Physics

Robert Coleman Richardson; Betty McCarthy Richardson; Alan Giambattista

Chapter 16 Electric Forces and Fields - all with Video Answers

Educators

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Chapter Questions

03:32

Problem 1

Find the total positive charge of all the protons in $1.0 \mathrm{mol}$ of water.

Zulfiqar Ali

Numerade Educator

02:10

Problem 2

Suppose a $1.0 \mathrm{g}$ nugget of pure gold has zero net charge. What would be its net charge after it has $1.0 \%$ of its electrons removed?

Zulfiqar Ali

Numerade Educator

03:20

A balloon, initially neutral, is rubbed with fur until it acquires a net charge of $-0.60 \mathrm{nC}$. (a) Assuming that only electrons are transferred, were electrons removed from the balloon or added to it? (b) How many electrons were transferred?

Zulfiqar Ali

Numerade Educator

04:09

Problem 4

A metallic sphere has a charge of $+4.0 \mathrm{nC}$. A negatively charged rod has a charge of $-6.0 \mathrm{nC}$. When the rod touches the sphere, $8.2 \times 10^{9}$ electrons are transferred. What are the charges of the sphere and the rod now?

Mukesh Devi

Numerade Educator

01:01

Problem 5

A hollow metal sphere carries a charge of $6.0 \mu \mathrm{C}$. An identical sphere carries a charge of $18.0 \mu$ C. The two spheres are brought into contact with each other, then separated. How much charge is on each?

Narayan Hari

Numerade Educator

02:49

Problem 6

A positively charged rod is brought near two uncharged conducting spheres of the same size that are initially touching each other (diagram a). The spheres are moved apart, and then the charged rod is removed (diagram b). (a) What is the sign of the net charge on sphere 1 in diagram b? (b) In comparison with the charge on sphere 1, how much and what sign of charge is on sphere $2 ?$

Zulfiqar Ali

Numerade Educator

04:28

Problem 7

A metal sphere A has charge $Q$. Two other spheres, $B$ and $C,$ are identical to $A$ except they have zero net charge, A touches $B$, then the two spheres are separated. B touches $C$, then those spheres are separated. Finally, $C$ touches A and those two spheres are separated. How much charge is on each sphere?

Zulfiqar Ali

Numerade Educator

04:40

Problem 8

Repeat Problem 7 with a slight change. The difference this time is that sphere $C$ is grounded while it is touching $\mathrm{B}$, but $\mathrm{C}$ is not grounded at any other time. What is the final charge on each sphere?

Vishal Gupta

Numerade Educator

03:27

Problem 9

Five conducting spheres are charged as shown. All have the same magnitude net charge except $E$, whose net charge is zero. Which pairs are attracted to each other and which are repelled by each other when they are brought near each other, but well away from the other spheres?

Zulfiqar Ali

Numerade Educator

03:14

Problem 10

In each of five situations, two point charges $\left(Q_{1}, Q_{2}\right)$ are separated by a distance $d$. Rank them in order of the magnitude of the electric force on $Q_{1}$, from largest to smallest.
(a) $Q_{1}=1 \mu C, Q_{2}=2 \mu C, d=1 \mathrm{m}$
(b) $Q_{1}=2 \mu \mathrm{C}, Q_{2}=-1 \mu \mathrm{C}, d=1 \mathrm{m}$
(c) $Q_{1}=2 \mu C, Q_{2}=-4 \mu C, d=4 m$
(d) $Q_{1}=-2 \mu \mathrm{C}, Q_{2}=2 \mu \mathrm{C}, d=2 \mathrm{m}$
(e) $Q_{1}=4 \mu \mathrm{C}, Q_{2}=-2 \mu \mathrm{C}, d=4 \mathrm{m}$

Narayan Hari

Numerade Educator

01:01

Problem 11

If the electric forces of repulsion between two 1.0 charges have magnitude $10 \mathrm{N}$, how far apart are they?

Narayan Hari

Numerade Educator

01:37

Problem 12

Two small metal spheres are $25.0 \mathrm{cm}$ apart. The spheres have equal amounts of negative charge and repel each other with forces of magnitude $0.036 \mathrm{N}$. What is the charge on each sphere?

Narayan Hari

Numerade Educator

02:05

Problem 13

What is the ratio of the electric force to the gravitational force due to a proton on an electron separated by $5.3 \times 10^{-11} \mathrm{m}$ (the radius of a hydrogen atom)?

Narayan Hari

Numerade Educator

01:15

Problem 14

How many electrons must be removed from each of two $5.0 \mathrm{kg}$ copper spheres to make the electric forces of repulsion between them equal in magnitude to the gravitational forces of attraction between them? Assume the distance between the spheres is large compared with their diameters.

Narayan Hari

Numerade Educator

04:45

Problem 15

$\mathrm{A}+2.0 \mathrm{nC}$ point charge is $3.0 \mathrm{cm}$ away from a $-3.0 \mathrm{nC}$ point charge. (a) What are the magnitude and direction of the electric force acting on the $+2.0 \mathrm{nC}$ charge?
(b) What are the magnitude and direction of the electric force acting on the $-3.0 \mathrm{nC}$ charge?

