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Sears and Zemansky’s University Physics with Modern Physics

Hugh D Young, Roger A Freedman, Ragbir Bhathal

Chapter 22

Gauss's law - all with Video Answers

Educators

Chapter Questions

05:32

Problem 1

A flat sheet of paper of area $0.250 \mathrm{~m}^2$ is oriented so that the normal to the sheet is at an angle of $60^{\circ}$ to a uniform electric field of magnitude $14 \mathrm{~N} / \mathrm{C}$. (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.

Artemisa Mazón
Artemisa Mazón
Numerade Educator
04:03

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 $75.0 \mathrm{~N} / \mathrm{C}$ that is directed at $20^{\circ}$ from the plane of the sheet (Fig. 22.30). Find the magnitude of the electric flux through the sheet.
(FIGURE CANT COPY)

Youssef Eweis
Youssef Eweis
Numerade Educator
04:36

Problem 3

You measure an electric field of $1.25 \times 10^6 \mathrm{~N} / \mathrm{C}$ at a distance of $0.150 \mathrm{~m}$ from a point charge. (a) What is the electric flux through a sphere whose centre is at that distance and whose radius is less than $0.150 \mathrm{~m}$ from the charge? (b) What is the magnitude of the charge?

Vishal Gupta
Vishal Gupta
Numerade Educator
07:58

Problem 4

A cube has sides of length $L=0.300 \mathrm{~m}$. It is placed with one corner at the origin as shown in Fig. 22.31. The electric field is not uniform but is given by $\vec{E}=(-5.00 \mathrm{~N} / \mathrm{C} \cdot \mathrm{m}) x \hat{\imath}+$
(FIGURE CANT COPY)
$(3.00 \mathrm{~N} / \mathrm{C} \cdot \mathrm{m}) z \hat{k}$. (a) Find the electric flux through each of the six cube faces $S_1, S_2, S_3, S_4, S_5$, and $S_6$. (b) Find the total electric charge inside the cube.

JW
Jordan Wray
Numerade Educator
02:48

Problem 5

A hemispherical surface with radius $r$ in a region of uniform electric field $\vec{E}$ has its axis aligned parallel to the direction of the field. Calculate the flux through the surface.

Vishal Gupta
Vishal Gupta
Numerade Educator
07:20

Problem 6

The cube in Fig. 22.31 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 $36.9^{\circ}$ measured from the $+x$-axis towards 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 $S_6$ ? (b) What is the total electric flux through all faces of the cube?

Cody Johnston
Cody Johnston
Vanderbilt University
04:09

Problem 7

It was shown in Example 21.11 (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 $r=0.250 \mathrm{~m}$ and length $l=0.400 \mathrm{~m}$ that has an infinite line of positive charge running along its axis. The charge per unit length on the line is $\lambda=6.00 \mu \mathrm{C} / \mathrm{m}$. (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 $r=0.500 \mathrm{~m}$ ? (c) What is the flux through the cylinder if its length is increased to $l=0.800 \mathrm{~m}$ ?

Youssef Eweis
Youssef Eweis
Numerade Educator
08:32

Problem 8

The three small spheres shown in Fig. 22.32 carry charges $q_1=4.00 \mathrm{nC}, q_2=-7.80 \mathrm{nC}$, and $q_3=2.40 \mathrm{nC}$. 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?
(FIGURE CANT COPY)

Vishal Gupta
Vishal Gupta
Numerade Educator
04:10

Problem 9

A charged paint is spread in a very thin uniform layer over the surface of a plastic sphere of diameter $12.0 \mathrm{~cm}$, giving it a charge of $-15.0 \mu \mathrm{C}$. 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.

