Solutions to Homework problems Chapters 29. Including a
Solutions to Homework problems Chapters 29. Including a
Solutions to Homework problems Chapters 29. Including a
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936 I CHAPTER 29 I PARTICLES AND WAVES<br />
11. A s<strong>to</strong>ne is dropped from the <strong>to</strong>p of a building. As the s<strong>to</strong>ne falls,<br />
does its de Broglie wavelength increase, decrease, or remain the<br />
same? Provide a reason for your answer.<br />
12. An electron and a neutron have different masses. Is it possible,<br />
according <strong>to</strong> Equation <strong>29.</strong>8, that they can have the same de Broglie<br />
wavelength? Account foryour answer.<br />
PROBLEMS<br />
Note <strong>to</strong> Instruc<strong>to</strong>rs: Most of the homework <strong>problems</strong> in this chapter are available for assignment via an online homework management program sllchas<br />
WileyPLUS or WebAssign. and those marked with the icon @ are presented in a guided IlI<strong>to</strong>rialformatthat provides enhanced interactivity. See Preface<br />
for additional details.<br />
In working these <strong>problems</strong>, ignore relativistic effects.<br />
ssm Solution is in the Student <strong>Solutions</strong> Manual.<br />
www Solution is available on the World Wide Web at www.wiley.comlcollege/cutnell<br />
Section <strong>29.</strong>3 Pho<strong>to</strong>ns and the Pho<strong>to</strong>electric Effect<br />
1. ssm Ultraviolet light with a frequency of 3.00 X 1015Hz strikes<br />
a metal surface and ejects electrons that have a maximum kinetic energy<br />
of 6.1 eV. What is the work function (in eV) of the metal?<br />
2. Two sources produce electromagnetic waves. Source B produces<br />
a wavelength that is three times the wavelength produced by source<br />
A. Each pho<strong>to</strong>n from source A has an energy of 2.1 X 10-181. What<br />
is the energy of a pho<strong>to</strong>n from source B?<br />
3. An FM radio station broadcasts at a frequency of 98.1 MHz. The<br />
power radiated from the antenna is 5.0 X 104 W. How many pho<strong>to</strong>ns<br />
per second does the antenna emit?<br />
4. The maximum wavelength that an electromagnetic wave can have<br />
and still eject electrons from a metal surface is 485 nm. What is the<br />
work function Wo of this metal? Express your answer in electron volts.<br />
5. ssm Ultraviolet light is responsible for sun tanning. Find the wavelength<br />
(in nm) of an ultraviolet pho<strong>to</strong>n whose energy is 6.4 X 10-191.<br />
6. Light is shining perpendicularly on the surface of the earth with<br />
an intensity of 680 W/m2• Assuming all the pho<strong>to</strong>ns in the light have<br />
the same wavelength (in vacuum) of.730 nm, determine the number<br />
of pho<strong>to</strong>ns per second per square meter that reach the earth.<br />
o Radiation of a certain wavelength causes electrons with a maximum<br />
kinetic energy of 0.68 eV <strong>to</strong> be ejected from a metal whose<br />
work function is 2.75 eV. What will be the maximum kinetic energy<br />
(in eV) with which this same radiation ejects electrons from another<br />
metal whose work function is 2.17 eV?<br />
~Multiple-Concept Example 3 provides pertinent information for<br />
<strong>problems</strong> such as this. The maximum wavelength for which an electromagnetic<br />
wave can eject electrons from a platinum surface is<br />
196 nm. When radiation with a wavelength of 141 nm shines on the<br />
surface, what is the maximum speed of the ejected electrons?<br />
* 9. ssm Consult Interactive LearningWare <strong>29.</strong>1 at www.wiley.com/<br />
college/cutnell for background material relating <strong>to</strong> this problem. An<br />
owl has good night vision because its eyes can detect a light intensity<br />
as small as 5.0 X 10-13 W/m2• What is the minimum number of<br />
pho<strong>to</strong>ns per second that an owl eye can detect if its pupil has a diameter<br />
of 8.5 mm and the light has a wavelength of 510 nm?<br />
* 10. A pro<strong>to</strong>n is located at a distance of 0.420 m from a point charge<br />