Zulfiqar Ali

Numerade Educator

03:22

Problem 16

Two metal spheres separated by a distance much greater than either sphere's radius have equal mass $m$ and equal electric charge $q .$ What is the ratio of charge to mass $q / m$ in $\mathrm{C} / \mathrm{kg}$ if the electrical and gravitational forces balance?

Zulfiqar Ali

Numerade Educator

04:32

Problem 17

In the figure, a third point charge $-q$ is placed at point $P .$ What is the electric force on $-q$ due to the other two point charges?

Zulfiqar Ali

Numerade Educator

01:09

Problem 18

Two point charges are separated by a distance $r$ repel each other with forces of magnitude $F$. If $t$ separation is reduced to 0.25 times the original va what is the magnitude of the forces of repulsion?

Narayan Hari

Numerade Educator

01:02

Problem 19

$\mathrm{A} \mathrm{K}^{+}$ ion and $\mathrm{a} \mathrm{Cl}^{-}$ ion are directly across from each other on opposite sides of a cell membrane $9.0 \mathrm{nm}$ thick. What is the electric force on the $\mathrm{K}^{+}$ ion due to the $\mathrm{Cl}^{-}$ ion? Ignore the presence of other charges.

Narayan Hari

Numerade Educator

06:24

Problem 20

In a DNA molecule, the base pair adenine and thymine is held together by two hydrogen bonds (see Fig. 16.5 ). Let's model one of these hydrogen bonds as four point charges arranged along a straight line. Using the information in the figure, calculate the magnitude of the net electric force exerted by one base on the other.

Debra Mangion

Numerade Educator

03:27

Problem 21

Three point charges are fixed in place in a right triangle, as shown in the figure. What is the electric force on the $-0.60 \mu \mathrm{C}$ charge due to the other two charges?

Narayan Hari

Numerade Educator

09:54

Problem 22

Three point charges are fixed in place in a right triangle, as shown in the figure. What is the electric force on the $+1.0 \mu \mathrm{C}$ charge due to the other two charges?

Vishal Gupta

Numerade Educator

05:30

Problem 23

Three point charges are fixed in place in a right triangle, as shown in the figure. What is the electric force on the $+1.0 \mu C$ charge due to the other two charges?

Vishal Gupta

Numerade Educator

07:48

Problem 24

Two Styrofoam balls with the same mass $m=9.0 \times 10^{-8} \mathrm{kg}$ and the same positive charge $Q$ are suspended from the same point by insulating threads of length $L=0.98 \mathrm{m} .$ The separation of the balls is $d=0.020 \mathrm{m} .$ What is the charge $Q ?$

Zulfiqar Ali

Numerade Educator

08:48

Problem 25

Using the three point charges of Example $16.3,$ find the magnitude of the force on $q_{1}$ due to the other two charges, $q_{2}$ and $q_{3}$. [ Hint: After finding the force on $q_{1}$ due to $q_{2}$ separate that force into $x$ - and $y$ -components.]

Vishal Gupta

Numerade Educator

06:10

Problem 26

An equilateral triangle has a point charge $+q$ at each of the three vertices $(A, B, C)$. Another point charge $Q$ is placed at $D$, the midpoint of the side $B C$. Solve for $Q$ if the total electric force on the charge at $A$ due to the charges at $B, C,$ and $D$ is zero.

Zulfiqar Ali

Numerade Educator

01:05

Problem 27

A small sphere with a charge of $-0.60 \mu \mathrm{C}$ is placed in a uniform electric field of magnitude $1.2 \times 10^{6} \mathrm{N} / \mathrm{C}$ pointing to the west. What is the magnitude and direc tion of the electric force on the sphere?

Narayan Hari

Numerade Educator

01:10

Problem 28

The electric field across a cell membrane is $1.0 \times$ $10^{7} \mathrm{N} / \mathrm{C}$ directed into the cell. (a) If a pore opens, which way do sodium ions $\left(\mathrm{Na}^{+}\right)$ flow $-$ into the cell or out of the cell?
(b) What is the magnitude of the electric force on the sodium ion? The charge on the sodium ion is +e.

Narayan Hari

Numerade Educator

04:01

Problem 29

What are the magnitude and direction of the acceleration of a proton at a point where the electric field has magnitude $33 \mathrm{kN} / \mathrm{C}$ and is directed straight up?

Zulfiqar Ali

Numerade Educator

03:03

Problem 30

What are the magnitude and direction of the electric field midway between two point charges, $-15 \mu C$ and $+12 \mu C,$ that are $8.0 \mathrm{cm}$ apart?

Zulfiqar Ali

Numerade Educator

03:03

Problem 31

What are the magnitude and direction of the electric field midway between two point charges, $-15 \mu \mathrm{C}$ and $+12 \mu C,$ that are $8.0 \mathrm{cm}$ apart?

Zulfiqar Ali

Numerade Educator

01:29

Problem 32

An electron traveling horizontally from west to east enters a region where a uniform electric field is directed upward. What is the direction of the electric force exerted on the electron once it has entered the field?

Zulfiqar Ali

Numerade Educator

02:44

Problem 33

Rank points $A-E$ in order of the magnitude of the electric field, from largest to smallest.

