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University Physics with Modern Physics In SI Units

Hugh D Young; Roger A Freedman

Chapter 22

Gauss's Law - all with Video Answers

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Chapter Questions

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Problem 1

A flat sheet of paper of area $0.320 \mathrm{~m}^{2}$ is oriented so that the normal to the sheet is at an angle of $64^{\circ}$ to a uniform electric field of magnitude $12 \mathrm{~N} / \mathrm{C}$.
(a) Find the magnitude of the electric flux through the sheet. (b) Does the answer to part (a) depend on the shape of the sheet? Why or why not? (c) For what angle $\phi$ between the normal to the sheet and the electric field is the magnitude of the flux through the sheet
(i) largest and (ii) smallest? Explain your answers.

Gregory Devenport
Gregory Devenport
Numerade Educator
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Problem 2

A flat sheet is in the shape of a rectangle with sides of lengths $0.400 \mathrm{~m}$ and $0.600 \mathrm{~m}$. The sheet is immersed in a uniform electric field of magnitude $76.0 \mathrm{~N} / \mathrm{C}$ that is directed at $20^{\circ}$ from the plane of the sheet (Fig. E22.2). Find the magnitude of the electric flux through the sheet.

Gregory Devenport
Gregory Devenport
Numerade Educator
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Problem 3

You measure an electric field of $1.37 \times 10^{6} \mathrm{~N} / \mathrm{C}$ at a distance of $0.165 \mathrm{~m}$ from a point charge. There is no other source of electric field in the region other than this point charge. (a) What is the electric flux through the surface of a sphere that has this charge at its center and that has radius $0.165 \mathrm{~m} ?$ (b) What is the magnitude of this charge?

Gregory Devenport
Gregory Devenport
Numerade Educator
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Problem 4

It was shown in Example 21.10 (Section 21.5 ) that the electric field due to an infinite line of charge is perpendicular to the line and has magnitude $E=\lambda / 2 \pi \epsilon_{0} r .$ Consider an imaginary cylinder with radius $r=0.185 \mathrm{~m}$ and length $l=0.500 \mathrm{~m}$ that has an infinite line of positive charge running along its axis. The charge per unit length on the line is $\lambda=4.25 \mu \mathrm{C} / \mathrm{m} .$ (a) What is the electric flux through the cylinder due to this infinite line of charge? (b) What is the flux through the cylinder if its radius is increased to $r=0.575 \mathrm{~m} ?$ (c) What is the flux through the cylinder if its length is increased to $l=0.905 \mathrm{~m} ?$

Gregory Devenport
Gregory Devenport
Numerade Educator
01:46

Problem 5

A uniform electric field makes an angle of $60.0^{\circ}$ with a flat surface. The area of the surface is $6.66 \times 10^{-4} \mathrm{~m}^{2}$. The resulting electric flux through the surface is $4.44 \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}$. Calculate the magnitude of the electric field.

Keshav Singh
Keshav Singh
Numerade Educator
06:30

Problem 6

The cube in Fig. E22.6 has sides of length $L=10.0 \mathrm{~cm}$. The electric field is uniform, has magnitude $E=4.00 \times 10^{3} \mathrm{~N} / \mathrm{C},$ and is parallel to the $x y$ -plane at an angle of $53.1^{\circ}$ measured from the $+x$ -axis toward the $+y$ -axis. (a) What is the electric flux through each of the six cube faces $S_{1}, S_{2}, S_{3}, S_{4}, S_{5},$ and $S_{6} ?(\mathrm{~b})$ What is the total electric flux through all faces of the cube?

Youssef Eweis
Youssef Eweis
Numerade Educator
02:30

Problem 7

A charge of $87.6 \mathrm{pC}$ is uniformly distributed on the surface of a thin sheet of insulating material that has a total area of $29.2 \mathrm{~cm}^{2}$. A Gaussian surface encloses a portion of the sheet of charge. If the flux through the Gaussian surface is $5.00 \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}$, what area of the sheet is enclosed by the Gaussian surface?

Keshav Singh
Keshav Singh
Numerade Educator
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Problem 8

The three small spheres shown in Fig. E22.8 carry charges $q_{1}=4.30 \mathrm{nC}, q_{2}=-7.50 \mathrm{nC},$ and $q_{3}=2.60 \mathrm{nC}$. Find the net electric
flux through each of the following closed surfaces shown in cross section in the figure:
(a) $S_{1} ;$ (b) $S_{2} ;$ (c) $S_{3} ;$ (d) $S_{4} ;$ (e) $S_{5}$. (f) Do your answers to parts (a)-(e) depend on how the charge is distributed over each small sphere? Why or why not?

Lainey Roebuck
Lainey Roebuck
Numerade Educator
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Problem 9

A charged paint is spread in a very thin uniform layer over the surface of a plastic sphere of diameter $20.0 \mathrm{~cm},$ giving it a charge of $-14.0 \mu \mathrm{C}$. Find the electric field (a) just inside the paint layer; (b) just outside the paint layer; (c) $5.00 \mathrm{~cm}$ outside the surface of the paint layer.

Gregory Devenport
Gregory Devenport
Numerade Educator
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Problem 10

A point charge $q_{1}=4.15 \mathrm{nC}$ is located on the $x$ -axis at $x=1.80 \mathrm{~m},$ and a second point charge $q_{2}=-6.15 \mathrm{nC}$ is on the $y$ -axis at $y=1.15 \mathrm{~m}$. What is the total electric flux due to these two point charges through a spherical surface centered at the origin and with ra-
dius (a) $0.755 \mathrm{~m},$ (b) $1.40 \mathrm{~m},$ (c) $2.95 \mathrm{~m}$ ?