of +8.30 /Le. The repulsive electric force moves the pro<strong>to</strong>n until it<br />
is at a distance of 1.58 m from the charge. Suppose that the electric<br />
potential energy lost by the system is carried off by a pho<strong>to</strong>n that is<br />
emitted during the process. What is its wavelength?<br />
** 11. ssm www A laser emits 1.30 X 1018 pho<strong>to</strong>ns per second in a<br />
beam of light that has a diameter of 2.00 mm and a wavelength of<br />
13. ssm In Figure <strong>29.</strong>1 replace the electrons with pro<strong>to</strong>ns that have<br />
the same speed. With the aid of Equation 27.1 for the bright fringes<br />
in Young's double-slit experiment and Equation <strong>29.</strong>8, decide<br />
whether the angular separation between the fringes would increase,<br />
decrease, or remain the same when compared <strong>to</strong> that produced by<br />
the electrons.<br />
TThiS icon represents a biomedical application.<br />
514.5 nm. Determine (a) the average electric field strength and<br />
(b) the average magnetic field strength for the electromagnetic wave<br />
that constitutes the beam.<br />
** 12. (a) How many pho<strong>to</strong>ns (wavelength = 620 nm) must be absorbed<br />
<strong>to</strong> melt a 2.0-kg block of ice at 0 °C in<strong>to</strong> water at 0 °C?<br />
(b) On the average, how many H20 molecules does one pho<strong>to</strong>n convert<br />
from the ice phase <strong>to</strong> the water phase?<br />
Section <strong>29.</strong>4 The Momentum of a Pho<strong>to</strong>n<br />
and the Comp<strong>to</strong>n Effect<br />
13. A light source emits a beam of pho<strong>to</strong>ns, each of which has a<br />
momentum of 2.3 X 10-z9 kg' m/so (a) What is the frequency of the<br />
pho<strong>to</strong>ns? (b) To what region of the electromagnetic spectrum do the<br />
pho<strong>to</strong>ns belong? Consult Figure 24.10 if necessary.<br />
14. A pho<strong>to</strong>n of red light (wavelength = 720 nm) and a Ping-Pong<br />
ball (mass = 2.2 X 10-3 kg) have the same momentum. At what<br />
speed is the ball moving?<br />
15. ssm In a Comp<strong>to</strong>n scattering experiment, the incident X-rays<br />
have a wavelength of 0.2685 nm, and the scattered X-rays have a<br />
wavelength of 0.2703 nm. Through what angle e in Figure <strong>29.</strong>10 are<br />
the X-rays scattered?<br />
ti(;) Incident X-rays have a wavelength of 0.3120 nm and are scat<br />
~d by the "free" electrons in graphite. The scattering angle in Figure<br />
<strong>29.</strong>10 is e = 135.0°. What is the magnitude of the momentum of<br />
(a) the incident pho<strong>to</strong>n and (b) the scattered pho<strong>to</strong>n? (For accuracy,<br />
use h = 6.626 X 10-34 J . sand c = 2.998 X 108 m/s.)<br />
17. Refer <strong>to</strong> Interactive Solution <strong>29.</strong>17 at www.wiley.com/college/<br />
cutnell for help in solving this problem. An incident X-ray pho<strong>to</strong>n of<br />
wavelength 0.2750 nm is scattered from an electron that is initially at<br />
rest. The pho<strong>to</strong>n is scattered at an angle of e = 180.0° in Figure<br />
<strong>29.</strong>10 and has a wavelength of 0.2825 nm. Use the conservation of<br />
linear momentum <strong>to</strong> find the momentum gained by the electron.<br />
* 18. The X-rays detected at a scattering angle of e = 163° in Figure<br />
<strong>29.</strong>10 have a wavelength of 0.1867 nm. Find (a) the wavelength of an<br />
incident pho<strong>to</strong>n, (b) the energy of an incident pho<strong>to</strong>n, (c) the energy<br />
of a scattered pho<strong>to</strong>n, and (d) the kinetic energy of the recoil electron.<br />
(For accuracy, use h = 6.626 X 10-34 J. sand c = 2.998 X 108 m/s.)<br />
* 19. ssm www What is the maximum amount by which the wavelength<br />
of an incident pho<strong>to</strong>n could change when it undergoes Comp<strong>to</strong>n<br />
scattering from a nitrogen molecule (Nz)?<br />
** 20. An X-ray pho<strong>to</strong>n is scattered at an angle of e = 180.0° from an<br />
electron that is initially at rest. After scattering, the electron has a speed<br />
of 4.67 X 106 m/so Find the wavelength of the incident X-ray pho<strong>to</strong>n.