Vishal Gupta

Numerade Educator

03:10

Problem 34

What is the electric field at $x=d$ (point $P$ )?

Vishal Gupta

Numerade Educator

02:34

Problem 35

What is the electric field at $x=2 d$ (point $S$ )?

Vishal Gupta

Numerade Educator

04:25

Problem 36

Are there any points not on the $x$ -axis where $\overrightarrow{\mathbf{E}}=0 ?$ Explain

Vishal Gupta

Numerade Educator

06:26

Problem 37

On the $x$ -axis, in which of the three regions $x<0$, $0<x<3 d,$ and $x>3 d$ is there a point where $\overrightarrow{\mathbf{E}}=0 ?$ Explain.

Vishal Gupta

Numerade Educator

15:29

Problem 38

(a) Find the $x$ -coordinates of the point(s) on the $x$ -axis where $\overrightarrow{\mathbf{E}}=0 .$ (b) Sketch a graph of $E_{x}$ vs. $x$ for points on the $x$ -axis.

Mark J

Numerade Educator

04:42

Problem 39

Sketch the electric field lines in the plane of the page due to the charges shown in the diagram.

Zulfiqar Ali

Numerade Educator

03:08

Problem 40

Sketch the electric field lines near two isolated and equal negative point charges. Include arrowheads to show the field directions.

Narayan Hari

Numerade Educator

07:47

Problem 41

Find the electric field at point $B$, midway between the upper left and right corners.

Vishal Gupta

Numerade Educator

07:19

Problem 42

Find the electric field at point $C$ the center of the square.

Debra Mangion

Numerade Educator

08:50

Problem 43

Find the electric field at point $A$, the upper left corner.

Vishal Gupta

Numerade Educator

12:17

Problem 44

Where would you place a third small object with the same charge so that the electric field is zero at the corner of the square labeled $A$ ?

Debra Mangion

Numerade Educator

05:05

Problem 45

Three point charges are placed on the $x$ -axis. A charge of $3.00 \mu \mathrm{C}$ is at the origin. A charge of $-5.00 \mu \mathrm{C}$ is at $20.0 \mathrm{cm},$ and a charge of $8.00 \mu \mathrm{C}$ is at $35.0 \mathrm{cm} .$ What is the force on the charge at the origin?

Zulfiqar Ali

Numerade Educator

06:00

Problem 46

Two equal charges $(Q=+1.00 \mathrm{nC})$ are situated at the diagonal corners $A$ and $B$ of a square of side $1.0 \mathrm{m}$ What is the magnitude of the electric field at point $D ?$

Zulfiqar Ali

Numerade Educator

09:05

Problem 47

Suppose a charge $q$ is placed at point $x=0, y=0 .$ A second charge $q$ is placed at point $x=8.0 \mathrm{m}, y=0 .$ What charge must be placed at the point $x=4.0 \mathrm{m}, y=0$ in order that the field at the point $x=4.0 \mathrm{m}, y=3.0 \mathrm{m}$ be zero?

Vishal Gupta

Numerade Educator

04:10

Problem 48

Two point charges, $q_{1}=+20.0 \mathrm{nC}$ and $q_{2}=+10.0 \mathrm{nC}$ are located on the $x$ -axis at $x=0$ and $x=1.00 \mathrm{m},$ respectively. Where on the $x$ -axis is the electric field equal to zero?

Zulfiqar Ali

Numerade Educator

06:37

Problem 49

Two electric charges, $q_{1}=+20.0 \mathrm{nC}$ and $q_{2}=+10.0 \mathrm{nC}$, are located on the $x$ -axis at $x=0 \mathrm{m}$ and $x=1.00 \mathrm{m},$ respectively. What is the magnitude of the electric field at the point $x=0.50 \mathrm{m}, y=0.50 \mathrm{m} ?$

Zulfiqar Ali

Numerade Educator

04:54

Problem 50

In each of six situations, a particle (mass $m$, charge $q$ ) is located at a point where the electric field has magnitude $E .$ No other forces act on the particles. Rank them in order of the magnitude of the particle's acceleration, from largest to smallest.
(a) $m=6$ pg, $q=5 \mathrm{nC}, E=40 \mathrm{N} / \mathrm{C}$
(b) $m=3$ pg, $q=-5$ nC, $E=40 \mathrm{N} / \mathrm{C}$
(c) $m=3$ pg, $q=-10 \mathrm{nC}, E=80 \mathrm{N} / \mathrm{C}$
(d) $m=6$ pg, $q=-1$ nC, $E=200 \mathrm{N} / \mathrm{C}$
(e) $m=1$ pg $, q=3 \mathrm{nC}, E=300 \mathrm{N} / \mathrm{C}$
(f) $m=3$ pg, $q=-1$ nC, $E=100 \mathrm{N} / \mathrm{C}$

Vishal Gupta

Numerade Educator

04:47

Problem 51

An electron is placed in a uniform electric field of strength $232 \mathrm{N} / \mathrm{C}$. If the electron is at rest at the origin of a coordinate system at $t=0$ and the electric field is in the positive $x$ -direction, what are the $x$ - and $y$ -coordinates of the electron at $t=2.30 \mathrm{ns} ?$

Zulfiqar Ali

Numerade Educator

03:12

Problem 52

An electron is projected horizontally into the space between two oppositely charged metal plates. The electric field between the plates is $500.0 \mathrm{N} / \mathrm{C},$ directed up.
(a) While in the field, what is the force on the electron?
(b) If the vertical deflection of the electron as it leaves the plates is $3.00 \mathrm{mm},$ how much has its kinetic energy increased due to the electric field?