Prabhu Ramji
Prabhu Ramji
Numerade Educator
06:45

Problem 10

A point charge $q_1=4.00 \mathrm{nC}$ is located on the $x$-axis at $x=2.00 \mathrm{~m}$, and a second point charge $q_2=-6.00 \mathrm{nC}$ is on the $y$-axis at $y=1.00 \mathrm{~m}$. What is the total electric flux due to these two point charges through a spherical surface centred at the origin and with radius (a) $0.500 \mathrm{~m}$, (b) $1.50 \mathrm{~m}$, (c) $2.50 \mathrm{~m}$ ?

Vishal Gupta
Vishal Gupta
Numerade Educator
06:14

Problem 11

In a certain region of space, the electric field $\overrightarrow{\boldsymbol{E}}$ is uniform. (a) Use Gauss's law to prove that this region of space must be electrically neutral; that is, the volume charge density $\rho$ must be zero. (b) Is the converse true? That is, in a region of space where there is no charge, must $\vec{E}$ be uniform? Explain.

Vishal Gupta
Vishal Gupta
Numerade Educator
06:42

Problem 12

(a) In a certain region of space, the volume charge density $\rho$ has a uniform positive value. Can $\overrightarrow{\boldsymbol{E}}$ be uniform in this region? Explain. (b) Suppose that in this region of uniform positive $\rho$ there is a 'bubble' within which $\rho=0$. Can $\overrightarrow{\boldsymbol{E}}$ be uniform within this bubble? Explain.

Vishal Gupta
Vishal Gupta
Numerade Educator
02:08

Problem 13

A $9.60 \mu \mathrm{C}$ point charge is at the centre of a cube with sides of length $0.500 \mathrm{~m}$. (a) What is the electric flux through one of the six faces of the cube? (b) How would your answer to part (a) change if the sides were $0.250 \mathrm{~m}$ long? Explain.

Prabhu Ramji
Prabhu Ramji
Numerade Educator
07:09

Problem 14

The nuclei of large atoms, such as uranium, with 92 protons, can be modelled 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 $1.0 \times 10^{-10} \mathrm{~m}$ ? (c) The electrons can be modelled as forming a uniform shell of negative charge. What net electric field do they produce at the location of the nucleus?

Umar Sohail Qureshi
Umar Sohail Qureshi
Numerade Educator
04:06

Problem 15

A point charge of $+5.00 \mu \mathrm{C}$ is located on the $x$-axis at $x=4.00 \mathrm{~m}$, next to a spherical surface of radius $3.00 \mathrm{~m}$ centred at the origin. (a) Calculate the magnitude of the electric field at $x=3.00 \mathrm{~m}$. (b) Calculate the magnitude of the electric field at $x=-3.00 \mathrm{~m}$. (c) According to Gauss's law, the net flux through the sphere is zero because it contains no charge. Yet the field due to the external charge is much stronger on the near side of the sphere (i.e., at $x=3.00 \mathrm{~m}$ ) than on the far side (at $x=-3.00 \mathrm{~m}$ ). How, then, can the flux into the sphere (on the near side) equal the flux out of it (on the far side)? Explain. A sketch will help.

Youssef Eweis
Youssef Eweis
Numerade Educator
03:24

Problem 16

A solid metal sphere with radius $0.450 \mathrm{~m}$ carries a net charge of $0.250 \mathrm{nC}$. 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.

Ren Jie Tuieng
Ren Jie Tuieng
Numerade Educator
05:42

Problem 17

On a humid day, an electric field of $2.00 \times 10^4 \mathrm{~N} / \mathrm{C}$ is enough to produce sparks about an inch long. Suppose that in your physics class, a van de Graaff generator (see Fig. 22.27) with a sphere radius of $15.0 \mathrm{~cm}$ is producing sparks 6 inches long. (a) Use Gauss's law to calculate the amount of charge stored on the surface of the sphere before you bravely discharge it with your hand. (b) Assume all the charge is concentrated at the centre of the sphere, and use Coulomb's law to calculate the electric field at the surface of the sphere.