Section <strong>29.</strong>5 The De Broglie Wavelength'<br />
and the Wave Nature of Matter<br />
21. The intera<strong>to</strong>mic spacing in a crystal of table salt is 0.282 nm.<br />
This crystal is being studied in a neutron diffraction experiment,<br />
similar <strong>to</strong> the one that produced the pho<strong>to</strong>graph in Figure <strong>29.</strong>13a.<br />
How fast must a neutron (mass = 1.67 X 10-27 kg) be moving <strong>to</strong><br />
have a de Broglie wavelength of 0.282 nm?<br />
22. A bacterium (mass = 2 X 10-15 kg) in the blood is moving at<br />
0,33 mfs. What is the de Broglie wavelength of this bacterium?<br />
23. ssm As Section 27.5 discusses, sound waves diffract, or bend,<br />
around the edges of a doorway. Larger wavelengths diffract more<br />
than smaller wavelengths. (a) The speed of sound is 343 m/so With<br />
what speed would a 55.0-kg person have <strong>to</strong> move through a doorway<br />
<strong>to</strong> diffract <strong>to</strong> the same extent as a 128-Hz bass <strong>to</strong>ne? (b) At the speed<br />
calculated in part (a), how long (in years) would it take the person <strong>to</strong><br />
move a distance of one meter?<br />
@ How fast does a pro<strong>to</strong>n have <strong>to</strong> be moving in order <strong>to</strong> have the<br />
same de Broglie wavelength as an electron that is moving with a<br />
25 A pho<strong>to</strong>n has the same momentum as an electron moving with a<br />
~eed peed of 4.50 2.0 X 105 106mm/s? Is. What is the wavelength of the pho<strong>to</strong>n?<br />
* 26. A particle has a de Broglie wavelength of 2.7 X 10-10 m. Then<br />
its kinetic energy doubles. What is the particle's new de Broglie<br />
wavelength, assuming that relativistic effects can be ignored?<br />
* 27. ssm From a cliff that is 9.5 m above a lake, a young woman<br />
(mass = 41 kg) jumps from rest, straight down in<strong>to</strong> the water. At<br />
the instant she strikes the water, what is her de Broglie wavelength?<br />
@ The width of the central bright fringe in a diffraction pattern on<br />
a screen is identical when either electrons or red light (vacuum<br />
wavelength = 661 nm) pass through a single slit. The distance<br />
between the screen and the slit is the same in each case and is large<br />
compared <strong>to</strong> the slit width. How fast are the electrons moving?<br />
* <strong>29.</strong> Consult Interactive Solution <strong>29.</strong>29 at www.wiley.comlcollege/<br />
cutnell <strong>to</strong> explore a model for solving this problem. In a television<br />
picture tube the electrons are accelerated from rest through a potential<br />
difference V. Just before an electron strikes the screen, its de Broglie<br />
wavelength is 0.900 X 10-11 m. What is the potential difference?<br />
** 30. The kinetic energy of a particle is equal <strong>to</strong> the energy of a pho<strong>to</strong>n.<br />
The particle moves at 5.0% of the speed of light. Find the ratio of the<br />
pho<strong>to</strong>n wavelength <strong>to</strong> the de Broglie wavelength of the particle.<br />
IADDITIONAL PROBLEMS<br />
37. A particle has a speed of 1.2 X 106 m/so Its de Broglie wavelength<br />
is 8.4 X 10-14 m. What is the mass of the particle?<br />
38. The dissociation energy of a molecule is the energy required <strong>to</strong><br />