Zulfiqar Ali

Numerade Educator

03:49

Problem 53

A horizontal beam of electrons initially moving at $4.0 \times 10^{7} \mathrm{m} / \mathrm{s}$ is deflected vertically by the vertical electric field between oppositely charged parallel plates. The magnitude of the field is $2.00 \times 10^{4} \mathrm{N} / \mathrm{C}$.
(a) What is the direction of the field between the plates? (b) What is the charge per unit area on the plates? (c) What is the vertical deflection $d$ of the electrons as they leave the plates?

Zulfiqar Ali

Numerade Educator

08:02

Problem 54

A particle with mass $2.30 \mathrm{g}$ and charge $+10.0 \mu \mathrm{C}$ enters through a small hole in a metal plate with a speed of $8.50 \mathrm{m} / \mathrm{s}$ at an angle of $55.0^{\circ} .$ The uniform $\overrightarrow{\mathbf{E}}$ field in the region above the plate has magnitude $6.50 \times 10^{3} \mathrm{N} / \mathrm{C}$ and is directed downward. The region above the metal plate is essentially a vacuum, so there is no air resistance. (a) Can you ignore the force of gravity when solving for the horizontal distance traveled by the particle? Why or why not? (b) How far will the $\Delta x,$ before it hits the metal plate?

Vishal Gupta

Numerade Educator

07:08

Problem 55

Consider the same situation as in Problem $54,$ but with a proton entering through the small hole at the same angle with a speed of $v=8.50 \times 10^{5} \mathrm{m} / \mathrm{s}$
(a) Can you ignore the force of gravity when solving this problem for the horizontal distance traveled by the proton? Why or why not?
(b) How far will the proton travel, $\Delta x$ before it hits the metal plate?

Vishal Gupta

Numerade Educator

04:42

Problem 56

Some forms of cancer can be treated using proton therapy in which proton beams are accelerated to high energies, then directed to collide into a tumor, killing the malignant cells. Suppose a proton accelerator is $4.0 \mathrm{m}$ long and must accelerate protons from rest to a speed of $1.0 \times 10^{7} \mathrm{m} / \mathrm{s} .$ Ignore any relativistic effects (Chapter 26 ) and determine the magnitude of the average electric field that could accelerate these protons.

Zulfiqar Ali

Numerade Educator

08:19

Problem 57

If the electric field between plates (A) is $1.0 \times 10^{3} \mathrm{N} / \mathrm{C}$ in the $+x$ -direction, what is the horizontal deflection $(\Delta x)$ of the beam as it exits the region between plates (A)?

Vishal Gupta

Numerade Educator

08:42

Problem 58

The electric field between plates (A) is zero. As the beam exits the space between plates (B), it has been deflected $2.0 \mathrm{mm}$ downward $(\Delta y=-2.0 \mathrm{mm}) .$ What is the electric field between plates (B)?

Vishal Gupta

Numerade Educator

08:42

Problem 59

The electric field between plates (A) is zero. As the beam exits the space between plates (B), it has been deflected $2.0 \mathrm{mm}$ downward $(\Delta y=-2.0 \mathrm{mm}) .$ In what direction is the beam moving now?

Vishal Gupta

Numerade Educator

06:41

Problem 60

The inner sphere has a net charge of $+6 \mu \mathrm{C}$ and the shell has a net charge of $+6 \mu C$.

Jose Carlos

Numerade Educator

05:02

Problem 61

The inner sphere has a net charge of $+6 \mu C$ and the shell has a net charge of $-6 \mu \mathrm{C}$.

Jose Carlos

Numerade Educator

06:17

Problem 62

The inner sphere has a net charge of $-6 \mu C$ and the shell has a net charge of $+2 \mu C$.

Jose Carlos

Numerade Educator

01:45

Problem 63

A negative point charge $-Q$ is situated near a large metal plate that has a total charge of $+Q .$ Sketch the electric field lines.

Zulfiqar Ali

Numerade Educator

01:03

Problem 64

A conductor in electrostatic equilibrium has a cavity that contains a point charge $q_{1}=+5 \mu \mathrm{C}$. Outside the conductor is another point charge $q_{2}=-12 \mu \mathrm{C}$. The conductor itself carries a net charge $-4 \mu \mathrm{C}$. Find the net charge on (a) the inner surface of the conductor and
(b) the outer surface of the conductor.

Narayan Hari

Numerade Educator

01:12

Problem 65

A conductor in electrostatic equilibrium has a cavity that contains two point charges: $q_{1}=+5 \mu \mathrm{C}$ and $q_{2}=$ $-12 \mu C .$ The conductor itself carries a net charge $-4 \mu C$ Find the net charge on (a) the inner surface of the conductor and (b) the outer surface of the conductor.