Vishal Gupta
Vishal Gupta
Numerade Educator
04:27

Problem 18

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 $3.63 \times 10^{16} \mathrm{~N} \cdot \mathrm{m}^2 / \mathrm{C}$ 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 inside the back cover); (c) the charge density on Mars, assuming all the charge is uniformly distributed over the planet's surface.

Andrew Eddins
Andrew Eddins
Emory University
03:09

Problem 19

How many excess electrons must be added to an isolated spherical conductor $32.0 \mathrm{~cm}$ in diameter to produce an electric field of $1150 \mathrm{~N} / \mathrm{C}$ just outside the surface?

Andrew Eddins
Andrew Eddins
Emory University
03:11

Problem 20

The electric field $0.400 \mathrm{~m}$ from a very long uniform line of charge is $840 \mathrm{~N} / \mathrm{C}$. How much charge is contained in a $2.00-\mathrm{cm}$ section of the line?

Vishal Gupta
Vishal Gupta
Numerade Educator
04:12

Problem 21

Engineering. 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) $y=0.600 \mathrm{~m}$ ?

Prashant Bana
Prashant Bana
Numerade Educator
07:33

Problem 22

(a) At a distance of $0.200 \mathrm{~cm}$ from the centre of a charged conducting sphere with radius $0.100 \mathrm{~cm}$, the electric field is $480 \mathrm{~N} / \mathrm{C}$. What is the electric field $0.600 \mathrm{~cm}$ from the centre of the sphere? (b) At a distance of $0.200 \mathrm{~cm}$ from the axis of a very long charged conducting cylinder with radius $0.100 \mathrm{~cm}$, the electric field is $480 \mathrm{~N} / \mathrm{C}$. What is the electric field $0.600 \mathrm{~cm}$ from the axis of the cylinder? (c) At a distance of $0.200 \mathrm{~cm}$ from a large uniform sheet of charge, the electric field is $480 \mathrm{~N} / \mathrm{C}$. What is the electric field $1.20 \mathrm{~cm}$ from the sheet?

Vishal Gupta
Vishal Gupta
Numerade Educator
12:26

Problem 23

A hollow, conducting sphere with an outer radius of $0.250 \mathrm{~m}$ and an inner radius of $0.200 \mathrm{~m}$ has a uniform surface charge density of $+6.37 \times 10^{-6} \mathrm{C} / \mathrm{m}^2$. A charge of $-0.500 \mu \mathrm{C}$ is now introduced into 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?

Andrew Eddins
Andrew Eddins
Emory University
06:45

Problem 24

A point charge of $-2.00 \mu \mathrm{C}$ is located in the centre of a spherical cavity of radius $6.50 \mathrm{~cm}$ inside an insulating charged solid. The charge density in the solid is $\rho=7.35 \times 10^{-4} \mathrm{C} / \mathrm{m}^3$. Calculate the electric field inside the solid at a distance of $9.50 \mathrm{~cm}$ from the centre of the cavity.

Linda Winkler
Linda Winkler
Numerade Educator
07:22

Problem 25

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 centre.

Vishal Gupta
Vishal Gupta
Numerade Educator
03:29

Problem 26

A conductor with an inner cavity, like that shown in Fig. $22.23 \mathrm{c}$, carries a total charge of $+5.00 \mathrm{nC}$. The charge within the cavity, insulated from the conductor, is $-6.00 \mathrm{nC}$. How much charge is on (a) the inner surface of the conductor and (b) the outer surface of the conductor?

Andrew Eddins
Andrew Eddins
Emory University
02:40

Problem 27

Apply Gauss's law to the Gaussian surfaces $S_2, S_3$ and $S_4$ in Fig. $22.21 \mathrm{~b}$ to calculate the electric field between and outside the plates.

Averell Hause
Averell Hause
Carnegie Mellon University
03:16

Problem 28

A square insulating sheet $80.0 \mathrm{~cm}$ on a side is held horizontally. The sheet has $7.50 \mathrm{nC}$ of charge spread uniformly over its area. (a) Calculate the electric field at a point $0.100 \mathrm{~mm}$ above the centre of the sheet. (b) Estimate the electric field at a point $100 \mathrm{~m}$ above the centre 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
04:11

Problem 29

Engineering. 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).

