Fundamentals of Physics

David Halliday, Robert Resnick , Jearl Walker

Chapter 23

Gauss' Law - all with Video Answers

Educators

+ 3 more educators

Chapter Questions

Problem 1

SSM The square surface shown
in Fig. $23-30$ measures 3.2 $\mathrm{mm}$ on
each side. It is immersed in a uniform clectric ficld with magnitude
$E=1800 \mathrm{N} / \mathrm{C}$ and with field lines at
an angle of $\theta=35^{\circ}$ with a normal to
the surface, as shown. Take that
normal to be directed "outward," as
though the surface were one face of
a box. Calculate the clectric flux
through the surface.

Supratim Pal

Supratim Pal

Numerade Educator

Problem 2

An electric field given by
$\vec{E}=4.0 \hat{\mathrm{i}}-3.0\left(y^{2}+2.0\right) \hat{\mathrm{j}}$ pierces $\mathrm{oy}$
Gaussian cube of edge length 2.0 $\mathrm{m}$
and positioned as shown in Fig. $23-7$ .
(The magnitude $E$ is in newtons per
coulomb and the position $x$ is in meters.) What is the electric flux through
the (a) top face, ( b ) bottom face, (c) left
face, and (d) back face? (e) What is the
net electric flux through the cube?

Ben Nicholson

Numerade Educator

Problem 3

The cube in Fig. $23-31$ has edge
length 1.40 $\mathrm{m}$ and is oriented as shown
in a region of uniform electric field. Find
the electric flux through the right face if
the electric field, in newtons per coulomb, is given by (a) 6.00 $\mathrm{i}$ , (b)
$-2.00 \hat{\mathrm{j}},$ and $(\mathrm{c})-3.00 \hat{\mathrm{i}}+4.00 \hat{\mathrm{k}}$ . (d) What is the total flux through the
cube for each field?

Katie Mcalpine

Numerade Educator

Problem 4

In Fig. $23-32,$ a butterfly net is
in a uniform electric field of magnitude $E=3.0 \mathrm{mN} / \mathrm{C}$ . The rim, a circle of radius $a=11 \mathrm{cm},$ is aligned
perpendicular to the field. The net
contains no net charge. Find the
electric flux through the netting.

Ben Nicholson

Numerade Educator

Problem 5

In Fig. $23-33,$ a proton is a distance $d / 2$ directly above the center of a square of side $d .$ What is the
magnitude of the electric flux through the square? (Hint: Think of the
square as one face of a cube with edge $d . )$

Katie Mcalpine

Numerade Educator

Problem 6

At each point on the surface of the cube shown in Fig. $23-31$
the electric field is parallel to the $z$ axis. The length of each edge
of the cube is 3.0 $\mathrm{m} .$ On the top face of the cube the field is
$\vec{E}=-34 \hat{k} \mathrm{N} / \mathrm{C},$ and on the bottom face it is $\vec{E}=+20 \hat{\mathrm{k}} \mathrm{N} / \mathrm{C}$ .
Determine the net charge contained within the cube.

Ben Nicholson

Numerade Educator

Problem 7

A particle of charge 1.8$\mu \mathrm{C}$ is at the center of a Gaussian cube
55 $\mathrm{cm}$ on edge. What is the net electric flux through the surface?

Katie Mcalpine

Numerade Educator

Problem 8

When a shower is turned on in a closed bathroom, the
splashing of the water on the bare tub can fill the room's air with
negatively charged ions and produce an electric field in the air as
great as 1000 $\mathrm{N} / \mathrm{C}$ . Consider a bathroom with dimensions 2.5 $\mathrm{m} \times$
3.0 $\mathrm{m} \times 2.0 \mathrm{m}$ . Along the ceiling, floor, and four walls, approximate
the electric field in the air as being directed perpendicular to the sur-
face and as having a uniform magnitude of 600 $\mathrm{N} / \mathrm{C}$ Also, treat those
surfaces as forming a closed Gaussian surface around the room's air.
What are (a) the volume charge density $\rho$ and (b) the number of
excess elementary charges e per cubic meter in the room's air?

Ben Nicholson

Numerade Educator

Problem 9

ILW Fig. $23-31$ shows a Gaussian surface in the shape of a cube
with edge length 1.40 $\mathrm{m} .$ What are (a) the net flux $\Phi$ through the
surface and (b) the net charge $q_{\mathrm{cnc}}$ enclosed by the surface if
$\vec{E}=(3.00 y \hat{\mathrm{j}}) \mathrm{N} / \mathrm{C},$ with $y$ in meters? What
are $(\mathrm{c}) \quad \Phi$ and $(\mathrm{d}) \quad q_{\mathrm{enc}}$ if $\vec{E}=[-4.00 \hat{\mathrm{i}}+$
$(6.00+3.00 y) j ] \mathrm{N} / \mathrm{C} ?$

Katie Mcalpine

Numerade Educator

Problem 10

Figure $23-34$ shows a closed Gaussian
surface in the shape of a cube of edge length
2.00 $\mathrm{m}$ . It lies in a region where the nonuni-
form electric field is given by $\vec{E}=(3.00 x+$
$4.00) \hat{\mathrm{i}}+6.00 \hat{\mathrm{j}}+7.00 \hat{\mathrm{k}} \mathrm{N} / \mathrm{C},$ with $x$ in meters.
What is the net charge contained by the cube?

