University Physics

Samuel J. Ling, Jeff Sanny, William Moebs

Chapter 5

Electric Charges and Fields - all with Video Answers

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Chapter Questions

Problem 1

There are very large numbers of charged particles in most objects. Why, then, don't most objects exhibit static electricity?

Katie Mcalpine

Katie Mcalpine

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Problem 2

Why do most objects tend to contain nearly equal numbers of positive and negative charges?

Katie Mcalpine

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Problem 3

A positively charged rod attracts a small piece of cork.
(a) Can we conclude that the cork is negatively charged?
(b) The rod repels another small piece of cork. Can we conclude that this piece is positively charged?

Katie Mcalpine

Numerade Educator

Problem 4

Two bodies attract each other electrically. Do they both have to be charged? Answer the same question if the bodies repel one another.

Katie Mcalpine

Numerade Educator

Problem 5

How would you determine whether the charge on a particular rod is positive or negative?

Katie Mcalpine

Numerade Educator

Problem 6

An eccentric inventor attempts to levitate a cork ball by wrapping it with foil and placing a large negative charge on the ball and then putting a large positive charge on the ceiling of his workshop. Instead, while attempting to place a large negative charge on the ball, the foil flies off. Explain.

Katie Mcalpine

Numerade Educator

Problem 7

When a glass rod is rubbed with silk, it becomes positive and the silk becomes negative -yet both attract dust. Does the dust have a third type of charge that is attracted to both positive and negative? Explain.

Katie Mcalpine

Numerade Educator

Problem 8

Why does a car always attract dust right after it is polished? (Note that car wax and car tires are insulators.)

Katie Mcalpine

Numerade Educator

Problem 9

Does the uncharged conductor shown below experience a net electric force?

Katie Mcalpine

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Problem 10

While walking on a rug, a person frequently becomes charged because of the rubbing between his shoes and the nug. This charge then causes a spark and a slight shock when the person gets close to a metal object. Why are these shocks so much more common on a dry day?

Katie Mcalpine

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Problem 11

Compare charging by conduction to charging by induction.

Katie Mcalpine

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Problem 12

Small pieces of tissue are attracted to a charged comb. Soon after sticking to the comb, the pieces of tissue are repelled from it. Explain.

Katie Mcalpine

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Problem 13

Trucks that carry gasoline often have chains dangling from their undercarriages and brushing the ground. Why?

Katie Mcalpine

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Problem 14

Why do electrostatic experiments work so poorly in humid weather?

Katie Mcalpine

Numerade Educator

Problem 15

Why do some clothes cling together after being removed from the clothes dryer? Does this happen if they're still damp?

Katie Mcalpine

Numerade Educator

Problem 16

Can induction be used to produce charge on an insulator?

Katie Mcalpine

Numerade Educator

Problem 17

Suppose someone tells you that rubbing quartz with cotton cloth produces a third kind of charge on the quartz. Describe what you might do to test this claim.

Katie Mcalpine

Numerade Educator

Problem 18

A handheld copper rod does not acquire a charge when you rub it with a cloth. Explain why.

Katie Mcalpine

Numerade Educator

Problem 19

Suppose you place a charge $q$ near a large metal plate.
(a) If $q$ is attracted to the plate, is the plate necessarily charged? (b) If $q$ is repelled by the plate, is the plate necessarily charged?

Katie Mcalpine

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Problem 20

Would defining the charge on an electron to be positive have any effect on Coulomb's law?

Katie Mcalpine

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Problem 21

An atomic nucleus contains positively charged protons and uncharged neutrons. since nuclei do stay together, what must we conclude about the forces between these nuclear particles?

Katie Mcalpine

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Problem 22

Is the force between two fixed charges influenced by the presence of other charges?

Katie Mcalpine

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Problem 23

When measuring an electric field, could we use a negative rather than a positive test charge?

Katie Mcalpine

Numerade Educator

Problem 24

During fair weather, the electric field due to the net charge on Earth points downward. Is Earth charged positively or negatively?

Katie Mcalpine

Numerade Educator

Problem 25

If the electric field at a point on the line between two charges is zero, what do you know about the charges?

