Fundamentals of Physics, Volume 2

David Halliday & Robert Resnick & Jearl Walker

Chapter 23

Gauss' Law - all with Video Answers

Educators

Chapter Questions

Problem 1

E ssm The square surface shown in Fig. 23.9 measures $3.2 \mathrm{~mm}$ on each side. It is immersed in a uniform electric field 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 electric flux through the surface.

Keshav Singh

Keshav Singh

Numerade Educator

Problem 2

M CALC An electric field given by $\vec{E}=4.0 \hat{\mathrm{i}}-3.0\left(y^2+2.0\right) \hat{\mathrm{j}}^{\hat{\mathrm{H}}}$ pierces a Gaussian cube of edge length $2.0 \mathrm{~m}$ and positioned as shown in Fig. 23.1.7. (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?

Keshav Singh

Numerade Educator

Problem 3

M The cube in Fig. 23.10 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) $6.00 \hat{\mathrm{i}}$, (b) $-2.00 \hat{\mathrm{j}}$, and (c) $-3.00 \hat{\mathrm{i}}+$ $4.00 \mathrm{k}$. (d) What is the total flux through the cube for each field?

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 4

E In Fig. 23.11, 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.

Raj Bala

Numerade Educator

Problem 5

E In Fig. 23.12, 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 $d$.)

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 6

Et each point on the surface of the cube shown in Fig. 23.10, 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 $\vec{E}=$ $-34 \hat{\mathrm{k}} \mathrm{N} / \mathrm{C}$, 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.

Raj Bala

Numerade Educator

Problem 7

EE 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

FCP 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 surface 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?

Keshav Singh

Numerade Educator

Problem 9

M Fig. 23.10 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 $q_{\text {enc }}$ enclosed by the surface if $\vec{E}=$ $\left(3.00 y^*\right) \mathrm{N} / \mathrm{C}$, with $y$ in meters? What are (c) $\Phi$ and (d) $q_{\mathrm{enc}}$ if $\left.\vec{E}=\left[-4.00 \hat{h}^4+(6.00+3.00 y)\right\}^*\right] \mathrm{N} / \mathrm{C}$ ?

Keshav Singh

Numerade Educator

Problem 10

$ \mathrm{M}$ Figure 23.13 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 nonuniform electric field is given by $\vec{E}=$ $(3.00 x+4.00) \mathrm{i}+6.00 \mathrm{j}+7.00 \mathrm{k} \mathrm{N} / \mathrm{C}$, with $x$ in meters. What is the net charge contained by the cube?

Keshav Singh

Numerade Educator

Problem 11

$ \mathbf{M}$ Figure 23.14 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 $\vec{E}=-3.00 \hat{\mathrm{i}}-4.00 \mathrm{y}^2 \hat{\mathrm{j}}+3.00 \hat{\mathrm{k}} \mathrm{N} / \mathrm{C}$, with $y$ in meters. What is the net charge contained by the cube?

Keshav Singh

Numerade Educator

Problem 12

Migure 23.15 shows two nonconducting 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 $x=2.0 \mathrm{~cm}$ ?

Raj Bala

Numerade Educator

Problem 13

$ M$ 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

M 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 $23.16 a$ shows a cross section. Figure $23.16 b$ 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 $B$ ?

Ben Nicholson

Numerade Educator

Problem 15

$ \mathrm{M}$ 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

H CALC (6) The box-like Gaussian surface shown in Fig. 23.17 encloses a net charge of $+24.0 \varepsilon_0 \mathrm{C}$ and lies in an electric field given by $\vec{E}=[(10.0+2.00 x) \hat{i}-3.00 \hat{j}+b z \hat{k}] \mathrm{N} / \mathrm{C}$, 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 $b$ ?

Keshav Singh

Numerade Educator

Problem 17

Esm 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.

Bettina Hanlon

Numerade Educator

Problem 18

The electric field just above the surface of the charged conducting 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?

Bettina Hanlon

Numerade Educator

Problem 19

E 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.

Raj Bala

Numerade Educator

Problem 20

E 6 Flux and conducting shells. A charged particle is held at the center of two concentric conducting spherical shells. Figure $23.18 a$ shows a cross section. Figure $23.18 b$ 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 (c) shell $B$ ?

Keshav Singh

Numerade Educator

Problem 21

$\mathrm{M}$ 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

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

Bettina Hanlon

Numerade Educator

Problem 23

E (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 surface 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

E Figure 23.19 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) $r=2.00 R$ ? (c) Graph $E$ versus $r$ for the range $r=0$ to $2.00 R$

Raj Bala

Numerade Educator

Problem 25

E 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.