Ben Nicholson

Numerade Educator

Problem 11

Figure $23-35$ shows a closed Gaussian surface in the shape of
a cube of edge length 2.00 $\mathrm{m}$ , with one corner at $x_{1}=5.00 \mathrm{m}, y_{1}=4.00$
$\mathrm{m} .$ The cube lies in a region where the electric field vector is given by
$\overline{E}=-3.00 \mathrm{i}-4.00 y^{2} \mathrm{j}+3.00 \hat{\mathrm{k}} \mathrm{N} / \mathrm{C},$ with $y$ in meters. What is the net
charge contained by the cube?

Katie Mcalpine

Numerade Educator

Problem 12

Figure $23-36$ shows two non-
conducting spherical shells fixed in
place. Shell 1 has uniform surface
charge density $+6.0 \mu \mathrm{C} / \mathrm{m}^{2}$ on its
outer surface and radius 3.0 $\mathrm{cm}$
shell 2 has uniform surface charge
density $+4.0 \mu \mathrm{C} / \mathrm{m}^{2}$ on its outer
surface and radius $2.0 \mathrm{cm} ;$ the shell
centers are separated by $L=10 \mathrm{cm} .$
In unit-vector notation, what is the
net electric field at $x=2.0 \mathrm{cm} ?$

Ben Nicholson

Numerade Educator

Problem 13

SSM The electric field in a certain region of Earth's atmosphere is directed vertically down. At an altitude of 300 $\mathrm{m}$ the field
has magnitude $60.0 \mathrm{N} / \mathrm{C} ;$ at an altitude of $200 \mathrm{m},$ the magnitude is
100 $\mathrm{N} / \mathrm{C}$ . Find the net amount of charge contained in a cube 100 $\mathrm{m}$
on edge, with horizontal faces at altitudes of 200 and 300 $\mathrm{m} .$

Katie Mcalpine

Numerade Educator

Problem 14

Flux and nonconducting shells. A charged particle is suspended at the center of two concentric spherical shells that are
very thin and made of nonconducting material. Figure $23-37 a$
shows a cross section. Figure $23-37 b$ gives the net flux $\Phi$ through a
Gaussian sphere centered on the particle, as a function of the radius $r$ of the sphere. The scale of the vertical axis is set by $\Phi_{s}=$
$5.0 \times 10^{5} \mathrm{N} \cdot \mathrm{m}^{2} / \mathrm{C}$ (a) What is the charge of the central particle?
What are the net charges of (b) shell $A$ and (c) shell $B$ ?

Keshav Singh

Numerade Educator

Problem 15

A particle of charge $+q$ is placed at one corner of a Gaussian
cube. What multiple of $q / \varepsilon_{0}$ gives the flux through (a) each cube face
forming that corner and (b) each of the other cube faces?

Katie Mcalpine

Numerade Educator

Problem 16

The box-like Gaussian surface shown in Fig. $23-38$ encloses a net charge of $+24.0 \varepsilon_{0} \mathrm{C}$ and lies in an electric field given
by $\vec{E}=[(10.0+2.00 x) \hat{\mathrm{i}}-3.00 \hat{\mathrm{j}}+b z \hat{\mathrm{k}}] \mathrm{N} / \mathrm{C},$ with $x$ and $z$ in meters and $b$ a constant. The bottom face is in the $x z$ plane; the top face
is in the horizontal plane passing through $y_{2}=1.00 \mathrm{m} .$ For $x_{1}=$
$1.00 \mathrm{m}, x_{2}=4.00 \mathrm{m}, z_{1}=1.00 \mathrm{m},$ and $z_{2}=3.00 \mathrm{m},$ what is $b$ ?

Ben Nicholson

Numerade Educator

Problem 17

SSM A uniformly charged conducting sphere of 1.2 $\mathrm{m}$ diameter has surface charge density 8.1$\mu \mathrm{C} / \mathrm{m}^{2} .$ Find (a) the net charge
on the sphere and (b) the total electric flux leaving the surface.

Katie Mcalpine

Numerade Educator

Problem 18

The electric field just above the surface of the charged con-
ducting drum of a photocopying machine has a magnitude $E$ of
$2.3 \times 10^{5} \mathrm{N} / \mathrm{C} .$ What is the surface charge density on the drum?

Ben Nicholson

Numerade Educator

Problem 19

Space vehicles traveling through Earth's radiation belts can
intercept a significant number of electrons. The resulting charge
buildup can damage electronic components and disrupt operations.
Suppose a spherical metal satellite 1.3 $\mathrm{m}$ in diameter accumulates
2.4$\mu \mathrm{C}$ of charge in one orbital revolution. (a) Find the resulting surface charge density. (b) Calculate the magnitude of the electric field
just outside the surface of the satellite, due to the surface charge.

Katie Mcalpine

Numerade Educator

Problem 20

Flux and conducting shells. A charged particle is held at the
center of two concentric conducting spherical shells. Figure $23-39 a$
shows a cross section. Figure $23-39 b$ gives the net flux $\Phi$ through a
Gaussian sphere centered on the particle, as a function of the radius $r$
of the sphere. The scale of the vertical axis is set by $\Phi_{s}=5.0 \times$
$10^{5} \mathrm{N} \cdot \mathrm{m}^{2} / \mathrm{C}$ What are (a) the charge of the central particle and the
net charges of (b) shell $A$ and $(\mathrm{c})$ shell $B$ ?