Katie Mcalpine

Numerade Educator

Problem 26

Two charges lie along the $x$ -axis. Is it true that the net electric field always vanishes at some point (other than infinity) along the $x$ -axis?

Katie Mcalpine

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Problem 27

Give a plausible argument as to why the electric field outside an infinite charged sheet is constant.

Katie Mcalpine

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Problem 28

Compare the electric fields of an infinite sheet of charge, an infinite, charged conducting plate, and infinite, oppositely charged parallel plates.

Katie Mcalpine

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Problem 29

Describe the electric fields of an infinite charged plate and of two infinite, charged parallel plates in terms of the electric field of an infinite sheet of charge.

Katie Mcalpine

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Problem 30

A negative charge is placed at the center of a ring of uniform positive charge. What is the motion (if any) of the charge? What if the charge were placed at a point on the axis of the ring other than the center?

Katie Mcalpine

Numerade Educator

Problem 31

If a point charge is released from rest in a uniform electric field, will it follow a field line? Will it do so if the electric field is not uniform?

Katie Mcalpine

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Problem 32

Under what conditions, if any, will the trajectory of a charged particle not follow a field line?

Katie Mcalpine

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Problem 33

How would you experimentally distinguish an electric field from a gravitational field?

Katie Mcalpine

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Problem 34

A representation of an electric field shows 10 field lines perpendicular to a square plate. How many field lines should pass perpendicularly through the plate to depict a field with twice the magnitude?

Katie Mcalpine

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Problem 35

What is the ratio of the number of electric field lines leaving a charge $10 q$ and a charge $q$ ?

Katie Mcalpine

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Problem 36

What are the stable orientation(s) for a dipole in an external electric field? What happens if the dipole is slightly perturbed from these orientations?

Katie Mcalpine

Numerade Educator

Problem 37

Common static electricity involves charges ranging from nanocoulombs to microcoulombs. (a) How many electrons are needed to form a charge of $-2.00 \mathrm{nC}$ ? (b) How many electrons must be removed from a neutral object to leave a net charge of $0.500 \mu \mathrm{C} ?$

Chasen Shaw

Numerade Educator

Problem 38

If $1.80 \times 10^{20}$ electrons move through a pocket calculator during a full day's operation, how many coulombs of charge moved through it?

cm

Charles Magnusen

Numerade Educator

Problem 39

To start a car engine, the car battery moves $3.75 \times 10^{21}$ electrons through the starter motor. How many coulombs of charge were moved?

Kevin Shryock

Numerade Educator

Problem 40

A certain lightning bolt moves 40.0 C of charge. How many fundamental units of charge is this?

Katie Mcalpine

Numerade Educator

Problem 41

A $2.5-\mathrm{g}$ copper penny is given a charge of $-2.0 \times 10^{-9} \mathrm{C} .$ (a) How many excess electrons are on the penny? (b) By what percent do the excess electrons change the mass of the penny?

Krystal K

Numerade Educator

Problem 42

A $2.5-\mathrm{g}$ copper penny is given a charge of $4.0 \times 10^{-9} \mathrm{C} .(\text { a })$ How many electrons are removed from the penny? (b) If no more than one electron is removed from an atom, what percent of the atoms are ionized by this charging process?

Prabhat Tyagi

Numerade Educator

Problem 43

Suppose a speck of dust in an electrostatic precipitator has $1.0000 \times 10^{12}$ protons in it and has a net charge of $-5.00 \mathrm{nC}$ (a very large charge for a small speck). How many electrons does it have?

Kevin Shryock

Numerade Educator

Problem 44

An amoeba has $1.00 \times 10^{16}$ protons and a net charge =of $0.300 \mathrm{pC}$. (a) How many fewer electrons are there than protons? (b) If you paired them up, what fraction of the protons would have no electrons?

Katie Mcalpine

Numerade Educator

Problem 45

A 50.0-g ball of copper has a net charge of 2.00 $\mu \mathrm{C}$ What fraction of the copper's electrons has been removed? (Each copper atom has 29 protons, and copper has an atomic mass of 63.5.)