Narayan Hari

Numerade Educator

05:38

Problem 66

Two oppositely charged parallel plates produce a uniform electric field between them. An uncharged metal sphere is placed between the plates. Assume that the sphere is small enough that it does not affect the charge distribution on the plates. Sketch the electric field lines between the plates once electrostatic equilibrium is reached.

Vishal Gupta

Numerade Educator

04:53

Problem 67

Two metal spheres of the same radius $R$ are given charges of equal magnitude and opposite sign. No other charges are nearby. Sketch the electric field lines when the center-to-center distance between the spheres is approximately $3 R$.

Vishal Gupta

Numerade Educator

01:47

Problem 68

A hollow conducting sphere of radius $R$ carries a negative charge $-q .$ (a) Write expressions for the electric field $\overrightarrow{\mathbf{E}}$ inside $(r<R)$ and outside $(r>R)$ the sphere. Also indicate the direction of the field.
(b) Sketch a graph of the field strength as a function of $r .[$Hint: See Conceptual Example 16.8 .1

Narayan Hari

Numerade Educator

02:07

Problem 69

A conducting sphere is placed within a conducting spherical shell. The conductors are in electrostatic equilibrium. The inner sphere has a radius of $1.50 \mathrm{cm}$ the inner radius of the spherical shell is $2.25 \mathrm{cm},$ and the outer radius of the shell is $2.75 \mathrm{cm}$. If the inner sphere has a charge of $230 \mathrm{nC}$ and the spherical shell has zero net charge, (a) what is the magnitude of the electric field at a point $1.75 \mathrm{cm}$ from the center?
(b) What is the electric field at a point $2.50 \mathrm{cm}$ from the center?
(c) What is the electric field at a point $3.00 \mathrm{cm}$ from the center? [Hint: What must be true about the electric field inside a conductor in electrostatic equilibrium?]

Zulfiqar Ali

Numerade Educator

01:18

Problem 70

In fair weather, over flat ground, there is a downward electric field of about $150 \mathrm{N} / \mathrm{C}$. Assume that Earth is a conducting sphere with charge on its surface. If the electric field just outside is $150 \mathrm{N} / \mathrm{C}$ pointing radially inward, calculate the total charge on Earth and the charge per unit area.

Narayan Hari

Numerade Educator

03:34

Problem 71

(a) Find the electric flux through each side of a cube of edge length $a$ in a uniform electric field of magnitude $E$. The field direction is perpendicular to two of the faces.
(b) What is the total flux through the cube?

Zulfiqar Ali

Numerade Educator

01:43

Problem 72

In a uniform electric field of magnitude $E$, the field lines cross through a rectangle of area $A$ at an angle of $60.0^{\circ}$ with respect to the plane of the rectangle. What is the flux through the rectangle?

Zulfiqar Ali

Numerade Educator

01:36

Problem 73

An object with a charge of $0.890 \mu \mathrm{C}$ is placed at the center of a cube. What is the electric flux through one surface of the cube?

Zulfiqar Ali

Numerade Educator

06:15

Problem 74

In this problem, you can show from Coulomb's law that the constant of proportionality in Gauss's law must be $1 / \epsilon_{0}$. Imagine a sphere with its center at a point charge
q. (a) Write an expression for the electric flux in terms of the field strength $E$ and the radius $r$ of the sphere. $[$Hint: The field strength $E$ is the same everywhere on the sphere and the field lines cross the sphere perpendicular to its surface. $]$ (b) Use Gauss's law in the form $\Phi_{\mathrm{L}}=c q$ (where $c$ is the constant of proportionality) and the electric field strength given by Coulomb's law to show that $c=1 / \epsilon_{0}$

Vishal Gupta

Numerade Educator

08:10

Problem 75

(a) Use Gauss"s law to prove that the electric field outside any spherically symmetrical charge distribution is the same as if all of the charge were concentrated into a point charge. (b) Now use Gauss's law to prove that the electric field inside a spherically symmetrical charge distribution is zero if none of the charge is at a distance from the center less than that of the point where we determine the field.

Vishal Gupta

Numerade Educator

09:32

Problem 76

Using the results of Problem $75,$ we can find the electric field at any radius for any spherically symmetrical charge distribution. A solid sphere of charge of radius $R$ has a total charge of $q$ uniformly spread throughout the sphere.
(a) Find the magnitude of the electric field for $r \geq R$
(b) Find the magnitude of the electric field for $r \leq R .$ (c) Sketch a graph of $E(r)$ for $0 \leq r \leq 3 R$

Vishal Gupta

Numerade Educator

03:03

Problem 77

An electron is suspended at a distance of $1.20 \mathrm{cm}$ above a uniform line of charge. What is the linear charge density of the line of charge? Ignore end effects.

Zulfiqar Ali

Numerade Educator

02:39

Problem 78

A thin, flat sheet of charge has a uniform surface charge density $\sigma($ ol 2 on each side). (a) Sketch the field lines due to the sheet.
(b) Sketch the field lines for an infinitely large sheet with the same charge density.
(c) For the infinite sheet, how does the field strength depend on the distance from the sheet? [Hint: Refer to your field line sketch.] (d) For points close to the finite sheet and far from its edges, can the sheet be approximated by an infinitely large sheet? [Hint: Again, refer to the field line sketches.] (e) Use Gauss's law to show that the magnitude of the electric field near a sheet of uniform charge density $\sigma$ is $E=\sigma /\left(2 \epsilon_{0}\right)$.