Ben Nicholson

Numerade Educator

Problem 21

An isolated conductor has net charge $+10 \times 10^{-6} \mathrm{C}$ and a cavity with a particle of charge $q=+3.0 \times 10^{-6} \mathrm{C}$ What is the charge on
(a) the cavity wall and (b) the outer surface?

Katie Mcalpine

Numerade Educator

Problem 22

An electron is released 9.0 $\mathrm{cm}$ from a very long nonconduct-
ing rod with a uniform 6.0$\mu \mathrm{C} / \mathrm{m} .$ What is the magnitude of the
electron's initial acceleration?

Ben Nicholson

Numerade Educator

Problem 23

(a) (a) The drum of a photocopying machine has a length of 42 $\mathrm{cm}$
and a diameter of 12 $\mathrm{cm}$ . The electric field just above the drum's sur-
face is $2.3 \times 10^{5} \mathrm{N} / \mathrm{C}$ . What is the total charge on the drum? (b) The
manufacturer wishes to produce a desktop version of the machine.
This requires reducing the drum length to 28 $\mathrm{cm}$ and the diameter to
8.0 $\mathrm{cm} .$ The electric field at the drum surface must not change. What
must be the charge on this new drum?

Katie Mcalpine

Numerade Educator

Problem 24

Figure $23-40$ shows a section of a
long, thin-walled metal tube of radius
$R=3.00 \mathrm{cm},$ with a charge per unit
length of $\lambda=2.00 \times 10^{-8} \mathrm{C} / \mathrm{m} .$ What
is the magnitude $E$ of the electric field
at radial distance (a) $r=R / 2.00$ and
(b) $r=2.00 R ?(\mathrm{c})$ Graph $E$ versus $r$
for the range $r=0$ to 2.00$R .$

Anand Jangid

Numerade Educator

Problem 25

SSM An infinite line of charge
produces a field of magnitude $4.5 \times$
$10^{4} \mathrm{N} / \mathrm{C}$ at distance 2.0 $\mathrm{m} .$ Find the
linear charge density.

Katie Mcalpine

Numerade Educator

Problem 26

Figure $23-41 a$ shows a narrow charged solid cylinder that is
coaxial with a larger charged cylindrical shell. Both are noncon-
ducting and thin and have uniform surface charge densities on
their outer surfaces. Figure $23-41 b$ gives the radial component $E$ of
the electric field versus radial distance $r$ from the common axis,
and $E_{x}=3.0 \times 10^{3} \mathrm{N} / \mathrm{C}$ . What is the shell's linear charge density?

Suzanne W.

Numerade Educator

Problem 27

A long, straight wire has fixed negative charge with a linear charge density of magnitude 3.6 $\mathrm{nC} / \mathrm{m}$ . The wire is to be enclosed by a coaxial, thin-walled nonconducting cylindrical shell of
radius 1.5 $\mathrm{cm}$ . The shell is to have positive charge on its outside surface with a surface charge density $\sigma$ that makes the net external
electric field zero. Calculate $\sigma$ .

Katie Mcalpine

Numerade Educator

Problem 28

A charge of uniform linear density 2.0 $\mathrm{nClm}$ is distributed
along a long, thin, nonconducting rod. The rod is coaxial with a long
conducting cylindrical shell (inner radius $=5.0 \mathrm{cm},$ outer radius $=$
10 $\mathrm{cm} ) .$ The net charge on the shell is zero. (a) What is the magnitude of the electric field 15 $\mathrm{cm}$ from the axis of the shell? What is
the surface charge density on the (b) inner and (c) outer surface of
the shell?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 29

SSM WWW Figure $23-42$ is
a section of a conducting rod of ra-
dius $R_{1}=1.30 \mathrm{mm}$ and length $L=$
11.00 $\mathrm{m}$ inside a thin-walled coax-
ial conducting cylindrical shell of
radius $R_{2}=10.0 R_{1}$ and the (same)
length $L$ . The net charge on the rod
is $Q_{1}=+3.40 \times 10^{-12} \mathrm{C} ;$ that on
the shell is $Q_{2}=-2.00 Q_{1} .$ What
are the (a) magnitude $E$ and (b) direction (radially inward or out-
ward) of the clectric field at radial
distance $r=2.00 R_{2} ?$ What are (c) $E$ and (d) the direction at $r=$
5.00$R_{1} ?$ What is the charge on the (c) interior and (f) exterior surface of the shell?

Katie Mcalpine

Numerade Educator

Problem 30

In Fig. $23-43,$ short sections of
two very long parallel lines of
charge are shown, fixed in place,
separated by $L=8.0 \mathrm{cm} .$ The uni-
form linear charge densities are
$+6.0 \mu \mathrm{C} / \mathrm{m}$ for line 1 and $-2.0$
$\mu \mathrm{C} / \mathrm{m}$ for line $2 .$ Where along the $x$
axis shown is the net electric field
from the two lines zero?