Averell Hause
Averell Hause
Carnegie Mellon University
11:03

Problem 30

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, as shown in Fig. 22.33. These surface charge densities have the values $\sigma_1=$ $-6.00 \mu \mathrm{C} / \mathrm{m}^2, \quad \sigma_2=+5.00 \mu \mathrm{C} / \mathrm{m}^2$, $\sigma_3=+2.00 \mu \mathrm{C} / \mathrm{m}^2$ 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
05:47

Problem 31

A negative charge $-Q$ is placed inside the cavity of a hollow metal solid. The outside of the solid is grounded by connecting a conducting wire between it and the earth. (a) Is there any excess charge induced on the inner surface of the piece of metal? If so, find its sign and magnitude. (b) Is there any excess charge on the outside of the piece of metal? Why or why not? (c) Is there an electric field in the cavity? Explain. (d) Is there an electric field within the metal? Why or why not? Is there an electric field outside the piece of metal? Explain why or why not. (e) Would someone outside the solid measure an electric field due to the charge $-Q$ ? Is it reasonable to say that the grounded conductor has shielded the region from the effects of the charge $-Q$ ? In principle, could the same thing be done for gravity? Why or why not?

Andrew Eddins
Andrew Eddins
Emory University
07:20

Problem 32

A cube has sides of length $L$. It is placed with one corner at the origin as shown in Fig. 22.31. The electric field is uniform and given by $\overrightarrow{\boldsymbol{E}}=-B \hat{i}+C \hat{j}-D \hat{\boldsymbol{k}}$, where $B, C$ and $D$ are positive constants. (a) Find the electric flux through each of the six cube faces $S_1, S_2, S_3, S_4, S_5$, and $S_6$. (b) Find the electric flux through the entire cube.

Cody Johnston
Cody Johnston
Vanderbilt University
04:32

Problem 33

The electric field $\overrightarrow{\boldsymbol{E}}$ in Fig. 22.34 is everywhere parallel to the $x$-axis, so the components $E_y$ and $E_z$ are zero. The $x$-component of the field $E_s$ depends on $x$ but not on $y$ and $z$. 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. 22.34? (b) What is the electric flux through surface II? (c) The volume shown in the figure 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 $\overrightarrow{\boldsymbol{E}}$ at the face opposite surface I? (d) Is the electric field produced only by charges within the slab, or is the field also due to charges outside the slab? How can you tell?

Keshav Singh
Keshav Singh
Numerade Educator
05:12

Problem 34

A flat, square surface with sides of length $L$ is described by the equations
$$
x=L \quad(0 \leq y \leq L, 0 \leq z \leq L)
$$
(a) Draw this square and show the $x$-, $y$ - and $z$-axes. (b) Find the electric flux through the square due to a positive point charge $q$ located at the origin $(x=0, y=0$, $z=0$ ).

Sanat Mukherjee
Sanat Mukherjee
Numerade Educator
08:27

Problem 35

The electric field $\overrightarrow{\boldsymbol{E}}_1$ at one face of a parallelepiped is uniform over the entire face and is directed out of the face. At the opposite face, the electric field $\overrightarrow{\boldsymbol{E}}_2$ is also uniform over the entire face and is directed into that face (Fig. 22.35). The two
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.
(FIGURE CANT COPY)

Keshav Singh
Keshav Singh
Numerade Educator
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
07:16

Problem 37

Engineering. 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.

Averell Hause
Averell Hause
Carnegie Mellon University
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
08:28

Problem 39

Repeat Problem 22.38, but now let the conducting tube have charge per unit length $-\alpha$. As in Problem 22.38, the line of charge has charge per unit length $+\alpha$.

Andrew Eddins
Andrew Eddins
Emory University
04:05

Problem 40

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 $r=3 R$.