Gregory Devenport
Gregory Devenport
Numerade Educator
00:59

Problem 11

As discussed in Section 22.5 , human nerve cells have a net negative charge and the material in the interior of the cell is a good conductor. If a cell has a net charge of $-8.65 \mathrm{pC}$, what are the magnitude and direction (inward or outward) of the net flux through the cell boundary?

Averell Hause
Averell Hause
Carnegie Mellon University
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Problem 12

Electric Fields in an Atom. The nuclei of large atoms, such as uranium, with 92 protons, can be modeled as spherically symmetric spheres of charge. The radius of the uranium nucleus is approximately $7.4 \times 10^{-15} \mathrm{~m}$. (a) What is the electric field this nucleus produces just outside its surface? (b) What magnitude of electric field does it produce at the distance of the electrons, which is about $1.9 \times 10^{-10} \mathrm{~m} ?$ (c) The electrons can be modeled as forming a uniform shell of negative charge. What net electric field do they produce at the location of the nucleus?

Gregory Devenport
Gregory Devenport
Numerade Educator
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Problem 13

Two very long uniform lines of charge are parallel and are separated by $0.220 \mathrm{~m}$. Each line of charge has charge per unit length $+5.00 \mu \mathrm{C} / \mathrm{m} .$ What magnitude of force does one line of charge exert on a $4.90 \times 10^{-2}-\mathrm{m}$ section of the other line of charge?

Gregory Devenport
Gregory Devenport
Numerade Educator
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Problem 14

A solid metal sphere with radius $0.500 \mathrm{~m}$ carries a net charge of $0.280 \mathrm{nC}$. Find the magnitude of the electric field (a) at a point $0.100 \mathrm{~m}$ outside the surface of the sphere and $(b)$ at a point inside the sphere, $0.100 \mathrm{~m}$ below the surface.

Gregory Devenport
Gregory Devenport
Numerade Educator
04:34

Problem 15

(a) A conducting sphere has charge $Q$ and radius $R$. If the electric field of the sphere at a distance $r=2 R$ from the center of the sphere is $1400 \mathrm{~N} / \mathrm{C},$ what is the electric field of the sphere at $r=4 R ?$
(b) A very long conducting cylinder of radius $R$ has charge per unit length $\lambda$. Let $r$ be the perpendicular distance from the axis of the cylinder. If the electric field of the cylinder at $r=2 R$ is $1400 \mathrm{~N} / \mathrm{C},$ what is the electric field at $r=4 R ?$ (c) A very large uniform sheet of charge has surface charge density $\sigma$. If the electric field of the sheet has a value of $1400 \mathrm{~N} / \mathrm{C}$ at a perpendicular distance $d$ from the sheet, what is the electric field of the sheet at a distance of $2 d$ from the sheet?

Keshav Singh
Keshav Singh
Numerade Educator
03:45

Problem 16

Some planetary scientists have suggested that the planet Mars has an electric field somewhat similar to that of the earth, producing a net electric flux of $-3.67 \times 10^{16} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}$ at the planet's surface. Calculate: (a) the total electric charge on the planet; (b) the electric field at the planet's surface (refer to the astronomical data on the inside back cover); (c) the charge density on Mars, assuming all the charge is uniformly distributed over the planet's surface.

Ely Crowder
Ely Crowder
Numerade Educator
04:39

Problem 17

A very long uniform line of charge has charge per unit length $4.80 \mu \mathrm{C} / \mathrm{m}$ and lies along the $x$ -axis. A second long uniform line of charge has charge per unit length $-2.40 \mu \mathrm{C} / \mathrm{m}$ and is parallel to the $x$ -axis at $y=0.400 \mathrm{~m}$. What is the net electric field (magnitude and direction) at the following points on the $y$ -axis:
(a) $y=0.200 \mathrm{~m}$ and
(b) $y=0.600 \mathrm{~m} ?$

Averell Hause
Averell Hause
Carnegie Mellon University
07:25

Problem 18

The electric field $0.355 \mathrm{~m}$ from a very long uniform line of charge is $900 \mathrm{~N} / \mathrm{C}$. How much charge is contained in a $2.80 \mathrm{~cm}$ section of the line?

Suman Saurav Thakur
Suman Saurav Thakur
Numerade Educator
07:56

Problem 19

A hollow, conducting sphere with an outer radius of $0.248 \mathrm{~m}$ and an inner radius of $0.208 \mathrm{~m}$ has a uniform surface charge density of $+6.44 \times 10^{-6} \mathrm{C} / \mathrm{m}^{2}$. A charge of $-0.560 \mu \mathrm{C}$ is now introduced at the center of the cavity inside the sphere. (a) What is the new charge density on the outside of the sphere? (b) Calculate the strength of the electric field just outside the sphere.
(c) What is the electric flux through a spherical surface just inside the inner surface of the sphere?