Ben Nicholson

Numerade Educator

Problem 31

ILW Two long, charged,
thin-walled, concentric cylindrical shells have radii of 3.0 and
6.0 $\mathrm{cm} .$ The charge per unit length is $5.0 \times 10^{-6} \mathrm{Clm}$ on the inner
shell and $-7.0 \times 10^{-6} \mathrm{Clm}$ on the outer shell. What are the (a)
magnitude $E$ and (b) direction (radially inward or outward) of the
electric field at radial distance $r=4.0 \mathrm{cm} ?$ What are $(\mathrm{c}) E$ and
(d) the direction at $r=8.0 \mathrm{cm} ?$

Katie Mcalpine

Numerade Educator

Problem 32

A long. nonconducting, solid cvlinder of radius 4.0 $\mathrm{cm}$ has a
nonuniform volume charge density $\rho$ that is a function of radial distance $r$ from the cylinder axis: $\rho=$
$A r^{2} .$ For $A=2.5 \mu \mathrm{C} / \mathrm{m}^{5},$ what is the
magnitude of the electric field at
(a) $r=3.0 \mathrm{cm}$ and $(\mathrm{b}) r=5.0 \mathrm{cm} ?$

Ben Nicholson

Numerade Educator

Problem 33

In Fig. $23-44,$ two large, thin
metal plates are parallel and close
to each other. On their inner faces,
the plates have excess surface charge densities of opposite signs and
magnitude $7.00 \times 10^{-22} \mathrm{Clm}^{2} .$ In unit-vector notation, what is the
electric field at points (a) to the left of the plates, (b) to the right of
them, and (c) between them?

Katie Mcalpine

Numerade Educator

Problem 34

In Fig. $23-45,$ a small circular hole of radius $R=1.80 \mathrm{cm}$ has
been cut in the middle of an infinite, flat, nonconducting surface
that has uniform charge density $\sigma=4.50 \mathrm{pC} / \mathrm{m}^{2} . \mathrm{A} z$ axis, with its its
origin at the hole's center, is perpendicular to the surface. In unit-
vector notation, what is the electric ficld at point $P$ at $z=2.56 \mathrm{cm}$ ?
(Hint: See Eq. $22-26$ and use superposition.)

Ben Nicholson

Numerade Educator

Problem 35

Figure $23-46 a$ shows three plastic sheets that are large, parallel, and uniformly charged. Figure $23-46 b$ gives the component of the
net electric field along an $x$ axis through the sheets. The scale of the
vertical axis is set by $E=6.0 \times 10^{5} \mathrm{N} / \mathrm{C}$ . What is the ratio of the charge
density on sheet 3 to that on sheet 2 ?

Katie Mcalpine

Numerade Educator

Problem 36

Figure $23-47$ shows cross sections through two large, parallel, non-
conducting sheets with identical distributions of positive charge with surface
charge density $\sigma=1.77 \times 10^{-22} \mathrm{C} / \mathrm{m}^{2}$
In unit-vector notation, what is $\vec{E}$ at
points (a) above the sheets, (b) between them, and (c) below them?

Ben Nicholson

Numerade Educator

Problem 37

SSM WWW A square metal plate of edge length 8.0 $\mathrm{cm}$ and
negligible thickness has a total charge of $6.0 \times 10^{-6} \mathrm{C}$ (a) Estimate
the magnitude $E$ of the electric field just off the center of the plate (at,
say, a distance of 0.50 $\mathrm{mm}$ from the center) by assuming that the
charge is spread uniformly over the two faces of the plate. (b)
Estimate $E$ at a distance of 30 $\mathrm{m}($ large relative to the plate size by assuming that the plate is a charged particle.

Katie Mcalpine

Numerade Educator

Problem 38

In Fig. $23-48 a,$ an electron is shot directly away from a uniformly charged plastic sheet, at speed $v_{s}=2.0 \times 10^{5} \mathrm{m} / \mathrm{s}$ .The sheet is
nonconducting flat, and very large. Figure $23-48 b$ gives the electron's
vertical velocity component $v$ versus time $t$ until the return to the
launch point. What is the sheet's surface charge density?

Ben Nicholson

Numerade Educator

Problem 39

SSM In Fig. $23-49,$ a small, nonconducting
ball of mass $m=1.0 \mathrm{mg}$ and charge $q=2.0 \mathrm{x}$
$10^{-8} \mathrm{C}($ distributed uniformly through its vol-
ume) hangs from an insulating thread that makes
an angle $\theta=30^{\circ}$ with a vertical, uniformly
charged nonconducting sheet (shown in cross sec- -
tion). Considering the gravitational force on the
ball and assuming the sheet extends farce on the
ball and and out of the page, calculate the surface
charge density $\sigma$ of the page, calculate the surface
charge density $\sigma$ of the sheet.

Katie Mcalpine

Numerade Educator

Problem 40

Figure $23-50$ shows a very large nonconducting sheet that has a uniform surface charge density
of $\sigma=-2.00 \mu \mathrm{C} / \mathrm{m}^{2}$ ; it also shows a particle of
charge $Q=6.00 \mu \mathrm{C},$ at distance $d$ from the sheet.
Both are fixed in place. If $d=0.200 \mathrm{m}$
at what (a) positive and (b) negative
coordinate on the $x$ axis (other than in-
finity) is the net electric field $\vec{E}_{\text { net }}$ of
the sheet and particle zero? (c) If $d=$
$0.800 \mathrm{m},$ at what coordinate on the $x$
axis is $\overline{E}_{\mathrm{net}}=0 ?$

Ben Nicholson

Numerade Educator

Problem 41

An electron is shot directly
toward the center of a large metal
plate that hat has surface charge density $-2.0 \times 10^{-6} \mathrm{C} / \mathrm{m}^{2} .$ If the initial
kinetic energy of the electron is $1.60 \times 10^{-17} \mathrm{J}$ and if the electron is to
stop (due to electrostatic repulsion from the plate) just as it reaches
the plate, how far from the plate must the launch point be?