break apart the molecule in<strong>to</strong> its separate a<strong>to</strong>ms. The dissociation<br />
energy for the cyanogen molecule is 1.22 X 10-18 J. Suppose that<br />
this energy is provided by a single pho<strong>to</strong>n. Determine the (a) wavelength<br />
and (b) frequency of the pho<strong>to</strong>n. (c) In what region of the<br />
electromagnetic spectrum (see Figure 24.10) does this pho<strong>to</strong>n lie?<br />
39. ssm The de Broglie wavelength of a pro<strong>to</strong>n in a particle accelera<strong>to</strong>r<br />
is 1.30 X 10-14 m. Determine the kinetic energy (in joules) of<br />
the pro<strong>to</strong>n.<br />
40. What is (a) the wavelength of a 5.0-eV pho<strong>to</strong>n and (b) the<br />
de Broglie wavelength of a 5.0-eV electron?<br />
ADDITIONAL PROBLEMS I 937<br />
Section <strong>29.</strong>6 The Heisenberg Uncertainty Principle<br />
/31) Consider a line that is 2.5 m long. A moving object is some<br />
~ere along this line, but its position is not known. (a) Find the minimum<br />
uncertainty in the momentum of the object. Find the minimum<br />
uncertainty in the object's velocity, assuming that the object is<br />
(b) a golf ball (mass = 0.045 kg) and (c) an electron.<br />
d32) An electron is trapped wit~in a sphere that has a diameter<br />
L;,(6.0 X 10-15 m (about the size of the nucleus of an oxygen<br />
a<strong>to</strong>m). What is the minimum uncertainty in the electron's<br />
momentum?<br />
33. ssm www In the lungs there are tiny sacs of air, which are<br />
called alveoli. The average diameter of one of these sacs is 0.25 mm.<br />
Consider an oxygen molecule (mass = 5.3 X 10-26 kg) trapped<br />
within a sac. What is the minimum uncertainty in the velocity of this<br />
oxygen molecule?<br />
34. Review Conceptual Example 7 as background for this problem.<br />
When electrons'pass through a single slit, as in Figure <strong>29.</strong>15, they<br />
form a diffraction pattern. As Section <strong>29.</strong>6 discusses, the central<br />
bright fringe extends <strong>to</strong> either side of the midpoint, according <strong>to</strong> an<br />
angle ()given by sin () = MW, where A is the de Broglie wavelength<br />
of the electron and W is the width of the slit. When A is the same<br />
size as W, A = 90°, and the central fringe fills the entire observation<br />
screen. In this case, an electron passing through the slit has roughly<br />
the same probability of hitting the screen either straight ahead or<br />
anywhere off <strong>to</strong> one side or the other. Now, imagine yourself in a<br />
world where Planck's constant is large enough so you exhibit similar<br />
effects when you walk through a 0.90-m-wide doorway. Your mass<br />
is 82 kg and you walk at a speed of 0.50 m/so How large would<br />
Planck's constant have <strong>to</strong> be in this hypothetical world?<br />
35. ssm www Particles pass through a single slit of width<br />
0.200 mm (see Figure <strong>29.</strong>15). The de Broglie wavelength of each<br />
particle is 633 nm. After the particles pass through the slit, they<br />
spread out over a range of angles. Use the Heisenberg uncertainty<br />
principle <strong>to</strong> determine the minimum range of angles.<br />
* 36. The minimum uncertainty /:::"y in the position y of a particle is<br />
equal <strong>to</strong> its de Broglie wavelength. Determine the minimum<br />
