Physics for Scientists and Engineers with Modern Physics

Raymond A. Serway, John W. Jewett, Jr.

Chapter 24

Gauss’s Law - all with Video Answers

Educators

Chapter Questions

Problem 1

A 40.0-cm-diameter circular loop is rotated in a uni- form electric field until the position of maximum electric flux is found. The flux in this position is measured to be $5.20 \times 10^{5} \mathrm{N} \cdot \mathrm{m}^{2} / \mathrm{C} .$ What is the magnitude of the electric field?

Nathan Silvano

Nathan Silvano

Numerade Educator

Problem 2

A vertical electric field of magnitude $2.00 \times 10^{4} \mathrm{N} / \mathrm{C}$ exists above the Earth’s surface on a day when a thunderstorm is brewing. A car with a rectangular size of 6.00 m by 3.00 m is traveling along a dry gravel roadway sloping downward at $10.0^{\circ}$. Determine the electric flux through the bottom of the car.

Nathan Silvano

Numerade Educator

Problem 3

A flat surface of area 3.20 $\mathrm{m}^{2}$ is rotated in a uniform electric field of magnitude $E=6.20 \times 10^{5} \mathrm{N} / \mathrm{C}$ . Determine the electric flux through this area (a) when the electric field is perpendicular to the surface and (b) when the electric field is parallel to the surface.

Nathan Silvano

Numerade Educator

Problem 4

Consider a closed triangular box resting within a horizontal electric field of magnitude $E=7.80 \times 10^{4} \mathrm{N} / \mathrm{C}$ as shown in Figure P 24.4. Calculate the electric flux through ( a the vertical rectangular surface, (b) the slanted surface, and (c) the entire surface of the box.

Nathan Silvano

Numerade Educator

Problem 5

An electric field of magnitude 3.50 $\mathrm{kN} / \mathrm{C}$ is applied along the $x$ axis. Calculate the electric flux through a rectangular plane 0.350 $\mathrm{m}$ wide and 0.700 $\mathrm{m}$ long (a) if the plane is parallel to the $y$ z plane, (b) if the plane is parallel to the $x y$ plane, and (c) if the plane contains the $y$ axis and its normal makes an angle of $40.0^{\circ}$ with the $x$ axis.

Nathan Silvano

Numerade Educator

Problem 6

Find the net electric flux through the spherical closed surface shown in Figure P 24.6. The two charges on the right are inside the spherical surface.

Nathan Silvano

Numerade Educator

Problem 7

An uncharged, nonconducting, hollow sphere of radius 10.0 cm surrounds a $10.0-\mu \mathrm{C}$ charge located at the origin of a Cartesian coordinate system. A drill with a radius of 1.00 mm is aligned along the $z$ axis, and a hole is drilled in the sphere. Calculate the electric flux through the hole.

Nathan Silvano

Numerade Educator

Problem 8

A charge of 170$\mu \mathrm{C}$ is at the center of a cube of edge 80.0 cm. No other charges are nearby. (a) Find the flux through each face of the cube. (b) Find the flux through the whole surface of the cube. (c) What If? Would your answers to either part (a) or part (b) change if the charge were not at the center? Explain.

Nathan Silvano

Numerade Educator

Problem 9

The following charges are located inside a submarine: $5.00 \mu \mathrm{C},-9.00 \mu \mathrm{C}, 27.0 \mu \mathrm{C},$ and $-84.0 \mu \mathrm{C}$ . (a) Calculate the net electric flux through the hull of the submarine. (b) Is the number of electric field lines leaving the submarine greater than, equal to, or less than the number entering it?

Nathan Silvano

Numerade Educator

Problem 10

The electric field everywhere on the surface of a thin, spherical shell of radius 0.750 m is of magnitude 890 N/C and points radially toward the center of the sphere. (a) What is the net charge within the sphere’s surface? (b) What is the distribution of the charge inside the spherical shell?

