CHAPTER 21: Electric Charges and Electric Field
Responses to Questions
1. Rub a glass rod with silk and use it to charge an electroscope. The electroscope will end up with a
net positive charge. Bring the pocket comb close to the electroscope. If the electroscope leaves move
farther apart, then the charge on the comb is positive, the same as the charge on the electroscope. If
the leaves move together, then the charge on the comb is negative, opposite the charge on the
electroscope.
2. The shirt or blouse becomes charged as a result of being tossed about in the dryer and rubbing
against the dryer sides and other clothes. When you put on the charged object (shirt), it causes
charge separation within the molecules of your skin (see Figure 21-9), which results in attraction
between the shirt and your skin.
3. Fog or rain droplets tend to form around ions because water is a polar molecule, with a positive
region and a negative region. The charge centers on the water molecule will be attracted to the ions
(positive to negative).
4. See also Figure 21-9 in the text. The negatively +++++++ -+
charged electrons in the paper are attracted to the -+
positively charged rod and move towards it within -+
their molecules. The attraction occurs because the -+
negative charges in the paper are closer to the
positive rod than are the positive charges in the
paper, and therefore the attraction between the
unlike charges is greater than the repulsion
between the like charges.
5. A plastic ruler that has been rubbed with a cloth is charged. When brought near small pieces of
paper, it will cause separation of charge in the bits of paper, which will cause the paper to be
attracted to the ruler. On a humid day, polar water molecules will be attracted to the ruler and to the
separated charge on the bits of paper, neutralizing the charges and thus eliminating the attraction.
6. The net charge on a conductor is the difference between the total positive charge and the total
negative charge in the conductor. The “free charges” in a conductor are the electrons that can move
about freely within the material because they are only loosely bound to their atoms. The “free
electrons” are also referred to as “conduction electrons.” A conductor may have a zero net charge
but still have substantial free charges.
7. Most of the electrons are strongly bound to nuclei in the metal ions. Only a few electrons per atom
(usually one or two) are free to move about throughout the metal. These are called the “conduction
electrons.” The rest are bound more tightly to the nucleus and are not free to move. Furthermore, in
the cases shown in Figures 21-7 and 21-8, not all of the conduction electrons will move. In Figure
21-7, electrons will move until the attractive force on the remaining conduction electrons due to the
incoming charged rod is balanced by the repulsive force from electrons that have already gathered at
the left end of the neutral rod. In Figure 21-8, conduction electrons will be repelled by the incoming
rod and will leave the stationary rod through the ground connection until the repulsive force on the
remaining conduction electrons due to the incoming charged rod is balanced by the attractive force
from the net positive charge on the stationary rod.
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Physics for Scientists & Engineers with Modern Physics, 4th Edition Instructor Solutions Manual
8. The electroscope leaves are connected together at the top. The horizontal component of this tension
force balances the electric force of repulsion. (Note: The vertical component of the tension force
balances the weight of the leaves.)
9. Coulomb’s law and Newton’s law are very similar in form. The electrostatic force can be either
attractive or repulsive; the gravitational force can only be attractive. The electrostatic force constant
is also much larger than the gravitational force constant. Both the electric charge and the
gravitational mass are properties of the material. Charge can be positive or negative, but the
gravitational mass only has one form.
10. The gravitational force between everyday objects on the surface of the Earth is extremely small.
(Recall the value of G: 6.67 x 10-11 Nm2/kg2.) Consider two objects sitting on the floor near each
other. They are attracted to each other, but the force of static fiction for each is much greater than the
gravitational force each experiences from the other. Even in an absolutely frictionless environment,
the acceleration resulting from the gravitational force would be so small that it would not be
noticeable in a short time frame. We are aware of the gravitational force between objects if at least
one of them is very massive, as in the case of the Earth and satellites or the Earth and you.
The electric force between two objects is typically zero or close to zero because ordinary objects are
typically neutral or close to neutral. We are aware of electric forces between objects when the
objects are charged. An example is the electrostatic force (static cling) between pieces of clothing
when you pull the clothes out of the dryer.
11. Yes, the electric force is a conservative force. Energy is conserved when a particle moves under the
influence of the electric force, and the work done by the electric force in moving an object between
two points in space is independent of the path taken.
12. Coulomb observed experimentally that the force between two charged objects is directly
proportional to the charge on each one. For example, if the charge on either object is tripled, then the
force is tripled. This is not in agreement with a force that is proportional to the sum of the charges
instead of to the product of the charges. Also, a charged object is not attracted to or repelled from a
neutral object, which would be the case if the numerator in Coulomb’s law were proportional to the
sum of the charges.
13. When a charged ruler attracts small pieces of paper, the charge on the ruler causes a separation of
charge in the paper. For example, if the ruler is negatively charged, it will force the electrons in the
paper to the edge of the paper farthest from the ruler, leaving the near edge positively charged. If the
paper touches the ruler, electrons will be transferred from the ruler to the paper, neutralizing the
positive charge. This action leaves the paper with a net negative charge, which will cause it to be
repelled by the negatively charged ruler.
14. The test charges used to measure electric fields are small in order to minimize their contribution to
the field. Large test charges would substantially change the field being investigated.
15. When determining an electric field, it is best, but not required, to use a positive test charge. A
negative test charge would be fine for determining the magnitude of the field. But the direction of
the electrostatic force on a negative test charge will be opposite to the direction of the electric field.
The electrostatic force on a positive test charge will be in the same direction as the electric field. In
order to avoid confusion, it is better to use a positive test charge.
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2
Chapter 21 Electric Charges and Electric Field
16. See Figure 21-34b. A diagram of the electric field lines around two negative charges would be just
like this diagram except that the arrows on the field lines would point towards the charges instead of
away from them. The distance between the charges is l.
17. The electric field will be strongest to the right of the positive charge (between the two charges) and
weakest to the left of the positive charge. To the right of the positive charge, the contributions to the
field from the two charges point in the same direction, and therefore add. To the left of the positive
charge, the contributions to the field from the two charges point in opposite directions, and therefore
subtract. Note that this is confirmed by the density of field lines in Figure 21-34a.
18. At point C, the positive test charge would experience zero net force. At points A and B, the direction
of the force on the positive test charge would be the same as the direction of the field. This direction
is indicated by the arrows on the field lines. The strongest field is at point A, followed (in order of
decreasing field strength) by B and then C.
19. Electric field lines can never cross because they give the direction of the electrostatic force on a
positive test charge. If they were to cross, then the force on a test charge at a given location would be
in more than one direction. This is not possible.
20. The field lines must be directed radially toward or away from the point charge (see rule 1). The
spacing of the lines indicates the strength of the field (see rule 2). Since the magnitude of the field
due to the point charge depends only on the distance from the point charge, the lines must be
distributed symmetrically.
21. The two charges are located along a line as shown in the Q 2Q
diagram.
(a) If the signs of the charges are opposite then the point on
the line where E = 0 will lie to the left of Q. In that region ℓ
the electric fields from the two charges will point in
opposite directions, and the point will be closer to the
smaller charge.
(b) If the two charges have the same sign, then the point on the line where E = 0 will lie between
the two charges, closer to the smaller charge. In this region, the electric fields from the two
charges will point in opposite directions.
22. The electric field at point P would point in the negative x-direction. The magnitude of the field
would be the same as that calculated for a positive distribution of charge on the ring:
1 Qx
4o x2 a2 3/2
E
23. The velocity of the test charge will depend on its initial velocity. The field line gives the direction of
the change in velocity, not the direction of the velocity. The acceleration of the test charge will be
along the electric field line.
24. The value measured will be slightly less than the electric field value at that point before the test
charge was introduced. The test charge will repel charges on the surface of the conductor and these
charges will move along the surface to increase their distances from the test charge. Since they will
then be at greater distances from the point being tested, they will contribute a smaller amount to the
field.
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Physics for Scientists & Engineers with Modern Physics, 4th Edition Instructor Solutions Manual
25. The motion of the electron in Example 21-16 is projectile motion. In the case of the gravitational
force, the acceleration of the projectile is in the same direction as the field and has a value of g; in
the case of an electron in an electric field, the direction of the acceleration of the electron and the
field direction are opposite, and the value of the acceleration varies.
26. Initially, the dipole will spin clockwise. It will “overshoot” the equilibrium position (parallel to the
field lines), come momentarily to rest and then spin counterclockwise. The dipole will continue to
oscillate back and forth if no damping forces are present. If there are damping forces, the amplitude
will decrease with each oscillation until the dipole comes to rest aligned with the field.
27. If an electric dipole is placed in a nonuniform electric field, the charges of the dipole will experience
forces of different magnitudes whose directions also may not be exactly opposite. The addition of
these forces will leave a net force on the dipole.
Solutions to Problems
1. Use Coulomb’s law to calculate the magnitude of the force.
F
k Q1Q2 8.988109 N m2 C2 1.6021019 C 261.6021019 C 2.7 103 N
r2 1.51012 m 2
2. Use the charge per electron to find the number of electrons.
38.0 106C 1 electron C 2.37 1014 electrons
1.602 1019
3. Use Coulomb’s law to calculate the magnitude of the force.
F
k Q1Q2 8.988 109 Nm2 C2 25 106C 2.5 103C 7200 N
r2
0.28 m2
4. Use Coulomb’s law to calculate the magnitude of the force.
1.602 1019 C 2
4.0 1015 m 2 14 N
F Q1Q2 C2
k r2 8.988 109 N m2
5. The charge on the plastic comb is negative, so the comb has gained electrons.
m 1e 9.109 1031kg
3.0 106C 1.602 1019 C 1e 4.9 1016
m 0.035kg 4.9 1014%
6. Since the magnitude of the force is inversely proportional to the square of the separation distance,
F 1 , if the distance is multiplied by a factor of 1/8, the force will be multiplied by a factor of 64.
r2
F 64F0 64 3.2102 N 2.0 N
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4
Chapter 21 Electric Charges and Electric Field
7. Since the magnitude of the force is inversely proportional to the square of the separation distance,
F 1 , if the force is tripled, the distance has been reduced by a factor of 3.
r2
r r0 8.45 cm 4.88 cm
33
8. Use the charge per electron and the mass per electron.
46 106C 1 electron C 2.871 1014 2.9 1014 electrons
1.602 1019
2.8711014 e 9.109 1031kg 2.6 1016 kg
1 e
9. To find the number of electrons, convert the mass to moles, the moles to atoms, and then multiply by
the number of electrons in an atom to find the total electrons. Then convert to charge.
15 kg Au 15 kg Au 1mole Al 6.022 1023atoms 79 electrons 1.602 1019 C
0.197 kg 1 mole 1molecule electron
5.8 108C
The net charge of the bar is 0C , since there are equal numbers of protons and electrons.
10. Take the ratio of the electric force divided by the gravitational force.
k Q1Q2 8.988109 N m2 C2 1.602 1019 C 2
G 6.67 1011 N m2 kg2 9.111031 kg 1.67 1027 kg
FE r2
FG m1m2
kQ1Q2 2.31039
r2 Gm1m2
The electric force is about 2.31039 times stronger than the gravitational force for the given scenario.
11. (a) Let one of the charges be q , and then the other charge is QT q. The force between the
q QT q k
r2
r2
charges .
is FE k qQT q2 To find the maximum and minimum force, set the
first derivative equal to 0. Use the second derivative test as well.
F k ; dFE k 0
E r2 dq r2
qQ q2 QT 2q q 1 QT
T 2
d 2FE 2k 0
r2
dq2
q Q1 gives F
E max
2T
So q1 q2 Q1 gives the maximum force.
2T
(b) If one of the charges has all of the charge, and the other has no charge, then the force between
them will be 0, which is the minimum possible force. So q1 0, q2 QT gives the minimum
force.
