charged particles on an x axis: Ϫq ϭ
Ϫ3.20 ϫ 10
Ϫ19
C at x ϭ Ϫ3.00 m and
q ϭ 3.20 ϫ 10
Ϫ19
C at x ϭ ϩ3.00 m.
What are the (a) magnitude and
(b) direction (relative to the positive
direction of the x axis) of the net electric field produced at point
P at y ϭ 4.00 m?
653
PROB LE M S
•2 In Fig. 22-34 the electric field
lines on the left have twice the separation of those on the right. (a) If the
magnitude of the field at A is 40 N/C,
what is the magnitude of the force
on a proton at A?(b) What is the
magnitude of the field at B?
Module 22-2 The Electric Field Due to a Charged Particle
•3
The nucleus of a plutonium-239 atom contains 94 protons.
Assume that the nucleus is a sphere with radius 6.64 fm and with
the charge of the protons uniformly spread through the sphere. At
the surface of the nucleus, what are the (a) magnitude and (b) direction (radially inward or outward) of the electric field produced by
the protons?
•4 Two charged particles are attached to an x axis: Particle 1 of
charge Ϫ2.00 ϫ 10
Ϫ7
C is at position x ϭ 6.00 cm and particle 2 of
charge ϩ2.00 ϫ 10
Ϫ7
C is at position x ϭ 21.0 cm. Midway between
the particles, what is their net electric field in unit-vector notation?
•5
A charged particle produces an electric field with a magnitude of 2.0 N/C at a point that is 50 cm away from the particle.
What is the magnitude of the particle’s charge?
•6 What is the magnitude of a point charge that would create an
electric field of 1.00 N/C at points 1.00
m away?
••7
In Fig. 22-35, the
four particles form a square of edge
length a ϭ 5.00 cm and have charges
q 1 ϭ ϩ10.0 nC, q 2 ϭ Ϫ20.0 nC, q 3 ϭ
ϩ20.0 nC, and q 4 ϭ Ϫ10.0 nC. In unitvector notation, what net electric field
do the particles produce at the square’s
center?
••8
In Fig. 22-36, the four particles are fixed in place and have charges
q 1 ϭ q 2 ϭ ϩ5e, q 3 ϭ ϩ3e, and q 4 ϭ
Ϫ12e. Distance d ϭ 5.0 mm. What is the
magnitude of the net electric field at
point P due to the particles?
••9
Figure 22-37 shows two
WWW
ILW
SSM
SSM
SSM
A
B
Figure 22-34 Problem 2.
Figure 22-35 Problem 7.
a
a
q 4
q 3
q 1
q 2
x
y
••11
Two charged particles are fixed to an x axis: Particle 1
SSM
••10
Figure 22-38a shows two charged particles fixed in place
on an x axis with separation L. The ratio q 1 /q 2 of their charge magnitudes is 4.00. Figure 22-38b shows the x component E net,x of their
net electric field along the x axis just to the right of particle 2. The x
axis scale is set by x s ϭ 30.0 cm. (a) At what value of x Ͼ 0 is E net,x
maximum? (b) If particle 2 has charge Ϫq 2 ϭ Ϫ3e, what is the
value of that maximum?
Figure 22-36 Problem 8.
x
y
P
–q
q
Figure 22-37 Problem 9.
of charge q 1 ϭ 2.1 ϫ 10
Ϫ8
C is at position x ϭ 20 cm and particle 2 of
charge q 2 ϭ Ϫ4.00q 1 is at position x ϭ 70 cm. At what coordinate on
the axis (other than at infinity) is the net electric field produced by
the two particles equal to zero?
E
net,x (10
–8
N/C)
–2
–4
0
2
(b)
(a)
0
x (cm)
y
x
+q 1
–q 2
L
x s
Figure 22-38 Problem 10.
q 2
d
P d
d
d
q 3
q 4
q 1
p
p
e
p
e
y
x
1
θ
2
θ
3
θ
4
θ
Figure 22-39 Problem 12.
e s
e s
e c
p
R
R
z
Figure 22-40 Problem 13.
y
L
x
q 1
q 2
Figure 22-41 Problem 14.
moved to z ϭ R/10.0. What then are the magnitudes of
(c)
at the proton’s location? (e) From (a) and (c)
we see that as the proton gets nearer to the disk, the magnitude of
E c
: and (d) E
:
s,net
increases, as expected. Why does the
magnitude of
from the two side
electrons decrease, as we see from
(b) and (d)?
••14 In Fig. 22-41, particle 1 of charge
q 1 ϭ Ϫ5.00q and particle 2 of charge q 2
ϭ ϩ2.00q are fixed to an x axis. (a) As a
multiple of distance L, at what coordiE
:
s,net
E
:
c
nate on the axis is the net electric field of the particles zero? (b)
Sketch the net electric field lines between and around the particles.
••12
Figure 22-39 shows an uneven arrangement of electrons (e)
and protons (p) on a circular arc of
radius r ϭ 2.00 cm, with angles
u 1 ϭ 30.0Њ, u 2 ϭ 50.0Њ, u 3 ϭ 30.0Њ, and
u 4 ϭ 20.0Њ. What are the (a) magnitude and (b) direction (relative to the
positive direction of the x axis) of the
net electric field produced at the center of the arc?
••13
Figure 22-40 shows a proton
(p) on the central axis through a disk
with a uniform charge density due to
excess electrons. The disk is seen from
an edge-on view. Three of those electrons are shown: electron e c at the
disk center and electrons e s at opposite sides of the disk, at radius R from
the center. The proton is initially at
distance z ϭ R ϭ 2.00 cm from the disk. At that location, what are
the magnitudes of (a) the electric field due to electron e c and (b)
the net electric field
due to electrons e s ? The proton is then
E
:
s,net
E
:
c
Ϫ3.20 ϫ 10
Ϫ19
C at x ϭ Ϫ3.00 m and
q ϭ 3.20 ϫ 10
Ϫ19
C at x ϭ ϩ3.00 m.