uncertainty in the speed of the particle, where this minimum uncer<br />
tainty /:::"vy is expressed as a percentage of the particle's speed vy<br />
(Percentage = /:::"vy X 100%). Assume that relativistic effects can<br />
vy<br />
be ignored.<br />
41. Recall from Section 14.3 that the average kinetic energy of an a<strong>to</strong>m<br />
in a mona<strong>to</strong>mic ideal gas is given by KE = ~kT, where k = 1.38 X<br />
10-23 J/K and T is the Kelvin temperanlre of the gas. Determine the de<br />
Broglie wavelength of a helium a<strong>to</strong>m (mass = 6.65 X 10-27 kg) that<br />
has the average kinetic energy at room temperature (293 K).<br />
42. A magnesium surface has a work function of 3.68 eV. Electro<br />
magnetic waves with a wavelength of 215 nm strike the surface and<br />
eject electrons. Find the maximum kinetic energy of the ejected<br />
electrons. Express your answer in electron volts.<br />
* 43. ssm An electron, starting from rest, accelerates through a potential<br />
difference of 418 V. What is the final de Broglie wavelength<br />
of the electron, assuming that its final speed is much less than the<br />
speed of light?
e e<br />
/L=-=f<br />
(E I h)<br />
Chapter 29 Problems 1483<br />
he (6.63x10-34 J·s)(3.0x108 m/s) = 3.1 x 10-7 m = j310nm I<br />
= E = 6.4x 10-19 J<br />
6. REASONING Light intensity I is the power that passes perpendicularly through a surface<br />
divided by the area of that surface (Equation 16.8). The number N of pho<strong>to</strong>ns per second per<br />
square meter that reaches the earth is the light intensity divided by the energy E of a single<br />
pho<strong>to</strong>n; N = liE. The energy of a single pho<strong>to</strong>n is E = hf (Equation <strong>29.</strong>2), where h is<br />
Planck's constant and f is the frequency of the pho<strong>to</strong>n. The frequency of the pho<strong>to</strong>n is<br />
related <strong>to</strong> its wavelength /L by Equation 16.1 asf = e//L, where e is the speed of light in a<br />
vacuum. Therefore, we have<br />
SOLUTION The number of pho<strong>to</strong>ns per second per square meter that reach the earth is<br />
N=I/L= (680W/m2)(730x10-9m) =!2.5X1021 pho<strong>to</strong>ns/(s.m2)1<br />
he (6.63x10-34 J ·s)(3.00x10s m/s) --------<br />
7. REASONING AND SOLUTION In the first case, the energy of the incident pho<strong>to</strong>n is given<br />
by Equation <strong>29.</strong>3 as<br />
hf = KE max + Wo = 0.68 eV + 2.75 eV = 3.43 eV<br />
In the second case, a rearrangement of Equation <strong>29.</strong>3 yields<br />
KEmax=hf-Wo=3.43eV-2.17eV= 11.26eVI<br />
8. REASONING AND SOLUTION The work function of the material (using /L = 196 nm) is<br />
found from<br />
The maximum kinetic energy of the ejected electron is (using /L= 141 nm)<br />
The speed of the electron is then<br />
KE = hf- W = he//L- W = 3.96 X 10-19J<br />
mu 0 0<br />
2(3.96X103~191) = !9.32X105 m/s [<br />
9.11 x 10 kg
Chapter 29 Problems 1487<br />
15. I SSM I REASONING The angle e through which the X-rays are scattered is related <strong>to</strong> the<br />
difference between the wavelength A' of the scattered X-rays and the wavelength /L of the<br />
incident X-rays by Equation <strong>29.</strong>7 as .<br />
A' - /L= ~ (1- cos e)<br />
me<br />
where h is Planck's constant, m is the mass of the electron, and e is the speed of light in a<br />