Ely Crowder
Ely Crowder
Numerade Educator
02:33

Problem 20

A solid insulating sphere has total charge $Q$ and radius $R$. The sphere's charge is distributed uniformly throughout its volume. Let $r$ be the radial distance measured from the center of the sphere. If $E=800 \mathrm{~N} / \mathrm{C}$ at $r=R / 2,$ what is $E$ at $r=2 R ?$

Keshav Singh
Keshav Singh
Numerade Educator
04:18

Problem 21

The electric field at a distance of $0.145 \mathrm{~m}$ from the surface of a solid insulating sphere with radius $0.355 \mathrm{~m}$ is $1750 \mathrm{~N} / \mathrm{C}$. (a) Assuming the sphere's charge is uniformly distributed, what is the charge density inside it? (b) Calculate the electric field inside the sphere at a distance of $0.200 \mathrm{~m}$ from the center.

Averell Hause
Averell Hause
Carnegie Mellon University
10:10

Problem 22

A point charge of $-2.00 \mu \mathrm{C}$ is located in the center of a spherical cavity of radius $6.55 \mathrm{~cm}$ that, in turn, is at the center of an insulating charged solid sphere. The charge density in the solid is $\rho=7.36 \times 10^{-4} \mathrm{C} / \mathrm{m}^{3} .$ Calculate the electric field inside the solid at a distance of $9.49 \mathrm{~cm}$ from the center of the cavity.

Deborah Israel
Deborah Israel
Numerade Educator
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Problem 23

An electron is released from rest at a distance of $0.540 \mathrm{~m}$ from a large insulating sheet of charge that has uniform surface charge density $+3.00 \times 10^{-12} \mathrm{C} / \mathrm{m}^{2}$. (a) How much work is done on the electron by the electric field of the sheet as the electron moves from its initial position to a point $7.00 \times 10^{-2} \mathrm{~m}$ from the sheet? (b) What is the speed of the electron when it is $7.00 \times 10^{-2} \mathrm{~m}$ from the sheet?

Eduard Sanchez
Eduard Sanchez
Numerade Educator
07:21

Problem 24

Charge $Q$ is distributed uniformly throughout the volume of an insulating sphere of radius $R=4.00 \mathrm{~cm} .$ At a distance of $r=8.00 \mathrm{~cm}$ from the center of the sphere, the electric field due to the charge distribution has magnitude $E=940 \mathrm{~N} / \mathrm{C}$. What are (a) the volume charge density for the sphere and (b) the electric field at a distance of $2.00 \mathrm{~cm}$ from the sphere's center?

Mohit Khurana
Mohit Khurana
Texas A&M University
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Problem 25

A conductor with an inner cavity, like that shown in Fig. $22.23 \mathrm{c}$, carries a total charge of $+4.60 \mathrm{nC}$. The charge within the cavity, insulated from the conductor, is $-6.10 \mathrm{nC}$. How much charge is on (a) the inner surface of the conductor and (b) the outer surface of the conductor?

Gregory Devenport
Gregory Devenport
Numerade Educator
03:16

Problem 26

A square insulating sheet $80.0 \mathrm{~cm}$ on a side is held horizontally. The sheet has $4.50 \mathrm{nC}$ of charge spread uniformly over its area.
(a) Estimate the electric field at a point $0.100 \mathrm{~mm}$ above the center of the sheet. (b) Estimate the electric field at a point $100 \mathrm{~m}$ above the center of the sheet. (c) Would the answers to parts (a) and (b) be different if the sheet were made of a conducting material? Why or why not?

Keshav Singh
Keshav Singh
Numerade Educator
08:42

Problem 27

An infinitely long cylindrical conductor has radius $R$ and uniform surface charge density $\sigma$. (a) In terms of $\sigma$ and $R,$ what is the charge per unit length $\lambda$ for the cylinder? (b) In terms of $\sigma,$ what is the magnitude of the electric field produced by the charged cylinder at a distance $r>R$ from its axis? (c) Express the result of part (b) in terms of $\lambda$ and show that the electric field outside the cylinder is the same as if all the charge were on the axis. Compare your result to the result for a line of charge in Example 22.6 (Section 22.4 ).

Andrew Eddins
Andrew Eddins
Emory University
11:03

Problem 28

Two very large, nonconducting plastic sheets, each $10.0 \mathrm{~cm}$ thick, carry uniform charge densities $\sigma_{1}, \sigma_{2}, \sigma_{3},$ and $\sigma_{4}$ on their surfaces (Fig. E22.28). These surface charge densities have the values $\sigma_{1}=-6.00 \mu \mathrm{C} / \mathrm{m}^{2}, \quad \sigma_{2}=+5.00 \mu \mathrm{C} / \mathrm{m}^{2}, \quad \sigma_{3}=+2.00 \mu \mathrm{C} / \mathrm{m}^{2},$ and
$\sigma_{4}=+4.00 \mu \mathrm{C} / \mathrm{m}^{2} .$ Use Gauss's law to find the magnitude and direction of the electric field at the following points, far from the edges of these sheets: (a) point $A, 5.00 \mathrm{~cm}$ from the left face of the left-hand sheet; (b) point $B, 1.25 \mathrm{~cm}$ from the inner surface of the right-hand sheet; (c) point $C$, in the middle of the right-hand sheet.