Katie Mcalpine

Numerade Educator

Problem 42

Two large metal plates of area 1.0 $\mathrm{m}^{2}$ face each other, 5.0
cm apart, with equal charge magnitudes $|q|$ but opposite signs.
The field magnitude $E$ between them (neglect fringing) is 55 $\mathrm{N} / \mathrm{C}$ .
Find $|q|$

Ben Nicholson

Numerade Educator

Problem 43

Figure $23-51$ shows a cross section through a very large nonconducting
slab of thickness $d=9.40 \mathrm{mm}$ and uni-
form volume charge density $\rho=5.80$
$\mathrm{fC} / \mathrm{m}^{3} .$ The origin of an $x$ axis is at the
slab's center. What is the magnitude of
the slab's electric field at an $x$ coordinate of $(a) 0,(b) 2.00 \mathrm{mm},(\mathrm{c}) 4.70 \mathrm{mm},$
and $(\mathrm{d}) 26.0 \mathrm{mm}$ ?

Katie Mcalpine

Numerade Educator

Problem 44

Figure $23-52$ gives the magni-
tude of the electric ficld inside and
outside a sphere with a positive charge
distributed uniformly throughout its
volume. The scale of the vertical axis is
set by $E_{g}=5.0 \times 10^{7} \mathrm{N} / \mathrm{C}$ . What is the
charge on the sphere?

Ben Nicholson

Numerade Educator

Problem 45

Two charged concentric spherical shells have radii 10.0 $\mathrm{cm}$ and 15.0 $\mathrm{cm} .$ The charge on the inner
shell is $4.00 \times 10^{-8} \mathrm{C},$ and that on the outer shell is $2.00 \times 10^{-8} \mathrm{C}$ .
Find the electric field (a) at $r=12.0 \mathrm{cm}$ and $(\mathrm{b})$ at $r=20.0 \mathrm{cm} .$

Katie Mcalpine

Numerade Educator

Problem 46

Assume that a ball of charged particles has a uniformly
distributed negative charge density except for a narrow radial
tunnel through its center, from the surface on one side to the
surface on the opposite side. Also assume that we can position a
proton anywhere along the tunnel or outside the ball. Let $F_{R}$ be
the magnitude of the electrostatic force on the proton when it is
located at the ball's surface, at radius $R$ . As a multiple of $R,$ how
far from the surface is there a point where the force magnitude is
0.50$F_{R}$ if we move the proton (a) away from the ball and (b) into
the tunnel?

Ben Nicholson

Numerade Educator

Problem 47

An unknown charge sits on a conducting solid sphere of
radius 10 $\mathrm{cm} .$ If the electric field 15 $\mathrm{cm}$ from the center of the
sphere has the magnitude $3.0 \times 10^{3} \mathrm{N} / \mathrm{C}$ and is directed radially in-
ward, what is the net charge on the sphere?

Katie Mcalpine

Numerade Educator

Problem 48

A charged particle is held at the center of a spherical
shell. Figure $23-53$ gives the magnitude $E$ of the electric field ver-
sus radial distance $r .$ The scale of the vertical axis is set by $E_{s}=$
$10.0 \times 10^{7} \mathrm{N} / \mathrm{C}$ . Approximately, what is the net charge on the
shell?

Ben Nicholson

Numerade Educator

Problem 49

In Fig. $23-54,$ a solid sphere of
radius $a=2.00 \mathrm{cm}$ is concentric with a
spherical conducting shell of inner ra-
dius $b=2.00 a$ and outer radius $c=$
2.40$a .$ The sphere has a net uniform
charge $q_{1}=+5.00 \mathrm{fC}$ ; the shell has a
net charge $q_{2}=-q_{1} .$ What is the mag-
nitude of the electric field at radial
distances (a) $r=0,$ (b) $r=a / 2.00$ , (c)
$r=a,(\mathrm{d}) r=1.50 a,(\mathrm{e}) r=2.30 a,$ and
(f) $r=3.50 \mathrm{a}$ ? What is the net charge
on the (g) inner and (h) outer surface
of the shell?

Katie Mcalpine

Numerade Educator

Problem 50

Figure $23-55$ shows two nonconducting spherical shells fixed in
place on an $x$ axis. Shell 1 has uniform
surface charge density $+4.0 \mu \mathrm{C} / \mathrm{m}^{2}$
on its outer surface and radius 0.50
$\mathrm{cm},$ and shell 2 has uniform surface
charge density $-2.0 \mu \mathrm{C} / \mathrm{m}^{2}$ on its
outer surface and radius 2.0 $\mathrm{cm}$ ; the
centers are separated by $L=6.0 \mathrm{cm} .$
Other than at $x=\infty,$ where on the $x$
axis is the net electric field equal to
zero?

