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c field at
3R
2R
tion 3.
c
d
ion 4.
(+)
+
+
–
–
(–)
tion 5.
rods
extendsurfaces
into andofout
of the
page
(we see only their cross secGaussian
radii
R, 2R,
and
tions). The value below each cross section gives that particular
3R, all with the same center.The unirod’s uniform charge density in microcoulombs per meter.
The
R
form
charges
on
the
three
objects
rods are separated by either d or 2d as drawn, and a central point
ball,midway
Q; smaller
shell,the
3Q;
larger
: Rank the situations
isare:
shown
between
inner
rods.
acA
! (2î " 3ĵ) m2. What
1
A
surface
has
the
area
vector
is the
shell,
5Q.
Rank
the
Gaussian
surcording to the magnitude of the net electric
field
at
that
central
Gaussian
2R is
flux of
a uniform
electric
field of
through
the area if the field
faces
according
to the
magnitude
point,
first.
surface
:greatest
:
E ! 4k̂
(a) E
(b) point
! 4î N/C
the
electric
fieldand
at any
on N/C?
the
Fig. 23-21 Question 3.
surface,
greatest
2 Figure
23-20 first.
shows, in cross section, three solid cylinders, each of
(a)
length
L and
uniform
charge
Concentric
4 Figure
23-22
shows,
in
sec+3
+2 crossQ.
–2
–3with each cylinder is a
cylindrical
Gaussianspheres
surface,and
with
all three surfaces having the same
tion,
two Gaussian
two
radius.
thethat
Gaussian
surfaces
Gaussian
cubes
are centered
on according to the electric field at
(b)Rank
point on the
surface,
greatest first.
+2
–4
+2
aanypositively
charged
particle.
(a) –4
+q
Rank the net flux through the four
(c ) surfaces, greatest first. (b)
Gaussian
+8
–2
+2
+8 a
c
RankCylinder
the magnitudes of the electric
b
fields on the surfaces, greatest first,
d
(d )
and indicate
whether
the
magni–6
+5
+5
–6
Gaussian
tudessurface
are uniformFig.
or variable
along
23-24 Question 7.
each surface.
Fig. 23-22 Question
4.
(a)
(b)
(c)
5 In Fig. 23-23, an electron is reFig. 23-20
Question
2. with charge
between
infinite
8leased
Figure
23-25 two
shows
four nonconsolid spheres,
each
σ(+)
ducting
sheets
that are horizontal
Q
uniformly
distributed
through itsand
volume. (a) Rank the spheres
+ + + + + + +
according
to23-21
their
volume
charge
density,
have
uniform
surface
charge
densities
3 Figure
shows,
in cross
sec- greatest first. The figure 3R
also
shows
point
P ball,
for each
sphere, allShell
at the same distance
s
s($)a, as
indicated.The
tion,
a central
metal
twoelectron
spheri(") and
e
from
the
center
of
the
sphere.
(b)
Rank
the
spheres
according to
is
subjected
to
the
following
three
situcal metal shells, and three spherical
the
magnitude
of surface
the electric
field
they produce at point P, greatations
involving
charge
densiGaussian
surfaces
of radii
R, 2R,
and
–
–
–
–
–
–
–
est first.
ties
andwith
sheet
Rank unithe
3R, all
theseparations.
same center.The
σ(–)
R
magnitudes
of
the
electron’s
acceleraform charges on the three objects
tion,greatest
are: ball, Q; first.
smaller shell, 3Q; larger Fig. 23-23 Question 5.
P
P
P
P
shell, 5Q. Rank the Gaussian sur- Gaussian
2R
faces according to the
magnitude ofs($)
surface
Situation
s(")
Separation
the electric field at any point on the
3.
1 greatest first."4s
$4sFig. 23-21 Question
d
surface,
2
"7sin cross sec- $s
4d
4 Figure
23-22(b)
shows,
(a)
(c)
(d)
"3s and two$5s
9d
tion,3two Gaussian spheres
Fig.