Andrew Eddins
Andrew Eddins
Emory University
05:00

Problem 29

At time $t=0$ a proton is a distance of $0.360 \mathrm{~m}$ from a very large insulating sheet of charge and is moving parallel to the sheet with speed $9.70 \times 10^{2} \mathrm{~m} / \mathrm{s}$. The sheet has uniform surface charge density $2.34 \times 10^{-9} \mathrm{C} / \mathrm{m}^{2}$. What is the speed of the proton at $t=5.00 \times 10^{-8} \mathrm{~s} ?$

Andrew Eddins
Andrew Eddins
Emory University
04:27

Problem 30

A very small object with mass $8.20 \times 10^{-9} \mathrm{~kg}$ and positive charge $6.50 \times 10^{-9} \mathrm{C}$ is projected directly toward a very large insulating sheet of positive charge that has uniform surface charge density $5.90 \times 10^{-8} \mathrm{C} / \mathrm{m}^{2}$. The object is initially $0.400 \mathrm{~m}$ from the sheet. What initial speed must the object have in order for its closest distance of approach to the sheet to be $0.100 \mathrm{~m}$ ?

Andrew Eddins
Andrew Eddins
Emory University
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Problem 31

A small sphere with mass $2.00 \times 10^{-3} \mathrm{~kg}$ and charge $4.80 \times 10^{-8} \mathrm{C}$
hangs from a thread near a very large, charged insulating sheet (Fig. $\mathbf{P 2 2 . 3 1}$ ). The charge density on the surface of the sheet is uniform and equal to $-2.20 \times 10^{-9} \mathrm{C} / \mathrm{m}^{2}$
Find the angle of the thread.

Gregory Devenport
Gregory Devenport
Numerade Educator
10:28

Problem 32

A cube has sides of length $L=0.350 \mathrm{~m}$. One corner is at the origin (Fig. E22.6). The nonuniform electric field is given by $\overrightarrow{\boldsymbol{E}}=(-5.64 \mathrm{~N} / \mathrm{C} \cdot \mathrm{m}) x \hat{\imath}+(2.54 \mathrm{~N} / \mathrm{C} \cdot \mathrm{m}) z \hat{\boldsymbol{k}} .$ (a) Find the electric flux
(b) Find the through each of the six cube faces $S_{1}, S_{2}, S_{3}, S_{4}, S_{5},$ and $S_{6}$ total electric charge inside the cube.

Sanat Mukherjee
Sanat Mukherjee
Numerade Educator
01:27

Problem 33

The electric field $\overrightarrow{\boldsymbol{E}}$ in Fig. $\mathbf{P 2 2} .33$ is everywhere parallel to the $x$ -axis, so the components $E_{y}$ and $E_{z}$ are zero. The $x$ -component of the field $E_{x}$ depends on $x$ but not on $y$ or $z$. At points in the $y z$ -plane (where $x=0$ ), $E_{x}=125 \mathrm{~N} / \mathrm{C}$. (a) What is the electric flux through surface I in Fig. P22.33? (b) What is the electric flux through surface II?
(c) The volume shown is a small section of a very large insulating slab $1.0 \mathrm{~m}$ thick. If there is a total charge of $-24.0 \mathrm{nC}$ within the volume shown, what are the magnitude and direction of $\vec{E}$ at the face opposite surface I?
(d) Is the electric field produced by charges only within the slab, or is the field also due to charges outside the slab? How can you tell?

Dominador Tan
Dominador Tan
Numerade Educator
03:12

Problem 34

In a region of space there is an electric field $\overrightarrow{\boldsymbol{E}}$ that is in the $z$ -direction and that has magnitude $E=[963 \mathrm{~N} /(\mathrm{C} \cdot \mathrm{m})] x .$ Find the flux for this field through a square in the $x y$ -plane at $z=0$ and with side length $0.480 \mathrm{~m}$. One side of the square is along the $+x$ -axis and another side is along the $+y$ -axis.

Ely Crowder
Ely Crowder
Numerade Educator
02:59

Problem 35

Negative charge $-Q$ is distributed uniformly over the surface of a thin spherical insulating shell with radius $R .$ Calculate the force (magnitude and direction) that the shell exerts on a positive point charge $q$ located a distance (a) $r>R$ from the center of the shell (outside the shell); (b) $r<R$ from the center of the shell (inside the shell).

Averell Hause
Averell Hause
Carnegie Mellon University
04:16

Problem 36

A long line carrying a uniform linear charge density $+50.0 \mu \mathrm{C} / \mathrm{m}$ runs parallel to and $10.0 \mathrm{~cm}$ from the surface of a large, flat plastic sheet that has a uniform surface charge density of $-100 \mu \mathrm{C} / \mathrm{m}^{2}$ on one side. Find the location of all points where an $\alpha$ particle would feel no force due to this arrangement of charged objects.

Andrew Eddins
Andrew Eddins
Emory University
07:41

Problem 37

The Coaxial Cable. A long coaxial cable consists of an inner cylindrical conductor with radius $a$ and an outer coaxial cylinder with inner radius $b$ and outer radius $c$. The outer cylinder is mounted on insulating supports and has no net charge. The inner cylinder has a uniform positive charge per unit length $\lambda$. Calculate the electric field (a) at any point between the cylinders a distance $r$ from the axis and $(b)$ at any point outside the outer cylinder. (c) Graph the magnitude of the electric field as a function of the distance $r$ from the axis of the cable, from $r=0$ to $r=2 c$. (d) Find the charge per unit length on the inner surface and on the outer surface of the outer cylinder.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
09:33

Problem 38

A very long conducting tube (hollow cylinder) has inner radius $a$ and outer radius $b$. It carries charge per unit length $+\alpha,$ where $\alpha$ is a positive constant with units of $\mathrm{C} / \mathrm{m}$. A line of charge lies along the axis of the tube. The line of charge has charge per unit length $+\alpha$.
(a) Calculate the electric field in terms of $\alpha$ and the distance $r$ from the axis of the tube for (i) $r<a ;$ (ii) $a<r<b ;$ (iii) $r>b$. Show your results in a graph of $E$ as a function of $r$. (b) What is the charge per unit length on (i) the inner surface of the tube and (ii) the outer surface of the tube?