What are the (a) magnitude and
(b) direction (relative to the positive
direction of the x axis) of the net electric field produced at point
P at y ϭ 4.00 m?
653
PROB LE M S
•2 In Fig. 22-34 the electric field
lines on the left have twice the separation of those on the right. (a) If the
magnitude of the field at A is 40 N/C,
what is the magnitude of the force
on a proton at A?(b) What is the
magnitude of the field at B?
Module 22-2 The Electric Field Due to a Charged Particle
•3
The nucleus of a plutonium-239 atom contains 94 protons.
Assume that the nucleus is a sphere with radius 6.64 fm and with
the charge of the protons uniformly spread through the sphere. At
the surface of the nucleus, what are the (a) magnitude and (b) direction (radially inward or outward) of the electric field produced by
the protons?
•4 Two charged particles are attached to an x axis: Particle 1 of
charge Ϫ2.00 ϫ 10
Ϫ7
C is at position x ϭ 6.00 cm and particle 2 of
charge ϩ2.00 ϫ 10
Ϫ7
C is at position x ϭ 21.0 cm. Midway between
the particles, what is their net electric field in unit-vector notation?
•5
A charged particle produces an electric field with a magnitude of 2.0 N/C at a point that is 50 cm away from the particle.
What is the magnitude of the particle’s charge?
•6 What is the magnitude of a point charge that would create an
electric field of 1.00 N/C at points 1.00
m away?
••7
In Fig. 22-35, the
four particles form a square of edge
length a ϭ 5.00 cm and have charges
q 1 ϭ ϩ10.0 nC, q 2 ϭ Ϫ20.0 nC, q 3 ϭ
ϩ20.0 nC, and q 4 ϭ Ϫ10.0 nC. In unitvector notation, what net electric field
do the particles produce at the square’s
center?
••8
In Fig. 22-36, the four particles are fixed in place and have charges
q 1 ϭ q 2 ϭ ϩ5e, q 3 ϭ ϩ3e, and q 4 ϭ
Ϫ12e. Distance d ϭ 5.0 mm. What is the
magnitude of the net electric field at
point P due to the particles?
••9
Figure 22-37 shows two
WWW
ILW
SSM
SSM
SSM
A
B
Figure 22-34 Problem 2.
Figure 22-35 Problem 7.
a
a
q 4
q 3
q 1
q 2
x
y
••11
Two charged particles are fixed to an x axis: Particle 1
SSM
••10
Figure 22-38a shows two charged particles fixed in place
on an x axis with separation L. The ratio q 1 /q 2 of their charge magnitudes is 4.00. Figure 22-38b shows the x component E net,x of their
net electric field along the x axis just to the right of particle 2. The x
axis scale is set by x s ϭ 30.0 cm. (a) At what value of x Ͼ 0 is E net,x
maximum? (b) If particle 2 has charge Ϫq 2 ϭ Ϫ3e, what is the
value of that maximum?
Figure 22-36 Problem 8.
x
y
P
–q
q
Figure 22-37 Problem 9.
of charge q 1 ϭ 2.1 ϫ 10
Ϫ8
C is at position x ϭ 20 cm and particle 2 of
charge q 2 ϭ Ϫ4.00q 1 is at position x ϭ 70 cm. At what coordinate on
the axis (other than at infinity) is the net electric field produced by
the two particles equal to zero?
E
net,x (10
–8
N/C)
–2
–4
0
2
(b)
(a)
0
x (cm)
y
x
+q 1
–q 2
L
x s
Figure 22-38 Problem 10.
q 2
d
P d
d
d
q 3
q 4
q 1
p
p
e
p
e
y
x
1
θ
2
θ
3
θ
4
θ
Figure 22-39 Problem 12.
e s
e s
e c
p
R
R
z
Figure 22-40 Problem 13.
y
L
x
q 1
q 2
Figure 22-41 Problem 14.
moved to z ϭ R/10.0. What then are the magnitudes of
(c)
at the proton’s location? (e) From (a) and (c)
we see that as the proton gets nearer to the disk, the magnitude of
E c
: and (d) E
:
s,net
increases, as expected. Why does the
magnitude of
from the two side
electrons decrease, as we see from
(b) and (d)?
••14 In Fig. 22-41, particle 1 of charge
q 1 ϭ Ϫ5.00q and particle 2 of charge q 2
ϭ ϩ2.00q are fixed to an x axis. (a) As a
multiple of distance L, at what coordiE
:
s,net
E
:
c
nate on the axis is the net electric field of the particles zero? (b)
Sketch the net electric field lines between and around the particles.
••12
Figure 22-39 shows an uneven arrangement of electrons (e)
and protons (p) on a circular arc of
radius r ϭ 2.00 cm, with angles
u 1 ϭ 30.0Њ, u 2 ϭ 50.0Њ, u 3 ϭ 30.0Њ, and
u 4 ϭ 20.0Њ. What are the (a) magnitude and (b) direction (relative to the
positive direction of the x axis) of the
net electric field produced at the center of the arc?
••13
Figure 22-40 shows a proton
(p) on the central axis through a disk
with a uniform charge density due to
excess electrons. The disk is seen from
an edge-on view. Three of those electrons are shown: electron e c at the
disk center and electrons e s at opposite sides of the disk, at radius R from
the center. The proton is initially at
distance z ϭ R ϭ 2.00 cm from the disk. At that location, what are
the magnitudes of (a) the electric field due to electron e c and (b)
the net electric field
due to electrons e s ? The proton is then
E
:
s,net
E
:
c