Question
8.
Gaussian cubes that
are23-25
centered
on
a positively charged particle. (a)
the net
flux through
four the hollow of a+qmetallic
9Rank
A small
charged
ball liesthe
within
Gaussianshell
surfaces,
greatest
spherical
of radius
R. Forfirst.
three(b)
situations, the net charges on
a
the
ballthe
andmagnitudes
shell, respectively,
are (1) "4q, 0; (2) $6q, "10q; (3) c
Rank
of the electric
(c)
(d)
6 Three
face char
the two sh
if the elec
8E !Figure
0 in o
Q uniform
7 Figur
according
rods exte
also show
tions). Th
from the c
rod’s uni
the magni
rods are s
est first.
is shown
cording t
point, gre
P
(a)
(b)
(a)
(c)
9 A sma
spherical
(d)s
the ball an
"16q, $1
the inner
tive first.
10
Rank
8 Figur
of
elec
Qthe
uniform
2R
from th
according
also show
from the
the magn
est first.
** View All Solutions H
b
•7 A point charge of 1.8 mC is at the center of a Gaussian cube 55
cle
of radius
aaFlux
!
cm,
is
WWW
Worked-out
solution
is
HALLIDAY
REVISED
sec.
23-3
of11
an Electric
Field
cle
radius
!at11
cm,
is aligned
aligned
the the
surface
and i
cm on edge. What is the net electric flux through
surface? An
••21
Takeofthat
normal
to
be
directed
http://www.wiley.com/college/halliday
•1 SSM The square surface
a
Normal
a
ILW
Interactive
solution
is
at
perpendicular
to
the
field.
–Φ
••8
When a shower is turned on in a closed bathroom, the
perpendicular
to23-26
the
field.
Thenet
net
shown in Fig.
measures
3.2 The
s
treat room’s
those
surfa
“outward,” as though the surface
with
splashing of the water on the bare tub can fill the ity
air witha p
the
θ
23-26
Problem
1. produce an the
negatively
charged ions and
electricroom’s
field in the air
as W
contains
no
charge.
Find the Fig.
air.
a uniform
electric
with magniwere one
facenet
of afield
box.
Calculate
the cavity
(a)tude
(b)great as 1000 N/C. Consider a bathroom with dimensions
2.5 m $ w
E ! 1800 N/C and with field
electric
flux
through
netting.
the
number
of ex
3.0 m $ 2.0 m. Along the ceiling, floor, and
four walls,
approximate
electric
flux
the
netting.
linesflux
atthrough
an angle
of u the
35°the
with
the A
electric
through
•7
point
charge
of !1.8
mCasurface.
is at
the
center
of
a
Gaussian
cube
55
the
electric
field
in
the
air
as
being
directed
perpendicular
to
normal
to the surface,
as shown.
:
Fig.
Fig.
23-33
Problem
14.2the"surface
room’s
Fig
and as
having a uniform
magnitude
ofsec.
600air?
N/C.23-7
Also,
Take23-29,
that normal
toproton
be directedby
••2
An
electric
field
given
pierces
a
!
4.0î
#
3.0(y
2.0)ĵ
E
•5
In
Fig.
a
is
a
discm
What
is the
net electric
flux through the
PA
RT
3
•5 onInedge.
Fig.
23-29,
a proton
is a distreatsurface?
those surfaces as forming a closed Gaussian
surface
around
“outward,” as though the surface
Fig.
23-26
Problempositioned
1.
the room’s as
air. What
are (a) the
charge density
(b) 23
Gaussian
cube
ofofedge
length
2.0
m and
shown
in volume ••9
•22r and
An
e
one face
a box. Calculate
ILW
Fig.
tance
d/2
directly
above
the
center
tance
d/2were
directly
above
the
center
the number
of excess elementary
charges e per cubic meter in the
••8
When
a
shower
is
turned
on
in
a
closed
bathroom,
the
the electric
flux through
the surface.
••15
A particle
of
charge
%q
is
placed
at per
oneroom’s
corner
ofand
a Gaussian
Fig.