Andrew Eddins
Andrew Eddins
Emory University
06:30

Problem 39

A very long, solid cylinder with radius $R$ has positive charge uniformly distributed throughout it, with charge per unit volume $\rho$.
(a) Derive the expression for the electric field inside the volume at a distance $r$ from the axis of the cylinder in terms of the charge density $\rho$.
(b) What is the electric field at a point outside the volume in terms of the charge per unit length $\lambda$ in the cylinder? (c) Compare the answers to parts (a) and (b) for $r=R$. (d) Graph the electric-field magnitude as a function of $r$ from $r=0$ to $r=3 R$.

CD
Colin Devine
Numerade Educator
03:08

Problem 40

A Sphere in a Sphere. A solid conducting sphere carrying charge $q$ has radius $a$. It is inside a concentric hollow conducting sphere with inner radius $b$ and outer radius $c .$ The hollow sphere has no net charge. (a) Derive expressions for the electric-field magnitude in terms of the distance $r$ from the center for the regions $r<a$, $a<r<b, b<r<c,$ and $r>c .$ (b) Graph the magnitude of the electric field as a function of $r$ from $r=0$ to $r=2 c .$ (c) What is the charge on the inner surface of the hollow sphere? (d) On the outer surface? (e) Represent the charge of the small sphere by four plus signs. Sketch the field lines of the system within a spherical volume of radius $2 c.$

Keshav Singh
Keshav Singh
Numerade Educator
05:04

Problem 41

A solid conducting sphere with radius $R$ that carries positive charge $Q$ is concentric with a very thin insulating shell of radius $2 R$ that also carries charge $Q$. The charge $Q$ is distributed uniformly over the insulating shell. (a) Find the electric field (magnitude and direction) in each of the regions $0<r<R, R<r<2 R,$ and $r>2 R .$ (b) Graph the electric-field magnitude as a function of $r$.

Rashmi Sinha
Rashmi Sinha
Numerade Educator
07:27

Problem 42

A conducting spherical shell with inner radius $a$ and outer radius $b$ has a positive point charge $Q$ located at its center. The total charge on the shell is $-3 Q$, and it is insulated from its surroundings (Fig. P22.42). (a) Derive expressions for the electric-field magnitude $E$ in terms of the distance $r$ from the center for the regions $r<a, a<r<b,$ and $r>b$. What is the surface charge density (b) on the inner surface of the conducting shell; (c) on the outer surface of the conducting shell?
(d) Sketch the electric field lines and the location of all charges.
(e) Graph $E$ as a function of $r$.

Mohit Khurana
Mohit Khurana
Texas A&M University
02:13

Problem 43

Concentric Spherical Shells. A small conducting spherical shell with inner radius $a$ and outer radius $b$ is concentric with a larger conducting spherical shell with inner radius $c$ and outer radius $d$ (Fig. P22.43). The inner shell has total charge $+2 q,$ and the outer shell has charge $+4 q$ (a) Calculate the electric field $\overrightarrow{\boldsymbol{E}}$ (magnitude and direction) in terms of $q$ and the distance $r$ from the common center of the two shells for (1) $r<a ;$ (ii) $a<r<b ;$ (iii) $b<r<c ;$ (iv) $c<r<d$ (v) $r>d$. Graph the radial component of $\overrightarrow{\boldsymbol{E}}$ as a function of $r$. (b) What is the total charge on the (i) inner surface of the small shell; (ii) outer surface of the small shell; (iii) inner surface of the large shell; (iv) outer surface of the large shell?

Dominador Tan
Dominador Tan
Numerade Educator
06:24

Problem 44

Repeat Problem $22.43,$ but now let the outer shell have charge $-2 q$. The inner shell still has charge $+2 q$.

Linda Winkler
Linda Winkler
Numerade Educator
08:25

Problem 45

An insulating hollow sphere has inner radius $a$ and outer radius $b$. Within the insulating material the volume charge density is given by $\rho(r)=\alpha / r,$ where $\alpha$ is a positive constant. (a) In terms of $\alpha$ and $a$, what is the magnitude of the electric field at a distance $r$ from the center of the shell, where $a<r<b ?$ (b) A point charge $q$ is placed at the center of the hollow space, at $r=0$. In terms of $\alpha$ and $a$, what value must $q$ have (sign and magnitude) in order for the electric field to be constant in the region $a<r<b$, and what then is the value of the constant field in this region?