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Physics for Scientists & Engineers with Modern Physics, 4th Edition Instructor Solutions Manual
12. Let the right be the positive direction on the line of charges. Use the fact that like charges repel and
unlike charges attract to determine the direction of the forces. In the following expressions,
k 8.988109 N m2 C2 .
k 75C 48C ˆi k 75C 85C ˆi 147.2 N ˆi 150 N ˆi
F75 0.35 m2 0.70 m2
k 75C 48C ˆi k 48C 85C ˆi 563.5 N ˆi 560 N ˆi
F48 0.35 m2 0.35 m2
k 85C 75C ˆi k 85C 48C ˆi 416.3 N ˆi 420 N ˆi
F85 0.70 m2 0.35 m2
13. The forces on each charge lie along a line connecting the charges. Let the
variable d represent the length of a side of the triangle. Since the triangle
is equilateral, each angle is 60o. First calculate the magnitude of each F13 F12
individual force. d Q1 d
F12
k Q1Q2 8.988 109 N m2 C2 7.0 106C 8.0 106 C
d2 F23 Q2 Q3 F32
1.20 m2
d
0.3495 N F31
F21
F13
k Q1Q3 8.988 109 N m2 C2 7.0 106 C 6.0 106 C
d2
1.20 m2
0.2622 N
F23 k
Q2Q3 8.988109 N m2 C2 8.0 106 C 6.0 106 C 0.2996 N F32
d2
1.20 m2
Now calculate the net force on each charge and the direction of that net force, using components.
F1x F12x F13x 0.3495 N cos 60o 0.2622 N cos 60o 4.365 102 N
F1y F12 y F13y 0.3495 N sin 60o 0.2622 N sin 60o 5.297 101 N
F1 F2 F2 0.53 N 1 tan1 F1 y tan1 5.297 101 N 265
1x 1y F1x 4.365 102 N
F2x F21x F23x 0.3495 N cos 60o 0.2996 N 1.249 101 N
F2 y F21y F23y 0.3495 N sin 60o 0 3.027 101 N
F2 F2 F2 0.33 N 2 tan1 F2 y tan1 3.027 101 N 112
2x 2y F2 x 1.249 101 N
F3x F31x F32x 0.2622 N cos 60o 0.2996 N 1.685 101 N
F3y F31y F32 y 0.2622 N sin 60o 0 2.271101 N
F3 F2 F2 0.26 N 3 tan1 F3 y tan1 2.271 101 N 53
3x 3y F3x 1.685 101 N
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6
Chapter 21 Electric Charges and Electric Field
14. (a) If the force is repulsive, both charges must be positive since the total charge is positive. Call the
total charge Q.
Q1 Q2 Q F kQ1Q2 kQ1 Q Q1 Q12 QQ1 Fd 2 0
d2 k
d2
Q Q2 4 Fd 2 Q Q2 4 Fd 2
Q1 k k
2 2
90.0 106 C 2 4 12.0N 1.16 m2
1 90.0 106 C 8.988 109 N m2 C2
2
60.1 106 C , 29.9 106 C
(b) If the force is attractive, then the charges are of opposite sign. The value used for F must then
be negative. Other than that, the solution method is the same as for part (a).
Q1 Q2 Q F kQ1Q2 kQ1 Q Q1 Q12 QQ1 Fd 2 0
d2 k
d2
Q Q2 4 Fd 2 Q Q2 4 Fd 2
Q1 k k
2 2
90.0 106 C 2 4 12.0N 1.16 m2
1 90.0 106 C 8.988 109 N m2 C2
2
106.8 106 C , 16.8 106 C
15. Determine the force on the upper right charge, and then use the
symmetry of the configuration to determine the force on the other three F43 F42
charges. The force at the upper right corner of the square is the vector
sum of the forces due to the other three charges. Let the variable d Q1 Q4
d F41
represent the 0.100 m length of a side of the square, and let the variable
Q represent the 4.15 mC charge at each corner.
F41 k Q2 F41x k Q2 , F41y 0 Q2 Q3
d2 d2
F42 k Q2 F42 x k Q2 cos45o k 2Q2 , F42 y k 2Q2
2d 2 2d 2 4d 2 4d 2
F43 k Q2 F43 x 0 , F43 y k Q2
d2 d2
Add the x and y components together to find the total force, noting that F4x F4y .
F4 x F41x F42x F43x k Q2 k 2Q2 0 k Q2 1 2 F4 y
d2 4d 2 d2 4
F4 F2 F2 k Q2 1 2 2 k Q2 2 1
4x 4y d2 d2 2
4
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Physics for Scientists & Engineers with Modern Physics, 4th Edition Instructor Solutions Manual
4.15 103C 2
0.100 m2
8.988 109 N m2 C2 2 1 2.96 107 N
2
tan1 F4y 45o above the x-direction.
F4 x
For each charge, the net force will be the magnitude determined above, and will lie along the line
from the center of the square out towards the charge.
16. Determine the force on the upper right charge, and then use the symmetry of the configuration to
determine the force on the other charges.
The force at the upper right corner of the square is the vector sum of the Q1 F42
forces due to the other three charges. Let the variable d represent the d F41 Q4
0.100 m length of a side of the square, and let the variable Q represent F43
the 4.15 mC charge at each corner.
F41 k Q2 F41x k Q2 , F41y 0
d2 d2
F42 k Q2 F42 x k Q2 cos45o k 2Q2 , F42 y k 2Q2 Q2 Q3
2d 2 2d 2 4d 2 4d 2
F43 k Q2 F43 x 0 , F43 y k Q2
d2 d2
Add the x and y components together to find the total force, noting that F4x F4y .
F4 x F41x F42x F43x k Q2 k 2Q2 0 k Q2 1 2 0.64645k Q2 F4 y
d2 4d 2 d2 4 d2
F4 F 2 F2 k Q2 k Q2 0.9142
4x 4y d2 d2
0.64645 2
8.988 109 N m2 C2 2
4.15 103C
0.9142 1.42 107 N
0.100 m2
tan1 F4y 225o from the x-direction, or exactly towards the center of the square.
F4 x
For each charge, there are two forces that point towards the adjacent corners, and one force that
points away from the center of the square. Thus for each charge, the net force will be the magnitude
of 1.42 107 N and will lie along the line from the charge inwards towards the center of the square.
17. The spheres can be treated as point charges since they are spherical, and so Coulomb’s law may be
used to relate the amount of charge to the force of attraction. Each sphere will have a magnitude Q
of charge, since that amount was removed from one sphere and added to the other, being initially
uncharged.
F k Q1Q2 k Q2 Qr F 0.12 m 1.7 102 N
r2 r2 k 8.988109 N m2 C2
1.650 107 C 1 electron C 1.01012 electrons
1.602 1019
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8
Chapter 21 Electric Charges and Electric Field
18. The negative charges will repel each other, and so the third charge Q0 Q 4Q0
must put an opposite force on each of the original charges.
Consideration of the various possible configurations leads to the x l–x
conclusion that the third charge must be positive and must be between
the other two charges. See the diagram for the definition of variables. l
For each negative charge, equate the magnitudes of the two forces on the charge. Also note that
0 x l.
left: k Q0Q k 4Q02 right: k 4Q0Q k 4Q02
x2 l2 l2
l x2
k Q0Q k 4Q0Q x 1 l
x2 3
l x2
Q0Q 4Q02 Q 4Q0 x2 4
kx2 k l2 l2 Q0 32 Q4
90
Thus the charge should be of magnitude Q4 , and a distance 1 l from Q0 towards 4Q0 .
3
90
19. (a) The charge will experience a force that is always pointing Q qQ
towards the origin. In the diagram, there is a greater force of
dx dx
Qq to the left, and a lesser force of Qq to
40 d x2 40 d x2
the right. So the net force is towards the origin. The same would be true if the mass were to the
left of the origin. Calculate the net force.
Fnet Qq x2 Qq x2 40 d Qq x2 d x2 d x2
40 d 40 d x2 d
40 4Qqd x2 x 0 d Qqd x2 x
d x2 d x2 d
We assume that x d.
Fnet 0 d Qqd x2 x Qq x
0d 3
x2 d
This has the form of a simple harmonic oscillator, where the “spring constant” is kelastic Qq .
0d 3
The spring constant can be used to find the period. See Eq. 14-7b.
T 2 m 2 m 2 m0d 3
kelastic Qq Qq
0d 3
(b) Sodium has an atomic mass of 23.
3 1010 m 3
T 2
m0d 3 2 29 1.66 1027 kg 8.85 1012 C2 Nm2
Qq
1.60 1019 C 2
2.4 1013 s 1012ps 0.24 ps 0.2 ps
1s
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Physics for Scientists & Engineers with Modern Physics, 4th Edition Instructor Solutions Manual
20. If all of the angles to the vertical (in both cases) are assumed to
be small, then the spheres only have horizontal displacement, FT1 1 2 FT2
and so the electric force of repulsion is always horizontal. m2g
Likewise, the small angle condition leads to tan sin m1g
FE1 FE2
for all small angles. See the free-body diagram for each sphere,
showing the three forces of gravity, tension, and the
electrostatic force. Take to the right to be the positive
horizontal direction, and up to be the positive vertical direction. Since the spheres are in equilibrium,
the net force in each direction is zero.
(a) F1x FT1 sin1 FE1 0 FE1 FT1 sin1
F1y FT1 cos1 m1g FT1 m1g FE1 m1g sin 1 m1g tan 1 m1g1
cos1 cos1
A completely parallel analysis would give FE2 m2 g2 . Since the electric forces are a
Newton’s third law pair, they can be set equal to each other in magnitude.
FE1 FE2 m1g1 m2 g2 1 2 m2 m1 1
(b) The same analysis can be done for this case.
FE1 FE2 m1g1 m2 g2 1 2 m1 m1 2
(c) The horizontal distance from one sphere to the other is
s by the small angle approximation. See the diagram. Use the
relationship derived above that FE mg to solve for the distance. l 1 2 l
Case 1: d l 1 2 2l1 d
1 2l
m1g1 FE1 kQ 2Q mg d d 4lkQ2 1/ 3 l sin1 l sin2
2l mg
d2
Case 2: d l 1 2 3 l1 2d
2 1 3l
m1g1 FE1 kQ 2Q mg 2d d 3lkQ2 1/ 3
3l mg
d2
Use EEq.2F1–3to positive x direction.
21. the 3.08 1016 N ˆi
q calculate the force. Take east to be C ˆi 3.08 1016 N west
1019 C 1920 N
F qE 1.602
22. Use EEq.2F1–3 to calculate the electric field. Take north to be the positive y direction.
q 2.18 1014 N ˆj 1.36 105 N C ˆj 1.36 105 N C south
1.602 1019 C
23. Use Eq. 21–4a to calculate the electric field due to a point charge.
E Q C2 33.0 106 C 1.10 107
k r2 8.988 109 N m2 N C up
0.164 m2
Note that the electric field points away from the positive charge.
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10
Chapter 21 Electric Charges and Electric Field
24. Use EEq.2F1–3 to calculate the electric field.
q 8.4 N down 9.5105 N C up
8.8106 C
25. Use the definition of the electric field, Eq. 21-3.
F 7.22 104 N ˆj
4.20 106 C
172 N C ˆj
E
q
26.
Use thEedeFqfiniti3o.n01ˆio.f253th.9e1ˆje0le6c1Ct0ric3 Nfield, Eq. 21-3.
2400 ˆi 3100
ˆj NC
27. Assuming the electric force is the only force on the electron, then Newton’s second law may be used
to find the acceleration.
Fnet
ma a q E 1.602 1019 C 576 N C 1.011014 m s2
qE m 9.109 1031kg
Since the charge is negative, the direction of the acceleration is opposite to the field .
28. The electric field due to the negative charge will point Q1 0 Q2 0
toward the negative charge, and the electric field due to the E1
positive charge will point away from the positive charge.
Thus both fields point in the same direction, towards the l2
negative charge, and so can be added. E2
Q1 Q2 Q1 Q2 4k
r12 r22 l2
l / 22 l / 22
E
E1 E2 k k k k Q1 Q2
4 8.988 109 N m2 C2
0.080m2
8.0 106C 5.8 106C 7.8 107 N C
The direction is towards the negative charge .
29.
30. Assuming the electric force is the only force on the electron, then Newton’s second law may be used
to find the electric field strength.
Fnet ma qE
E ma 1.673 1027 kg 1.8 106 9.80 m s2 0.18 N C
q 1.602 1019 C
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11
Physics for Scientists & Engineers with Modern Physics, 4th Edition Instructor Solutions Manual
31. The field at the point in question is the vector sum of the two fields shown in Figure 21-56. Use the
results of Example 21-11 to find the field of the long line of charge.