(The
magnitude
E
is#is
in
newtons
coulomb
the 623
cube ••12
with
edge
air?
ing
rod
wil
••2 An
electric
field given
by
3.0(y
" 2.0)ĵ pierces a
E ! 4.0î is
of
aa23-5.
square
of
side
d.d.
What
the
Figu
of
square
of
side
What
the
••12
Fig
splashing
of
the
water
on
the
bare
tub
can
fill
the
room’s
air
with
Fig.
23-28
Problem
4.
Fig.
23-28
Problem
4.
Gaussian
cube
of
edge
length
2.0
m
and
positioned
as
shown
in
ILW
••9
Fig.
23-27
shows
a
Gaussian
surface
in
the
shape
of
a
cube.
What
multiple
of q/'
theelectric
flux through
(a) eachthe
cube
position
xFig.is23-5.
in(The
meters.)
What
is the
flux
through
(a) facethe
surface
and i
0 gives
electron’s
magnitude
E is in newtonsflux
per
coulomb and the
cube
within
edge
lengthair
1.40 as
m. What are (a)
flux
% through
magnitude
of
the
electric
magnitude
of
the
electric
flux
conductin
:the netconducting
negatively
charged
ions
and
produce
an
electric
field
the
position
x is in meters.)
What
is the
electric
fluxthe
through
the
(a) cube
the surface
and face?
(b) the net(e)
charge q enclosed
by the surfaceN/C
if
forming
and
(b)
each
of
other
faces?
top face,that
(b)corner
bottom
face,
(c)
left
face,
and
(d)
back
E ! (3.00yĵ)
top face,
(b) bottom
face,
(c) left
and (d) back with
face? (e)dimensions
through
the
square?
(Hint:
with m
y in $
meters? What:
are (c) %
and (d) q(a)
if T
E ! (3.00yĵ) N/C,2.5
place.
She
through
the
square?
(Hint:
great
as 1000
N/C.
Consider
a face,
bathroom
place.
She
•23
What
is theand
net
electric
fluxflux
through
the cube?
What is the
net
electric
through
the
cube?
•20
Flux
conducting
shells.
A
charged
particle
is
held
at
the
E!
[#4.00î
"
(6.00
"
3.00y)ĵ]
N/C?
E
!
[#4.00î
"
(
+
+ approximate
•••16
The
box-like
Gaussian
surface
in Fig. 23-34 enThink
of
the
square
as
one
3.0
m$
2.0
m.
Along
the
ceiling,
floor,
and fourshown
walls,
Think
of
the
square
as23-27
one
charge
de
••3
The
cube in Fig.
has
and
a
diam
••10 Figure 23-30 shows a closed Gaussian surface charge
in
the shape
of ade
z
center
two
conducting
spherical
shells.
Figure
23-35a
edge
lengthconcentric
1.40
and is oriented
z
cube
edge length 2.00
m. to
It lies
in a region
••3 electric
The
cube
ininmedge
Fig.
23-27
••10 outer
Figure
23-3
z ofelectric
closes
aof
net
charge
ofof %24.0'
C anddirected
lies
in an
field
given
the
field
the
air
as 0has
being
perpendicular
face
of
cube
with
d.)
surf
face
of
a
cube
with
edge
d.)
outer
sur#
face
is
2.3
as
shown
in
a
region
uniform
where
the
nonuniform
electric
field
is given by
d/2
d/2
:
shows
a
cross
section.
Figure
23-35b
gives
the
net
flux
!
through
a
edge
length
1.40
m
and
is
oriented
electric
field.
Find
the
electric
flux
E
!