Artemisa Mazón
Artemisa Mazón
Numerade Educator
01:07

Problem 46

Thomson's Model of the Atom. Early in the 20 th century, a leading model of the structure of the atom was that of English physicist J. J. Thomson (the discoverer of the electron). In Thomson's model, an atom consisted of a sphere of positively charged material in which were embedded negatively charged electrons, like chocolate chips in a ball of cookie dough. Consider such an atom consisting of one electron with mass $m$ and charge $-e,$ which may be regarded as a point charge, and a uniformly charged sphere of charge $+e$ and radius $R$.
(a) Explain why the electron's equilibrium position is at the center of the nucleus. (b) In Thomson's model, it was assumed that the positive material provided little or no resistance to the electron's motion. If the electron is displaced from equilibrium by a distance less than $R$, show that the resulting motion of the electron will be simple harmonic, and calculate the frequency of oscillation. (Hint: Review the definition of SHM in Section 14.2. If it can be shown that the net force on the electron is of this form, then it follows that the motion is simple harmonic. Conversely, if the net force on the electron does not follow this form, the motion is not simple harmonic.) (c) By Thomson's time, it was known that excited atoms emit light waves of only certain frequencies. In his model, the frequency of emitted light is the same as the oscillation frequency of the electron(s) in the atom. What radius would a Thomson-model atom need for it to produce red light of frequency $4.57 \times 10^{14} \mathrm{~Hz}$ ? Compare your answer to the radii of real atoms, which are of the order of $10^{-10} \mathrm{~m}$ (see Appendix $\mathrm{G}$ ). (d) If the electron were displaced from equilibrium by a distance greater than $R,$ would the electron oscillate? Would its motion be simple harmonic? Explain your reasoning. (Historical note: In $1910,$ the atomic nucleus was discovered, proving the Thomson model to be incorrect. An atom's positive charge is not spread over its volume, as Thomson supposed, but is concentrated in the tiny nucleus of radius $10^{-14}$ to $10^{-15} \mathrm{~m}$.)

Dominador Tan
Dominador Tan
Numerade Educator
10:20

Problem 47

A nonuniform, but spherically symmetric, distribution of charge has a charge density $\rho(r)$ given as follows:
$$
\begin{array}{ll}
\rho(r)=\rho_{0}\left(1-\frac{r}{R}\right) & \text { for } r \leq R \\
\rho(r)=0 & \text { for } r \geq R
\end{array}
$$
where $\rho_{0}=3 Q / \pi R^{3}$ is a positive constant. (a) Show that the total charge contained in the charge distribution is $Q .$ (b) Show that the electric field in the region $r \geq R$ is identical to that produced by a point charge $Q$ at $r=0 .$ (c) Obtain an expression for the electric field in the region $r \leq R .$ (d) Graph the electric-field magnitude $E$ as a function of $r$.
(e) Find the value of $r$ at which the electric field is maximum, and find the value of that maximum field.

Jerrah Biggerstaff
Jerrah Biggerstaff
Numerade Educator
04:19

Problem 48

(a) How many excess electrons must be distributed uniformly within the volume of an isolated plastic sphere $26.0 \mathrm{~cm}$ in diameter to produce an electric field of magnitude $1500 \mathrm{~N} / \mathrm{C}$ just outside the surface of the sphere? (b) What is the electric field at a point $14.5 \mathrm{~cm}$ outside the surface of the sphere?

Ely Crowder
Ely Crowder
Numerade Educator
02:39

Problem 49

A very long insulating cylinder has radius $R$ and carries positive charge distributed throughout its volume. The charge distribution has cylindrical symmetry but varies with perpendicular distance from the axis of the cylinder. The volume charge density is $\rho(r)=\alpha(1-r / R),$ where $\alpha$ is a constant with units $\mathrm{C} / \mathrm{m}^{3}$ and $r$ is the perpendicular distance from the center line of the cylinder. Derive an expression, in terms of $\alpha$ and $R,$ for $E(r),$ the electric field as a function of $r$. Do this for $r<R$ and also for $r>R$. Do your results agree for $r=R ?$

Rashmi Sinha
Rashmi Sinha
Numerade Educator
09:18

Problem 50

A solid insulating sphere has radius $R$ and carries positive charge distributed throughout its volume. The charge distribution has spherical symmetry but varies with radial distance $r$ from the center of the sphere. The volume charge density is $\rho(r)=\rho_{0}(1-r / R)$, where $\rho_{0}$ is a constant with units of $\mathrm{C} / \mathrm{m}^{3}$. (a) Derive an expression for the electric field as a function of $r$ for $r<R .$ (b) Repeat part (a) for $r>R .$ (c) At what value of $r$, in terms of $R$, does the electric field have its maximum value?

Sheh Lit Chang
Sheh Lit Chang
University of Washington
01:05

Problem 51

A Nonuniformly Charged Slab. A slab of insulating material has thickness $2 d$ and is oriented so that its faces are parallel to the $y z$ -plane and given by the planes $x=d$ and $x=-d$. The $y$ - and $z$ -dimensions of the slab are very large compared to $d$, so treat them as infinite. The slab has charge density given by $\rho(x)=\rho_{0}(x / d)^{2},$ where $\rho_{0}$ is a positive constant. Using Gauss's law, find the electric field due to the slab (magnitude and direction) at all points in space.

Dominador Tan
Dominador Tan
Numerade Educator
21:34

Problem 52

A nonuniform, but spherically symmetric, distribution of charge has a charge densitv $\rho(r)$ given as follows:
$$
\begin{array}{ll}
\rho(r)=\rho_{0}\left(1-\frac{4 r}{3 R}\right) & \text { for } r \leq R \\
\rho(r)=0 & \text { for } r \geq R
\end{array}
$$
where $\rho_{0}$ is a positive constant.
(a) Find the total charge contained in the charge distribution. Obtain an expression for the electric field in the region (b) $r \geq R ;$ (c) $r \leq R$. (d) Graph the electric-field magnitude $E$ as a function of $r$. (e) Find the value of $r$ at which the electric field is maximum, and find the value of that maximum field.