vacuum. Wecan use this relation directly <strong>to</strong> find the angle, since all the other variables are<br />
known.<br />
SOLUTION Solving Equation <strong>29.</strong>7 for the angle e, we obtain<br />
cose=l- me (A' -/L)<br />
h<br />
(9.11x10-31 kg) (3.00x108 m1s) (0.2703x10-9 m _ 0.2685x10-9 m) = 0.26<br />
=1- 6.63x10-34 J.s<br />
16. REASONING AND SOLUTION<br />
a. According <strong>to</strong> Equation <strong>29.</strong>6, the magnitude of the momentum of the incident pho<strong>to</strong>n is<br />
P =12.= /L 6.626x10-34 0.3120x10-9 J,s=12.124XlO-24 m k g ·m/s I<br />
b. The wavelength of the scattered pho<strong>to</strong>n is, from Equation <strong>29.</strong>7,<br />
/LI = /L+ ~ (1- cos e)<br />
me<br />
where e is the scattering angle. Combining this expression with Equation <strong>29.</strong>6, we find that<br />
the magnitude of the momentum of the scattered pho<strong>to</strong>n is<br />
I h h<br />
p =;.;= /L+ (hme J (I-cos e)<br />
6.626x10-34 J·s<br />
=<br />
I 0.3120x10-9 m+ (9.109x10-31kg)(2.998x108m/s)<br />
6.626xI0-3' J·s }I-COS 135.0°)<br />
= !2.096X10-24 kg·m/s [
--~~-------------~-------------------~--------------------~-- ------ -----<br />
j<br />
s<br />
s<br />
)<br />
)<br />
i i<br />
SOLUTION a. Since the wavelengths are equal, we have that<br />
A =A<br />
sound person<br />
h<br />
A sound = m person v person<br />
Chapter 29 Problems 1491<br />
Solving for v , and using the relation A d = V d / f d (Equation 16.1), we have<br />
person soun soun soun<br />
V person<br />
h<br />
m person (v sound / fsound )<br />
(6.63 X 10-34 J. s)(128 Hz)<br />
(55.0 kg)(343 m/ s)<br />
h fsound<br />
m person V sound<br />
j4.50xlO-36 m/s I<br />
b. At the speed calculated in part (a), the time required for the person <strong>to</strong> move a distance of<br />
one meter is<br />
t<br />
x<br />
v<br />
1.0 m<br />
4.50x10-36 m/s ( 3600 10 h s) C24.0 day) h<br />
v<br />
( 365.25 1 year days J<br />
Fac<strong>to</strong>rs <strong>to</strong> convert<br />
seconds <strong>to</strong> years<br />
j7.05X1027 years I<br />
24. REASONING The speed v of a particle is related <strong>to</strong> the magnitude p of its momentum by<br />
v =p/m (Equation 7.2). The magnitude of the momentum is related <strong>to</strong> the particle's<br />
de Broglie wavelength A by p = h/A (Equation <strong>29.</strong>8), where h is Planck's constant. Thus,<br />
the speed of a particle can be expressed as v = h/(mA). We will use this relation <strong>to</strong> find the<br />
speed of the pro<strong>to</strong>n.<br />
SOLUTION The speeds of the pro<strong>to</strong>n and electron are<br />
v pro<strong>to</strong>n = mpro<strong>to</strong>n Apro<strong>to</strong>n<br />
h<br />
and<br />
velectron = melectron Aelectron<br />
Dividing<br />
obtain<br />
the first equation by the second equation, and noting that AeIect ron= At, pro on we<br />
vpro<strong>to</strong>n _ melectronAelectron = melectron<br />
velectron - mpro<strong>to</strong>n Apro<strong>to</strong>n mpro<strong>to</strong>n<br />
h
1492 PARTICLES AND WAVES<br />
Using values for melectron and mpro<strong>to</strong>n taken from the inside of the front cover, we find that<br />
the speed of the pro<strong>to</strong>n is<br />
pro<strong>to</strong>n - velectron m =. x m/s -27 = 2.45xlO m/S<br />
v - [melectron: pro<strong>to</strong>n (450 106 )(9.11X10-31 1.67 X 10 kg] kg I 3 I<br />