Anand Jangid

Numerade Educator

Problem 51

SSM WWW In Fig. $23-56,$ a
nonconducting spherical shell of in-
ner radius $a=2.00 \mathrm{cm}$ and outer ra-
dius $b=2.40 \mathrm{cm}$ has (within its thick-
ness) a positive volume charge
density $\rho=A / r,$ where $A$ is a constant
and $r$ is the distance from the center
of the shell. In addition, a small ball of
charge $q=45.0 \mathrm{fC}$ is located at that
center. What value should $A$ have if
the electric field in the shell $(a \leq r \leq$
b) is to be uniform?

Katie Mcalpine

Numerade Educator

Problem 52

Figure $23-57$ shows a spherical shell with uniform volume charge
density $\rho=1.84 \mathrm{nC} / \mathrm{m}^{3}$ , inner radius
$a=10.0 \mathrm{cm},$ and outer radius $b=$
2.00$a$ . What is the magnitude of the
electric field at radial distances (a) $r=$
$0 ;$ (b) $r=a / 2.00,$ (c) $r=a,$ (d) $r=$
$1.50 a,($ e) $r=b,$ and $(f) r=3.00 b ?$

Ben Nicholson

Numerade Educator

Problem 53

ILW The volume charge density of a solid nonconducting sphere
of radius $R=5.60 \mathrm{cm}$ varies with radial distance $r$ as given by $\rho=$
$\left(14.1 \mathrm{pC} / \mathrm{m}^{3}\right) r / R .$ (a) What is the sphere's total charge? What is the
field magnitude $E$ at $(\mathrm{b}) r=0,(\mathrm{c}) r=R / 2.00,$ and $(\mathrm{d}) r=R ?$ (e)
Graph $E$ versus $r .$

Katie Mcalpine

Numerade Educator

Problem 54

Figure $23-58$ shows, in cross
section, two solid spheres with uniformly distributed charge through-
out their volumes. Each has radius
R. Point $P$ lies on a line connecting
the centers of the spheres, at radial
distance $R / 2.00$ from the center of sphere $1 .$ If the net electric field
at point $P$ is zero, what is the ratio $q_{2} / q_{1}$ of the total charges?

Ben Nicholson

Numerade Educator

Problem 55

A charge distribution that is spherically symmetric but not
uniform radially produces an electric field of magnitude $E=K r^{4}$ ,
directed radially outward from the center of the sphere. Here $r$ is
the radial distance from that center, and $K$ is a constant. What is
the volume density $\rho$ of the charge distribution?

Katie Mcalpine

Numerade Educator

Problem 56

The electric field in a particular space is $\vec{E}=(x+2) \hat{\mathrm{i}} \mathrm{N} / \mathrm{C}$
with $x$ in meters Consider a cylindrical Gaussian surface of radius
20 $\mathrm{cm}$ that is coaxial with the $x$ axis. One end of the cylinder is at
$x=0 .$ (a) What is the magnitude of the electric flux through the
other end of the cylinder at $x=2.0 \mathrm{m} ?$ (b) What net charge is en-
closed within the cylinder?

Ben Nicholson

Numerade Educator

Problem 57

A thin-walled metal spherical shell has radius 25.0 $\mathrm{cm}$ and
charge $2.00 \times 10^{-7} \mathrm{C}$ Find $E$ for a point (a) inside the shell, (b)
just outside it, and $(\mathrm{c}) 3.00 \mathrm{m}$ from the center.

Katie Mcalpine

Numerade Educator

Problem 58

A uniform surface charge of density 8.0 $\mathrm{nC} / \mathrm{m}^{2}$ is distributed over
the entire $x y$ plane. What is the electric flux through a spherical
Gaussian surface centered on the origin and having a radius of 5.0 $\mathrm{cm} ?$

Ben Nicholson

Numerade Educator

Problem 59

Charge of uniform volume density $\rho=1.2 \mathrm{nClm}^{3}$ fills an infi-
nite slab between $x=-5.0 \mathrm{cm}$ and $x=+5.0 \mathrm{cm} .$ What is the mag-
nitude of the electric field at any point with the coordinate (a) $x=$
4.0 $\mathrm{cm}$ and $(\mathrm{b}) x=6.0 \mathrm{cm} ?$

Katie Mcalpine

Numerade Educator

Problem 60

The chocolate crumb mystery. Explosions ignited by
electrostatic discharges (sparks) constitute a serious danger in fa-
cilities handling grain or powder. Such an explosion occurred in
chocolate crumb powder at a biscuit factory in the 1970 s. Workers
usually emptied newly delivered sacks of the powder into a loading
bin, from which it was blown through electrically grounded plastic
pipes to a silo for storage. Somewhere along this route, two condi-
tions for an explosion were met: (1) The magnitude of an electric
field became $3.0 \times 10^{6} \mathrm{N} / \mathrm{C}$ or greater, so that electrical break-
down and thus sparking could occur. (2) The energy of a spark was
150 $\mathrm{mJ}$ or greater so that it could ignite the powder explosively. Let
us check for the first condition in the powder flow through the
plastic pipes.
Suppose a stream of negatively charged powder was blown
through a cylindrical pipe of radius $R=5.0 \mathrm{cm} .$ Assume that the
powder and its charge were spread uniformly through the pipe
with a volume charge density $\rho$ (a) Using Gauss' law, find an ex-
pression for the magnitude of the electric field $\vec{E}$ in the pipe as a
function of radial distance $r$ from the pipe center. (b) Does $E$ in-
crease or decrease with increasing $r ?$ (c) Is $\overline{E}$ directed radially in-
ward or outward? (d) For $\rho=1.1 \times 10^{-3} \mathrm{Clm}^{3}$ (a typical value at
the factory), find the maximum $E$ and determine where that maxi-
mum field occurs.(e) Could sparking occur, and if so, where? (The
story continues with Problem 70 in Chapter $24 . )$