Deborah Israel
Deborah Israel
Numerade Educator
09:44

Problem 53

(a) An insulating sphere with radius $a$ has a uniform charge density $\rho .$ The sphere is not centered at the origin but at $\overrightarrow{\boldsymbol{r}}=\overrightarrow{\boldsymbol{b}}$. Show that the electric field inside the sphere is given by $\overrightarrow{\boldsymbol{E}}=\rho(\overrightarrow{\boldsymbol{r}}-\overrightarrow{\boldsymbol{b}}) / 3 \epsilon_{0} .$ (b) An insulating sphere of
radius $R$ has a spherical hole of radius $a$ located within its volume and centered a distance $b$ from the center of the sphere, where $a<b<R$ (a cross section of the sphere is shown in Fig. P22.53). The solid part of the sphere has a uniform volume charge density $\rho$. Find the magnitude and direction of the electric field $\overrightarrow{\boldsymbol{E}}$ inside the hole, and show that $\overrightarrow{\boldsymbol{E}}$ is uniform over the entire hole. [Hint: Use the principle of superposition and the result of part (a).]

João Gabriel Alencar Caribé
João Gabriel Alencar Caribé
Numerade Educator
05:34

Problem 54

A very long, solid insulating cylinder has radius $R$; bored along its entire length is a cylindrical hole with radius $a$. The axis of the hole is a distance $b$ from the axis of the cylinder, where $a<b<R$ (Fig. P22.54). The solid material of the cylinder has a uniform volume charge density $\rho .$ Find the magnitude and direction of the electric field $\overrightarrow{\boldsymbol{E}}$ inside the hole, and show that $\overrightarrow{\boldsymbol{E}}$ is uniform over the entire hole. (Hint: See Problem 22.53.)

Mohit Khurana
Mohit Khurana
Texas A&M University
02:31

Problem 55

In one experiment the electric field is measured for points at distances $r$ from a uniform line of charge that has charge per unit length $\lambda$ and length $l$, where $l \gg r .$ In a second experiment the electric field is measured for points at distances $r$ from the center of a uniformly charged insulating sphere that has volume charge density $\rho$ and radius $R=8.00 \mathrm{~mm}$, where $r>R$. The results of the two measurements are listed in the table, but you aren't told which set of data applies to which experiment:
$$
\begin{array}{lcc}
\hline & {\boldsymbol{E}\left(\mathbf{1 0}^{\mathbf{5}} \mathrm{N} / \mathrm{C}\right)} & \\
\boldsymbol{r}(\mathbf{c m}) & \text { Measurement A } & \text { Measurement B } \\
\hline 1.00 & 2.72 & 5.45 \\
1.50 & 1.79 & 2.42 \\
2.00 & 1.34 & 1.34 \\
2.50 & 1.07 & 0.861 \\
3.00 & 0.902 & 0.605 \\
3.50 & 0.770 & 0.443 \\
4.00 & 0.677 & 0.335 \\
\hline
\end{array}
$$
For each set of data, draw two graphs: one for $E r^{2}$ versus $r$ and one for Er versus $r$. (a) Use these graphs to determine which data set, $\mathrm{A}$ or $\mathrm{B}$, is for the uniform line of charge and which set is for the uniformly charged sphere. Explain your reasoning. (b) Use the graphs in part (a) to calculate $\lambda$ for the uniform line of charge and $\rho$ for the uniformly charged sphere.

Dominador Tan
Dominador Tan
Numerade Educator
04:58

Problem 56

The electric field is measured for points at distances $r$ from the center of a uniformly charged insulating sphere that has volume charge density $\rho$ and radius $R,$ where $r<R$ (Fig. P22.56). Calculate $\rho$.

Mohit Khurana
Mohit Khurana
Texas A&M University
View

Problem 57

The volume charge density $\rho$ for a spherical charge distribution of radius $R=6.00 \mathrm{~mm}$ is not uniform. Figure $\mathbf{P 2 2 . 5 7}$ shows $\rho$ as a function of the distance $r$ from the center of the distribution. Calculate the electric field at these values of $r:$ (i) $1.00 \mathrm{~mm} ;$
(ii) $3.00 \mathrm{~mm} ;$
(iii) $\begin{array}{ll}5.00 & \mathrm{~mm} ;\end{array}$
(iv) $7.00 \mathrm{~mm}$.

Lainey Roebuck
Lainey Roebuck
Numerade Educator
02:13

Problem 58

As a honeybee flies, the passing air strips electrons from its hairs, giving the bee a net positive charge. Since flowers are negatively charged, pollen then jumps onto a bee even if the bee does not physically touch the pollen particles.
(a) Estimate the diameter of the central disk of a daisy. (b) If a bee has had 75,000 electrons stripped by the air, what is its net charge? (c) If this bee lands at the edge of the daisy's central disk, determine its electric field at the far edge of the disk. Treat the bee as a thin-walled hollow sphere with its net charge distributed uniformly over its surface. (d) A pollen particle requires a force of $10 \mathrm{pN}$ to dislodge from a stamen. Estimate the net charge on a pollen particle at the far end of the disk required for the particle to dislodge and jump to the bee.