1 ˆj 1 Q
20 y EQ 40 d2
Ethread
; cosˆi sinˆj
1 Q cos ˆi 1 1 Q sin ˆj
E 40 d2 2 y 40 d2
0
d 2 0.070 m2 0.120 m2 0.0193m2 ; y 0.070 m ; tan1 12.0cm 59.7
7.0 cm
1 Q 2.0 C
d2
0.0193 m2
40
Ex
cos 8.988 109 N m2 C2 cos 59.7 4.699 1011 N C
Ey 1 1 Q sin 1 2 Q sin
20 y 40 d2 40 y d2
22.5C m 2.0 C sin 59.7
8.988 109 N m2 C2 1.622 1011 N C
0.070cm 0.0193 m2
E
4.7 1011 N C ˆi 1.6 1011 N C ˆj
4.699 1011 N C 2 1.622 1011 N C 2 5.0 1011 N C
E Ex2 E 2
y
E tan1
1.622 1011 N C 199
4.699 1011 N C
32. The field due to the negative charge will point towards Q Q
the negative charge, and the field due to the positive charge l2 EQ
will point towards the negative charge. Thus the
magnitudes of the two fields can be added together to find EQ
the charges.
2EQ Q 8kQ El2 586 N C 0.160 m2 2.09 1010 C
l2 8k
/ 22 8 8.988 109 N m2 C2
Enet 2k l Q
33. The field at the upper right corner of the square is the vector sum of
E3 E2
the fields due to the other three charges. Let the variable l represent
the 1.0 m length of a side of the square, and let the variable Q represent Q1
l E1
the charge at each of the three occupied corners.
E1 k Q E1x k Q , E1y 0
l2 l2
E2 k Q E2 x k Q cos45o k 2Q , E2 y k 2Q Q2 Q3
2l2 2l2 4l2 4l2
E3 k Q E3x 0 , E1 y k Q
l2 l2
Add the x and y components together to find the total electric field, noting that Ex Ey .
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12
Chapter 21 Electric Charges and Electric Field
Ex E1x E2 x E3x k Q k 2Q 0 k Q 1 2 Ey
l2 4l2 l2 4
E Ex2 E 2 k Q 1 2 2 k Q 2 1
y l2 l2 2
4
8.988 109 N m2 C2
2.25 106 C 2 1 2.60 104 N C
2
1.22 m2
tan1 Ey 45.0 from the x-direction.
Ex
34. The field at the center due to the two 27.0C negative charges
on opposite corners (lower right and upper left in the diagram) Q2 Q2 27.0C
will cancel each other, and so only the other two charges need to
be considered. The field due to each of the other charges will E1
E2
point directly toward the charge. Accordingly, the two fields are l
in opposite directions and can be combined algebraically.
E E1 E2 k Q1 k Q2 k Q1 Q2
l2 2 l2 2 l2 2
Q2
38.6 27.0 106C Q1 38.6C
0.525m2 2
8.988 109 N m2 C2
7.57 106 N C, towards the 38.6C charge
35. Choose the rightward direction to be positive. Then the field due to +Q will be positive, and the
field due to –Q will be negative.
k Q k Q kQ 1 1 4kQxa
a x2 a2 2
x a2 x a2 x a2
E x 2
The negative sign means the field points to the left .
36. For the net field to be zero at point P, the magnitudes of the fields created by Q1 and Q2 must be
equal. Also, the distance x will be taken as positive to the left of Q1 . That is the only region where
the total field due to the two charges can be zero. Let the variable l represent the 12 cm distance,
and note that Q1 1 Q2 .
2
k Q1 k x Q2
E1 E2 x2
l2
x l Q1 12cm 25C 35cm
Q2 Q1 45C 25C
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13
Physics for Scientists & Engineers with Modern Physics, 4th Edition Instructor Solutions Manual
37. Make use of Example 21-11. From that, we see that the electric field due to the line charge along the
y axis is 1 ˆi. In particular, the field due to that line of charge has no y dependence. In a
E1 20 x
similar fashion, the electric field due to the line charge along the x axis is 1 ˆj. Then the
E2 20 y
total field at x, y is the vector sum of the two fields.
1 ˆi 1 ˆj 1 ˆi 1 ˆj
E E1 E2 20 x 20 y 20 x y
1
E 11 x2 y2 ; tan1 Ey tan1 20 y tan1 x
20 x2 y2 20xy Ex 1 y
20 x
38. (a) The field due to the charge at A will point straight downward, and A Q
the field due to the charge at B will point along the line from A to l l
the origin, 30o below the negative x axis. l
EB
EA k Q EAx 0 , EAx k Q Q
l2 l2 EA B
EB k Q EBx k Q cos 30o k 3Q ,
l2 l2 2l2
EBy k Q sin 30o k Q
l2 2l2
Ex EAx EBx k 3Q Ey EAy EBy k 3Q
2l2 2l2
E Ex2 E 2 3k 2Q2 9k 2Q2 12k 2Q2 3kQ
y 4l4 4l4 4l4 l2
tan1 Ey k 3Q tan1 3 tan1
2l2
tan1 3 240o
Ex 3Q 3
k 2l2
(b) Now reverse the direction of EA
EA k Q EAx 0 , EAx k Q
l2 l2
EB k Q EBx k Q cos 30o k 3Q , EBy k Q sin 30o k Q
l2 l2 2l2 l2 2l2
Ex EAx EBx k 3Q Ey EAy EBy k Q
2l2 2l2
E Ex2 E 2 3k 2Q2 k 2Q2 4k 2Q2 kQ
y 4l4 4l4 4l4 l2
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14
Chapter 21 Electric Charges and Electric Field
tan1 Ey tan1 kQ tan1 1 330o
2l2
Ex k 3Q 3
2l2
39. Near the plate, the lines should come from it almost vertically, + ++ + +
because it is almost like an infinite line of charge when the
observation point is close. When the observation point is far
away, it will look like a point charge.
40. Consider Example 21-9. We use the result from this example, but y
shift the center of the ring to be at x 1 l for the ring on the right,
2
and at x 1 l for the ring on the left. The fact that the original
2
expression has a factor of x results in the interpretation that the sign RR
of the field expression will give the direction of the field. No special x
consideration needs to be given to the location of the point at which 1 l 1 l
2 2
the field is to be calculated. O
E Eright Eleft
1 Q x 1 l ˆi 1 Q x 1 l ˆi
40 2 40 2
x 1 l2 R2 / 2 x 1 l2 R2 / 2
2 2
ˆi Q x 1 l x 1 l
40 2 2
x 1 l 2 R2 / 2 x 1 l 2 R2 / 2
2 2
41. Both charges must be of the same sign so that the electric fields created by the two charges oppose
each other, and so can add to zero. The magnitudes of the two electric fields must be equal.
E1 E2 k Q1 k Q2 9Q1 9Q2 Q1 1
4 Q2 4
l 32 2l 32
42. In each case, find the vector sum of the field caused by the charge on the left Eleft and the field
Eright
caused by the charge on the right Eright Eleft
Point A: From the symmetry of the geometry, in A
calculating the electric field at point A only the vertical Q d d
components of the fields need to be considered. The
horizontal components will cancel each other. Q
tan1 5.0 26.6
10.0
d 5.0cm2 10.0cm2 0.1118m
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15
Physics for Scientists & Engineers with Modern Physics, 4th Edition Instructor Solutions Manual
EA kQ C2 5.7 106 C 3.7 106 N
2 d2 sin 2 8.988 109 N m2 sin 26.6 C A 90
0.1118 m2
Point B: Now the point is not symmetrically placed, and Eleft
so horizontal and vertical components of each individual Eright
field need to be calculated to find the resultant electric
field. d right
right
left tan1 5.0 45 right tan1 5.0 18.4 dleft
5.0 15.0
Q left Q
dleft 5.0cm2 5.0cm2 0.0707 m
dright 5.0cm2 15.0 cm2 0.1581m
Q Q
Eleft Eright d2 d2
Ex x k cosleft k cosright
left right
x
8.988 109 N m2 C2 cos45 cos18.4 106
5.7 106 C 5.30 N C
0.0707 m 2 0.1581m2
Q Q
Eleft Eright d2 d2
Ey y k sinleft k sin right
left right
y
8.988 109 N m2 C2 sin45 sin18.4
5.7 106 C 7.89 106 N C
0.0707 m 2 0.1581m2
EB Ex2 E 2 9.5 106 N C B tan1 Ey 56
y Ex
The results are consistent with Figure 21-34b. In the figure, the field at Point A points straight up,
matching the calculations. The field at Point B should be to the right and vertical, matching the
calculations. Finally, the field lines are closer together at Point B than at Point A, indicating that the
field is stronger there, matching the calculations.
43. (a) See the diagram. From the symmetry of the charges, we see that y
the net electric field points along the y axis. E
Q Qy 3/ 2 ˆj E2
l2 y2 l2 y2 y
E2
40 ¬ +Q x
sin ˆj 20 E1
(b) To find the position where the magnitude is a maximum, set the r
first derivative with respect to y equal to 0, and solve for the y Q+
¬
value.
Qy
l2 y2
E 3/2
20
dE Qy
l2 y2
dy 20
Q 3 20 5/2 2y 0
2
l2 y2 3/ 2
1 3y2 y2 1 l2 y l 2
2
l2 y2 3/ 2 l2 y2 5/ 2
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16
Chapter 21 Electric Charges and Electric Field
This has to be a maximum, because the magnitude is positive, the field is 0 midway between the
charges, and E 0 as y .
From Example 21-9, the electric field along the x-axis is E 1 Qx . To find the position
40 3
x2 a2 2
44.
where the magnitude is a maximum, we differentiate and set the first derivative equal to zero.
31
dE Q x2 a2 x x2 a2
dx 40
2 3 2 2x Q
2 x2 a2
40 5 x2 a2 3x2
x2 a2 3
2
Q 5 a2 2x2 0 xM a
x2 a2 2
2
40
Note that E 0 at x 0 and x , and that E 0 for 0 x . Thus the value of the magnitude
of E at x xM must be a maximum. We could also show that the value is a maximum by using the
second derivative test.
45. Because the distance from the wire is much smaller than the length of the wire, we can approximate
the electric field by the field of an infinite wire, which is derived in Example 21-11.
2 4.75 106 C
2.0 m
1 1 2 109 Nm2 1.8 106 N C,
20 x 40 x C2 away from the
E 8.988
2.4 102 m wire
46. This is essentially Example 21-11 again, but with different limits of y
integration. From the diagram here, we see that the maximum
dy dq P
angle is given by sin l 2 . We evaluate the results at y r
x2 l 22 l dE
that angle. x x
E sin sin l 2
x2 l 22
4 0 x sin l 2
x2 l 22
l2 l2 l l
x2 l 2 2 20 x 4x2 l2 1/ 2
40x
x2 l 22 x2 l 22 40x
47. If we consider just one wire, then from the answer to problem 46, we Eright Eleft
would have the following. Note that the distance from the wire to the wire wire
point in question is x z2 l 22 .
Ewire l z2 l 22
20 z
z2 l 22 4 z2 l 22 l2 1/2
l2 l2
But the total field is not simply four times the above expression,
because the fields due to the four wires are not parallel to each other.
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17
Physics for Scientists & Engineers with Modern Physics, 4th Edition Instructor Solutions Manual
Consider a side view of the problem. The two dots represent two parallel wires, on opposite sides of
the square. Note that only the vertical component of the field due to each wire will actually
contribute to the total field. The horizontal components will cancel.
Ewire 4 Ewire cos 4 Ewire z
z2 l 22
Ewire l z
4
1/ 2 z2 l 22
2 z2 l 22
0 4 z2 l 22 l2
8lz
0 4z2 l2
4z2 2l2 1/2
The direction is vertical, perpendicular to the loop.
48. From the diagram, we see that the x components of the two fields will cancel each other at the point
P. Thus the net electric field will be in the negative Q
y-direction, and will be twice the y-component of
either electric field vector.
Enet 2E sin 2 kQ sin a
x2 a2
x
2kQ a a EQ EQ
x2 a2 x2 a2 1/ 2
2kQa in the negative y direction Q
x2 a2 3/2
49. Select a differential element of the arc which makes an
angle of with the x axis. The length of this element Rd
is Rd , and the charge on that element is dq Rd . R x
The magnitude of the field produced by that element is
dEbottom 0
dE 1 Rd . From the diagram, considering
4 0 R2
dEtop
pieces of the arc that are symmetric with respect to the x
axis, we see that the total field will only have an x
component. The vertical components of the field due to
symmetric portions of the arc will cancel each other.
So we have the following.
dEhorizontal 1 Rd cos
4 0 R2
0 1 Rd 0 2 sin0
0 4 0 R2 4 0 R
cos d
Ehorizontal
cos 4 0R 0 4 0 R sin0 sin 0
The field points in the negative x direction, so E 2 sin0 ˆi
4 0 R
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18
Chapter 21 Electric Charges and Electric Field
50. (a) Select a differential element of the arc which makes an
angle of with the x axis. The length of this element dQ Rd
is Rd , and the charge on that element is dq Rd .