Averell Hause
Averell Hause
Carnegie Mellon University
02:36

Problem 41

A small sphere with a mass of $0.002 \mathrm{~g}$ and carrying a charge of $5.00 \times 10^{-8} \mathrm{C}$ hangs from a thread near a very large, charged insultating sheet, as shown in Fig. 22.36. The charge density on the sheet is $2.50 \times 10^{-9} \mathrm{C} / \mathrm{m}^2$. Find the angle of the thread.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
03:08

Problem 42

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
Figure $\mathbf{2 2 . 3 6}$
(FIGURE CANT COPY)
(a) Derive expressions for the electric-field magnitude in terms of the distance, fiven the centse for the regions, $<u, a<t<b$, $b<r<c$ 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
01:37

Problem 43

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 $r$.

Averell Hause
Averell Hause
Carnegie Mellon University
View

Problem 44

A conducting spherical shell with inner radius $a$ and outer radius $b$ has a positive point charge $Q$ located at its centre. The total charge on the shell is $-3 Q$, and it is insulated from its surroundings (Fig. 22.37). (a) Derive expressions for the electric-field magnitude in terms
Figure 22.37
Problem 22.44. of the distance $r$ from the centre for the regions $r<a, a<r<b$ and $r>b$. (b) What is the surface charge density on the inner surface of the conducting shell? (c) What is the surface charge density on the outer surface of the conducting shell? (d) Sketch the electric field lines and the location of all charges. (e) Graph the electric-field magnitude as a function of $r$.
(FIGURE CANT COPY)

Emily Anderson
Emily Anderson
Numerade Educator
09:34

Problem 45

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. 22.38). The inner shell has total charge $+2 q$, and the outer shell has charge $+4 q$. (a) Calculate the electric field (magnitude and direction) in terms of $q$ and the distance $r$ from the common centre of the two shells for (i) $r<a$; (ii) $a<r<b$; (iii) $b<r<c$; (iv) $c<r<d$; (v) $r>d$. Show your results in a graph of the radial component of $\vec{E}$ 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?
(FIGURE CANT COPY)

Andrew Eddins
Andrew Eddins
Emory University
07:37

Problem 46

Repeat Problem 22.45, but now let the outer shell have charge $-2 q$. As in Problem 22.45, the inner shell has charge $+2 q$.

Mohit Khurana
Mohit Khurana
Texas A&M University
07:37

Problem 47

Repeat Problem 22.45, but now let the outer shell have charge $-4 q$. As in Problem 22.45, the inner shell has charge $+2 q$.

Mohit Khurana
Mohit Khurana
Texas A&M University
08:17

Problem 48

A solid conducting sphere with radius $R$ carries a positive total charge $Q$. The sphere is surrounded by an insulating shell with inner radius $R$ and outer radius $2 R$. The insulating shell has a uniform charge density $\rho$. (a) Find the value of $\rho$ so that the net charge of the entire system is zero. (b) If $\rho$ has the value found in part (a), find the electric field (magnitude and direction) in each of the regions $0<r<R, R<r<2 R$ and $r>2 R$. Show your results in a graph of the radial component of $\vec{E}$ as a function of $r$. (c) As a general rule, the electric field is discontinuous only at locations where there is a thin sheet of charge. Explain how your results in part (b) agree with this rule.

Mohit Khurana
Mohit Khurana
Texas A&M University
02:59

Problem 49

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) a distance $r>R$ from the centre of the shell (outside the shell) and (b) a distance $r<R$ from the centre of the shell (inside the shell).

Averell Hause
Averell Hause
Carnegie Mellon University
02:36

Problem 50

(a) How many excess electrons must be distributed uniformly within the volume of an isolated plastic sphere $30.0 \mathrm{~cm}$ in diameter to produce an electric field of $1150 \mathrm{~N} / \mathrm{C}$ just outside the surface of the sphere? (b) What is the electric field at a point $10.0 \mathrm{~cm}$ outside the surface of the sphere?