Nathan Silvano

Numerade Educator

Problem 11

Four closed surfaces, $S_{1}$ through $S_{4},$ together with the charges $-2 Q, Q,$ and $-Q$ are sketched in Figure P 24.11. (The colored lines are the intersections of the surfaces with the page.) Find the electric flux through each surface.

Nathan Silvano

Numerade Educator

Problem 12

A particle with charge of 12.0$\mu \mathrm{C}$ is placed at the center of a spherical shell of radius $22.0 \mathrm{cm} .$ What is the total electric flux through (a) the surface of the shell and (b) any hemispherical surface of the shell? (c) Do the results depend on the radius? Explain.

Nathan Silvano

Numerade Educator

Problem 13

In the air over a particular region at an altitude of 500 m above the ground, the electric field is 120 N/C directed downward. At 600 m above the ground, the electric field is 100 N/C downward. What is the average volume charge density in the layer of air between these two elevations? Is it positive or negative?

Nathan Silvano

Numerade Educator

Problem 14

(a) Find the net electric flux through the cube shown in Figure P 24.14.
(b) Can you use Gauss’s law to find the electric field on the surface of this cube? Explain.

Nathan Silvano

Numerade Educator

Problem 15

An infinitely long line charge having a uniform charge per unit length $\lambda$ lies a distance $d$ from point $O$ as shown in Figure P 24.15 Determine the total electric flux through the surface of a sphere of radius $R$ centered at $O$ resulting from this line charge. Consider both cases, where (a) $R < d$ and (b) $R >d$ .

Nathan Silvano

Numerade Educator

Problem 16

(a) A particle with charge $q$ is located a distance $d$ from an infinite plane. Determine the electric flux through the plane due to the charged particle. (b) What If? A particle with charge q is located a very small distance from the center of a very large square on the line perpendicular to the square and going through its center. Determine the approximate electric flux through the square due to the charged particle. (c) How do the answers to parts (a) and (b) compare? Explain.

Nathan Silvano

Numerade Educator

Problem 17

A particle with charge $Q$ is located a small distance $\delta$ immediately above the center of the flat face of a hemisphere of radius $R$ as shown in Figure P 24.17. What is the electric flux (a) through the curved surface and (b) through the flat face as $\delta \rightarrow 0 ?$

Nathan Silvano

Numerade Educator

Problem 18

Find the net electric flux through (a) the closed spherical surface in a uniform electric field shown in Figure P 24.18a and (b) the closed cylindrical surface shown in Figure P 24.18b. (c) What can you conclude about the charges, if any, inside the cylindrical surface?

Nathan Silvano

Numerade Educator

Problem 19

A particle with charge $Q=5.00 \mu \mathrm{C}$ is located at the center of a cube of edge $L=0.100 \mathrm{m} .$ In addition, six other identical charged particles having $q=-1.00 \mu \mathrm{C}$ are positioned symmetrically around $Q$ as shown in Figure P 24.19. Determine the electric flux through one face of the cube.

Nathan Silvano

Numerade Educator

Problem 20

A particle with charge $Q$ is located at the center of a cube of edge $L .$ In addition, six other identical charged particles $q$ are positioned symmetrically around $Q$ as shown in Figure P 24.19. For each of these particles, $q$ is a negative number. Determine the electric flux through one face of the cube.

Nathan Silvano

Numerade Educator

Problem 22

The charge per unit length on a long, straight filament is $-90.0 \mu \mathrm{C} / \mathrm{m} .$ Find the electric field (a) 10.0 cm, (b) 20.0 cm, and (c) 100 cm from the filament, where distances are measured perpendicular to the length of the filament.

Nathan Silvano

Numerade Educator

Problem 23

A large, flat, horizontal sheet of charge has a charge per unit area of $9.00 \mu \mathrm{C} / \mathrm{m}^{2}$. Find the electric field just above the middle of the sheet.