(3.00x
"
4.00)î
"
6.00ĵ
"
7.00k̂
N/C,
cube
of
edge
leng
the
surface
andthe%
as
having
a&uniform
600 N/C.
by E
with
x andAlso,
z in me" [(10.0
2.00x)î
3.00ĵ %magnitude
bzk̂] N/C,of
shell
shell 22 has
ha
manufactu
throughpoint
right face
if thethe
electric
with x in meters. What is the net charge con•6
At
each
on
•6
At
each
point
on
the
as
shown
in
a
region
of
uniform
Gaussian
sphere
centered
on
the
particle,
as
a
function
of
the
radius
r
y
where
the
nonuni
field,
in
newtons
per
coulomb,
is
tained
by
the
cube?
treat
those
surfaces
as
forming
a
closed
Gaussian
surface
around
dd face:
ters and bgiven
a constant.
The bottom face is in the xz plane; the top
density
"
density
"
This
requir
by (a)
6.00î, (b)
# 2.00ĵ, and
5!
x
surface
of
the
cube
shown
surface
of
the
cube
shown
••11set Figure
23-31"shows
a#
closed
y
electric
field.
Find
the
electric
flux
E
(3.00x
"
4
of
the
sphere.
The
scale
of
the
vertical
axis
is
by
!
5.0
10
the
room’s
air.
What
are
(a)
the
volume
charge
density
r
and
(b)
(c)
(d)
What
is
the
#
3.00î
"
4.00k̂.
s
is in the total
horizontal
plane
passing through yGaussian
For
x1 of"edge surface
surface inm.
the shape
of a cube
Fig.
23-30
2 " 1.00
an
surface
an
8.0
cm.
The
flux
through
the
cube
for
each
2 23-27,
in
the
electric
through
the
right
face
if the
in
Fig.
the
electric
length 2.00
m, with one
corner
x ! 5.00net
m,
with
xProblem
in meters.
ded per
10.
the
number
elementary
charges
cubic
meter
in
theat the
x
N
$Fig.
m
/C.x23-27,
What
are
(a)
the electric
charge
of the
central
particle
and
field?of excess
1.00
m,
what
y ! 4.00
m.Theis
cubeb?
lies in a region where the
2 " 4.00 m, z1 " 1.00 m, and z2 " 3.00 m,
centers
are
centers
ar
must
be
the
field,
in
newtons
per
coulomb,
is
field
parallel
to
the
z(c)
field isisair?
parallel
toA
the
z shell B?
tained
the
cube
electric field vector is given by E ! #3.00î
# 4.00y ĵ by
" 3.00k̂
N/C,
sec.
23-4shell
Gauss’
Law
room’s
charges
of
(b)
and
Fig.23-29
23-29
Problem
5.charge contained by the
23-27 Fig.
Problems
3,
with y in Problem
meters.What
is the 5.
net
cube?
In
unit-vec
In 6.00î,
Fig. 23-28, a(b)
butterfly
net is Fig.
In
unit-ve
givenThe
by •4in(a)
and
#
2.00ĵ,
y
axis.
length
of
each
axis.
The
length
of
each
6, and 9.
•24
Figur
a
uniform
electric
field
of
magni••11
Figure
y
••9 ILW tude
Fig.E !23-27
shows
a
Gaussian
surface
in
the
shape
of
a
z
net
electric
net
electri
3.0 mN/C. (d)
The rim,
a cir- is the
(c)
What
#
3.00î
"
4.00k̂.
edge
of
m.
On
the
top
face
the
edgewith
ofthe
the
cube
is3.0
3.0
m.
theare
top
faceof
ofthe
thecube
cube
thefield
fieldisisGaussian
of asurface
long, t
Φ(a)
cle
ofcube
radius
a !is
11
cm,
aligned
cube
edge
length
1.40
m.
What
% through
sa the net flux::
y 2On
Bis The
:
:
total
flux
through
cube
for
each
perpendicular
tothe
the field.
net
y
and
onthethe
the bottom
bottom
face itit
face
EEsurface
SSM
E!!surface
# 34k̂
N/C,
!"20k̂
" 20k̂
N/C.length••13
SSM
E
#34k̂
N/C,
!
••13
radius
Rwi"
2.00 ym,
contains
no(b)
netand
charge.