Prabhu Ramji
Prabhu Ramji
Numerade Educator
05:48

Problem 51

A single isolated, large conducting plate (Fig. 22.39) Figure 22.39 Problem 22.51. has a charge per unit area $\sigma$ on its surface. Because the plate is a conductor, the electric field at its surface is perpendicular to the surface and has magnitude $E=\sigma / \epsilon_0$ (a) In Example 22.7 (Section 22.4) it was shown that the field caused by a large, uniformly charged sheet with charge per unit area $\sigma$ has magnitude $E=\sigma / 2 \epsilon_{00}$ exactly half as much as for a charged conducting plate. Why is there a difference? (b) Regarding the charge distribution on the conducting plate as being two sheets of charge (one on each surface), each with charge per unit area $\sigma$, use the result of Example 22.7 and the principle of superposition to show that $E=0$ inside the plate and $E=\sigma / \epsilon_0$ outside the plate.
(FIGURE CANT COPY)

Sri Datta Vikas Buchemmavari
Sri Datta Vikas Buchemmavari
Numerade Educator
07:56

Problem 52

In the early years of the 20th century, a leading model of the structure of the atom was that of the 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 biscuit mixture. 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 equilibrium position of the electron is at the centre of the nucleus. (b) In Thomson's model, it was assumed that the positive material provided little or no resistance to the motion of the electron. 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 simple harmonic motion in Section 13.2. 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 or electrons in the atom. What would the radius of a Thomson-model atom have to be 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 F for data about the electron). (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 $10^{-15} \mathrm{~m}$.)
(FIGURE CANT COPY)

Keshav Singh
Keshav Singh
Numerade Educator
04:08

Problem 53

Using Thomson's (outdated) model
Figure $\mathbf{2 2 . 4 0}$
of the atom described in Problem 22.52, consider an atom consisting of two electrons, each of charge $-e$, embedded in a sphere of charge $+2 e$ and radius $R$. In equilibrium, each electron is a distance $d$ from the centre of the atom (Fig. 22.40). Find the distance $d$ in terms of the other properties of the atom.

Averell Hause
Averell Hause
Carnegie Mellon University
05:04

Problem 54

A slab of insulating material has Problem 22.53. 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 $d$ and may be treated as essentially infinite. The slab has a uniform positive charge density $\rho$. (a) Explain why the electric field due to the slab is zero at the centre of the slab $(x=0)$. (b) Using Gauss's law, find the electric field due to the slab (magnitude and direction) at all points in space.
(FIGURE CANT COPY)

Aparna Shakti
Aparna Shakti
Numerade Educator
05:49

Problem 55

Repeat Problem 22.54, but now let the charge density of the slab be given by $\rho(x)=\rho_0(x / d)^2$, where $\rho_0$ is a positive constant.

Salamat Ali
Salamat Ali
Numerade Educator
06:20

Problem 56

In Chapter 21 , several examples were given of calculating the force exerted on a point charge by other point charges in its surroundings. (a) Consider a positive point charge $+q$. Give an example of how you would place two other point charges of your choosing so that the net force on charge $+q$ will be zero. (b) If the net force on charge $+q$ is zero, then that charge is in equilibrium. The equilibrium will be stable if, when the charge $+q$ is displaced slightly in any direction from its position of equilibrium, the net force on the charge pushes it back towards the equilibrium position. For this to be the case, what must the direction of the electric field $\overrightarrow{\boldsymbol{E}}$ be due to the other charges at points surrounding the equilibrium position of $+q$ ? (c) Imagine that the charge $+q$ is moved very far away, and imagine a small Gaussian surface centred on the position where $+q$ was in equilibrium. By applying Gauss's law to this surface, show that it is impossible to satisfy the condition for stability described in part (b). In other words, a charge $+q$ cannot be held in stable equilibrium by electrostatic forces alone. This result is known as Earnshaw's theorem. (d) Parts (a)-(c) referred to the equilibrium of a positive point charge $+q$. Prove that Earnshaw's theorem also applies to a negative point charge $-q$.