Tatiana Graham

Numerade Educator

01:36

Problem 79

A flat conducting plate of area $A$ has a charge $q$ on each surface. (a) What is the electric field within the material of the plate?
(b) Use Gauss"s law to show that the electric field just outside the plate is $E=q /\left(\epsilon_{0} A\right)=$ $\sigma / \epsilon_{0},$ (c) Does this contradict the result of Problem $78 ?$ Compare the field line diagrams for the two situations.

Tatiana Graham

Numerade Educator

03:24

Problem 80

A parallel-plate capacitor consists of two flat metal plates of area $A$ separated by a small distance $d$. The plates are given equal and opposite net charges $\pm q$
(a) Sketch the field lines and use your sketch to explain why almost all of the charge is on the inner surfaces of the plates.
(b) Use Gauss's law to show that the electric field between the plates and away from the edges is $E=$ $q /\left(\epsilon_{0} A\right)=\sigma / \epsilon_{0} .$ (c) Does this agree with or contradict the result of Problem $79 ?$ Explain.
(d) Use the principle of superposition and the result of Problem 78 to arrive at this same answer. [Hint: The inner surfaces of the two plates are thin, flat sheets of charge. $.$

Tatiana Graham

Numerade Educator

00:51

Problem 81

A coaxial cable consists of a wire of radius $a$ surrounded by a thin metal cylindrical shell of radius $b$. The wire has a uniform linear charge density $\lambda>0$ and the outer shell has a uniform linear charge density $-\lambda$.
(a) Sketch the field lines for this cable.
(b) Find expressions for the magnitude of the electric field in the regions $r \leq a, a<r<b,$ and $b \leq r$

Mayukh Banik

Numerade Educator

00:38

Problem 82

Use Gauss's law to derive an expression for the electric field outside the thin spherical shell of Conceptual Example 16.8

Mayukh Banik

Numerade Educator

02:35

Problem 83

In a thunderstorm, charge is separated through a complicated mechanism that is ultimately powered by the Sun. A simplified model of the charge in a thundercloud represents the positive charge accumulated at the top and the negative charge at the bottom as a pair of point charges.
(a) What is the magnitude and direction of the electric field produced by the two point charges at point $P$, which is just above Earth's surface? (b) Treating Earth as a conductor, what sign of charge would accumulate on the surface near point $P ?$ (This accumulated charge increases the magnitude of the electric field near point $P_{0}$ )

Tatiana Graham

Numerade Educator

01:43

Problem 84

Two otherwise identical conducting spheres carry charges of $+5.0 \mu \mathrm{C}$ and $-1.0 \mu \mathrm{C} .$ They are initially a distance $L$ apart. The distance $L$ is much larger than the radii of the spheres. The spheres are brought together, touched together, and then returned to their original separation $L$. What is the ratio of the magnitude of the force on either sphere after they are touched to that before they were touched?

Narayan Hari

Numerade Educator

01:55

Problem 85

Two metal spheres of radius $5.0 \mathrm{cm}$ carry net charges of $+1.0 \mu \mathrm{C}$ and $+0.2 \mu \mathrm{C} .$ (a) What (approximately) is the magnitude of the electrical repulsion on either sphere when their centers are $1.00 \mathrm{m}$ apart?
(b) Why cannot Coulomb's law be used to find the force of repulsion when their centers are 12 cm apart?
(c) Would the actual force be larger or smaller than the result of using Coulomb's law with $r=12 \mathrm{cm} ?$ Explain.

Mayukh Banik

Numerade Educator

03:40

Problem 86

In the diagram, regions $A$ and $C$ extend far to the left and right, respectively. The electric field due to the two point charges is zero at some point in which region or regions? Explain.

Supratim Pal

Numerade Educator

07:03

Problem 87

In Problem $86,$ the $+2.0 \mu \mathrm{C}$ charge is at $x=0$ and the $-4.0 \mu \mathrm{C}$ charge is at $x=d .$ Find the $x$ -coordinates of the point(s) where the electric field is zero.

Vishal Gupta

Numerade Educator

03:25

Problem 88

(a) What would the net charges on the Sun and Earth have to be if the electric force instead of the gravitational force were responsible for keeping Earth in its orbit? There are many possible answers, so restrict yourself to the case where the magnitude of the charges is proportional to the masses. (b) If the magnitude of the charges of the proton and electron were not exactly equal, astronomical bodies would have net charges that are approximately proportional to their masses. Could this possibly be an explanation for Earth"s orbit?

Tatiana Graham

Numerade Educator

03:40

Problem 89

What is the electric force on the chloride ion in the lower right-hand corner in the diagram? Since the ions are in water, the "effective charge" on the chloride ions $\left(\mathrm{Cl}^{-}\right)$ is $-2 \times 10^{-21} \mathrm{C}$ and that of the sodium ions $\left(\mathrm{Na}^{+}\right)$
is $+2 \times 10^{-21} \mathrm{C}$. (The effective charge is a way to account for the partial shielding due to nearby water molecules.) Assume that all four ions are coplanar.

