University Physics with Modern Physics

Hugh D. Young

Chapter 22

Gauss's Law - all with Video Answers

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Chapter Questions

Problem 1

A flat sheet of paper of area 0.250 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 N/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.

Averell Hause

Averell Hause

Carnegie Mellon University

Problem 2

A flat sheet is in the shape of a rectangle with sides of lengths 0.400 m and 0.600 m. The sheet is immersed in a uniform electric field of magnitude 90.0 N/C that is directed at 20$^\circ$ from the plane of the sheet ($\textbf{Fig. E22.2}$). Find the magnitude of the electric flux through the sheet.

Cody Johnston

Vanderbilt University

Problem 3

You measure an electric field of 1.25 $\times$ 10$^6$ N/C at a distance of 0.150 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 0.150 m? (b) What is the magnitude of this charge?

Ankur S

Numerade Educator

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 $E = \lambda/2\pi\varepsilon_0r$. Consider an imaginary cylinder with radius $r =$ 0.250 m and length $l =$ 0.400 m that has an infinite line of positive charge running along its axis. The charge per unit length on the line is $\lambda =$ 3.00 $\mu$C/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 m? (c) What is the flux through the cylinder if its length is increased to $l =$ 0.800 m?

Cody Johnston

Vanderbilt University

Problem 5

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

Averell Hause

Carnegie Mellon University

Problem 6

The cube in $\textbf{Fig. E22.6}$ has sides of length $L =$ 10.0 cm. The electric field is uniform, has magnitude $E =$ 4.00 $\times$ 10$^3$ N/C, and is parallel to the $xy$-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 $S_6$? (b) What is the total electric flux through all faces of the cube?

Cody Johnston

Vanderbilt University

Problem 7

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 pC, what are the magnitude and direction (inward or outward) of the net flux through the cell boundary?

Averell Hause

Carnegie Mellon University

Problem 8

The three small spheres shown in $\textbf{Fig. E22.8}$ carry charges $q_1 =$ 4.00 nC, $q_2 = -$7.80 nC, and $q_3 =$ 2.40 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?

Vishal Gupta

Numerade Educator

Problem 9

A charged paint is spread in a very thin uniform layer over the surface of a plastic sphere of diameter 12.0 cm, giving it a charge of -49.0 $\mu$C. Find the electric field (a) just inside the paint layer; (b) just outside the paint layer; (c) 5.00 cm outside the surface of the paint layer.

Averell Hause

Carnegie Mellon University

Problem 10

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

Cody Johnston

Vanderbilt University

Problem 11

A 6.20-$\mu$C point charge is at the center of a cube with sides of length 0.500 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 m long? Explain.

Averell Hause

Carnegie Mellon University

Problem 12

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$^{-15}$ 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$^{-10}$ 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?

Cody Johnston

Vanderbilt University

Problem 13

Two very long uniform lines of charge are parallel and are separated by 0.300 m. Each line of charge has charge per unit length +5.20 $\mu$C/m. What magnitude of force does one line of charge exert on a 0.0500-m section of the other line of charge?

Averell Hause

Carnegie Mellon University

Problem 14

A solid metal sphere with radius 0.450 m carries a net charge of 0.250 nC. Find the magnitude of the electric field (a) at a point 0.100 m outside the surface of the sphere and (b) at a point inside the sphere, 0.100 m below the surface.

Cody Johnston

Vanderbilt University

Problem 15

How many excess electrons must be added to an isolated spherical conductor 26.0 cm in diameter to produce an electric field of magnitude 1150 N/C just outside the surface?

Averell Hause

Carnegie Mellon University

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 -3.63 $\times$ 10$^{16}$ N $\cdot$ m2/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.

Cody Johnston

Vanderbilt University

Problem 17

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

Averell Hause

Carnegie Mellon University

Problem 18

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

Cody Johnston

Vanderbilt University

Problem 19

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

Averell Hause

Carnegie Mellon University

Problem 20

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

Mohit Khurana

Texas A&M University

Problem 21

The electric field at a distance of 0.145 m from the surface of a solid insulating sphere with radius 0.355 m is 1750 N/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 m from the center.

Averell Hause

Carnegie Mellon University

Problem 22

A point charge of -3.00 $\mu$C is located in the center of a spherical cavity of radius 6.50 cm that, in turn, is at the center of an insulating charged solid sphere. The charge density in the solid is $\rho =$ 7.35 $\times$ 10$^{-4}$ C/m$^3$. Calculate the electric field inside the solid at a distance of 9.50 cm from the center of the cavity.