Prashant Bana

Numerade Educator

Problem 26

$ \mathrm{M}$ Figure $23.20 a$ shows a narrow charged solid cylinder that is coaxial with a larger charged cylindrical shell. Both are nonconducting and thin and have uniform surface charge densities on their outer surfaces. Figure $23.20 b$ gives the radial component $E$ of the electric field versus radial distance $r$ from the common axis, and $E_s=3.0 \times 10^3 \mathrm{~N} / \mathrm{C}$. What is the shell's linear charge density?

Keshav Singh

Numerade Educator

Problem 27

$ \mathrm{M}$ 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

$ \mathrm{M}$ A charge of uniform linear density $2.0 \mathrm{nC} / \mathrm{m}$ 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?

Keshav Singh

Numerade Educator

Problem 29

M SSM Figure 23.21 is a section of a conducting rod of radius $R_1=1.30 \mathrm{~mm}$ and length $L=11.00 \mathrm{~m}$ inside a thin-walled coaxial 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 outward) of the electric 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 (e) interior and (f) exterior surface of the shell?

Keshav Singh

Numerade Educator

Problem 30

M In Fig. 23.22, short sections of two very long parallel lines of charge are shown, fixed in place, separated by $L=8.0 \mathrm{~cm}$. The uniform 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?

Bettina Hanlon

Numerade Educator

Problem 31

T Two long, charged, thin-walled, concentric cylindrical shells have radii of 3.0 and $6.0 \mathrm{~cm}$. The charge per unit length is $5.0 \times 10^{-6} \mathrm{C} / \mathrm{m}$ on the inner shell and $-7.0 \times 10^{-6} \mathrm{C} / \mathrm{m}$ 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 (c) $E$ and (d) the direction at $r=8.0 \mathrm{~cm} ?$

Keshav Singh

Numerade Educator

Problem 32

H CALC A long, nonconducting, solid cylinder 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 (b) $r=5.0 \mathrm{~cm}$ ?

Keshav Singh

Numerade Educator

Problem 33

E In Fig. 23.23, 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 $7.00 \times 10^{-22} \mathrm{C} / \mathrm{m}^2$. 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?

Keshav Singh

Numerade Educator

Problem 34

In Fig. 23.24, 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 $\sigma=4.50 \mathrm{pC} / \mathrm{m}^2$. A $z$ axis, with its origin at the hole's center, is perpendicular to the surface. In unit-vector notation, what is the electric field at point $P$ at $z=2.56 \mathrm{~cm}$ ? (Hint: See Eq. 22.5 .5 and use superposition.)

Keshav Singh

Numerade Educator

Problem 35

Figure $23.25 a$ shows three plastic sheets that are large, parallel, and uniformly charged. Figure $23.25 b$ gives the component of the net electric field along an $x$ axis through the sheets. The scale of the vertical axis is set by $E_x=6.0 \times 10^5 \mathrm{~N} / \mathrm{C}$. What is the ratio of the charge density on sheet 3 to that on sheet 2 ?

Katie Mcalpine

Numerade Educator

Problem 36

E Figure 23.26 shows cross sections through two large, parallel, nonconducting sheets with identical distributions of positive charge with surface charge density $\sigma=1.77 \times 10^{-22} \mathrm{C}^2 \mathrm{~m}^2$. In unit-vector notation, what is $\vec{E}$ at points (a) above the sheets, (b) between them, and (c) below them?

Raj Bala

Numerade Educator

Problem 37

E SSM 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.

Keshav Singh

Numerade Educator

Problem 38

M (60 In Fig. 23.27a, 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 $23.27 b$ 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?

Keshav Singh

Numerade Educator

Problem 39

M SSM In Fig. 23.28, a small, nonconducting ball of mass $m=1.0 \mathrm{mg}$ and charge $q=2.0 \times 10^{-8} \mathrm{C}$ (that is distributed uniformly through its volume) hangs from an insulating thread that makes an angle $\theta=30^{\circ}$ with a vertical, uniformly charged nonconducting sheet (shown in cross section). Considering the gravitational force on the ball and assuming the sheet extends far vertically and into and out of the page, calculate the surface charge density $\sigma$ of the sheet.

Keshav Singh

Numerade Educator

Problem 40

M Figure 23.29 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 infinity) 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 $\vec{E}_{\mathrm{net}}=0$ ?

Keshav Singh

Numerade Educator

Problem 41

$ \mathrm{M}$ An electron is shot directly toward the center of a large metal plate that 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?