Mayukh Banik

Numerade Educator

00:54

Problem 90

Consider two protons (charge $+e$ ), separated by a distance of $2.0 \times 10^{-15} \mathrm{m}$ (as in a typical atomic nucleus). The electric force between these protons is equal in magnitude to the gravitational force on an object of what mass near Earth's surface?

Mayukh Banik

Numerade Educator

00:55

Problem 91

In lab tests it was found that rats can detect electric fields of about $5.0 \mathrm{kN} / \mathrm{C}$ or more. If a point charge of $1.0 \mu \mathrm{C}$ is sitting in a maze, how close must the rat come to the charge in order to detect it?

Zulfiqar Ali

Numerade Educator

00:54

Problem 92

A raindrop inside a thundercloud has charge - $8 e .$ What is the electric force on the raindrop if the electric field at its location (due to other charges in the cloud) has magnitude $2.0 \times 10^{6} \mathrm{N} / \mathrm{C}$ and is directed upward?

Zulfiqar Ali

Numerade Educator

00:52

Problem 93

An electron beam in an oscilloscope is deflected by the electric field produced by oppositely charged metal plates. If the electric field between the plates is $2.00 \times$ $10^{5} \mathrm{N} / \mathrm{C}$ directed downward, what is the force on each electron when it passes between the plates?

Zulfiqar Ali

Numerade Educator

04:21

Problem 94

A point charge $q_{1}=+5.0 \mu \mathrm{C}$ is fixed in place at $x=0$ and a point charge $q_{2}=-3.0 \mu \mathrm{C}$ is fixed at $x=-20.0 \mathrm{cm}$ Where can we place a point charge $q_{3}=-8.0 \mu \mathrm{C}$ so that the net electric force on $q_{1}$ due to $q_{2}$ and $q_{3}$ is zero?

Zulfiqar Ali

Numerade Educator

09:59

Problem 95

Two point charges are located on a coordinate system as follows: $Q_{1}=-4.5 \mu \mathrm{C}$ at $x=1.00 \mathrm{cm}$ and $y=1.00 \mathrm{cm}$
and $Q_{2}=6.0 \mu \mathrm{C}$ at $x=3.00 \mathrm{cm}$ and $y=1.00 \mathrm{cm}$
(a) What is the electric field at point $P$ located at $x=$ $1.00 \mathrm{cm}$ and $y=4.00 \mathrm{cm} ?$ (b) When a tiny $5.0 \mathrm{g}$ particle with a charge of $-2.0 \mu \mathrm{C}$ is placed at point $P$ and released, what is its initial acceleration?

Vishal Gupta

Numerade Educator

04:10

Problem 96

Object $A$ has mass $90.0 \mathrm{g}$ and hangs from an insulated thread. When object $B$, which has a charge of $+130 \mathrm{nC}$. is held nearby, $A$ is attracted to it. In equilibrium, $A$ hangs at an angle $\theta=7.20^{\circ}$ with respect to the vertical and is $5.00 \mathrm{cm}$ to the left of B. (a) What is the charge on $A ?$
(b) What is the tension in the thread?

Narayan Hari

Numerade Educator

06:20

Problem 97

An electron with a velocity of $10.0 \mathrm{m} / \mathrm{s}$ in the positive y-direction enters a region where there is a uniform electric field of $200 \mathrm{N} / \mathrm{C}$ in the positive $x$ -direction. What are the $x$ - and $y$ -components of the electron's displacement $2.40 \mu$ s after entering the electric-field region if no other forces act on it?

Vishal Gupta

Numerade Educator

01:07

Problem 98

Two point charges are located on the $x$ -axis: a charge of $+6.0 \mathrm{nC}$ at $x=0$ and an unknown charge $q$ at $x=0.50 \mathrm{m}$ No other charges are nearby. If the electric field is zero at the point $x=1.0 \mathrm{m},$ what is $q ?$

Mayukh Banik

Numerade Educator

01:59

Problem 99

Three equal charges are placed on $Q_{a}$ three corners of a square. If the force that $Q_{a}$ exerts on $Q$, has magnitude $F_{\text {ba }}$ and the force that $Q_{2}$ exerts on $Q_{c}$ has magnitude $F_{\text {eat }}$ what is the ratio of $F_{\mathrm{sa}}$ to $F_{\mathrm{ba}} ?$

Narayan Hari

Numerade Educator

03:09

Problem 100

A charge of $63.0 \mathrm{nC}$ is located at a distance of $3.40 \mathrm{cm}$ from a charge of $-47.0 \mathrm{nC}$. What are the $x$ - and y-components of the electric field at a point $P$ that is directly above the $63.0 \mathrm{nC}$ charge at a distance of $1.40 \mathrm{cm} ?$ Point $P$ and the two charges are on the vertices of a right triangle.

Mayukh Banik

Numerade Educator

04:11

Problem 101

In a cathode ray tube, clectrons initially at rest are accelerated by a uniform electric ficld of magnitude $4.0 \times 10^{5} \mathrm{N} / \mathrm{C}$ during the first $5.0 \mathrm{cm}$ of the tube's length; then they move at essentially constant velocity another $45 \mathrm{cm}$ before hitting the screen.
(a) Find the speed of the electrons when they hit the screen.
(b) How long does it take them to travel the length of the tube?