Kevin Shryock

Numerade Educator

Problem 46

What net charge would you place on a 100 -g piece of sulfur if you put an extra electron on 1 in $10^{12}$ of its atoms? (Sulfur has an atomic mass of 32.1 u.)

Katie Mcalpine

Numerade Educator

Problem 47

How many coulombs of positive charge are there in $4.00 \mathrm{kg}$ of plutonium, given its atomic mass is 244 and that each plutonium atom has 94 protons?

Kevin Shryock

Numerade Educator

Problem 48

Two point particles with charges $+3 \mu \mathrm{C}$ and $+5 \mu \mathrm{C}$
are held in place by $3-\mathrm{N}$ forces on each charge in appropriate directions. (a) Draw a free-body diagram for each particle. (b) Find the distance between the charges.

Katie Mcalpine

Numerade Educator

Problem 49

Two charges $+3 \mu \mathrm{C}$ and $+12 \mu \mathrm{C}$ are fixed $1 \mathrm{m}$ apart, with the second one to the right. Find the magnitude and direction of the net force on a $-2-\mathrm{nC}$ charge when placed at the following locations: (a) halfway between the two (b) half a meter to the left of the $+3 \mu$ C charge (c) half a meter above the $+12 \mu \mathrm{C}$ charge in a direction
perpendicular to the line joining the two fixed charges

Katie Mcalpine

Numerade Educator

Problem 50

In a salt crystal, the distance between adjacent sodium and chloride ions is $2.82 \times 10^{-10} \mathrm{m}$. What is the force of attraction between the two singly charged ions?

Katie Mcalpine

Numerade Educator

Problem 51

Protons in an atomic nucleus are typically $10^{-15} \mathrm{m}$ apart. What is the electric force of repulsion between nuclear protons?

Katie Mcalpine

Numerade Educator

Problem 52

Suppose Earth and the Moon each carried a net negative charge $-Q .$ Approximate both bodies as point masses and point charges.
(a) What value of $Q$ is required to balance the gravitational attraction between Earth and the Moon?
(b) Does the distance between Earth and the Moon affect your answer? Explain.
(c) How many electrons would be needed to produce this charge?

Katie Mcalpine

Numerade Educator

Problem 53

Point charges $q_{1}=50 \mu \mathrm{C}$ and $q_{2}=-25 \mu \mathrm{C}$ are placed $1.0 \mathrm{m}$ apart. What is the force on a third charge $q_{3}=20 \mu \mathrm{C}$ placed midway between $q_{1}$ and $q_{2} ?$

Katie Mcalpine

Numerade Educator

Problem 54

Where must $q_{3}$ of the preceding problem be placed so that the net force on it is zero?

Katie Mcalpine

Numerade Educator

Problem 55

Two small balls, each of mass 5.0 g, are attached to silk threads $50 \mathrm{cm}$ long, which are in turn tied to the same point on the ceiling, as shown below. When the balls are given the same charge $Q$, the threads hang at $5.0^{\circ}$ to the vertical, as shown below. What is the magnitude of $Q$ ? What are the signs of the two charges?

Katie Mcalpine

Numerade Educator

Problem 56

Point charges $Q_{1}=2.0 \mu \mathrm{C}$ and $Q_{2}=4.0 \mu \mathrm{C}$ are located at $\quad \overrightarrow{\mathbf{r}}_{1}=(4.0 \hat{\mathbf{i}}-2.0 \hat{\mathbf{j}}+5.0 \hat{\mathbf{k}}) \mathrm{m} \quad$ and $\overrightarrow{\mathbf{r}}_{2}=(8.0 \hat{\mathbf{i}}+5.0 \hat{\mathbf{j}}-9.0 \hat{\mathbf{k}}) \mathrm{m} .$ What is the force of $Q_{2}$ on $Q_{1} ?$

Katie Mcalpine

Numerade Educator

Problem 57

The net excess charge on two small spheres (small enough to be treated as point charges) is $Q .$ Show that the force of repulsion between the spheres is greatest when each sphere has an excess charge $Q / 2 .$ Assume that the distance between the spheres is so large compared with their radii that the spheres can be treated as point charges.