Urvashi Arora
Urvashi Arora
Numerade Educator
10:20

Problem 57

A nonuniform, but spherically symmetric, distribution of charge has a charge density $\rho(r)$ given as follows:
$$
\begin{array}{ll}
\rho(r)=\rho_0(1-r / R) & \text { for } r \leq R \\
\rho(r)=0 & \text { for } r \geq R
\end{array}
$$
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
08:58

Problem 58

A nonuniform, but spherically symmetric, distribution of charge has a charge density $\rho(r)$ given as follows:
$$
\begin{array}{ll}
\rho(r)=\rho_0(1-4 r / 3 R) & \text { for } r \leq R \\
\rho(r)=0 & \text { for } r \geq R
\end{array}
$$
where $\rho_0$ is a positive constant. (a) Find the total charge contained in the charge distribution. (b) Obtain an expression for the electric field in the region $r \geq R$. (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.

Linda Winkler
Linda Winkler
Numerade Educator
04:22

Problem 59

The gravitational force between two point masses separated by a distance $r$ is proportional to $1 / r^2$, just like the electric force between two point charges. Because of this similarity between gravitational and electric interactions, there is also a Gauss's law for gravitation. (a) Let $\vec{g}$ be the acceleration due to gravity caused by a point mass $m$ at the origin, so that $\vec{g}=-\left(G m / r^2\right) \hat{r}$. Consider a spherical Gaussian surface with radius $r$ centred on this point mass, and show that the flux of $\vec{g}$ through this surface is given by
$$
\oint \vec{g} \cdot d \vec{A}=-4 \pi G m
$$
(b) By following the same logical steps used in Section 22.3 to obtain Gauss's law for the electric field, show that the flux of $\vec{g}$ through any closed surface is given by
$$
\oint \vec{g} \cdot d \vec{A}=-4 \pi G M_{\text {encl }}
$$
where $M_{\text {and }}$ is the total mass enclosed within the closed surface.

Andrew Eddins
Andrew Eddins
Emory University
02:36

Problem 60

Using Gauss's law for gravitation (derived in part (b) of Problem 22.59), show that the following statements are true: (a) For any spherically symmetric mass distribution with total mass $M$, the acceleration due to gravity outside the distribution is the same as though all the mass were concentrated at the centre. (Hint: See Example 22.5 in Section 22.4.) (b) At any point inside a spherically symmetric shell of mass, the acceleration due to gravity is zero. (Hint: See Example 22.5.) (c) If we could drill a hole through a spherically symmetric planet to its centre, and if the density were uniform, we would find that the magnitude of $\vec{g}$ is directly proportional to the distance $r$ from the centre. (Hint: See Example 22.9 in Section 22.4.) We proved these results in Section 12.6 using some fairly strenuous analysis; the proofs using Gauss's law for gravitation are much easier.

Prashant Bana
Prashant Bana
Numerade Educator
05:36

Problem 61

Engineering. (a) An insulating sphere with radius $a$ has a uniform charge density $\rho$. The sphere is not centred at the origin but at $\vec{r}=\vec{b}$. Show that the electric field inside the sphere is given by $\vec{E}=$ $\rho(\overrightarrow{\boldsymbol{r}}-\overrightarrow{\boldsymbol{b}}) / 3 \epsilon_0$. (b) An insulating sphere of radius $R$ has a spherical hole of radius $a$ located within its volume and centred a distance $b$ from the centre of the sphere, where
Problem 22.61. $a<b<R$ (a cross-section of the sphere is shown in Fig. 22.41). 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.
(FIGURE CANT COPY)