Narayan Hari

Numerade Educator

09:26

Problem 102

A thin wire with positive charge $Q$ evenly spread along its length is shaped into a semicircle of radius $R$.
(a). What is the direction of the electric field at the center of curvature of the semicircle? Explain. (b) Is the magnitude of the field at the center less than, equal to, or greater than $k Q / R^{2}$ ? Explain.

Vishal Gupta

Numerade Educator

03:30

Problem 103

(a) Write an expression for the electric field at a point $(0, y)$ on the dipole axis for $y>d / 2$ What is the direction of the field?
(b) Show that when $y \gg d, E \approx 2 k q d l y^{3}$
[Hint: Use the binomial approximation from Appendix A.9.] (c) The field

Sam Stansfield

Numerade Educator

05:36

Problem 104

(a) Write an expression for the magnitude of the electric field at a point $(x, 0)$ on a line perpendicular to the dipole axis. State the direction of the field for $x>0$ and for $x<0$
(b) Show that when $x \gg d, E \approx k q d / x^{3}$
(c) The field is inversely proportional to the distance cubed. Does this conflict with Coulomb's law? Explain.

Sam Stansfield

Numerade Educator

01:37

Problem 105

A very small charged block with a mass of $2.35 \mathrm{g}$ is placed on an insulated, frictionless plane inclined at an angle of $17.0^{\circ}$ with respect to the horizontal. The block does not slide down the plane because of a $465 \mathrm{N} / \mathrm{C}$ uniform electric field that points parallel to the surface downward along the plane. What is the sign and magnitude of the charge on the block?

Mayukh Banik

Numerade Educator

01:08

Problem 106

The Bohr model of the hydrogen atom proposed that the electron orbits around the proton in a circle of radius $5.3 \times 10^{-11} \mathrm{m} .$ The electric force is responsible for the radial acceleration of the electron. What is the speed of the electron in this model?

Narayan Hari

Numerade Educator

03:29

Problem 107

In gel electrophoresis, the mobility $\mu$ of a molecule in a particular gel matrix is defined as $\mu=v / E$, where $v_{1}$ is the terminal speed of the molecule and $E$ is the applied electric field strength. In one case, a molecule has mobility $3.0 \times 10^{-8} \mathrm{C} \cdot \mathrm{m} /(\mathrm{N} \cdot \mathrm{s})$ and charge $-12 e$
(a) Estimate the electric field that should be applied to give this molecule a terminal speed of $2.0 \times 10^{-5} \mathrm{m} / \mathrm{s}$
(b) How long does it take the molecule to move $2.0 \mathrm{cm}$ through the gel?
(c) Suppose the same molecule had a charge of $-8 e$ instead of $-12 e .$ Considering the forces exerted on the molecule, would its terminal speed be smaller or larger (for the same applied field)? Would its mobility be smaller or larger?

Tatiana Graham

Numerade Educator

02:48

Problem 108

In an experiment to measure the Coulomb constant, a tiny sphere with charge $+7.0 \mathrm{nC}$ is suspended from a spring. When two other tiny charged spheres, each with a charge of $-4.0 \mu C,$ are placed in the positions shown in the figure, the spring stretches $0.50 \mathrm{mm}$ from its previous equilibrium position. Calculate the spring constant.

Narayan Hari

Numerade Educator

02:43

Problem 109

A spherical rain drop of radius $1.0 \mathrm{mm}$ has a charge of $+2.0 \mathrm{nC}$. The electric field in the vicinity is $2.0 \mathrm{kN} / \mathrm{C}$ downward. The terminal speed of an identical but uncharged drop is $6.5 \mathrm{m} / \mathrm{s}$. The drag force is related to the drop's speed by $F_{\mathrm{d}}=b v^{2}$ (turbulent drag rather than viscous drag). Calculate the terminal speed of the charged rain drop.

Narayan Hari

Numerade Educator

06:52

Problem 110

(a) Calculate the net electric force acting on the dipole.
(b) Show that the magnitude of the torque on the dipole is $\tau=q E d \sin \theta .$ (c) Calculate the torque acting on the dipole for $\theta=0,36.9^{\circ},$ and $90.0^{\circ} .$

Vishal Gupta

Numerade Educator

06:11

Problem 111

What is the angular acceleration of the dipole at $\theta=135^{\circ} ?$

Vishal Gupta

Numerade Educator

07:14

Problem 112

The dipole is released from rest at $\theta=90.0^{\circ} .$ What is its angular speed when it reaches $\theta=0 ?[$ Hint: First find the work done on each point charge. .1

Vishal Gupta

Numerade Educator

02:51

Problem 113

What is the maximum possible torque on the molecule due to the electric field?

Vishal Gupta

Numerade Educator

03:58

Problem 114

If the molecule is initially at rest at $\theta=90.0^{\circ},$ what is its angular speed when it reaches $\theta=0,$ assuming no other forces or torques? [Hint: First find the work done on each point charge.]

Vishal Gupta

Numerade Educator