Katie Mcalpine

Numerade Educator

Problem 58

Two small, identical conducting spheres repel each other with a force of $0.050 \mathrm{N}$ when they are $0.25 \mathrm{m}$ apart. After a conducting wire is connected between the spheres and then removed, they repel each other with a force of 0.060 N. What is the original charge on each sphere?

Nicholas Mogoi

Numerade Educator

Problem 59

A charge $q=2.0 \mu \mathrm{C}$ is placed at the point $P$ shown below. What is the force on $q ?$

Katie Mcalpine

Numerade Educator

Problem 60

What is the net electric force on the charge located at the lower right-hand corner of the triangle shown here?

Supratim Pal

Numerade Educator

Problem 61

Two fixed particles, each of charge $5.0 \times 10^{-6} \mathrm{C}$ are $24 \mathrm{cm}$ apart. What force do they exert on a third particle of charge $-2.5 \times 10^{-6} \mathrm{C}$ that is $13 \mathrm{cm}$ from each of them?

Katie Mcalpine

Numerade Educator

Problem 62

The charges $q_{1}=2.0 \times 10^{-7} \mathrm{C}, q_{2}=-4.0 \times 10^{-7} \mathrm{C}$ and $q_{3}=-1.0 \times 10^{-7} \mathrm{C}$ are placed at the corners of the triangle shown below. What is the force on $q_{1} ?$

Katie Mcalpine

Numerade Educator

Problem 63

What is the force on the charge $q$ at the lower-right-hand corner of the square shown here?

Katie Mcalpine

Numerade Educator

Problem 64

Point charges $q_{1}=10 \mu \mathrm{C}$ and $q_{2}=-30 \mu \mathrm{C}$ are fixed at $r_{1}=(3.0 \hat{\mathbf{i}}-4.0 \hat{\mathbf{j}}) \mathrm{m}$ and $r_{2}=(9.0 \hat{\mathrm{i}}+6.0 \hat{\mathrm{j}}) \mathrm{m} .$ What is the force of $q_{2}$ on $q_{1} ?$

Katie Mcalpine

Numerade Educator

Problem 65

A particle of charge $2.0 \times 10^{-8} \mathrm{C}$ experiences an upward force of magnitude $4.0 \times 10^{-6} \mathrm{N}$ when it is placed in a particular point in an electric field. (a) What is the electric field at that point? (b) If a charge $q=-1.0 \times 10^{-8} \mathrm{C}$ is placed there, what is the force on it?

Katie Mcalpine

Numerade Educator

Problem 66

a single electron charge. What is the acceleration (both
magnitude and direction) of the dust particle?

Supratim Pal

Numerade Educator

Problem 67

Consider an electron that is $10^{-10} \mathrm{m}$ from an alpha particle $\left(q=3.2 \times 10^{-19} \mathrm{C}\right) .$ (a) What is the electric field due to the alpha particle at the location of the electron?
(b) What is the electric field due to the electron at the location of the alpha particle? (c) What is the electric force on the alpha particle? On the electron?

Katie Mcalpine

Numerade Educator

Problem 68

Each the balls shown below carries a charge $q$ and has a mass $m$. The length of each thread is $l$, and at equilibrium, the balls are separated by an angle $2 \theta .$ How does $\theta$ vary with $q$ and $l ?$ Show that $\theta$ satisfies $\sin (\theta)^{2} \tan (\theta)=\frac{q^{2}}{16 \pi \varepsilon_{0} g l^{2} m}$

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 69

What is the electric field at a point where the force on a $-2.0 \times 10^{-6}-\mathrm{C}$ charge is $(4.0 \hat{\mathrm{i}}-6.0 \hat{\mathrm{j}}) \times 10^{-6} \mathrm{N} ?$

Katie Mcalpine

Numerade Educator

Problem 70

A proton is suspended in the air by an electric field at the surface of Earth. What is the strength of this electric field?

Katie Mcalpine

Numerade Educator

Problem 71

The electric field in a particular thundercloud is $2.0 \times 10^{5} \mathrm{N} / \mathrm{C} .$ What is the acceleration of an electron in this field?