25. REASONING AND SOLUTION The momentum of the pho<strong>to</strong>n is p = h/A and that of the<br />
electron is p = mv. Equating and solving for the wavelength of the pho<strong>to</strong>n,<br />
26. REASONING The de Broglie wavelength A is related <strong>to</strong> Planck's constant h and the<br />
magnitude p of the particle's momentum. The magnitude of the momentum can be related<br />
<strong>to</strong> the particle's kinetic energy. Thus, using the given wavelength and the fact that the<br />
kinetic energy doubles, we will be able <strong>to</strong> obtain the new wavelength.<br />
SOLUTION The de Broglie wavelength is<br />
The kinetic energy and the magnitude of the momentum are<br />
1 2<br />
KE = - mv<br />
2<br />
(6.2)<br />
p=mv<br />
or<br />
(7.2)<br />
(<strong>29.</strong>8)<br />
where m and v are the mass and speed of the particle. Substituting Equation 7.2 in<strong>to</strong><br />
Equation 6.2, we can relate the kinetic energy and momentum as follows:<br />
Substituting this result for p in<strong>to</strong> Equation <strong>29.</strong>8 gives<br />
h h<br />
A = p = ~2m(KE)
and<br />
Dividing the two equations and rearranging reveals that<br />
h<br />
A, ~ = ~2m(KE)f h = /KE), (KE)f<br />
~2m(KE\<br />
Using the given value for Aiand the fact that KEf = 2 (KEi ) , we find<br />
or<br />
Chapter 29 Problems 1493<br />
Applying this expression for the final and initial wavelengths Af and Ai' we obtain<br />
27. I SSMI REASONING AND SOLUTION The de Broglie wavelength A of the woman is<br />
given by Equation <strong>29.</strong>8 as A = h / p, where p is the magnitude of her momentum. The<br />
magnitude of the momentum is p = mv , where m is the woman's mass and v is her speed.<br />
According <strong>to</strong> Equation 3.6b of the equations of kinematics, the speed v is given by<br />
v = ~~UyY ~, since the woman jumps from rest. In this expression, a y = -9.80 m/ S2 and<br />
Y = -9.5 m. With these considerations we find that<br />
= 6.63 X 10-34 J ·s = 11.2 X 10-36 m I<br />
(41 kg)~2( -9.80 m/ S2)( -9.5 m) ----<br />
28. REASONING The width of the central bright fringe in the diffraction patterns will be<br />
identical when the electrons have the same de Broglie wavelength as the wavelength of the<br />
pho<strong>to</strong>ns in the red light. The de Broglie wavelength of one electron in the beam is given by<br />
Equation <strong>29.</strong>8, A eIect ron\<br />
= h / p, where p = mv.<br />
SOLUTION Following the reasoning described above, we find<br />
A =A<br />
red light electron
~~<br />
"<br />
";',~nl~,<br />
.;-.,~~<br />
1494 PARTICLES AND WAVES<br />
<strong>29.</strong> REASONING When the electron is at rest, it has electric potential energy, but no kinetic<br />
energy. The electric potential energy EPE is given by EPE = eV (Equation 19.3), where e is<br />
the magnitude of the charge on the electron and V is the potential difference. When the<br />
electron reaches its maximum speed, it has no potential energy, but its kinetic energy is<br />
30.<br />
Solving for the speed of the electron, we have<br />
velectron = n1 electron A red light<br />
h<br />
1- mv2. The conservation of energy states that the final <strong>to</strong>tal energy of the electron equals<br />
the initial <strong>to</strong>tal energy:<br />
I 2<br />
"2mv<br />
'------y------J<br />