The magnitude of the field produced by that element is
dE 1 Rd . From the diagram, considering
4 0 R2
dE
pieces of the arc that are symmetric with respect to the dE
x axis, we see that the total field will only have a y
component, because the magnitudes of the fields due
to those two pieces are the same. From the diagram
we see that the field will point down. The horizontal components of the field cancel.
dEvertical 1 Rd sin 0 sin2 d
4 0 R2 4 0 R
/ 2 0 sin2 d 0 /2 0 /2
/ 2 4 0R 4 0 R 4 0 R / 2
sin2 d
/ 2
Evertical
1 1 sin 2
2 4
0 1 0 0 ˆj
4 0 R 2 8 0 R E
8 0 R
(b) The force on the electron is given by Eq. 21-3. The acceleration is found from the force.
ma q0 ˆj
F qE
8 0 R
a q0 ˆj e0 ˆj
8m0R 8m0R 8
1.60 1019 C 1.0 106 C m ˆj
9.11 1031 kg 8.85 1012 C2 Nm2 0.010 m
2.5 1017 m s2 ˆj
51. (a) If we follow the first steps of Example 21-11, and refer to Figure 21-29, then the differential
electric field due to the segment of wire is still dE 1 dy . But now there is no
4 0 x2 y2
symmetry, and so we calculate both components of the field.
1 dy cos 1 x dy
4 0 x2 y2 4 0 x2 y2 3/2
dEx
dE cos
dE sin 1 dy sin 1 y dy
4 0 x2 y2 4 0 x2 y2 3/2
dEy
The anti-derivatives needed are in Appendix B4.
l 1 x dy x l dy x y l
0 4 0 x2 y2 3/ 2 40 x2 x2 y 2 0
Ex
x2 y2 3/2
4 0
0
l
40 x x2 l2
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19
Physics for Scientists & Engineers with Modern Physics, 4th Edition Instructor Solutions Manual
l 1 y dy l y dy 1 l
4 0 x2 y 2 3/ 2 40 x2 y2 0
Ey
x2 y2 3/2
0 4 0
0
1 1
x x2 l2
4 0 x2 l2 x 40 x x2 l2
Note that Ey 0, and so the electric field points to the right and down.
(b) The angle that the electric field makes with the x axis is given as follows.
tan Ey 40 x x x2 l2 x2 l2 x x2
Ex x2 l2 x l2
l 1
ll
40 x x2 l2
As l , the expression becomes tan 1, and so the field makes an angle of
45 below the x axis .
52. Please note: the first printing of the textbook gave the length of the charged wire as 6.00 m, but it
should have been 6.50 m. That error has been corrected in later printings, and the following solution
uses a length of 6.50 m.
(a) If we follow the first steps of Example 21-11, and refer to Figure 21-29, then the differential
dy
x2 y2
electric field due to the segment of wire is still dE 1 . But now there is no
4 0
symmetry, and so we calculate both components of the field.
1 dy cos 1 x dy
4 0 x2 y2 4 0 x2 y2 3/2
dExdEcos
dE cos 1 dy sin 1 y dy
4 0 x2 y2 4 0 x2 y2 3/2
dEy
The anti-derivatives needed are in Appendix B4.
ymax x ymax y ymax
x2 y 2 ymin
E 1x
4ymin
x dy dy x
4 0 x2 y2 3/ 2 40 x2
x2 y2 3/2
0 ymin
ymax ymin
4 0 x
x2 y2 x2 y2
max min
8.99 109 Nm2 3.15 106 C 6.50 m
C2 0.250 m
2.50 m 4.00 m
0.250 m2 2.50 m2 0.250 m2 4.00 m2
3.473 104 N C 3.5 104 N C
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20
Chapter 21 Electric Charges and Electric Field
ymax 1 ymax
ymax x2 y 2 ymin
E 1y
4ymin 0
y dy y dy
4 0 x2 y2 3/ 2 40
x2 y2 3/2
ymin
1 1
4 0
x2 y2 x2 y2
max min
8.99 109 Nm2 3.15 106 C
C2
6.50 m
1 1
0.250 m2 2.50 m2 0.250 m2 4.00 m2
647 N C 650 N C
(b) We calculate the infinite line of charge result, and calculate the errors.
E
2 2 8.99 109 Nm2 3.15 106 C 3.485 104 N m
20 x 40 x
C2 6.50 m 0.250 m
Ex E
E
3.473 104 N C 3.485 104 N m 0.0034
3.485 104 N m
Ey
E
647 N C 0.019
3.485 104 N m
And so we see that Ex is only about 0.3% away from the value obtained from the infinite line of
charge, and Ey is only about 2% of the value obtained from the infinite line of charge. The
field of an infinite line of charge result would be a good approximation for the field due to this
wire segment.
53. Choose a differential element of the rod dx a dq y x
distance x from the origin, as shown in the O dE
diagram. The charge on that differential element is x
dx x
dq Q dx. The variable x is treated as positive,
l
so that the field due to this differential element is dE 1 dq Q dx . Integrate
4 0 4 0 l
x x2 x x2
along the rod to find the total field.
l Q dx Q l dx Q 1 l Q 1 1
0 4 0 l 4 0 l 0 4 x x 0 40l x
x x2 x x2
E dE l x l
0
Q
40x x l
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21
Physics for Scientists & Engineers with Modern Physics, 4th Edition Instructor Solutions Manual
54. As suggested, we divide the plane into long narrow strips of width dy and length l. The charge on
the strip is the area of the strip times the charge per unit area: dq ldy. The charge per unit length
on the strip is dq dy. From Example 21-11, the field due to that narrow strip is
l
dE dy . From Figure 21-68 in the textbook, we see that this field
20 y2 z2 20 y2 z2
does not point vertically. From the symmetry of the plate, there is another long narrow strip a
distance y on the other side of the origin, which would create the same magnitude electric field. The
horizontal components of those two fields would cancel each other, and so we only need calculate
the vertical component of the field. Then we integrate along the y direction to find the total field.
dE dy ; dEz dE cos zdy
20 y 2 z2
20 y2 z2
zdy z dy z 1 1 y
20 y2 y2 z2 2 0 z z
2 0
EEz z2 tan
tan 1 tan 1
2 0 2 0 2 2 2 0
55. Take Figure 21-28 and add the angle , measured from the –z axis, y
as indicated in the diagram. Consider an infinitesimal length of the ad
ring ad. The charge on that infinitesimal length is dq ad a
rP
Q ad Q d. The charge creates an infinitesimal electric
a
x
Q d x
field, with magnitude dE 1 dq 1 . From the z
d E, 4 r2 4 x2 a2
dE
symmetry of the figure, we see that the z component of d E will be cancelled by the z component
due to the piece of the ring that is on the opposite side of the y axis. The trigonometric relationships
give dEx dE cos and dEy dE sin sin. The factor of sin can be justified by noting that
dEy 0 when 0, and dEy dE sin when 2.
dE cos Q d x Qx d
4 2 x2 a2 x2 a2 4 2 x2 a2 3/ 2
dEx
Qx Qx
x2 a2 x2 a2 3/2
3/ 2 d
0
Ex 4 2
4
dE sin sin Q d a sin Qa 3/2 sind
4 2 x2 a2 x2 a2
dEy
x2 a2 4 2
Qa Qa
x2 a2 x2 a2
Ey 4 2
3/2 sind 3/2 cos cos0
4 2
0
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22
Chapter 21 Electric Charges and Electric Field
2Qa
4 2
x2 a2 3/2
We can write the electric field in vector notation.
Qx 3/2 ˆi 2Qa 3/ 2 ˆj Q 3/ 2 xˆi 2a ˆj
x2 a2 x2 a2 x2 a2
E
4
4 2 40
56. (a) Since the field is uniform, the electron will experience a constant force in the direction opposite
to its velocity, so the acceleration is constant and negative. Use constant acceleration
relationships with a final velocity of 0.
F ma qE eE a eE ; v2 v02 2ax 0
m
27.5 106 m s 2
11.4 103 N C 0.189 m
x
v02 v02 mv02 9.11 1031 kg
2a 2eE 2 1.60 1019 C
2 eE
m
(b) Find the elapsed time from constant acceleration relationships. Upon returning to the original
position, the final velocity will be the opposite of the initial velocity.
v v0 at
t
v v0 2v0 2mv0 2 9.111031kg 27.5 106 m s 2.75 108s
a eE 1.60 1019 C 11.4 103 N C
eE
m
57. (a) The aFcncetelermaationqiEs produced by the electric force.
eE
a
e 1.60 1019 C 2.0ˆi 8.0ˆj 104 N C 3.5131015ˆi 1.405 1016 ˆj m s2
E 9.11 1031 kg
m
3.5 1015 m s2 ˆi 1.4 1016 m s2 ˆj
(b) The direction is found from the components of the velocity.
v v at 8.0 104 m s ˆj 3.5131015ˆi 1.4051016ˆj m s2 1.0 109s
3.513106ˆi 1.397 107 ˆj m s
tan1 vy tan 1 1.397 107 m s 256 or 104
vx 3.513 106 m s
This is the direction relative to the x axis. The direction of motion relative to the initial
direction is measured from the y axis, and so is 166 counter-clockwise from the initial
direction.
58. (a) The electron will experience a force in the opposite direction to the electric field. Since the
electron is to be brought to rest, the electric field must be in the same direction as the initial
velocity of the electron, and so is to the right .
(b) Since the field is uniform, the electron will experience a constant force, and therefore have a
constant acceleration. Use constant acceleration relationships to find the field strength.
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23
Physics for Scientists & Engineers with Modern Physics, 4th Edition Instructor Solutions Manual
F qE ma a qE v2 v02 2ax v02 2 qE x
m m
9.109 1031kg 7.5 105 m s 2
2 1.602 1019 C 0.040 m
m
E
v2 v02 mv02 40 N C 2 sig. fig.
2qx
2qx
59. The angle is determined by the velocity. The x component of the velocity is constant. The time to
pass through the plates can be found from the x motion. Then the y velocity can be found using
constant acceleration relationships.
x v0t t x ; vy vy0 ayt eE x
v0 m v0
eE x
tan vy
vx
m v0 eEx 1.60 1019 C 5.0 103 N C 0.049 m .4303
v0 mv02
9.111031kg 1.00 107 m s 2
tan1 0.4303 23
60. Since the field is constant, the force on the electron is constant, and so the acceleration is constant.
Thus constant acceleration relationships can be used. The initial conditions are x0 0, y0 0,
vx0 1.90 m s , and vy0 0.
ma a q e ; ax e Ex , ay e Ey
F qE m E m E m m
x x0 vx0t 1 axt2 vx0t eEx t2
2 2m
1.90m s2.0s
1.60 1019 C 2.00 1011 N C 2.0s2 3.2 m
2 9.111031kg
y
y0 vy0t 1 ayt2 eEy t2 1.60 1019 C 1.20 1011 N C 2.0s2 4.2 m
2 2m 2 9.111031kg
61. (a) The field along the axis of the ring is given in Example 21-9, with the opposite sign because this
ring is negatively charged. The force on the charge is the field times the charge q. Note that if
x is positive, the force is to the left, and if x is negative, the force is to the right. Assume that
x R.
Q x qQx
40
x2 R2 3/2
F qE q 1 3/ 2 qQx
4 0 x2 R2
4 0 R3
This has the form of a simple harmonic oscillator, where the “spring constant” is
kelastic Qq .
40R3
(b) The spring constant can be used to find the period. See Eq. 14-7b.
T 2 m 2 m 2 m40R3 4 m0R3
kelastic Qq Qq Qq
40R3
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24
Chapter 21 Electric Charges and Electric Field
62. (a) The dipole moment is given by the product of the positive charge and the separation distance.
p Ql 1.60 1019 C 0.68 109 m 1.088 1028Cm 1.11028Cm
(b) The torque on the dipole is given by Eq. 21-9a.
pE sin 1.088 1028Cm 2.2 104 N C sin 90 2.4 1024 Cm
(c) pE sin 1.088 1028Cm 2.2 104 N C sin 45 1.7 1024 Nm
(d) The work done by an external force is the change in potential energy. Use Eq. 21-10.