Nathan Silvano

Numerade Educator

Problem 23

A large, flat, horizontal sheet of charge has a charge per unit area of $9.00 \mu \mathrm{C} / \mathrm{m}^{2} .$ Find the electric field just above the middle of the sheet.

Nathan Silvano

Numerade Educator

Problem 24

Determine the magnitude of the electric field at the surface of a lead-208 nucleus, which contains 82 protons and 126 neutrons. Assume the lead nucleus has a volume 208 times that of one proton and consider a proton to be a sphere of radius $1.20 \times 10^{-15} \mathrm{m}$.

Nathan Silvano

Numerade Educator

Problem 25

A $10.0-\mathrm{g}$ piece of Styrofoam carries a net charge of $-0.700 \mu \mathrm{C}$ and is suspended in equilibrium above the center of a large, horizontal sheet of plastic that has a uniform charge density on its surface. What is the charge per unit area on the plastic sheet?

Nathan Silvano

Numerade Educator

Problem 26

Suppose you fill two rubber balloons with air, suspend both of them from the same point, and let them hang down on strings of equal length. You then rub each with wool or on your hair so that the balloons hang apart with a noticeable separation between them. Make order-of-magnitude estimates of (a) the force on each, (b) the charge on each, (c) the field each creates at the center of the other, and (d) the total flux of electric field created by each balloon. In your solution, state the quantities you take as data and the values you measure or estimate for them.

Nathan Silvano

Numerade Educator

Problem 27

Consider a thin, spherical shell of radius 14.0 cm with a total charge of 32.0$\mu \mathrm{C}$ distributed uniformly on its surface. Find the electric field (a) 10.0 cm and (b) 20.0 cm from the center of the charge distribution.

Nathan Silvano

Numerade Educator

Problem 28

A nonconducting wall carries charge with a uniform density of 8.60$\mu \mathrm{Cm}^{2}$. (a) What is the electric field 7.00 cm in front of the wall if 7.00 cm is small compared with the dimensions of the wall? (b) Does your result change as the distance from the wall varies? Explain.

Nathan Silvano

Numerade Educator

Problem 29

A uniformly charged, straight filament 7.00 m in length has a total positive charge of $2.00 \mu \mathrm{C} .$ An uncharged cardboard cylinder 2.00 cm in length and 10.0 cm in radius surrounds the filament at its center, with the filament as the axis of the cylinder. Using reasonable approximations, find (a) the electric field at the surface of the cylinder and (b) the total electric flux through the cylinder.

Nathan Silvano

Numerade Educator

Problem 30

Assume the magnitude of the electric field on each face of the cube of edge $L=1.00 \mathrm{m}$ in Figure P 24.30 is uniform and the directions of the fields on each face as indicated. Find (a) the net electric flux through the cube and (b) the net charge inside the cube. (c) Could the net charge be a single point charge?

Nathan Silvano

Numerade Educator

Problem 31

A solid sphere of radius 40.0 cm has a total positive charge of 26.0$\mu \mathrm{C}$ uniformly distributed throughout its volume. Calculate the magnitude of the electric field (a) 0 cm, (b) 10.0 cm, (c) 40.0 cm, and (d) 60.0 cm from the center of the sphere.

Nathan Silvano

Numerade Educator

Problem 32

A cylindrical shell of radius 7.00 cm and length 2.40 m has its charge uniformly distributed on its curved surface. The magnitude of the electric field at a point 19.0 cm radially outward from its axis (measured from the midpoint of the shell) is 36.0 kN/C. Find (a) the net charge on the shell and (b) the electric field at a point 4.00 cm from the axis, measured radially outward from the midpoint of the shell.