Darshan Maheshwari

Darshan Maheshwari

Numerade Educator

Problem 23

An electron is released from rest at a distance of 0.300 m from a large insulating sheet of charge that has uniform surface charge density +2.90 $\times$ 10$^{-12}$ C/m2. (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 0.050 m from the sheet? (b) What is the speed of the electron when it is 0.050 m from the sheet?

Averell Hause

Carnegie Mellon University

Problem 24

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

Linda Winkler

Numerade Educator

Problem 25

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?

Averell Hause

Carnegie Mellon University

Problem 26

A very large, horizontal, nonconducting sheet of charge has uniform charge per unit area $\sigma =$ 5.00 $\times$ 10$^{-6}$ C/m$^2$. (a) A small sphere of mass $m =$ 8.00 $\times$ 10$^{-6}$ kg and charge $q$ is placed 3.00 cm above the sheet of charge and then released from rest. (a) If the sphere is to remain motionless when it is released, what must be the value of $q$? (b) What is $q$ if the sphere is released 1.50 cm above the sheet?

Linda Winkler

Numerade Educator

Problem 27

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

Averell Hause

Carnegie Mellon University

Problem 28

A square insulating sheet 80.0 cm on a side is held horizontally. The sheet has 4.50 nC of charge spread uniformly over its area. (a) Calculate the electric field at a point 0.100 mm above the center of the sheet. (b) Estimate the electric field at a point 100 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?

Mohit Khurana

Texas A&M University

Problem 29

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 s, 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

Carnegie Mellon University

Problem 30

Two very large, nonconducting plastic sheets, each 10.0 cm thick, carry uniform charge densities $\sigma_1, \sigma_2, \sigma_3$, and $\sigma_4$ on their surfaces ($\textbf{Fig. E22.30}$). These surface charge densities have the values $\sigma_1 = -$6.00 $\mu$C/m2, $\sigma_2 = +$5.00 mC/m2, $\sigma_3 = +$2.00 $\mu$C/m2, and $\sigma_4 = +$4.00 $\mu$C/m2. 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 cm from the left face of the left-hand sheet; (b) point $B$, 1.25 cm from the inner surface of the right-hand sheet; (c) point $C$, in the middle of the right-hand sheet.

Eduard Sanchez

Numerade Educator

Problem 31

At time $t =$ 0 a proton is a distance of 0.360 m from a very large insulating sheet of charge and is moving parallel to the sheet with speed 9.70 $\times$ 10$^2$ m/s. The sheet has uniform surface charge density 2.34 $\times$ 10$^{-9}$ C/m2. What is the speed of the proton at $t =$ 5.00 $\times$ 10$^{-8}$ s?

Averell Hause

Carnegie Mellon University

Problem 32

A very small object with mass 8.20 $\times$ 10$^{-9}$ kg and positive charge 6.50 $\times$ 10$^{-9}$ C is projected directly toward a very large insulating sheet of positive charge that has uniform surface charge density 5.90 $\times$ 10$^{-8}$ C/m2. The object is initially 0.400 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 m?

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 33

A small sphere with mass 4.00 $\times$ 10$^{-6}$ kg and charge 5.00 $\times$ 10$^{-8}$ C hangs from a thread near a very large, charged insulating sheet ($\textbf{Fig. P22.33}$). The charge density on the surface of the sheet is uniform and equal to -2.50 $\times$ 10$^{-9}$ C/m2. Find the angle of the thread.

Averell Hause

Carnegie Mellon University

Problem 34

A cube has sides of length $L =$ 0.300 m. One corner is at the origin (Fig. E22.6). The nonuniform electric field is given by $\overrightarrow{E} =$ (-5.00 N/C $\cdot$ m)$x\hat{\imath}$ + (3.00 N/C $\cdot$ 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.

Mohit Khurana

Texas A&M University

Problem 35

The electric field $\overrightarrow{E}$ in $\textbf{Fig. P22.35}$ 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 yz-plane (where $x =$ 0), $E_x =$ 125 N/C. (a) What is the electric flux through surface I in Fig. P22.35? (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 m thick. If there is a total charge of -24.0 nC within the volume shown, what are the magnitude and direction of $\overrightarrow{E}$ 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?