Ben Nicholson

Numerade Educator

Problem 61

SSM A thin-walled metal spherical shell of radius $a$ has a charge
$q_{a}$ . Concentric with it is a thin-walled metal spherical shell of radius
$b>a$ and charge $q_{b}$ . Find the electric field at points a distance $r$ from
the common center, where (a) $r < a,($ b) $a < r < b,$ and $(c) r>b$
(d) Discuss the criterion you would use to determine how the charges
are distributed on the inner and outer surfaces of the shells.

Katie Mcalpine

Numerade Educator

Problem 62

A particle of charge $q=1.0 \times 10^{-7} \mathrm{C}$ is at the center of a
spherical cavity of radius 3.0 $\mathrm{cm}$ in a chunk of metal. Find the electric
field (a) 1.5 $\mathrm{cm}$ from the cavity center and (b) anyplace in the metal.

Ben Nicholson

Numerade Educator

Problem 63

A proton at speed $v=3.00 \times 10^{5} \mathrm{m} / \mathrm{s}$ orbits at radius $r=1.00 \mathrm{cm}$
outside a charged sphere. Find the sphere's charge.

Katie Mcalpine

Numerade Educator

Problem 64

Equation $23-11\left(E=\sigma / \varepsilon_{0}\right)$ gives the electric field at points near a
charged conducting surface. Apply this equation to a conducting
sphere of radius $r$ and charge $q$ and show the electric field outside
the sphere is the same as the field of a charged particle located at the
center of the sphere.

Ben Nicholson

Numerade Educator

Problem 65

Charge $Q$ is uniformly distributed in a sphere of radius $R$ . (a)
What fraction of the charge is contained within the radius
$r=R / 2.00 ?$ (b) What is the ratio of the electric field magnitude at
$r=R / 2.00$ to that on the surface of the sphere?

Katie Mcalpine

Numerade Educator

Problem 66

A charged particle causes an electric flux of $-750 \mathrm{N} \cdot \mathrm{m}^{2} / \mathrm{C}$ to
pass through a spherical Gaussian surface of 10.0 $\mathrm{cm}$ radius cen-
tered on the charge. (a) If the radius of the Gaussian surface were
doubled, how much flux would pass through the surface? (b) What
is the charge of the particle?

Ben Nicholson

Numerade Educator

Problem 67

SSM The electric field at point $P$ just outside the outer sur-
face of a hollow spherical conductor of inner radius 10 $\mathrm{cm}$ and
outer radius 20 $\mathrm{cm}$ has magnitude 450 $\mathrm{N} / \mathrm{C}$ and is directed outward.
When a particle of unknown charge $Q$ is introduced into the center
of the sphere, the electric field at $P$ is still directed outward but is
now 180 $\mathrm{NC}$ (a) What was the net charge enclosed by the outer
surface before $Q$ was introduced? (b) What is charge $Q$ ? After $Q$ is
introduced, what is the charge on the (c) inner and (d) outer sur-
face of the conductor?

Vishal Gupta

Numerade Educator

Problem 68

The net electric flux through each face of a die (singular of
dice) has a magnitude in units of $10^{3} \mathrm{N} \cdot \mathrm{m}^{2} / \mathrm{C}$ that is exactly equal
to the number of spots $N$ on the face $(1$ through 6$) .$ The flux is in-
ward for $N$ odd and outward for $N$ even. What is the net charge in-
side the die?

Ben Nicholson

Numerade Educator

Problem 69

Figure $23-59$ shows, in
cross section, three infinitely
large nonconducting sheets on
which charge is uniformly
spread. The surface charge
densities are $\sigma_{1}=+2.00$
$\mu \mathrm{C} / \mathrm{m}^{2}, \sigma_{2}=+4.00 \mu \mathrm{C} / \mathrm{m}^{2}$
and $\sigma_{3}=-5.00 \mu \mathrm{Cm}^{2},$ and
distance $L=1.50 \mathrm{cm}$ . In unit-
vector notation, what is the net
electric field at point $P ?$

Katie Mcalpine

Numerade Educator

Problem 70

Charge of uniform vol-
ume density $\rho=3.2 \mu \mathrm{C} / \mathrm{m}^{3}$ sills a nonconducting solid sphere of
radius 5.0 $\mathrm{cm} .$ What is the magnitude of the electric field (a) 3.5
$\mathrm{cm}$ and $(\mathrm{b}) 8.0 \mathrm{cm}$ from the sphere's center?

Ben Nicholson

Numerade Educator

Problem 71

A Gaussian surface in the form of a hemisphere of radius $R=$
5.68 cm lies in a uniform electric field of magnitude $E=2.50 \mathrm{N} / \mathrm{C}$ .
The surface encloses no net charge. At the (flat) base of the sur-
face, the field is perpendicular to the surface and directed into the
surface. What is the flux through (a) the base and (b) the curved
portion of the surface?