= e V<br />
'--v-----'<br />
Final <strong>to</strong>tal Initial <strong>to</strong>tal<br />
energy energy<br />
h2<br />
V-- - ..<br />
2meA 2<br />
h<br />
A red light = melectro n V electron<br />
= 6.63xlO-34 J·s<br />
(9.l1xlO 31 kg)(661xIO-9 m) =11.10XI03 m/s I<br />
Solving this equation for the potential difference gives V = mv2 I (2e) .<br />
The speed of the electron can be expressed in terms of the magnitude p of its momentum by<br />
v = plm (Equation 7.2). The magnitude of the electron's momentum is related <strong>to</strong> its<br />
de Broglie wavelength A by p = hi A (Equation <strong>29.</strong>8), where his Planck's constant. Thus, the<br />
speed can be written as v = hl(mA). Substituting this expression for v in<strong>to</strong> V = mv2 I (2e)<br />
gives V = h2 I (2meA2 ).<br />
SOLUTION The potential difference that accelerates the electron is<br />
(6.63x10-34 J .s)2<br />
2(9.11xlO-31 kg)(1.60xI0-19 C)(0.900xlO-11 m)2<br />
!I.86XI04 vi<br />
REASONING AND SOLUTION The energy of the pho<strong>to</strong>n is E = hi = he! A h ' while the<br />
p o<strong>to</strong>n<br />
kinetic energy of the particle is KE = (l12)mv2 = h2/(2mA2). Equating the two energies and<br />
rearranging the result gives AphotO/ A = (2me!h)A. Now the speed of the particle is v ==<br />
0.050e, so A = hl(0.050 me), and<br />
Ah IA=2/0.050= [40XI01<br />
p o<strong>to</strong>n .
Chapter 29 Problems 1495<br />
31. REASONING We know that the object is somewhere on the line. Therefore, the<br />
uncertainty in the object's position is ~y = 2.5 m. The minimum uncertainty in the object's<br />
momentum is ~Py and is specified by the Heisenberg uncertainty principle (Equation <strong>29.</strong>10)<br />
in the form (~p )(~y) = hl(4Jr). Since momentum is mass m times velocity v, the uncertainty<br />
y<br />
in the velocity ~v is related <strong>to</strong> the uncertainty in the momentum by ~v = (~p)lm.<br />
SOLUTION<br />
a. Using the uncertainty principle, we find the minimum uncertainty in the momentum as<br />
follows:<br />
h<br />
(~py )(~y) = 4Jr<br />
~p= _h__ 6.63X10-34].s<br />
y 4ff~- 4ff(2.5m) -12.1X10-35kg.m/sl<br />
b. For a golf ball this uncertainty in momentum corresponds <strong>to</strong> an uncertainty in velocity<br />
that is given by<br />
~py _ 2.lx10-35 kg·m/s =14.7X10-34 m/s I<br />
~vy = ---;;: - 0.045 kg<br />
c. For an electron this uncertainty in momentum corresponds <strong>to</strong> an uncertainty in velocity<br />
that is given by<br />
_ ~py = 2.1x10-35 kg·m/s -12.3X10-5 m/s I<br />
~vy - m 9.1lxlO-31 kg<br />
32. REASONING We assume that the electron is moving along the y direction, and that it can<br />
be anywhere within the sphere. Therefore, the uncertainty in the electron's position is equal<br />
<strong>to</strong> the diameter d of the sphere, so ~y = d. The minimum uncertainty ~p in the y component<br />
y<br />
of the electron's momentum is given by the Heisenberg uncertainty principle as<br />
~py = hi (4Jr~y) (Equation <strong>29.</strong>10).<br />
SOLUTION Setting ~y = d in the relation ~Py = hi (4Jr~Y) gives<br />
~p = _h __ h 6.63x10-34 ].<br />
y 4ff~y - 4ffd = 4ff(6.0x10-15 ~) =18.8X10-21 kg.m/sl<br />
33. I SSMII wwwl REASONING AND SOLUTION According <strong>to</strong> the uncertainty principle,<br />
the minimum uncertainty in the momentum can be determined from ~py~y = hi ( 4ff) .