Prashant Bana
Prashant Bana
Numerade Educator
04:55

Problem 59

A very long insulating cylinder with radius $R_{\text {cylinder }}$ has nonuniform positive charge density $\rho=\left(1-r / R_{\text {cylinder }}\right) \rho_{0}$ where $\rho_{0}$ is constant and $r$ is measured radially from the axis of the cylinder. A particle with charge $-Q$ and mass $M$ orbits the cylinder at a constant distance $R_{\text {orbit }}>R_{\text {cylinder }}$.
(a) What is the linear charge density $\lambda$ of the tube, in terms of $R_{\text {cylinder }}$ and $\rho_{0} ?$ (b) Determine the period of the motion in terms of $R_{\text {orbit }}$. Hint: Use Gauss's law to determine the electric field, and therefore the electric force felt by the particle, that acts centripetally.)

Keshav Singh
Keshav Singh
Numerade Educator
05:29

Problem 60

(a) Show that the component of the electric force normal to any flat surface with a uniform charge density $\sigma$ is given by $F_{\perp}=\sigma \Phi_{E},$ where $\Phi_{E}$ is the electric flux through that surface due to an external electric field. (b) An insulating hemisphere with radius $R$ and charge $Q$ distributed uniformly over its flat, circular surface lies above a large plane with uniform charge density $\sigma .$ The axis of the hemisphere is oriented vertically. For what mass $M$ could the hemisphere remain stationary? (c) If the hemisphere and the plane share the same charge density of $100 \mu \mathrm{C} / \mathrm{m}^{2}$ and the hemisphere has a radius of $3.00 \mathrm{~cm},$ what would be its upward acceleration if its mass were $100 \mathrm{~g}$ ?

Keshav Singh
Keshav Singh
Numerade Educator
13:38

Problem 61

A uniformly charged insulating sphere with radius $r$ and charge $+Q$ lies at the center of a thin-walled hollow cylinder with radius $R>r$ and length $L>2 r$. The cylinder is non-conducting and carries no net charge.
(a) Determine the outward electric flux through the rounded "side" of the cylinder, excluding the circular end caps. (Hint: Choose a cylindrical coordinate system with the axis of the cylinder as its $z$ -axis and the center of the charged sphere as its origin. Note that an area element on the cylinder has magnitude $d A=2 \pi R d z .$ ) (b) Determine the electric flux upward through the circular cap at the top of the cylinder. (c) Determine the electric flux downward through the circular cap at the bottom of the cylinder. (d) Add the results from parts
(a)-(c) to determine the outward electric flux through the closed cylinder. (e) Show that your result is consistent with Gauss's law.

Sheh Lit Chang
Sheh Lit Chang
University of Washington
01:19

Problem 62

A region in space contains a total positive }\end{array}$ charge $Q$ that is distributed spherically such that the volume charge density $\rho(r)$ is given by
$$
\begin{array}{ll}
\rho(r)=3 \alpha r / 2 R & \text { for } r \leq R / 2 \\
\rho(r)=\alpha\left[1-(r / R)^{2}\right] & \text { for } R / 2 \leq r \leq R \\
\rho(r)=0 & \text { for } r \geq R
\end{array}
$$
Here $\alpha$ is a positive constant having units of $\mathrm{C} / \mathrm{m}^{3}$. (a) Determine $\alpha$ in terms of $Q$ and $R$. (b) Using Gauss's law, derive an expression for the magnitude of the electric field as a function of $r$. Do this separately for all three regions. Express your answers in terms of $Q$.
(c) What fraction of the total charge is contained within the region $R / 2 \leq r \leq R ?$
(d) What is the magnitude of $\overrightarrow{\boldsymbol{E}}$ at $r=R / 2$ ? (e) If an electron with charge $q^{\prime}=-e$ is released from rest at any point in any of the three regions, the resulting motion will be oscillatory but not simple harmonic. Why?

Dominador Tan
Dominador Tan
Numerade Educator
01:31

Problem 63

Suppose that to repel electrons in the radiation from a solar flare, each sphere must produce an electric field $\overrightarrow{\boldsymbol{E}}$ of magnitude $1 \times 10^{6} \mathrm{~N} / \mathrm{C}$ at $25 \mathrm{~m}$ from the center of the sphere. What net charge on each sphere is needed?
(a) $-0.07 \mathrm{C} ;$ (b) $-8 \mathrm{mC}$
(c) $-80 \mu \mathrm{C}$
(d) $-1 \times 10^{-20} \mathrm{C}$

Keshav Singh
Keshav Singh
Numerade Educator
02:47

Problem 64

What is the magnitude of $\overrightarrow{\boldsymbol{E}}$ just outside the surface of such a sphere? (a) $0 ;$ (b) $10^{6} \mathrm{~N} / \mathrm{C} ;$ (c) $10^{7} \mathrm{~N} / \mathrm{C} ;$ (d) $10^{8} \mathrm{~N} / \mathrm{C}$

Mohit Khurana
Mohit Khurana
Texas A&M University
01:09

Problem 65

What is the direction of $\vec{E}$ just outside the surface of such a sphere? (a) Tangent to the surface of the sphere; (b) perpendicular to the surface, pointing toward the sphere; (c) perpendicular to the surface, pointing away from the sphere;
(d) there is no electric field just outside the surface.

Averell Hause
Averell Hause
Carnegie Mellon University
02:48

Problem 66

Which statement is true about $\overrightarrow{\boldsymbol{E}}$ inside a negatively charged sphere as described here? (a) It points from the center of the sphere to the surface and is largest at the center. (b) It points from the surface to the center of the sphere and is largest at the surface. (c) It is zero. (d) It is constant but not zero.

Mohit Khurana
Mohit Khurana
Texas A&M University