Katie Mcalpine

Numerade Educator

Problem 72

A small piece of cork whose mass is $2.0 \mathrm{g}$ is given a charge of $5.0 \times 10^{-7} \mathrm{C}$. What electric field is needed to place the cork in equilibrium under the combined electric and gravitational forces?

Prabhu Ramji

Numerade Educator

Problem 73

If the electric field is $100 \mathrm{N} / \mathrm{C}$ at a distance of $50 \mathrm{cm}$ from a point charge $q,$ what is the value of $q ?$

Katie Mcalpine

Numerade Educator

Problem 74

What is the electric field of a proton at the first Bohr orbit for hydrogen $\left(r=5.29 \times 10^{-11} \mathrm{m}\right) ?$ What is the force on the electron in that orbit?

Katie Mcalpine

Numerade Educator

Problem 75

(a) What is the electric field of an oxygen nucleus at a point that is $10^{-10} \mathrm{m}$ from the nucleus? (b) What is the force this electric field exerts on a second oxygen nucleus placed at that point?

Katie Mcalpine

Numerade Educator

Problem 76

Two point charges, $q_{1}=2.0 \times 10^{-7} \mathrm{C}$ and $q_{2}=-6.0 \times 10^{-8} \mathrm{C},$ are held $25.0 \mathrm{cm}$ apart. (a) What is the electric field at a point $5.0 \mathrm{cm}$ from the negative charge and along the line between the two charges? (b)What is the force on an electron placed at that point?

Katie Mcalpine

Numerade Educator

Problem 77

Point charges $q_{1}=50 \mu \mathrm{C}$ and $q_{2}=-25 \mu \mathrm{C}$ are placed 1.0 m apart. (a) What is the electric field at a point midway between them? (b) What is the force on a charge $q_{3}=20 \mu \mathrm{C}$ situated there?

Katie Mcalpine

Numerade Educator

Problem 78

Can you arrange the two point charges $q_{1}=-2.0 \times 10^{-6} \mathrm{C}$ and $q_{2}=4.0 \times 10^{-6} \mathrm{C}$ along the $x$ -axis so that $E=0$ at the origin?

Katie Mcalpine

Numerade Educator

Problem 79

Point charges $q_{1}=q_{2}=4.0 \times 10^{-6} \mathrm{C}$ are fixed on the $x$ -axis at $x=-3.0 \mathrm{m}$ and $x=3.0 \mathrm{m}$. What charge $q$ must be placed at the origin so that the electric field vanishes at $x=0, y=3.0 \mathrm{m} ?$

Katie Mcalpine

Numerade Educator

Problem 80

A thin conducting plate $1.0 \mathrm{m}$ on the side is given a charge of $-2.0 \times 10^{-6} \mathrm{C}$. An electron is placed $1.0 \mathrm{cm}$ above the center of the plate. What is the acceleration of the electron?

Narayan Hari

Numerade Educator

Problem 81

Calculate the magnitude and direction of the electric field $2.0 \mathrm{m}$ from a long wire that is charged uniformly at $\lambda=4.0 \times 10^{-6} \mathrm{C} / \mathrm{m}$

Katie Mcalpine

Numerade Educator

Problem 82

Two thin conducting plates, each $25.0 \mathrm{cm}$ on a side, are situated parallel to one another and $5.0 \mathrm{mm}$ apart. If $10^{-11}$ electrons are moved from one plate to the other, what is the electric field between the plates?

Katie Mcalpine

Numerade Educator

Problem 83

The charge per unit length on the thin rod shown below is $\lambda .$ What is the electric field at the point $P ?$ (Hint:
Solve this problem by first considering the electric field $d \space\overrightarrow{\mathbf{E}} \quad$ at $P$ due to a small segment $d x$ of the rod, which contains charge $d q=\lambda d x .$ Then find the net field by integrating $d \space\overrightarrow{\mathbf{E}}$ over the length of the rod.)