Nathan Silvano

Numerade Educator

Problem 33

Consider a long, cylindrical charge distribution of radius $R$ with a uniform charge density $\rho .$ Find the electric field at distance $r$ from the axis, where $r < R$

Nathan Silvano

Numerade Educator

Problem 34

A particle with a charge of $-60.0 \mathrm{nC}$ is placed at the center of a nonconducting spherical shell of inner radius 20.0 cm and outer radius 25.0 cm. The spherical shell carries charge with a uniform density of $-1.33 \mu \mathrm{C} / \mathrm{m}^{3}$ . A proton moves in a circular orbit just outside the spherical shell. Calculate the speed of the proton.

Nathan Silvano

Numerade Educator

Problem 35

A long, straight metal rod has a radius of 5.00 cm and a charge per unit length of 30.0 nC/m. Find the electric field (a) 3.00 cm, (b) 10.0 cm, and (c) 100 cm from the axis of the rod, where distances are measured perpendicular to the rod’s axis.

Nathan Silvano

Numerade Educator

Problem 36

A positively charged particle is at a distance $R / 2$ from the center of an uncharged thin, conducting, spherical shell of radius $R$ . Sketch the electric field lines set up by this arrangement both inside and outside the shell.

Nathan Silvano

Numerade Educator

Problem 37

A solid metallic sphere of radius $a$ carries total charge $Q$ . No other charges are nearby. The electric field just outside its surface is $k_{e} Q / a^{2}$ radially outward. At this close point, the uniformly charged surface of the sphere looks exactly like a uniform flat sheet of charge. Is the electric field here given by $\sigma / \epsilon_{0}$ or by $\sigma / 2 \epsilon_{0} ?$

Nathan Silvano

Numerade Educator

Problem 38

Why is the following situation impossible? A solid copper sphere of radius 15.0 cm is in electrostatic equilibrium and carries a charge of 40.0 nC. Figure P 24.38 shows the magnitude of the electric field as a function of radial position $r$ measured from the center of the sphere.

Nathan Silvano

Numerade Educator

Problem 39

A very large, thin, flat plate of aluminum of area A has a total charge $Q$ uniformly distributed over its surfaces. Assuming the same charge is spread uniformly over the upper surface of an otherwise identical glass plate, compare the electric fields just above the center of the upper surface of each plate.

Nathan Silvano

Numerade Educator

Problem 40

A square plate of copper with 50.0-cm sides has no net charge and is placed in a region of uniform electric field of 80.0 kN/C directed perpendicularly to the plate. Find (a) the charge density of each face of the plate and (b) the total charge on each face.

Nathan Silvano

Numerade Educator

Problem 41

Two identical conducting spheres each having a radius of 0.500 cm are connected by a light, 2.00-m-long conducting wire. A charge of 60.0$\mu \mathrm{C}$ is placed on one of the conductors. Assume the surface distribution of charge on each sphere is uniform. Determine the tension in the wire.

Nathan Silvano

Numerade Educator

Problem 42

In a certain region of space, the electric field is $\overrightarrow{\mathbf{E}}=$ $6.00 \times 10^{3} x^{2} \hat{\mathbf{i}},$ where $\overrightarrow{\mathbf{E}}$ is in newtons per coulomb and $x$ is in meters. Electric charges in this region are at rest and remain at rest. (a) Find the volume density of electric charge at $x=0.300 \mathrm{m} .$ Suggestion: Apply Gauss's law to a box between $x=0.300 \mathrm{m}$ and $x=0.300 \mathrm{m}+d x .$ (b) Could this region of space be inside a conductor?

Nathan Silvano

Numerade Educator

Problem 43

A long, straight wire is surrounded by a hollow metal cylinder whose axis coincides with that of the wire. The wire has a charge per unit length of $\lambda,$ and the cylinder has a net charge per unit length of 2$\lambda$ . From this information, use Gauss’s law to find (a) the charge per unit length on the inner surface of the cylinder, (b) the charge per unit length on the outer surface of the cylinder, and (c) the electric field outside the cylinder a distance r from the axis.