Keshav Singh
Keshav Singh
Numerade Educator
View

Problem 62

Engineering. A very long, solid insulating cylinder with radius $R$ has a cylindrical hole with radius $a$ bored along its entire length. The axis of the hole is a distance $b$ from the axis of the cylinder, where $a<b<R$ (Fig. 22.42). The solid material of the cylinder has a uniform volume charge density $\rho$. Find the magnitude and direction of the electric field $\vec{E}$ inside the hole, and show that $\overrightarrow{\boldsymbol{E}}$ is uniform over the entire hole.
(FIGURE CANT COPY)

Susan Hallstrom
Susan Hallstrom
Numerade Educator
07:40

Problem 63

Positive charge $Q$ is distributed uniformly over each of two spherical volumes with radius $R$. One sphere of charge is centred at the origin and the other at $x=2 R$ (Fig. 22.43). Find the magnitude and direction of the net electric field due to these two distributions of charge at the following points on the $x$-axis: (a) $x=0$; (b) $x=R / 2$; (c) $x=R$; (d) $x=3 R$.
(FIGURE CANT COPY)

Darman Khan
Darman Khan
Numerade Educator
07:37

Problem 64

Repeat Problem 22.63, but now let the left-hand sphere have positive charge $Q$ and let the right-hand sphere have negative charge $-Q$.

Mohit Khurana
Mohit Khurana
Texas A&M University
04:16

Problem 65

A hydrogen atom is made up of a proton of charge $+Q=$ $1.60 \times 10^{-19} \mathrm{C}$ and an electron of charge $-\mathrm{Q}=-1.60 \times 10^{-19} \mathrm{C}$. The proton may be regarded as a point charge at $r=0$, the centre of the atom. The motion of the electron causes its charge to be 'smeared out' into a spherical distribution around the proton, so that the electron is equivalent to a charge per unit volume of
$$
\rho(r)=-\frac{Q}{\pi a_0^3} e^{-2 r / a_0}
$$
where $a_0=5.29 \times 10^{-11} \mathrm{~m}$ is called the Bohr radius. (a) Find the total amount of the hydrogen atom's charge that is enclosed within a sphere with radius $r$ centred on the proton. Show that as $r \rightarrow \infty$, the enclosed charge goes to zero. Explain this result. (b) Find the electric field (magnitude and direction) caused by the charge of the hydrogen atom as a function of $r$. (c) Graph the electric-field magnitude $E$ as a function of $r$.

Suzanne W.
Suzanne W.
Numerade Educator
18:19

Problem 66

A region in space contains a total positive charge $Q$ that is distributed spherically such that the volume charge density $\rho(r)$ is given by
$$
\begin{array}{ll}
\rho(r)=\alpha & \text { for } r \leq R / 2 \\
\rho(r)=2 \alpha(1-r / R) & \text { for } R / 2 \leq r \leq R \\
\rho(r)=0 & \text { for } r \geq R
\end{array}
$$
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 $\vec{E}$ as a function of $r$. Do this separately for all three regions. Express your answers in terms of the total charge $Q$. $\mathrm{Be}$ sure to check that your results agree on the boundaries of the regions. (c) What fraction of the total charge is contained within the region $r \leq R / 2$ ? (d) If an electron with charge $q^{\prime}=-e$ is oscillating back and forth about $r=0$ (the centre of the distribution) with an amplitude less than $R / 2$, show that the motion is simple harmonic. (Hint: Review the discussion of simple harmonic motion in Section 13.2. If, and only if, the net force on the electron is proportional to its displacement from equilibrium, then the motion is simple harmonic.) (e) What is the period of the motion in part (d)? (f) If the amplitude of the motion described in part (e) is greater than $R / 2$, is the motion still simple harmonic? Why or why not?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
05:54

Problem 67

A region in space contains a total positive 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 the total charge $Q$. (c) What fraction of the total charge is contained within the region $R / 2 \leq r \leq R$ ? (d) What is the magnitude
of $\vec{E}$ 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? (See Challenge Problem 22.66.)

Manne Andergronde
Manne Andergronde
Numerade Educator