W U pE cosfinal pE cosinitial pE cosinitial cosfinal
1.088 1028Cm 2.2 104 N C 1 1 4.8 1024 J
63. (a) The dipole moment is the effective charge of each atom times the separation distance.
p Ql Q p 3.4 1030 Cm 3.4 1020 C
l 1.0 1010 m
(b) Q 3.4 1020 C 0.21 No, the net charge on each atom is not an integer multiple of e. This
e 1.60 1019 C
is an indication that the H and Cl atoms are not ionized – they haven’t fully gained or lost an
electron. But rather, the electrons spend more time near the Cl atom than the H atom, giving the
molecule a net dipole moment. The electrons are not distributed symmetrically about the two
nuclei.
(c) The torque is given by Eq. 21-9a.
pE sin max pE 3.4 1030Cm 2.5 104 N C 8.5 1026 Nm
(d) The energy needed from an external force is the change in potential energy. Use Eq. 21-10.
W U pE cosfinal pE cosinitial pE cosinitial cosfinal
3.4 1030Cm 2.5 104 N C 1 cos 45 2.5 1026 J
64. (a) From the symmetry in the diagram, we see that the resultant field y
will be in the y direction. The vertical components of the two
E E
fields add together, while the horizontal components cancel.
Q r
r2 l2 r2 l2 1/2
Enet 2E sin 2 40 r
2Qr 2Qr 2Q Q+ +Q x
40r2 ¬ ¬
40
r2 l2 3/ 2 r3
40
(b) Both charges are the same sign. A long distance away from the
charges, they will look like a single charge of magnitude 2Q, and so E k q 2Q .
r2 4 0 r 2
65. (a) There will be a torque on the dipole, in a direction to decrease . That torque will give the
dipole an angular acceleration, in the opposite direction of .
pE sin I d 2 pE sin
dt 2 I
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25
Physics for Scientists & Engineers with Modern Physics, 4th Edition Instructor Solutions Manual
If is small, so that sin , then the equation is in the same form as Eq. 14-3, the equation
of motion for the simple harmonic oscillator.
d 2 pE sin pE d 2 pE 0
dt 2 I I dt 2 I
(b) The frequency can be found from the coefficient of in the equation of motion.
2 pE f 1 pE
I 2 I
66. I1f0thgeivFdexispoUleddxiUsxof vpedrdyEx smxpa.llEeSxixntecnet,thtpheepdndoExtthenetpiaolteennteiraglyeniserkgnyowisna, function of position, and so Eq. 21-
we can use Eq. 8-7.
Since the field does not depend on the y or z coordinates, all other components of the force will be 0.
p dE
Thus F Fxˆi dx ˆi .
67. (a) Along the x axis the fields from the two charges are y
–Q
parallel so the magnitude is found as follows. +Q Er
–
Enet EQ EQ Q Q + r
¬
40 r 1 l2 40 r 1 l2
2 2
Q r 1 l2 r 1 l2
2 2
40
r 1 l2 r 1 l2
2 2
Q 2rl Q 2rl 2Ql 1 2p
40r3
40 r 1 l2 r l1 2 40r4 40 r3
2
2
The same result is obtained if the point is to the left of Q.
(b) The electric field points in the same direction as the dipole moment vector.
68. Set the magnitude of the electric force equal to the magnitude of the force of gravity and solve for
the distance.
FE FG k e2 mg
r2
r e k 1.6021019 C
mg
8.988109 N m2 C2
9.111031kg 9.80 m s2 5.08 m
69. Water has an atomic mass of 18, so 1 mole of water molecules has a mass of 18 grams. Each water
molecule contains 10 protons.
65kg 6.02 1023 H2O molecules 10 protons 1.60 1019 C 3.5 109 C
0.018 kg 1 molecule proton
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26
Chapter 21 Electric Charges and Electric Field
70. Calculate the total charge on all electrons in 3.0 g of copper, and compare 38C to that value.
Total electron charge 3.0 g 1mole 6.02 1023 atoms 29 e 1.602 1019C 1.32 105 C
63.5g mole atoms 1e
Fraction lost 38106 C 2.9 1010
1.32105 C
71. Use Eq. 21-4a to calculate the magnitude of the electric charge on the Earth.
E 2
Q Er 2 150 N C 6.38106 m
k r2 Q k 6.8105 C
8.988109 N m2 C2
Since the electric field is pointing towards the Earth’s center, the charge must be negative .
72. (a) From problem 71, we know that the electric field is pointed towards the Earth’s center. Thus an
electron in such a field would experience an upwards force of magnitude FE eE . The force
of gravity on the electron will be negligible compared to the electric force.
FE eE ma
a eE
m
1.6021019 C 150 N C 2.638 1013 m s2 2.61013 m s2 , up
9.111031 kg
(b) A proton in the field would experience a downwards force of magnitude FE eE . The force of
gravity on the proton will be negligible compared to the electric force.
FE eE ma
a
eE 1.6021019 C 150 N C 1.4391010 m s2 1.41010 m s2 , down
m
1.67 1027 kg
(c) Electron: a 2.638 1013 m s2 2.7 1012 ; Proton: a 1.4391010 m s2 1.5 109
g 9.80 m s2 g 9.80 m s2
73. For the droplet to remain stationary, the magnitude of the electric force on the droplet must be the
same as the weight of the droplet. The mass of the droplet is found from its volume times the density
of water. Let n be the number of excess electrons on the water droplet.
FE q E mg neE 4 r 3 g
3
1.8105 m 3 1.00103 kg m3 9.80 m
3 1.6021019 C 150 N C
n 4r3g4 s2 9.96106 1.0107 electrons
3eE
74. There are four forces to calculate. Call the rightward direction the positive direction. The value of k
is 8.988109 N m2 C2 and the value of e is 1.6021019 C .
FCN k 0.40e 0.20e 1 1 1 1
1 109 m 2 0.302 0.402 0.182 0.282
FnetFCH FOH FON
2.445 1010 N 2.4 1010 N
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27
Physics for Scientists & Engineers with Modern Physics, 4th Edition Instructor Solutions Manual
75. Set the Coulomb electrical force equal to the Newtonian gravitational force on one of the bodies (the
Moon).
FE FG 2
Q M MMoon Earth
k G2 2
r rorbit orbit
Q
GM MMoon Earth 6.67 1011 Nm2 kg2 7.351022 kg 5.981024 kg 5.711013 C
k 8.988109 N m2 C2
76. The electric force must be a radial force in order for the electron to move in a circular orbit.
FE Fradial k Q2 mv2
r2 rorbit
orbit
1.602 1019 C 2
9.109 1031kg 2.2 106 m
rorbitQ2 5.2 1011m
k mv2 8.988 109 N m2 C2 s 2
77. Because of the inverse square nature of the electric field, Q1 Q2
any location where the field is zero must be closer to the
weaker charge Q2 . Also, in between the two charges, dl
the fields due to the two charges are parallel to each other and cannot cancel. Thus the only places
where the field can be zero are closer to the weaker charge, but not between them. In the diagram,
this means that l must be positive.
E k Q2 k l Q1 2 0 Q2 l d 2 Q1l2
l2 d
l Q2 d 5.0 106C 2.0 m 1.6 m from Q2,
Q1 Q2 2.5 105C 5.0 106C 3.6 m from Q1
78. We consider that the sock is only acted on by two forces – the force of gravity, acting downward,
and the electrostatic force, acting upwards. If charge Q is on the sweater, then it will create an
electric field of E Q A , where A is the surface area of one side of the sweater. The same
20 20
magnitude of charge will be on the sock, and so the attractive force between the sweater and sock is
FE QE Q2 . This must be equal to the weight of the sweater. We estimate the sweater area as
20 A
0.10 m2, which is roughly a square foot.
FE QE Q2 mg
20 A
Q 20 Amg 2 8.85 1012 C2 Nm2 0.10 m2 0.040 kg 9.80 m s2 8 107C
79. The sphere will oscillate sinusoidally about the equilibrium point, with an amplitude of 5.0 cm. The
angular frequency of the sphere is given by k m 126 N m 0.650 kg 13.92 rad s . The
distance of the sphere from the table is given by r 0.150 0.0500cos13.92tm. Use this distance
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28
Chapter 21 Electric Charges and Electric Field
and the charge to give the electric field value at the tabletop. That electric field will point upwards at
all times, towards the negative sphere.
E k
Q 8.988 109 N m2 C2 3.00 106C 2.70 104 N C
r2
0.150 0.0500cos13.92t2 m2 0.150 0.0500cos13.92t2
1.08 107 N C, upwards
3.00 cos13.9t2
80. The wires form two sides of an isosceles triangle, and so the two charges are
FT
separated by a distance l 2 78cm sin 26 68.4 cm and are directly horizontal
FE
from each other. Thus the electric force on each charge is horizontal. From the free- mg
body diagram for one of the spheres, write the net force in both the horizontal and
vertical directions and solve for the electric force. Then write the electric force by
Coulomb’s law, and equate the two expressions for the electric force to find the
charge.
Fy FT cos mg 0 FT mg
cos
Fx FT sin FE 0 FE FT sin mg sin mg tan
cos
FE k Q 22 mg tan Q 2l mg tan
k
l2
20.684m
24 103kg 9.80 m s2 tan 26 4.887 106C 4.9 106C
8.988 109 N m2 C2
81. The electric field at the surface of the pea is given by Eq. 21-4a. Solve that equation for the
charge.
E 2
Q Q Er2 3106 N C 3.75103 m
k r2 k 8.988109 N m2 C2 5109 C
This corresponds to about 3 billion electrons.
82. There will be a rightward force on Q1 due to Q2 , given by Coulomb’s law. There will be a leftward
force on Q1 due to the electric field created by the parallel plates. Let right be the positive direction.
F k Q1Q2 Q1 E
x2
8.988109 N m2 C2
6.7 106 C 1.8106 C 6.7 106 C 7.3104 N C
0.34 m2
0.45 N, right
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29
Physics for Scientists & Engineers with Modern Physics, 4th Edition Instructor Solutions Manual
83. The weight of the sphere is the density times the volume. The electric force is given by Eq. 21-1,
with both spheres having the same charge, and the separation distance equal to their diameter.
mg k Q2 4 r3g kQ2
3
d 2 2r2
Q 5
16 gr5 16 35kg m3 9.80 m s2 1.0 102 m
3 8.99 109 Nm2 C2 8.0 109C
3k
84. From the symmetry, we see that the resultant field will be in the y y
direction. So we take the vertical component of each field.
E E
2E sin E 2 Q r 2Q
r2 l2 r2 l2 1/ 2 40r2
Enet
40
E
2Q r 1 r
r2 l2 r2
3/ 2 Q -2Q – Q
40
+ ¬¬ + x
2Q 3/ 2
r2 l2
40
3/ 2 r2 r3 r2 l2
1 l2 3/ 2
1 r2
2Qr3
l2 3/ 2
r2
40r5 1
Use the binomial expansion, assuming r l.
1 l2 3/ 2 1 l2 l2
1 r2 r2 r2
2Qr3 2Qr3 1 3 2Qr3 3 3Ql2
2 2 40r4
Enet l2 3/ 2 1 l2 40r5 1
r2 r2
40r5 1 4 0 r 5 3
2
Notice that the field points toward the negative charges.
85. This is a constant acceleration situation, similar to projectile motion in a uniform gravitational field.
Let the width of the plates be l, the vertical gap between the plates be h, and the initial velocity be
v0. Notice that the vertical motion has a maximum displacement of h/2. Let upwards be the positive
vertical direction. We calculate the vertical acceleration produced by the electric field and the time t
for the electron to cross the region of the field. We then use constant acceleration equations to solve
for the angle.
Fy may qE eE ay eE ; l v0 cos0 t t l
m v0 cos0
vy v0 y ayttop 0 v0 sin 0 eE 1 v0 l v02 eE l
m 2 cos0 2m sin0 cos0
top
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30
Chapter 21 Electric Charges and Electric Field
ytop y0 v0 ty top 1 ayt2 1 h v0 sin 0 1 v0 l 1 eE 1 v0 l 2
2 2 2 cos0 2 m 2 cos0
h l tan 0 eEl2 0 12 l tan 0 eEl2 0 1 l tan0 1 l tan 0
4m cos2 v02 4m cos2 eE l 2
2m sin0 cos0
h 1 l tan 0 tan1 2h tan1 2 1.0 cm 18
2 l
6.0 cm
86. (a) The electric field from the long wire is derived in Example 21-11.