Find
the
and
the on
net
charge
qencx enclosed
byisisthe
if N/C.
field?
:
electric flux through
the netting.
x
A
thenet
net
charge
contained
within
the
cube.
Determine
the
charge
contained
within
the
sphere
m.The
sphere
dcd
0 are (c)
per
unitisisle
with
y in meters?
What
%2cube.
and
(d) Problem
qenc if11. xx r y1 ! 4.00
!(3.00yĵ)
N/C,
EDetermine
Fig.x23-31
•5 In Fig. 23-29, a proton is a dis:
d/2 directly
center
electric
fieldWha
vect
sec.
23-4tanceGauss’
Law
! [#4.00î
"
(6.00above
"z 1the
3.00y)ĵ]
N/C?
C/m.
E
of a square of side d. What
is the
••12 zFigure
23-32 shows two nony
1
Fig. 23-28 Problem 4.
23-27
Problems
3,fixed in withShell
ythe
in meters.W
magnitude
of a
thebutterfly
electric flux net is
conducting
spherical shells
•4 InFigure
Fig.
23-28,
–Fig.
electri
Φs surface
••10
23-30
shows
in
the
of
a
1 Shell
through
the square?
(Hint: a closed Gaussian
place.
Shell 1shape
has uniform
surface
2
6,
and
9.
Think of
the square asfield
one
in a uniform
electric
of magni-+a region
charge density "6.0 mC/m on its
(a) r "x R/
z 2 edge
z
cube
of edge
2.00
z 2outer surface
face oflength
a cube with
d.) m. It lies ind/2
and radius 3.0 cm;
tude E
!•6nonuniform
3.0
mN/C. The
rim,field
a cirshell 2 has uniform surface charge
–2 Φsby
(c) Graph
At each point onelectric
the
where
the
is given
d
density "4.0 mC/m on its outer
z
x
x
:
L
surface of
the
cube
shown
1 is aligned
2
cle
of
radius
a
!
11
cm,
surface and radius 2.0 cm; the shell
r
" 0 to 2.0
E ! (3.00x
"(a)
4.00)î
" 6.00ĵ "
7.00k̂ N/C,
in Fig.
23-27,
the electric
d
(b)
a by L ! 10 cm.
centers are
separated
field is parallel
to thefield.
z
perpendicular
to
the
The
net
Fig.
23-29
Problem
5.
In unit-vector notation, what is the
with x in meters.
What
is the23-34
net chargeProblem
conFig.
16.
axis. The length
of each
Fig. 23-32•25
Problem
12.
SSM
net electric field at x ! 2.0 cm?y
23-35
Problem
20.
contains
no
Find
the
edge
of net
the cubecharge.
is 3.0Fig.
m. On the
top face
of the cube
the field is
tained by the
cube?
E ! # 34k̂ N/C, and on the bottom face it is E ! " 20k̂ N/C.
••13 SSM The electric field in a certain region of Earth’s atmocharge pro
electric flux
through
thecontained
netting.
Determine
the net charge
within the cube.
sphere is directed vertically down. At an altitude of 300 m the field
sec.
23-6
A
Charged
Isolated
Conductor
••11
Figure 23-31 shows a closed x
Fig. 4.5
23-3
tude
#
•5
In
Fig.
23-29,
a
proton
is
a
dis&6
SSM
•17 Ansurface
A uniformly
conducting
sphere
of 1.2Cm
diamGaussian
in the
shapecharged
of ahas
cube
of edge
••21
isolated
conductor
net
charge %10
# 23-30
10
and
a cav- m. Find the
Fig.
**
View
All
Solutions
Here
**
tance
d/2
directly
above
the
center
2
&6
length
2.00
m,surface
with
one
corner
at x1 !
m,C.mC/m
eter
has
charge
of 8.1
. is(a)
the on
net(a)
ity
with
a apoint
charge
q " density
%3.0
#5.00
10
What
theFind
charge
Problem
10.
of
a
square
of
side
d.
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•36 Figure
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Problem
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••31
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