Keshav Singh

Numerade Educator

Problem 42

M Two large metal plates of area $1.0 \mathrm{~m}^2$ face each other, $5.0 \mathrm{~cm}$ 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

H 6 Figure 23.30 shows a cross section through a very large nonconducting slab of thickness $d=9.40 \mathrm{~mm}$ and uniform 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) $2.00 \mathrm{~mm}$, (c) $4.70 \mathrm{~mm}$, and (d) $26.0 \mathrm{~mm} ?$

Keshav Singh

Numerade Educator

Problem 44

Eigure 23.31 gives the magnitude of the electric field inside and outside a sphere with a positive charge distributed uniformly throughout its volume. The scale of the vertical axis is set by $E_s=$ $5.0 \times 10^7 \mathrm{~N} / \mathrm{C}$. What is the charge on the sphere?

Keshav Singh

Numerade Educator

Problem 45

E 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 (b) at $r=20.0 \mathrm{~cm}$.

Raj Bala

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

E SSM 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 inward, what is the net charge on the sphere?

Keshav Singh

Numerade Educator

Problem 48

$ \mathrm{M}$ A positively charged particle is held at the center of a spherical shell. Figure 23.32 gives the magnitude $E$ of the electric field versus 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?

Keshav Singh

Numerade Educator

Problem 49

M In Fig. 23.33, a solid sphere of radius $a=2.00 \mathrm{~cm}$ is concentric with a spherical conducting shell of inner radius $b=2.00 a$ 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 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) $r=2.30 a$, and (f) $r=3.50 a$ ? What is the net charge on the ( $\mathrm{g}$ ) inner and (h) outer surface of the shell?

Morgan Cheatham

Morgan Cheatham

Numerade Educator

Problem 50

М (c) Figure 23.34 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?

Raj Bala

Numerade Educator

Problem 51

M CALC SSM In Fig. 23.35, a nonconducting spherical shell of inner radius $a=2.00 \mathrm{~cm}$ and outer radius $b=2.40 \mathrm{~cm}$ has (within its thickness) 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?

Keshav Singh

Numerade Educator

Problem 52

$ \mathrm{M}$ (co) Figure 23.36 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) $r=3.00 b$ ?

Raj Bala

Numerade Educator

Problem 53

H CALC 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 (b) $r=0$, (c) $r=R / 2.00$, and (d) $r=R$ ? (e) Graph $E$ versus $r$.

Keshav Singh

Numerade Educator

Problem 54

H Figure 23.37 shows, in cross section, two solid spheres with uniformly distributed charge throughout 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?

Raj Bala

Numerade Educator

Problem 55

H CALC 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 $\vec{E}=(x+2) \hat{i} \mathrm{~N} / \mathrm{C}$, 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 enclosed within the cylinder?

Vishal Gupta

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 (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 $\rho=1.2 \mathrm{nC} / \mathrm{m}^3$ fills an infinite slab between $x=-5.0 \mathrm{~cm}$ and $x=+5.0 \mathrm{~cm}$. What is the magnitude of the electric field at any point with the coordinate (a) $x=4.0 \mathrm{~cm}$ and (b) $x=6.0 \mathrm{~cm}$ ?

Bettina Hanlon

Numerade Educator

Problem 60

FCP The chocolate crumb mystery. Explosions ignited by electrostatic discharges (sparks) constitute a serious danger in facilities handling grain or powder. Such an explosion occurred in chocolate crumb powder at a biscuit factory in the $1970 \mathrm{~s}$. 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 conditions 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 breakdown 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 expression 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 $E$ increase or decrease with increasing $r$ ? (c) Is $\vec{E}$ directed radially inward or outward? (d) For $\rho=1.1 \times 10^{-3} \mathrm{C} / \mathrm{m}^3$ (a typical value at the factory), find the maximum $E$ and determine where that maximum field occurs. (e) Could sparking occur, and if so, where? (The story continues with Problem 70 in Chapter 24.)

Keshav Singh

Numerade Educator

Problem 61

SSM A thin-walled metal spherical shell of radius $a$ has a charge $q_\alpha$ - Concentric with it is a thin-walled metal spherical shell of radius $b>a$ and charge $q_t$. 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.

Keshav Singh

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 $23.3 .1\left(E=\sigma / \varepsilon_0\right)$ 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 that 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 centered 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?

Keshav Singh

Numerade Educator

Problem 67

SSM The electric field at point $P$ just outside the outer surface 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 $180 \mathrm{~N} / \mathrm{C}$. (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 surface of the conductor?

Keshav Singh

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 inward for $N$ odd and outward for $N$ even. What is the net charge inside the die?