Supratim Pal

Numerade Educator

Problem 84

The charge per unit length on the thin semicircular wire shown below is $\lambda .$ What is the electric field at the point $P ?$

Katie Mcalpine

Numerade Educator

Problem 85

Two thin parallel conducting plates are placed 2.0 $\mathrm{cm}$ apart. Each plate is $2.0 \mathrm{cm}$ on a side; one plate carries a net charge of $8.0 \mu \mathrm{C}, \quad$ and the other plate carries a net charge of $-8.0 \mu \mathrm{C} .$ What is the charge density on the inside surface of each plate? What is the electric field between the plates?

Katie Mcalpine

Numerade Educator

Problem 86

A thin conducing plate $2.0 \mathrm{m}$ on a side is given a total charge of $-10.0 \mu \mathrm{C}$. (a) What is the electric field $1.0 \mathrm{cm}$ above the plate? (b) What is the force on an electron at this point? (c) Repeat these calculations for a point $2.0 \mathrm{cm}$ above the plate. (d) When the electron moves from 1.0 to $2,0 \mathrm{cm}$ above the plate, how much work is done on it by the electric field?

Katie Mcalpine

Numerade Educator

Problem 87

A total charge $q$ is distributed uniformly along a thin, straight rod of length $L$ (see below). What is the electric field at $P_{1} ?$ At $P_{2} ?$

Mayukh Banik

Numerade Educator

Problem 88

Charge is distributed along the entire $x$ -axis with uniform density $\lambda .$ How much work does the electric field of this charge distribution do on an electron that moves along the $y$ -axis from $y=a$ to $y=b ?$

Katie Mcalpine

Numerade Educator

Problem 89

Charge is distributed along the entire $x$ -axis with uniform density $\lambda_{x}$ and along the entire $y$ -axis with uniform density $\lambda_{y} .$ Calculate the resulting electric field at $$\text { (a) } \overrightarrow{\mathbf{r}}=a \hat{\mathbf{i}}+b \hat{\mathbf{j}} \text { and }(\mathrm{b}) \quad \overrightarrow{\mathbf{r}}=c \hat{\mathbf{k}}$$

Mayukh Banik

Numerade Educator

Problem 90

A rod bent into the arc of a circle subtends an angle $2 \theta$ at the center $P$ of the circle (see below). If the rod is charged uniformly with a total charge $Q,$ what is the electric field at $P ?$

Katie Mcalpine

Numerade Educator

Problem 91

A proton moves in the electric field $\overrightarrow{\mathbf{E}}=200 \hat{\mathbf{i}} \mathrm{N} / \mathrm{C}$. (a) What are the force on and the acceleration of the proton? (b) Do the same calculation for an electron moving in this field.

Katie Mcalpine

Numerade Educator

Problem 92

An electron and a proton, each starting from rest, are accelerated by the same uniform electric field of $200 \mathrm{N} / \mathrm{C}$ Determine the distance and time for each particle to acquire a kinetic energy of $3.2 \times 10^{-16} \mathrm{J}$.

Katie Mcalpine

Numerade Educator

Problem 93

A spherical water droplet of radius $25 \mu \mathrm{m}$ carries an excess 250 electrons. What vertical electric field is needed to balance the gravitational force on the droplet at the surface of the earth?

Katie Mcalpine

Numerade Educator

Problem 94

A proton enters the uniform electric field produced by the two charged plates shown below. The magnitude of the electric field is $4.0 \times 10^{5} \mathrm{N} / \mathrm{C},$ and the speed of the proton when it enters is $1.5 \times 10^{7} \mathrm{m} / \mathrm{s}$. What distance $d$ has the proton been deflected downward when it leaves the plates?

Katie Mcalpine

Numerade Educator

Problem 95

Shown below is a small sphere of mass $0.25 \mathrm{g}$ that carries a charge of $9.0 \times 10^{-10} \mathrm{C}$. The sphere is attached to one end of a very thin silk string $5.0 \mathrm{cm}$ long. The other end of the string is attached to a large vertical conducting plate that has a charge density of $30 \times 10^{-6} \mathrm{C} / \mathrm{m}^{2}$. What is the angle that the string makes with the vertical?