E 1 , radially away from the wire
20 r
(b) The force on the electron will point radially in, producing a centripetal acceleration.
F qE e mv2
20 r r
v
2 1 e 2 8.99 109 Nm2 C2 1.60 1019 C 0.14 106 C m
40 m 9.11 1031 kg
2.1107 m s
Note that this speed is independent of r.
87. We treat each of the plates as if it were infinite, and A – B + C – D
then use Eq. 21-7. The fields due to the first and –
third plates point towards their respective plates, E1 – E1 + E1 – E1
–
and the fields due to the second plate point away – + – E2
from it. See the diagram. The directions of the E2 – E2 + E2 –
fields are given by the arrows, so we calculate the 3
magnitude of the fields from Eq. 21-7. Let the – + E3
positive direction be to the right. E3 E3 E3
1 2 1 –+
2 0 2 0 20
EA E1 E2 E3 1 2
0.50 0.25 0.35 106 C m2 3.4 104 N C, to the right
2 8.85 1012 C2 Nm2
EB E1 E2 E3 1 2 1
20 20 2 0
0.50 0.25 0.35 106 C m2 2.3 104 N C 2.3 104 N C to the left
2 8.85 1012 C2 Nm2
EC E1 E2 E3 1 2 1
2 0 20 20
0.50 0.25 0.35 106 C m2 5.6 103 N C to the right
2 8.85 1012 C2 Nm2
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31
Physics for Scientists & Engineers with Modern Physics, 4th Edition Instructor Solutions Manual
ED E1 E2 E3 1 2 1
20 20 2 0
0.50 0.25 0.35 106 C m2 3.4 104 N C 3.4 103 N C to the left
2 8.85 1012 C2 Nm2
88. Since the electric field exerts a force on the charge in the 43cm L 55cm
same direction as the electric field, the charge is FT FE
positive. Use the free-body diagram to write the mg
equilibrium equations for both the horizontal and vertical
directions, and use those equations to find the magnitude
of the charge.
cos1 43cm 38.6o
55 cm
Fx FE FT sin 0 FE FT sin QE
Fy FT cos mg 0 FT mg QE mg tan
cos
Q
mg tan 1.0 103kg 9.80 m s2 tan 38.6o 5.2 107 C
E 1.5 104 N C
89. A negative charge must be placed at the center of the square. Let
F43 F42
Q 8.0 C be the charge at each corner, let q be the magnitude of Q4
F4q F41
negative charge in the center, and let d 9.2 cm be the side length of Q1
q
the square. By the symmetry of the problem, if we make the net force d
on one of the corner charges be zero, the net force on each other Q2
corner charge will also be zero.
F41 k Q2 F41x k Q2 , F41y 0 Q3
d2 d2
F42 k Q2 F42 x k Q2 cos45o k 2Q2 , F42 y k 2Q2
2d 2 2d 2 4d 2 4d 2
F43 k Q2 F43x 0 , F43 y k Q2
d2 d2
F4q k qQ F4qx k 2qQ cos 45o k 2qQ F4qy
d2 2 d2 d2
The net force in each direction should be zero.
Fx k Q2 k 2Q2 0k 2qQ 0
d2 4d 2 d2
q 1 1 1 1 7.66 106 C
Q 2 4 8.0 106 C 2 4
So the charge to be placed is q 7.7 106 C .
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32
Chapter 21 Electric Charges and Electric Field
This is an unstable equilibrium . If the center charge were slightly displaced, say towards the right,
then it would be closer to the right charges than the left, and would be attracted more to the right.
Likewise the positive charges on the right side of the square would be closer to it and would be
attracted more to it, moving from their corner positions. The system would not have a tendency to
return to the symmetric shape, but rather would have a tendency to move away from it if disturbed.
90. (a) The force of sphere B on sphere A is given by Coulomb’s law.
FAB kQ2 , away from B
R2
(b) The result of touching sphere B to uncharged sphere C is that the charge on B is shared between
the two spheres, and so the charge on B is reduced to Q 2 . Again use Coulomb’s law.
FAB k QQ 2 kQ2 , away from B
R2 2R2
(c) The result of touching sphere A to sphere C is that the charge on the two spheres is shared, and
so the charge on A is reduced to 3Q 4 . Again use Coulomb’s law.
FAB k 3Q 4Q 2 3kQ2 , away from B
8R2
R2
91. (a) The weight of the mass is only about 2 N. Since the tension in the string is more
FT
than that, there must be a downward electric force on the positive charge, which
means that the electric field must be pointed down . Use the free-body diagram to
write an expression for the magnitude of the electric field. mg
FE
F FT mg FE 0 FE QE FT mg
E s2
FT mg 5.18 N 0.210kg 9.80 m 9.18 106 N C
Q
3.40 107C
(b) Use Eq. 21-7.
E
20
2E0 2 9.18 106 N C 8.854 1012 1.63104 C m2
92. (a) The force will be attractive. Each successive charge is another distance d farther than the
previous charge. The magnitude of the charge on the electron is e.
F k eQ k eQ k eQ k eQ k eQ 1 1 1 1
d2 12 22 32 42
d 2 2d 2 3d 2 4d 2
k eQ 1 1 eQ 2 eQ
d2 n2 d2 6 240d 2
n1 40
(b) Now the closest Q is a distance of 3d from the electron.
F k eQ k eQ k eQ k eQ k eQ 1 1 1 1
d2 32 42 52 62
3d 2 4d 2 5d 2 6d 2
k eQ 1 k eQ 1 1 1 k eQ 2 5 eQ 2 5
d2 n3 n2 d2 n1 n2 12 22 d2 6 4 40d 2 6 4
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33
Physics for Scientists & Engineers with Modern Physics, 4th Edition Instructor Solutions Manual
93. (a) Take dE , set it equal to 0, and solve for the location of the maximum.
dx
E 1 Qx
4 x2 a2 3/2
3/ 2 1/2 2x
dE
Q x2 a2 x 3 x2 a2 Q a2 2x2 a2 2x2 0
2 x2 a2 5/2 0
x2 a2 3
dx 4
4
x a 10.0cm 7.07cm 2.5
22
(b) Yes, the maximum of the graph Electric field (106 N/C) 2.0
does coincide with the analytic 1.5
maximum. The spreadsheet used
for this problem can be found on 1.0
the Media Manager, with filename
“PSE4_ISM_CH21.XLS,” on tab 0.5
“Problem 21.93b.” 0.0
0 2 4 6 8 10 12
(c) The field due to the ring is x (cm)
1 Qx .3/ 2 6.0
4 x2 a2
Ering
(d) The field due to the point charge is Electric field (106 N/C)
Ering 1 Q . Both are plotted 4.5 Ring
4 x2 3.0 Point
on the graph. The graph shows that
the two fields converge at large 1.5
distances from the origin. The
spreadsheet used for this problem
can be found on the Media 0.0
Manager, with filename 0 10 20 30 40 50
“PSE4_ISM_CH21.XLS,” on tab x (cm)
“Problem 21.93cd.”
(e) According to the spreadsheet, Ering 0.9Epoint at about 37 cm. An analytic calculation gives the
same result.
1 Qx 0.9 1 Q
4 x2 a2 4 x2
Ering 0.9Epoint 3/2
x3 0.9 3/ 2 0.9x3 1 a2 3/ 2 a 10.0cm 37.07cm
x2 a2 x2 x
2 / 3 2/ 3
1 1
0.9 1 0.9 1
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34
Chapter 21 Electric Charges and Electric Field
94. (a) Let q1 8.00C, q2 2.00C, and q2 O q1
d 0.0500m. The field directions due to the dd E1
E 1 E2
charges are shown in the diagram. We take care E2 E 1
with the signs of the x coordinate used to E2
calculate the magnitude of the field.
1 q2 1 q1 1 q2 1 q1
40 x d d
x d2 x d2
40 40 40
Exd E2 E1 2 x 2
E2 1 q2 1 q1 1 q2 1 q1
40 2 40 x d
d x d x2 x d2
Edx0 E1 2 40 40
E0xd E2 E1 1 d q2 1 d q1
40
x2 x2
40
Ed x E1 E2 1 x q1 1 x q2
40 40
d2 d2
20
15
10
E (108 N/C) 5
0
-5
-10
-15
-20
-30 -20 -10 0 10 20 30
x (cm)
The spreadsheet used for this problem can be found on the Media Manager, with filename
“PSE4_ISM_CH21.XLS,” on tab “Problem 21.94a.”
(b) Now for points on the y axis. See the diagram for this case.
E1
Ex E1 cos E2 cos
1 q2 cos
d2 y2 E2
1 q1 cos
d2 y2 40
40
q1 q2 cos 1 q1 q2 y
40
d2 y2 d2 y2
1 d q2 q1
d2 y2 d d
40
1
40
q1 q2 d
d2 y2 3/2
1 q1 sin 1 q2 sin
40 d2 y2 d2 y2
40
Ey
E1 sin E2 sin
q1 q2 q1 q2 y 1
d2 y2 d2 y2
d 2 y2 40
1 q1 q2 y
40
d2 y2 3/2
1 sin
40
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35
Physics for Scientists & Engineers with Modern Physics, 4th Edition Instructor Solutions Manual
1.0E (107 N/C)
0.0
-1.0
-2.0 Ex
-3.0 Ey
-4.0
-30 -20 -10 0 10 20 30
y (cm)
The spreadsheet used for this problem can be found on the Media Manager, with filename
“PSE4_ISM_CH21.XLS,” on tab “Problem 21.94b.”
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36
CHAPTER 22: Gauss’s Law
Responses to Questions
1. No. If the net electric flux through a surface is zero, then the net charge contained in the surface is
zero. However, there may be charges both inside and outside the surface that affect the electric field
at the surface. The electric field could point outward from the surface at some points and inward at
others. Yes. If the electric field is zero for all points on the surface, then the net flux through the
surface must be zero and no net charge is contained within the surface.
2. No. The electric field in the expression for Gauss’s law refers to the total electric field, not just the
electric field due to any enclosed charge. Notice, though, that if the electric field is due to a charge
outside the Gaussian surface, then the net flux through the surface due to this charge will be zero.
3. The electric flux will be the same. The flux is equal to the net charge enclosed by the surface divided
by ε0. If the same charge is enclosed, then the flux is the same, regardless of the shape of the surface.
4. The net flux will be zero. An electric dipole consists of two charges that are equal in magnitude but
opposite in sign, so the net charge of an electric dipole is zero. If the closed surface encloses a zero
net charge, than the net flux through it will be zero.
5. Yes. If the electric field is zero for all points on the surface, then the integral of EdA over the
surface will be zero, the flux through the surface will be zero, and no net charge will be contained in
the surface. No. If a surface encloses no net charge, then the net electric flux through the surface will
be zero, but the electric field is not necessarily zero for all points on the surface. The integral of
EdA over the surface must be zero, but the electric field itself is not required to be zero. There may
be charges outside the surface that will affect the values of the electric field at the surface.
6. The electric flux through a surface is the scalar (dot) product of the electric field vector and the area
vector of the surface. Thus, in magnitude, E EAcos . By analogy, the gravitational flux
through a surface would be the product of the gravitational field (or force per unit mass) and the
area, or g gAcos . Any mass, such as a planet, would be a “sink” for gravitational field. Since
there is not “anti-gravity” there would be no sources.
7. No. Gauss’s law is most useful in cases of high symmetry, where a surface can be defined over
which the electric field has a constant value and a constant relationship to the direction of the
outward normal to the surface. Such a surface cannot be defined for an electric dipole.
8. When the ball is inflated and charge is distributed uniformly over its surface, the field inside is zero.
When the ball is collapsed, there is no symmetry to the charge distribution, and the calculation of the
electric field strength and direction inside the ball is difficult (and will most likely give a non-zero
result).
9. For an infinitely long wire, the electric field is radially outward from the wire, resulting from
contributions from all parts of the wire. This allows us to set up a Gaussian surface that is
cylindrical, with the cylinder axis parallel to the wire. This surface will have zero flux through the
top and bottom of the cylinder, since the net electric field and the outward surface normal are
perpendicular at all points over the top and bottom. In the case of a short wire, the electric field is not
radially outward from the wire near the ends; it curves and points directly outward along the axis of
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Physics for Scientists & Engineers with Modern Physics, 4th Edition Instructor Solutions Manual
the wire at both ends. We cannot define a useful Gaussian surface for this case, and the electric field
must be computed directly.