Katie Mcalpine

Numerade Educator

Problem 72

What net charge is enclosed by the Gaussian cube of
Problem 2$?$

Ben Nicholson

Numerade Educator

Problem 73

A nonconducting solid sphere has a uniform volume charge density $\rho$ Let $\vec{r}$ be the
vector from the center of the sphere to a gen-
eral point $P$ within the sphere. (a) Show that
the electric field at $P$ is given by $\overline{E}=\rho \vec{r} / 3 \varepsilon_{0}$
(Note that the result is independent of the ra-
dius of the sphere.) (b) A spherical cavity is
hollowed out of the sphere, as shown in Fig. $23-$
$60 .$ Using superposition concepts, show that
the electric field at all points within the cavity
is uniform and equal to $E=\rho \vec{a} / 3 \varepsilon_{0},$ where $\vec{a}$ is the position vector
from the center of the sphere to the center of the cavity.

Katie Mcalpine

Numerade Educator

Problem 74

A uniform charge density of 500 $\mathrm{nC} / \mathrm{m}^{3}$ is distributed throughout a spherical volume of radius 6.00 $\mathrm{cm} .$ Consider a cubical
Gaussian surface with its center at the center of the sphere. What is
the electric flux through this cubical surface if its edge length is
(a) 4.00 $\mathrm{cm}$ and $(\mathrm{b}) 14.0 \mathrm{cm} ?$

Ben Nicholson

Numerade Educator

Problem 75

Figure $23-61$ shows a Geiger counter, a device used to detect ionizing radiation, which causes ionization of atoms. A thin, positively charged central wire is surrounded by a concentric, circular, con- ducting cylindrical shell with an equal negative charge, creating a strong radial electric field. The shell contains a low-pressure inert gas. A particle of radiation entering the device through the shell wall ionizes a few of the gas atoms. The resulting free electrons (e) are drawn to the positive wire. However, the electric field is so intense that, between collisions with gas atoms, the free electrons gain energy sufficient to ionize these atoms also. More free electrons are thereby created, and the process is repeated until the electrons reach the wire. The resulting "avalanche" of electrons is collected by the wire, generating a signal that is used to record the passage of the original particle of radiation. Suppose that the radius of the central wire is $25 \mu \mathrm{m},$ the inner radius of the shell 1.4 $\mathrm{cm},$ and the length of the shell 16 $\mathrm{cm} .$ If the electric field at the
shell's inner wall is $2.9 \times 10^{4} \mathrm{N} / \mathrm{C}$ , what is the total positive charge on the central wire?

Katie Mcalpine

Numerade Educator

Problem 76

Charge is distributed uniformly throughout the volume of an in-
finitely long solid cylinder of radius $R$ . (a) Show that, at a distance $r<$
$R$ from the cylinder axis,
$$E=\frac{\rho r}{2 \varepsilon_{0}}$$
where $\rho$ is the volume charge density. (b) Write an expression for $E$
when $r>R .$

Ben Nicholson

Numerade Educator

Problem 77

SSM A spherical conducting shell has a charge of $-14 \mu C$ on
its outer surface and a charged particle in its hollow. If the net
charge on the shell is $-10 \mu C,$ what is the charge (a) on the inner
surface of the shell and (b) of the particle?

Katie Mcalpine

Numerade Educator

Problem 78

A charge of 6.00 $\mathrm{pC}$ is spread uniformly throughout the volume
of a sphere of radius $r=4.00 \mathrm{cm} .$ What is the magnitude of the electric
field at a radial distance of $(a) 6.00 \mathrm{cm}$ and $(\mathrm{b}) 3.00 \mathrm{cm} ?$

Ben Nicholson

Numerade Educator

Problem 79

Water in an irrigation ditch of width $w=3.22 \mathrm{m}$ and depth $d=$
1.04 $\mathrm{m}$ flows with a speed of 0.207 $\mathrm{m} / \mathrm{s}$ . The mass flux of the flowing
water through an imaginary surface is the product of the water's
density $(1000 \mathrm{kgh}$ and its volume flux through that surface. Find
the mass flux through the following imaginary surfaces: (a) a sur-
face of area $w d,$ entirely in the water, perpendicular to the flow;
(b) a surface with area $3 w d / 2,$ of which $w d$ is in the water, perpendi-
cular to the flow; (c) a surface of area $w d / 2$ , entirely in the water, per-
pendicular to the flow; (d) a surface of area $w d,$ half in the water and
half out, perpendicular to the flow; ( ) a surface of area $w d,$ entirely in
the water, with its normal $34.0^{\circ}$ from the direction of flow.

Katie Mcalpine

Numerade Educator

Problem 80

Charge of uniform surface density 8.00 $\mathrm{nC} / \mathrm{m}^{2}$ is distributed
over an entire $x y$ plane; charge of uniform surface density 3.00 $\mathrm{nClm}^{2}$
is distributed over the parallel plane defined by $z=2.00 \mathrm{m}$ .
Determine the magnitude of the electric field at any point having a $z$
coordinate of $(\mathrm{a}) 1.00 \mathrm{m}$ and $(\mathrm{b}) 3.00 \mathrm{m} .$

Ben Nicholson

Numerade Educator

Problem 81

A spherical ball of charged particles has a uniform charge
density. In terms of the ball's radius $R$ , at what radial distances
(a) inside and (b) outside the ball is the magnitude of the ball's
electric field equal to $\frac{1}{4}$ of the maximum magnitude of that field?

Katie Mcalpine

Numerade Educator