Ben Nicholson

Numerade Educator

Problem 69

Figure 23.38 shows, in cross section, three infinitely large nonconducting sheets on which charge is uniformly spread. The surface charge densities are $\sigma_1=+2.00 \mu \mathrm{Cm}^2, \sigma_2=$ $+4.00 \mu \mathrm{C}^2 \mathrm{~m}^2$, and $\sigma_3=$ $-5.00 \mu \mathrm{C} / \mathrm{m}^2$, 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 volume density $\rho=3.2 \mu \mathrm{C} / \mathrm{m}^3$ fills a nonconducting solid sphere of radius $5.0 \mathrm{~cm}$. What is the magnitude of the electric field (a) $3.5 \mathrm{~cm}$ and (b) $8.0 \mathrm{~cm}$ from the sphere's center?

Bettina Hanlon

Numerade Educator

Problem 71

A Gaussian surface in the form of a hemisphere of radius $R=5.68 \mathrm{~cm}$ 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 surface, 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 general point $P$ within the sphere. (a) Show that the electric field at $P$ is given by $\vec{E}=\rho \vec{r} / 3 e_{\text {is }}$. (Note that the result is independent of the radius of the sphere.)
(b) A spherical cavity is hollowed out of the sphere, as shown in Fig. 23.39. Using superposition concepts, show that the electric field at all points within the cavity is uniform and equal to $\vec{E}=\rho \vec{a} / 3 e_0$, where $\vec{a}$ is the position vector from the center of the sphere to the center of the cavity.

Keshav Singh

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 (b) $14.0 \mathrm{~cm}$ ?

Ben Nicholson

Numerade Educator

Problem 75

Figure 23.40 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, conducting 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?

Keshav Singh

Numerade Educator

Problem 76

Hydrogen atom model. In an early model of the hydrogen atom, that atom was considered as having a central pointlike proton of positive charge $+e$ and an electron of negative charge $-e$ that is distributed about the proton according to the volume charge density $\rho=A \exp \left(-2 r / a_0\right)$. Here $A$ is a constant, $a_0=0.53 \times 10^{-10} \mathrm{~m}$ is the Bohr radius, and $r$ is the distance from the center of the atom. (a) Using the fact that hydrogen is electrically neutral, find $A$. (b) Then find the electric field produced by the atom at the Bohr radius.

Suzanne W.

Numerade Educator

Problem 77

Rutherford atomic model. In 1911, Ernest Rutherford $\operatorname{sent} \alpha$ particles through atoms to determine the makeup of the atoms. He suggested: "In order to form some idea of the forces required to deflect an $\alpha$ particle through a large angle, consider an atom [as] containing a point positive charge $\mathrm{Ze}$ at its centre and surrounded by a distribution of negative electricity $-Z e$ uniformly distributed within a sphere of radius $R$. The electric field $E \ldots$ at a distance $r$ from the centre for a point inside the atom [is]
$$
E=\frac{Z e}{4 \pi \varepsilon_j}\left(\frac{1}{r^2}-\frac{r}{R^3}\right) . "
$$

Verify this equation for his model.

Keshav Singh

Numerade Educator

Problem 78

Airborne COVID-19 drops in electric fields. One of the major concerns with the COVID-19 pandemic is the transmission of the virus due to the water drops that are projected by sneezing, coughing, singing, talking, or even breathing. The larger drops soon settle out due to the gravitational force but drops with radii smaller than $5 \mu \mathrm{m}$ might remain suspended by air currents, posing a danger to anyone breathing them. However, common electric fields might remove some of them. Suppose a water drop with radius $r=2.0 \mu \mathrm{m}$ and charge $\left(-2.5 \times 10^4\right) e$ is near a flat plastic surface with surface charge density of $+7.0 \mathrm{nC} / \mathrm{m}^2$, as can commonly occur in homes. What are the magnitudes of (a) the Coulomb force and (b) the gravitational force that act on the drop?

Kai Chen

Princeton University

Problem 79

Particle in a shell. A particle with charge $+q$ is placed at the center of an electronically neutral, spherical conducting shell with inner radius $a$ and outer radius $b$. What charge appears on (a) the inner surface of the shell and (b) the outer surface? What is the net electric field magnitude at distance $r$ from the center of the shell if (c) $r\langle a$, (d) $b>r>a$, and (e) $r>b$ ? A particle with charge $-q$ is now placed outside the shell. Does the presence of the second particle alter the charge distribution on (f) the outer surface and (g) the inner surface? Is there an electrostatic force (h) on the second particle and (i) on the first particle?

Eduard Sanchez

Numerade Educator