Nathan Silvano

Numerade Educator

Problem 44

A thin, square, conducting plate 50.0 cm on a side lies in the $x y$ plane. A total charge of $4.00 \times 10^{-8} \mathrm{C}$ is placed on the plate. Find (a) the charge density on each face of the plate, (b) the electric field just above the plate, and (c) the electric field just below the plate. You may assume the charge density is uniform.

Nathan Silvano

Numerade Educator

Problem 45

Find the electric flux through the plane surface shown in Figure $\mathrm{P} 24.45$ if $\theta=60.0^{\circ}, E= 350 \mathrm{N} / \mathrm{C},$ and $d=5.00 \mathrm{cm} .$ The electric field is uniform over the entire area of the surface.

Nathan Silvano

Numerade Educator

Problem 46

Consider a plane surface in a uniform electric field as in Figure P 24.45, where $d=15.0 \mathrm{cm}$ and $\theta=70.0^{\circ} .$ If the net flux through the surface is $6.00 \mathrm{N} \cdot \mathrm{m}^{2} / \mathrm{C},$ find the magnitude of the electric field.

Nathan Silvano

Numerade Educator

Problem 47

A sphere of radius $R=1.00 \mathrm{m}$ surrounds a particle with charge $Q= 50.0\mu \mathrm{C}$ located at its center as shown in Figure P 24.47. Find the electric flux through a circular cap of half-angle $\theta=45.0^{\circ} .$

Nathan Silvano

Numerade Educator

Problem 48

A sphere of radius $R$ surrounds a particle with charge $Q$ located at its center as shown in Figure P 24.47. Find the electric flux through a circular cap of half-angle $\theta .$

Nathan Silvano

Numerade Educator

Problem 49

A nonuniform electric field is given by the expression
$$\overrightarrow{\mathbf{E}}=a y \hat{\mathbf{i}}+b z \hat{\mathbf{j}}+c x \hat{\mathbf{k}}$$
where $a, b,$ and $c$ are constants. Determine the electric flux through a rectangular surface in the $x y$ plane, extending from $x=0$ to $x=w$ and from $y=0$ to $y=h.$

Nathan Silvano

Numerade Educator

Problem 50

A hollow, metallic, spherical shell has exterior radius 0.750 m, carries no net charge, and is supported on an insulating stand. The electric field everywhere just outside its surface is 890 N/C radially toward the center of the sphere. Explain what you can conclude about (a) the amount of charge on the exterior surface of the sphere and the distribution of this charge, (b) the amount of charge on the interior surface of the sphere and its distribution, and (c) the amount of charge inside the shell and its distribution.

Nathan Silvano

Numerade Educator

Problem 51

A solid insulating sphere of radius $a=5.00 \mathrm{cm}$ carries a net positive charge of $Q=3.00 \mu \mathrm{C}$ uniformly distributed throughout its volume. Concentric with this sphere is a conducting spherical shell with inner radius $b=$ 10.0 $\mathrm{cm}$ and outer radius $c=15.0 \mathrm{cm}$ as shown in Figure P 24.51, having net charge $q=-1.00 \mu \mathrm{C} .$ Prepare a graph of the magnitude of the electric field due to this configuration versus $r$ for $0< r < 25.0 \mathrm{cm} .$

Nathan Silvano

Numerade Educator

Problem 52

A solid, insulating sphere of radius $a$ has a uniform charge density throughout its volume and a total charge $Q$ . Concentric with this sphere is an uncharged, conducting, hollow sphere whose inner and outer radii are $b$ and $c$ as shown in Figure P 24.51. We wish to understand completely the charges and electric fields at all locations. (a) Find the charge contained within a sphere of radius $r< a$ (b) From this value, find the magnitude of the electric field for $r< a$ (c) What charge is contained within a sphere of radius $r$ when $a< r < b ?$ (d) From this value, find the magnitude of the electric field for $r$ when $a< r < b$ . (e) Now consider $r$ when $b < r< $ c. What is the magnitude of the electric field for this range of values of $r ?(\mathrm{f})$ From this value, what must be the charge on the inner surface of the hollow sphere? (g) From part (f), what must be the charge on the outer surface of the hollow sphere? (h) Consider the three spherical surfaces of radii $a, b,$ and $c .$ Which of these surfaces has the largest magnitude of surface charge density?