10. In Example 22-6, there is no flux through the flat ends of the cylindrical Gaussian surface because
the field is directed radially outward from the wire. If instead the wire extended only a short distance
past the ends of the cylinder, there would be a component of the field through the ends of the
cylinder. The result of the example would be altered because the value of the field at a given point
would now depend not only on the radial distance from the wire but also on the distance from the
ends.
11. The electric flux through the sphere remains the same, since the same charge is enclosed. The
electric field at the surface of the sphere is changed, because different parts of the sphere are now at
different distances from the charge. The electric field will not have the same magnitude for all parts
of the sphere, and the direction of the electric field will not be parallel to the outward normal for all
points on the surface of the sphere. The electric field will be stronger on the side closer to the charge
and weaker on the side further from the charge.
12. (a) A charge of (Q – q) will be on the outer surface of the conductor. The total charge Q is placed
on the conductor but since +q will reside on the inner surface, the leftover, (Q – q), will reside
on the outer surface.
(b) A charge of +q will reside on the inner surface of the conductor since that amount is attracted
by the charge –q in the cavity. (Note that E must be zero inside the conductor.)
13. Yes. The charge q will induce a charge –q on the inside surface of the thin metal shell, leaving the
outside surface with a charge +q. The charge Q outside the sphere will feel the same electric force as
it would if the metal shell were not present.
14. The total flux through the balloon’s surface will not change because the enclosed charge does not
change. The flux per unit surface area will decrease, since the surface area increases while the total
flux does not change.
Solutions to Problems
1. The electric flux of a uniform field is given by Eq. 22-1b.
(a) E EA EAcos 580 N C 0.13m2 cos0 31 Nm2 C
(b) E 580 N C 0.13m2 cos 45 22 Nm2 C
EA EAcos
(c) E EAcos 580 N C 0.13m2 cos 90 0
EA
2. Use Eq. 22-1b for the electric flux of a uniform field. Note that the surface area vector points
radially outward, and the electric field vector points radially inward. Thus the angle between the two
is 180.
EA EAcos
E 2
150 N C 4 RE2 cos180 4 150 N C 6.38106 m
7.7 1016 Nm2 C
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38
Chapter 22 Gauss’s Law
3. (a) Since the field is uniform, no lines originate or terminate inside the cube, and so the net flux is
net 0 .
(b) There are two opposite faces with field lines perpendicular to the faces. The other four faces
have field lines parallel to those faces. For the faces parallel to the field lines, no field lines
enter or exit the faces. Thus parallel 0 .
Of the two faces that are perpendicular to the field lines, one will have field lines entering the
cube, and so the angle between the field lines and the face area vector is 180. The other will
have field lines exiting the cube, and so the angle between the field lines and the face area
vector is 0. Thus we have entering E A E0 Acos180 E0l2 and
leaving E A E0 Acos 0 E0l2 .
4. (a) From the diagram in the textbook, we see that the flux outward through the hemispherical
surface is the same as the flux inward through the circular surface base of the hemisphere. On
that surface all of the flux is perpendicular to the surface. Or, we say that on the circular base,
E A. Thus E EA r2E .
(b) E is perpendicular to the axis, then every field line would both enter through the hemispherical
surface and leave through the hemispherical surface, and so E 0 .
5. Use Gauss’s law to determine the enclosed charge.
EQencl
o Qencl Eo 1840 N m2 C 8.85 1012 C2 N m2 1.63108C
6. The net flux through each closed surface is determined by the net charge inside. Refer to the picture
in the textbook.
1 Q 3Q 0 2Q 0 ; 2 Q 2Q 3Q 0 0 ;
3 2Q 3Q 0 Q 0 ; 4 0 ; 5 2Q 0
7. (a) Use Gauss’s law to determine the electric flux.
E Qencl 1.0106 C 1.1105 N m2 C
o 8.851012 C2 N m2
(b) Since there is no charge enclosed by surface A2, E 0 .
8. The net flux is only dependent on the charge enclosed by the surface. Since both surfaces enclose
the same amount of charge, the flux through both surfaces is the same. Thus the ratio is 1: 1 .
9. The only contributions to the flux are from the faces perpendicular to the electric field. Over each of
these two surfaces, the magnitude of the field is constant, so the flux is just EA on each of these
two surfaces.
EA EA Qencl
0
right
E left lEright 2 lEleft 2
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Physics for Scientists & Engineers with Modern Physics, 4th Edition Instructor Solutions Manual
Qencl E Eright left l 20 410 N C 560 N C 25m 2 8.85 1012 C2 N m2 8.3 107 C
10. Because of the symmetry of the problem one sixth of the total flux will pass through each face.
Q Q1 1 encl encl
face 6 total 6 0
60
Notice that the side length of the cube did not enter into the calculation.
11. The charge density can be found from Eq. 22-4, Gauss’s law. The charge is the charge density times
the length of the rod.
Qencl l 0 7.3105 N m2 C 8.85 1012 C2 N m2 4.3 105 C m
0 0 l 0.15 m
12.
13. The electric field can be calculated by Eq. 21-4a, and that can be solved for the magnitude of the
charge.
E 2
Q Q Er2 6.25 102 N C 3.50 102 m
k r2 k 8.988 109 N m2 C2 8.52 1011C
This corresponds to about 5108 electrons. Since the field points toward the ball, the charge must
be negative. Thus Q 8.521011C .
14. The charge on the spherical conductor is uniformly distributed over the surface area of the sphere, so
Q . The field at the surface of the sphere is evaluated at r = R.
4 R2
Er R 1 Q 1 4 R2
40 R2 40 R2 0
15. The electric field due to a long thin wire is given in Example 22-6 as E 1 .
20 R
(a)
E 1 1 2 8.988 109 N m2 C2 2 7.2 106 C m 2.6 104 N C
20 R 40 R
5.0 m
The negative sign indicates the electric field is pointed towards the wire.
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40
Chapter 22 Gauss’s Law
(b)
E 1 1 2 8.988 109 N m2 C2 2 7.2 106 C m 8.6 104 N C
20 R 40 R
1.5 m
The negative sign indicates the electric field is pointed towards the wire.
16. Because the globe is a conductor, the net charge of -1.50 mC will
be arranged symmetrically around the sphere.
17. Due to the spherical symmetry of the problem, the electric field can be evaluated using Gauss’s law
and the charge enclosed by a spherical Gaussian surface of radius r.
EdA E 4 r2 Qencl E 1 Qencl
0 40 r2
Since the charge densities are constant, the charge enclosed is found by multiplying the appropriate
charge density times the volume of charge enclosed by the Gaussian sphere. Let r1 6.0 cm and
r2 12.0 cm.
(a) Negative charge is enclosed for r r1.
E
1 Qencl 1 4 r 3 r 5.0C m3 r
40 r2 40 3 30 8.85 1012 C2 N m2
r2 3
1.9 1011 N Cm r
(b) In the region r1 r r2, all of the negative charge and part of the positive charge is
enclosed.
E r4 3
1 Qencl 1 4 r3 r13 r13 r
40 r2 40 31 3 30r2 30
r2
5.0C m3 8.0C m3 0.060 m3 3 8.0C m3 r
3 8.85 1012 C2 8.85 1012 C2 N m2
N m2 r2
1.1108 Nm2 C 3.0 1011 N Cm r
r2
(c) In the region r2 r, all of the charge is enclosed.
E r4 3
1 Qencl 1 4 r23 r13 r13 r23
40 r2 40 31 3 30r2
r2
5.0C m3
8.0C m3 0.060 m3 8.0C m3 0.120m3 4.1108 Nm2 C
r2
3 8.85 1012 C2 N m2 r2
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Physics for Scientists & Engineers with Modern Physics, 4th Edition Instructor Solutions Manual
(d) See the adjacent plot. The field is Electric field (1010 N/C) 3.0
continuous at the edges of the layers. 2.0
The spreadsheet used for this problem 1.0
can be found on the Media Manager,
with filename “PSE4_ISM_CH22.XLS,”
on tab “Problem 22.17d.”
0.0
0 10 20 30 40 50
-1.0
-2.0 r (cm)
18. See Example 22-3 for a detailed discussion related to this problem.
(a) Inside a solid metal sphere the electric field is 0 .
(b) Inside a solid metal sphere the electric field is 0 .
(c) Outside a solid metal sphere the electric field is the same as if all the charge were concentrated
at the center as a point charge.
E
1 Q 8.988 109 N m2 C2 5.50 106C
r2
3.10 m2 5140 N C
40
The field would point towards the center of the sphere.
(d) Same reasoning as in part (c).
E
1 Q 8.988 109 N m2 C2 5.50 106C
40 r2
8.00 m2 772 N C
The field would point towards the center of the sphere.
(e) The answers would be no different for a thin metal shell.
(f) The solid sphere of charge is dealt with in Example 22-4. We see from that Example that the
field inside the sphere is given by E 1 Q r. Outside the sphere the field is no different.
r03
40
So we have these results for the solid sphere.
E r 0.250m 5.50 106C
8.988 109 N m2 C2 0.250 m 458 N C
3.00 m3
E r 2.90m 5.50 106C
8.988 109 N m2 C2 2.90 m 5310 N C
3.00 m3
E r 3.10m C2 5.50 106C
8.988 109 N m2 5140 N C
3.10 m2
E r 8.00m 5.50 106C
8.988 109 N m2 C2 772 N C
3.10 m2
All point towards the center of the sphere.
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42
Chapter 22 Gauss’s Law
19. For points inside the nonconducting Electric field (106 N/C) 4.0
3.0
spheres, the electric field will be
determined by the charge inside the
spherical surface of radius r.
4 r 3 r 3
3 r0
Qencl Q Q 2.0
r4 3
30
The electric field for r r0 can be 1.0
calculated from Gauss’s law. 0.0
0
E(r r0 ) Qencl 5 10 15 20 25 30
4 0 r 2
r (cm)
Q r 3 1 Q r
r0 4 0 r 2
4 r3
00
The electric field outside the sphere is calculated from Gauss’s law with Qencl Q.
E r r0 Qencl Q
4 0 r 2 4 0 r 2
The spreadsheet used for this problem can be found on the Media Manager, with filename
“PSE4_ISM_CH22.XLS,” on tab “Problem 22.19.”
20. (a) When close to the sheet, we approximate it as an infinite sheet, and use the result of Example
22-7. We assume the charge is over both surfaces of the aluminum.
275 109 C
0.25 m2 2.5 105 N C, away from the sheet
2o
8.85 1012 C2 N m2
E 2
(b) When far from the sheet, we approximate it as a point charge.
E 1 Q C2 275 109 C
40 r2 8.988 109 N m2 11 N C, away from the sheet
15 m2
21. (a) Consider a spherical gaussian surface at a radius of 3.00 cm. It encloses all of the charge.
EdA E Q
4 r2
0
1 Q 5.50 106C
40 r2 3.00 102 m
E C2 5.49 107
8.988 109 N m2 2 N C, radially outward
(b) A radius of 6.00 cm is inside the conducting material, and so the field must be 0. Note that
there must be an induced charge of 5.50 106C on the surface at r = 4.50 cm, and then an
induced charge of 5.50 106C on the outer surface of the sphere.
(c) Consider a spherical gaussian surface at a radius of 3.00 cm. It encloses all of the charge.
EdA E Q
4 r2
0
1 Q 5.50 106C
40 r2 30.0 102 m
E C2 5.49 105 N
8.988 109 N m2 2 C, radially outward
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Physics for Scientists & Engineers with Modern Physics, 4th Edition Instructor Solutions Manual
22. (a) Inside the shell, the field is that of the point charge, E 1 Q .
40 r2
(b) There is no field inside the conducting material: E 0 .
(c) Outside the shell, the field is that of the point charge, E 1 Q .
40 r2
(d) The shell does not affect the field due to Q alone, except in the shell material, where the field is
0. The charge Q does affect the shell – it polarizes it. There will be an induced charge of –Q
uniformly distributed over the inside surface of the shell, and an induced charge of +Q
uniformly distributed over the outside surface of the shell.
23. (a) There can be no field inside the conductor, and so there must be an induced charge of
8.00C on the surface of the spherical cavity.
(b) Any charge on the conducting material must reside on its boundaries. If the net charge of the
cube is 6.10C, and there is a charge of 8.00C on its inner surface, there must be a charge
of 1.90C on the outer surface.
24. Since the charges are of opposite sign, and since the charges are free to move since they are on
conductors, the charges will attract each other and move to the inside or facing edges of the plates.