Averell Hause

Carnegie Mellon University

Problem 36

In a region of space there is an electric field $\overrightarrow{E}$ that is in the z-direction and that has magnitude $E =$ [964 N/(C $\cdot$ m)]$x$. Find the flux for this field through a square in the $xy$-plane at $z =$ 0 and with side length 0.350 m. One side of the square is along the $+x$-axis and another side is along the $+y$-axis.

Darshan Maheshwari

Darshan Maheshwari

Numerade Educator

Problem 37

The electric field $\overrightarrow{E_1}$ 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 $\overrightarrow{E_2}$ is also uniform over the entire face and is directed into that face ($\textbf{Fig. P22.37}$). The two faces in question are inclined at 30.0$^\circ$ from the horizontal, while both $\overrightarrow{E_1}$ and $\overrightarrow{E_2}$ are horizontal; $\overrightarrow{E_1}$ has a magnitude of 2.50 $\times$ 10$^4$ N/C, and $\overrightarrow{E_2}$ has a magnitude of 7.00 $\times$ 10$^4$ N/C. (a) Assuming that no other electric field lines cross the surfaces of the parallelepiped, determine the net charge contained within. (b) Is the electric field produced by the charges only within the parallelepiped, or is the field also due to charges outside the parallelepiped? How can you tell?

Keshav Singh

Numerade Educator

Problem 38

A long line carrying a uniform linear charge density +50.0 $\mu$C/m runs parallel to and 10.0 cm from the surface of a large, flat plastic sheet that has a uniform surface charge density of -100 $\mu$C/m2 on one side. Find the location of all points where an $a$ particle would feel no force due to this arrangement of charged objects.

Mohit Khurana

Texas A&M University

Problem 39

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 $r = 2c$. (d) Find the charge per unit length on the inner surface and on the outer surface of the outer cylinder.

Averell Hause

Carnegie Mellon University

Problem 40

A very long conducting tube (hollow cylinder) has inner radius a and outer radius $b$. It carries charge per unit length $+a$, where $a$ is a positive constant with units of C/m. A line of charge lies along the axis of the tube. The line of charge has charge per unit length $+a$. (a) Calculate the electric field in terms of a 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?

Linda Winkler

Numerade Educator

Problem 41

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 $r = 3R$.

Averell Hause

Carnegie Mellon University

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 sphere has no net charge. (a) Derive expressions for the electricfield 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 $r =$ 2c. (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$c$.

Mohit Khurana

Texas A&M University

Problem 43

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

Averell Hause

Carnegie Mellon University

Problem 44

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 ($\textbf{Fig. P22.44}$). (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 $r$.

Emily Anderson

Numerade Educator

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 $d$ ($\textbf{Fig. P22.45}$). The inner shell has total charge +2$q$, and the outer shell has charge +4$q$. (a) Calculate the electric field $\overrightarrow{E}$ (magnitude and direction) in terms of $q$ and the distance $r$ from the common center of the two shells for (i)$ r < a; \mathrm{(ii)} a < r < b; \mathrm{(iii)} b < r < c; \mathrm{(iv)} c < r < d; \mathrm{(v)} r > d$. Graph the radial component of $\overrightarrow{E}$ 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?

Averell Hause

Carnegie Mellon University

Problem 46

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

Mohit Khurana

Texas A&M University

Problem 47

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

Carnegie Mellon University

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 $\overrightarrow{E}$ (magnitude and direction) in each of the regions 0 $< r < R, R < r < 2R$, and $r > 2R$. Graph the radial component of $\overrightarrow{E}$ 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

Texas A&M University

Problem 49

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?

Averell Hause

Carnegie Mellon University

Problem 50

Early in the 20th 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$^{14}$ Hz? Compare your answer to the radii of real atoms, which are of the order of 10$^{-10}$ m (see Appendix F). (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$^{-15}$ m.)

Linda Winkler

Numerade Educator

Problem 51

Using Thomson's (outdated) model of the atom described in Problem 22.50, consider an atom consisting of two electrons, each of charge $-e$, embedded in a sphere of charge $+2e$ and radius $R$. In equilibrium, each electron is a distance $d$ from the center of the atom ($\textbf{Fig. P22.51}$). Find the distance $d$ in terms of the other properties of the atom.