Nathan Silvano

Numerade Educator

Problem 53

A uniformly charged spherical shell with positive surface charge density $\sigma$ contains a circular hole in its surface. The radius $r$ of the hole is small compared with the radius $R$ of the sphere. What is the electric field at the center of the hole? Suggestion: This problem can be solved by using the principle of superposition.

Nathan Silvano

Numerade Educator

Problem 54

Two infinite, nonconducting sheets of charge are parallel to each other as shown in Figure P 24.54. The sheet on the left has a uniform surface charge density $\sigma,$ and the one on the right has a uniform charge density $-\sigma$ . Calculate the electric field at points (a) to the left of, $(b)$ in between, and $(c)$ to the right of the two sheets. (d) What If? Find the electric fields in all three regions if both sheets have positive uniform surface charge densities of value $\sigma$ .

Nathan Silvano

Numerade Educator

Problem 55

For the configuration shown in Figure P 24.51, suppose $a=5.00 \mathrm{cm}, b=20.0 \mathrm{cm},$ and $c=25.0 \mathrm{cm} .$ Furthermore, suppose the electric field at a point 10.0 $\mathrm{cm}$ from the center is measured to be $3.60 \times 10^{3} \mathrm{N} / \mathrm{C}$ radially inward and the electric field at a point 50.0 cm from the center is of magnitude 200 N/C and points radially outward. From this information, find (a) the charge on the insulating sphere, (b) the net charge on the hollow conducting sphere, (c) the charge on the inner surface of the hollow conducting sphere, and (d) the charge on the outer surface of the hollow conducting sphere.

Nathan Silvano

Numerade Educator

Problem 56

An insulating solid sphere of radius $a$ has a uniform volume charge density and carries a total positive charge $Q .$ A spherical gaussian surface of radius $r,$ which shares a common center with the insulating sphere, is inflated starting from $r=0 .$ (a) Find an expression for the electric flux passing through the surface of the gaussian sphere as a function of $r$ for $r < a$ . (b) Find an expression for the electric flux for $ r > a$ (c) Plot the flux versus $r .$

Nathan Silvano

Numerade Educator

Problem 57

An infinitely long, cylindrical, insulating shell of inner radius $a$ and outer radius $b$ has a uniform volume charge density $\rho .$ A line of uniform linear charge density $\lambda$ is placed along the axis of the shell. Determine the electric field for (a) $r< a,(\mathrm{b}) a< r< b,$ and $(\mathrm{c}) r > b.$

Nathan Silvano

Numerade Educator

Problem 58

An early (incorrect) model of the hydrogen atom, suggested by J. J. Thomson, proposed that a positive cloud of charge $+$ e was uniformly distributed throughout the volume of a sphere of radius $R,$ with the electron (an equal-magnitude negatively charged particle $-e )$ at the center.(a) Using Gauss's law, show that the electron would be in equilibrium at the center and, if displaced from the center a distance $r< R,$ would experience a restoring force of the form $F=-K r,$ where $K$ is a constant. (b) Show that $K=k_{e} e^{2} / R^{3} .$ (c) Find an expression for the frequency $f$ of simple harmonic oscillations that an electron of mass $m_{e}$ would undergo if displaced a small distance $(< R)$ from the center and released. (d) Calculate a numerical value for $R$ that would result in a frequency of $2.47 \times 10^{15} \mathrm{Hz}$ , the frequency of the light radiated in the most intense line in the hydrogen spectrum.