There will be no charge on the outside edges of the plates. And there cannot be charge in the plates
themselves, since they are conductors. All of the charge must reside on surfaces. Due to the
symmetry of the problem, all field lines must be perpendicular to the plates, as discussed in Example
22-7.
(a) To find the field between the plates, we choose a gaussian cylinder,
perpendicular to the plates, with area A for the ends of the cylinder. We + –
place one end inside the left plate (where the field must be zero), and the +
other end between the plates. No flux passes through the curved surface
of the cylinder. Ebetween –
EdA EdA EdA
Qencl +–
EdA +–
ends side right 0
end
Ebetween A A E between
0 0
The field lines between the plates leave the inside surface of the left plate, and terminate on the
inside surface of the right plate. A similar derivation could have been done with the right end of
the cylinder inside of the right plate, and the left end of the cylinder in the space between the
plates.
(b) If we now put the cylinder from above so that the right end is
inside the conducting material, and the left end is to the left of + –
+ –
the left plate, the only possible location for flux is through the –
Eoutside –
left end of the cylinder. Note that there is NO charge enclosed
by the Gaussian cylinder.
EdA
EdA
Qencl +
EdA EdA
0
ends side left
end
+
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44
Chapter 22 Gauss’s Law
E Aoutside 0 Eoutside 0
0 0
(c) If the two plates were nonconductors, the results would not change. The charge would be
distributed over the two plates in a different fashion, and the field inside of the plates would not
be zero, but the charge in the empty regions of space would be the same as when the plates are
conductors.
25. Example 22-7 gives the electric field from a positively charged
plate as E / 20 with the field pointing away from the plate.
The fields from the two plates will add, as shown in the figure.
(a) Between the plates the fields are equal in magnitude, but
point in opposite directions.
Ebetween 0
2 0
2 0
(b) Outside the two plates the fields are equal in magnitude and
point in the same direction.
Eoutside
2 0 2 0 0
(c) When the plates are conducting the charge lies on the surface of the plates. For nonconducting
plates the same charge will be spread across the plate. This will not affect the electric field
between or outside the two plates. It will, however, allow for a non-zero field inside each plate.
26. Because 3.0 cm << 1.0 m, we can consider the plates to be infinite in size, and ignore any edge
effects. We use the result from Problem 24(a).
E Q A
Q EA0 160 N C 1.0 m2 8.85 1012 C2 N m2 1.4 109 C
0 0
27. (a) In the region 0 r r1, a gaussian surface would enclose no charge. Thus, due to the spherical
symmetry, we have the following.
EdA E 4 r2 Qencl 0 E 0
0
(b) In the region r1 r r2 , only the charge on the inner shell will be enclosed.
Qencl 14 r12 E 1r12
EdA E 4 r2 0 0 0r2
(c) In the region r2 r, the charge on both shells will be enclosed.
Qencl 14 r12 2 4 r22 E 1r12 r2
EdA E 4 r2
22
0 0 0r2
(d) To make E 0 for r2 r, we must have 1r12 r2 0. This implies that the shells are of
22
opposite charge.
(e) To make E 0 for r1 r r2 , we must have 1 0 . Or, if a charge Q 41r12 were placed
at the center of the shells, that would also make E 0.
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Physics for Scientists & Engineers with Modern Physics, 4th Edition Instructor Solutions Manual
28. If the radius is to increase from r0 to 2r0 linearly during an elapsed time of T, then the rate of
increase must be r0 T. The radius as a function of time is then r r0 r0 t r0 1 t . Since the
T T
balloon is spherical, the field outside the balloon will have the same form as the field due to a point
charge.
(a) Here is the field just outside the balloon surface.
E 1 Q 1 Q
40 r2 40
r02 1 t 2
T
(b) Since the balloon radius is always smaller than 3.2r0, the total charge enclosed in a gaussian
surface at r 3.2r0 does not change in time.
E 1 Q 1 Q
40 r2 40
3.2r0 2
29. Due to the spherical symmetry of the problem, Gauss’s law using a sphere of radius r leads to the
following.
EdA E Qencl Qencl
4 r2 0 E 40r2
(a) For the region 0 r r1, the enclosed charge is 0.
E Qencl 0
40r2
(b) For the region r1 r r0, the enclosed charge is the product of the volume charge density times
the volume of charged material enclosed. The charge density is given by Q
4 r03 4 r13
3 3
3Q .
r03 r13
4
3Q 4 r 3 4 r13
r03 3 3
EQencl Vencl 4 r3 4 r13 4 r13 Q r3 r13
40r2 40r2 3 3 40r2 r03 r13
40r2 40r2
(c) For the region r r0, the enclosed charge is the total charge, Q.
E Q
40r2
30. By the superposition principle for electric fields (Section 21-6), we find the field for this problem by
adding the field due to the point charge at the center to the field found in Problem 29. At any
location r 0, the field due to the point charge is E q .
40r2
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46
Chapter 22 Gauss’s Law
(a) E Eq EQ q 0 q
40r2 40r2
(b)E Eq EQ q Q r3 r13 1 r3 r13
40r2 40r2 r03 r13 40r2 q Q r03 r13
(c) E Eq EQ q Q qQ
40r2 40r2 40r2
31. (a) Create a gaussian surface that just encloses the inner surface of the spherical shell. Since the
electric field inside a conductor must be zero, Gauss’s law requires that the enclosed charge be
zero. The enclosed charge is the sum of the charge at the center and charge on the inner surface
of the conductor.
Qenc q Qinner 0
Therefore Qinner q .
(b) The total charge on the conductor is the sum of the charges on the inner and outer surfaces.
Q Qouter Qinner Qouter Q Qinner Q q
(c) A gaussian surface of radius r r1 only encloses the center charge, q. The electric field will
therefore be the field of the single charge.
E(r r1 ) q
4 0 r 2
(d) A gaussian surface of radius r1 r r0 is inside the conductor so E 0 .
(e) A gaussian surface of radius r r0 encloses the total charge q Q . The electric field will then
be the field from the sum of the two charges.
E(r r0 ) qQ
4 0 r 2
32. (a) For points inside the shell, the field will be due to the point charge only.
E r r0 q
4 0 r 2
(b) For points outside the shell, the field will be that of a point charge, equal to the total charge.
E r r0 qQ
4 0 r 2
(c) If q Q, we have E r r0 Q and E r r0 2Q .
4 0 r 2 4 0 r 2
(d) If q Q, we have E r r0 Q and E r r0 0 .
4 0 r 2
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Physics for Scientists & Engineers with Modern Physics, 4th Edition Instructor Solutions Manual
33. We follow the development of Example 22-6. Because of the ¬
symmetry, we expect the field to be directed radially outward (no R
fringing effects near the ends of the cylinder) and to depend only on
the perpendicular distance, R, from the symmetry axis of the shell. R0
Because of the cylindrical symmetry, the field will be the same at all
points on a gaussian surface that is a cylinder whose axis coincides
with the axis of the shell. The gaussian surface is of radius r and
length l. E is perpendicular to this surface at all points. In order to apply Gauss’s law, we need a
closed surface, so we include the flat ends of the cylinder. Since E is parallel to the flat ends, there
is no flux through the ends. There is only flux through the curved wall of the gaussian cylinder.
EdA E Qencl Aencl E Aencl
2 Rl 0 0 20Rl
(a) For R R0, the enclosed surface area of the shell is Aencl 2 R0l.
E Aencl 2 R0l R0 , radially outward
20Rl 20Rl 0R
(b) For R R0, the enclosed surface area of the shell is Aencl 0, and so E 0 .
(c) The field for R R0 due to the shell is the same as the field due to the long line of charge, if we
substitute 2 R0 .
34. The geometry of this problem is similar to Problem 33, and so ¬
R
we use the same development, following Example 22-6. See R0
the solution of Problem 33 for details.
EdA E Qencl VE encl E VE encl
2 Rl 0 0 20Rl
(a) For R R0, the enclosed volume of the shell is
Vencl R02l.
E VE encl ER02 , radially outward
20Rl 20R
(b) For R R0, the enclosed volume of the shell is Vencl R2l.
E VE encl ER , radially outward
20Rl 20
35. The geometry of this problem is similar to Problem 33, and so we use the same development,
following Example 22-6. See the solution of Problem 33 for details. We choose the gaussian
cylinder to be the same length as the cylindrical shells.
EdA E 2 Rl Qencl E Qencl
0 20Rl
(a) For 0 R R1, no charge is enclosed, and so E Qencl 0.
20Rl
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Chapter 22 Gauss’s Law
(b) For R1 R R2, charge Q is enclosed, and so E Q , radially outward .
20Rl
(c) For R R2, both charges of Q and Q are enclosed, and so E Qencl 0.
20Rl
(d) The force on an electron between the cylinders points in the direction opposite to the electric
field, and so the force is inward. The electric force produces the centripetal acceleration for the
electron to move in the circular orbit.
Fcentrip eE eQ m v2 K 1 mv2 eQ
20Rl R 2 40l
Note that this is independent of the actual value of the radius, as long as R1 R R2.
36. The geometry of this problem is similar to Problem 33, and so we use the same development,
following Example 22-6. See the solution of Problem 33 for details. We choose the gaussian
cylinder to be the same length as the cylindrical shells.
EdA E 2 Rl Qencl E Qencl
0 20Rl
(a) At a distance of R 3.0cm, no charge is enclosed, and so E Qencl 0 .
20Rl
(b) At a distance of R 7.0cm, the charge on the inner cylinder is enclosed.
E Qencl 2 Qencl 2 8.988 109 N m2 C2 0.88 106C 4.5 104 N C
20Rl 40 Rl 0.070m 5.0m
The negative sign indicates that the field points radially inward.
(c) At a distance of R 12.0cm, the charge on both cylinders is enclosed.
E Qencl 2 Qencl 2 8.988 109 N m2 C2 1.56 0.88 106C 2.0 104 N C
20Rl 40 Rl 0.120 m 5.0m
The field points radially outward.
37. (a) The final speed can be calculated from the work-energy theorem, where the work is the integral
of between shells.
the force Fondtrhee12lemctvr2on the two
W
1 mv02
2
Setting the force equal to the electric field times the charge on the electron, and inserting the
electric field from Problem 36 gives the work done on the electron.
W R2 qQ dR qQ ln R2
R1 2 0 lR 2 0 l R1
2 1.60 1019 C 0.88C ln 9.0 cm 1.65 1016 J
8.85 1012 C2 /Nm2 5.0 6.5 cm
m
Solve for the velocity from the work-energy theorem.
v
2W 2 1.65 1016 J 1.9 107 m/s
m 9.1 1031 kg
© 2009 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they
currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
49
Physics for Scientists & Engineers with Modern Physics, 4th Edition Instructor Solutions Manual
(b) The electric force on the proton provides its centripetal acceleration.
Fc mv 2 qE qQ
R 20l R
The velocity can be solved for from the centripetal acceleration.
v
2 1.60 1019 C 0.88C 5.0 m 5.5 105 m/s
8.85 1012 C2/Nm2 1.67 1027 kg
Note that as long as the proton is between the two cylinders, the velocity is independent of the
radius.
38. The geometry of this problem is similar to Problem 33, and
so we use the same development, following Example 22-6. ¬
R
See the solution of Problem 33 for details. R3 R2
R1
EdA E 2 Rl Qencl E Qencl
0 20Rl E
(a) For 0 R R1, the enclosed charge is the volume of E
charge enclosed, times the charge density.
E Qencl E R2l ER
20Rl 20Rl 20
(b) For R1 R R2, the enclosed charge is all of the charge on the inner cylinder.
E Qencl E R12l ER12
20Rl 20Rl 20R
(c) For R2 R R3, the enclosed charge is all of the charge on the inner cylinder, and the part of
the charge on the shell that is enclosed by the gaussian cylinder.
E Qencl E R12l E R2l R22l E R12 R2 R22
20Rl 20Rl 20R
(d) For R R3, the enclosed charge is all of the charge on both the inner cylinder and the shell.
E Qencl E R12l E R32l R22l E R12 R32 R22
20Rl 20Rl 20R
(e) See the graph. The spreadsheet used for E (104 N/C) 10.0
this problem can be found on the Media 8.0
Manager, with filename 6.0
“PSE4_ISM_CH22.XLS,” on tab
“Problem 22.38e.”
4.0
2.0
0.0
0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0
R (cm)
© 2009 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they
currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
50
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