Averell Hause

Carnegie Mellon University

Problem 52

(a) How many excess electrons must be distributed uniformly within the volume of an isolated plastic sphere 30.0 cm in diameter to produce an electric field of magnitude 1390 N/C just outside the surface of the sphere? (b) What is the electric field at a point 10.0 cm outside the surface of the sphere?

Mohit Khurana

Texas A&M University

Problem 53

A nonuniform, but spherically symmetric, distribution of charge has a charge density $\rho$($r$) given as follows: $$\rho(r) = \rho_0 \bigg(1 - \frac{r}{R}\bigg) \space \space \space \mathrm{for} \space r \leq R$$ $$\rho(r) = 0 \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \mathrm{for} \space r \leq R$$ where $\rho_0 = 3Q/{\pi}R^3$ 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.

Salamat Ali

Numerade Educator

Problem 54

A slab of insulating material has thickness 2d and is oriented so that its faces are parallel to the yz-plane and given by the planes $x = d$ and $x = -d$. The $y$- and $z$-dimensions of the slab are very large compared to $d$; treat them 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 center 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.

Aparna Shakti

Numerade Educator

Problem 55

Repeat Problem 22.54, but now let the charge density of the slab be given by $r(x) = \rho_0(x/d)^2$, where r0 is a positive constant.

Salamat Ali

Numerade Educator

Problem 56

A nonuniform, but spherically symmetric, distribution of charge has a charge density $\rho(r)$ given as follows: $$\rho(r) = \rho_0 \bigg(1 - \frac{4r}{3R}\bigg) \space \space \space \mathrm{for} \space r \leq R$$ $$\rho(r) = 0 \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \mathrm{for} \space r \leq R$$ 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 electricfield 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.

Mohit Khurana

Texas A&M University

Problem 57

(a) An insulating sphere with radius a has a uniform charge density $\rho$. The sphere is not centered at the origin but at $\vec{r} = \vec{b}$. Show that the electric field inside the sphere is given by $\overrightarrow{$} = \rho(\vec{r} - \vec{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 $\textbf{Fig. P22.57}$). The solid part of the sphere has a uniform volume charge density $\rho$. Find the magnitude and direction of the electric field $\overrightarrow{E}$ inside the hole, and show that $\overrightarrow{E}$ is uniform over the entire hole.

Suzanne W.

Numerade Educator

Problem 58

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 $a < b < R$ ($\textbf{Fig. P22.58}$). The solid material of the cylinder has a uniform volume charge density $\rho$. Find the magnitude and direction of the electric field $\overrightarrow{E}$ inside the hole, and show that $\overrightarrow{E}$ is uniform over the entire hole. ($Hint:$ See Problem 22.57.)

Mohit Khurana

Texas A&M University

Problem 59

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 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:

For each set of data, draw two graphs: one for $Er^2$ versus r and one for $Er$ versus $r$. (a) Use these graphs to determine which data set, A or 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.

Dading Chen

Numerade Educator

Problem 60

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 $r < R$ ($\textbf{Fig. P22.60}$). Calculate $\rho$.

Mohit Khurana

Texas A&M University

Problem 61

The volume charge density $\rho$ for a spherical charge distribution of radius $R =$ 6.00 mm is not uniform. $\textbf{Figure P22.61}$ 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 mm; (ii) 3.00 mm; (iii) 5.00 mm; (iv) 7.00 mm.

Averell Hause

Carnegie Mellon University

Problem 62

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 $$\rho(r) = 3ar/2R \space \space \space \space \space \mathrm{for} \space r \leq R/2$$ $$\rho(r) = \alpha[1-(r/R)^2] \space \space \space \space \mathrm{for} \space R/2 \leq r \leq R$$ $$\rho(r) = 0 \space \space \space \space \space \space \mathrm{for} \space r \geq R$$ Here $\alpha$ is a positive constant having units of C/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 $\overrightarrow{E}$ at $r = R/2$? (e) If an electron with charge $q' = -e$ is released from rest at any point in any of the three regions, the resulting motion will be oscillatory but not simple harmonic. Why?

Sheh Lit Chang

University of Washington

Problem 63

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

Averell Hause

Carnegie Mellon University

Problem 64

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

Mohit Khurana

Texas A&M University

Problem 65

What is the direction of $\overrightarrow{E}$ 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

Carnegie Mellon University

Problem 66

Which statement is true about $\overrightarrow{E}$ 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

Texas A&M University