Nathan Silvano

Numerade Educator

Problem 59

A slab of insulating material has a nonuniform positive charge density $\rho=C x^{2},$ where $x$ is measured from the center of the slab as shown in Figure P 24.59 and $C$ is a constant. The slab is infinite in the $y$ and $z$ directions. Derive expressions for the electric field in (a) the exterior regions $(|x|>d / 2)$ and $(b)$ the interior region of the slab $(-d / 2< x<$ $d / 2 ) .$

Nathan Silvano

Numerade Educator

Problem 60

A sphere of radius 2$a$ is made of a nonconducting material that has a uniform volume charge density $\rho .$ Assume the material does not affect the electric field. A spherical cavity of radius $a$ is now removed from the sphere as shown in Figure P 24.60. Show that the electric field within the cavity is uniform and is given by $E_{x}=0$ and $E_{y}=\rho a / 3 \epsilon_{0}.$

Nathan Silvano

Numerade Educator

Problem 61

A closed surface with dimensions $a=b=0.400 \mathrm{m}$ and $c=0.600 \mathrm{m}$ is located as shown in Figure P 24.61. The left edge of the closed surface is located at position $x=a$ . The
electric field throughout the region is nonuniform and is given by $\overrightarrow{\mathbf{E}}=\left(3.00+2.00 x^{2}\right) \hat{\mathbf{i}} \mathrm{N} / \mathrm{C},$ where $x$ is in meters. (a) Calculate the net electric flux leaving the closed surface. (b) What net charge is enclosed by the surface?

Brandy Heflin

Numerade Educator

Problem 62

A solid insulating sphere of radius $R$ has a nonuniform charge density that varies with $r$ according to the expression $\rho=A r^{2},$ where $A$ is a constant and $r < R$ is measured from the center of the sphere. (a) Show that the magnitude of the electric field outside $(r>R)$ the sphere is $\vec{E}=A R^{5} / 5 \epsilon_{0} r^{2} .$ (b) Show that the magnitude of the electric field inside $(r< R)$ the sphere is $E=A r^{3} / 5 \epsilon_{0} .$ Note: The volume element $d V$ for a spherical shell of radius $r$ and thickness $d r$ is equal to $4 \pi r^{2} d r .$

Nathan Silvano

Numerade Educator

Problem 63

A spherically symmetric charge distribution has a charge density given by $\rho=a / r,$ where $a$ is constant. Find the electric field within the charge distribution as a function of $r .$ Note: The volume element $d V$ for a spherical shell of radius $r$ and thickness $d r$ is equal to $4 \pi r^{2} d r .$

Nathan Silvano

Numerade Educator

Problem 64

A particle with charge $Q$ is located on the axis of a circle of radius $R$ at a distance $b$ from the plane of the circle (Fig. P 24.64). Show that if one- fourth of the electric flux from the charge passes through the circle, then $R=\sqrt{3} b .$

Jacob Schulze

Numerade Educator

Problem 65

An infinitely long insulating cylinder of radius $R$ has a volume charge density that varies with the radius as
$$\rho=\rho_{0}\left(a-\frac{r}{b}\right)$$
where $\rho_{0}, a,$ and $b$ are positive constants and $r$ is the distance from the axis of the cylinder. Use Gauss's law to determine the magnitude of the electric field at radial distances (a) $r< R$ and (b) $r>R.$

Nathan Silvano

Numerade Educator

Problem 66

A slab of insulating material (infinite in the $y$ and $z$ directions $)$ has a thickness $d$ and a uniform positive charge density $\rho .$ An edge view of the slab is shown in Figure P 24.59. (a) Show that the magnitude of the electric field a distance $x$ from its center and inside the slab is $E=\rho x / \epsilon_{0} .$ (b) What If? Suppose an electron of charge $-e$ and mass $m_{e}$ can move freely within the slab. It is released from rest at a distance $x$ from the center. Show that the electron exhibits simple harmonic motion with a frequency
$$f=\frac{1}{2 \pi} \sqrt{\frac{\rho e}{m_{e} \epsilon_{0}}}$$

Nathan Silvano

Numerade Educator