/4.
Figure P40.63

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 (3k
B
T/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 difraction
efects when scattered by a crystal?
65. Show that the ratio of the Compton
wavelength
C
to the de Broglie wavelength
= h/p for a relativistic electron is
2 / 1
2
2
C
1

1
1
]
1

,
_

c m
E
e
where E is the total energy of the electron
and m
e
is its mass.
66. Johnny Jumpers 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) Johnnys 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
0
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 m/m in
its mass determination.
68. A photon of initial energy E
0

undergoes Compton scattering at an angle
from a free electron (mass m
e
) initially at
rest. Using relativistic equations for energy
and momentum conservation, derive the
following relationship for the fnal energy
E of the scattered photon:
( )
1
2
0
0
cos 1 1 '

1
1
]
1

,
_

+
c m
E
E E
e
69. Review problem. Consider an
extension of Youngs 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 fat array of charge-coupled detectors.
Each cell in the array registers individual
photons with high efciency, 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 flter or allowed to
pass through. The photons that come
through a polarizer have their electric feld
oscillating in the plane defned by their
direction of motion and the flter 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 efect 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 fat layer of liquid crystal, to which we can
apply an electric feld in either a horizontal
or a vertical direction. With the applied
feld we can very rapidly switch the liquid
crystal to transmit only photons with
horizontal electric feld, to act as a polarizer
with a vertical transmission axis, or to
transmit all photons with high efciency.
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 diferent
conditions, to end up with full sets of data
for all three of those trials. What are the
results?
70. A photon with wavelength
0
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 (Fig. P40.70b). Show that the
wavelength of the scattered photon is
( )
[ ]
( )
( )
( )

cos 1
/ 1
/ 1
/ 1
cos / 1
'
0

+
+

,
_

c u
c u
c m
h
c u
c u
e
Figure P40.70
Copyright 2004 Thomson. All r!h"s r#s#r$#%.