Willis James

Numerade Educator

Problem 96

Two infinite rods, each carrying a uniform charge density $\lambda, \quad$ are parallel to one another and perpendicular to the plane of the page. (See below.) What is the electrical field at $P_{1} ?$ At $P_{2} ?$

Vishal Gupta

Numerade Educator

Problem 97

Positive charge is distributed with a uniform density $\lambda$ along the positive $x$ -axis from $r$ to $\infty,$ along the positive $y$ -axis from $r$ to $\infty,$ and along a $90^{\circ}$ arc of a circle of radius $r,$ as shown below. What is the electric field at O?

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 98

From a distance of $10 \mathrm{cm},$ a proton is projected with a speed of $v=4.0 \times 10^{6} \mathrm{m} / \mathrm{s}$ directly at a large, positively charged plate whose charge density $\sigma=2.0 \times 10^{-5} \mathrm{C} / \mathrm{m}^{2} .$ (See below.) (a) Does the proton reach the plate? (b) If not, how far from the plate does it turn around?

Katie Mcalpine

Numerade Educator

Problem 99

A particle of mass $m$ and charge $-q$ moves along a straight line away from a fixed particle of charge $Q$. When the distance between the two particles is $r_{0},-q$ is moving with a speed $v_{0}$. (a) Use the work-energy theorem to calculate the maximum separation of the charges. (b) What do you have to assume about $v_{0}$ to make this calculation? (c) What is the minimum value of $v_{0}$ such that
$-q$ escapes from $Q ?$

Katie Mcalpine

Numerade Educator

Problem 100

Which of the following electric field lines are incorrect for point charges? Explain why.

Katie Mcalpine

Numerade Educator

Problem 101

In this exercise, you will practice drawing electric field lines. Make sure you represent both the magnitude and direction of the electric field adequately. Note that the number of lines into or out of charges is proportional to the charges. (a) Draw the electric field lines map for two charges $+20 \mu C$ and $-20 \mu C$ situated $5 \mathrm{cm}$ from each other. (b) Draw the electric field lines map for two charges $+20 \mu \mathrm{C}$ and $+20 \mu \mathrm{C}$ situated $5 \mathrm{cm}$ from each other. (c) Draw the electric field lines map for two charges $+20 \mu \mathrm{C}$ and $-30 \mu \mathrm{C}$ situated $5 \mathrm{cm}$ from each other.

Mayukh Banik

Numerade Educator

Problem 102

Draw the electric field for a system of three particles of charges $+1 \mu \mathrm{C}, \quad+2 \mu \mathrm{C}, \quad$ and $-3 \mu \mathrm{C}$ fixed at the corners of an equilateral triangle of side $2 \mathrm{cm}$.

Mayukh Banik

Numerade Educator

Problem 103

Two charges of equal magnitude but opposite sign make up an electric dipole. A quadrupole consists of two electric dipoles are placed anti-parallel at two edges of a square as shown.Draw the electric field of the charge distribution.

Katie Mcalpine

Numerade Educator

Problem 104

Suppose the electric field of an isolated point charge decreased with distance as $1 / r^{2+\delta}$ rather than as $1 / r^{2}$ Show that it is then impossible to draw continuous field lines so that their number per unit area is proportional to $E$

Mayukh Banik

Numerade Educator

Problem 105

Consider the equal and opposite charges shown below. (a) Show that at all points on the $x$ -axis for which $|x| \gg a, E \approx Q a / 2 \pi \varepsilon_{0} x^{3} .$ (b) Show that at all points on the $y$ -axis for which $|y| \gg a, E \approx Q a / \pi \varepsilon_{0} y^{3}$

Anand Jangid

Numerade Educator

Problem 106

(a) What is the dipole moment of the configuration shown above? If $Q=4.0 \mu \mathrm{C},$ (b) what is the torque on this dipole with an electric field of $4.0 \times 10^{5} \mathrm{N} / \mathrm{C} \hat{\mathrm{i}} ?(\mathrm{c})$ What is the torque on this dipole with an electric field of $-4.0 \times 10^{5} \mathrm{N} / \mathrm{C} \hat{\mathbf{i}} ?(\mathrm{d})$ What is the torque on this dipole with an electric field of $\pm 4.0 \times 10^{5} \mathrm{N} / \mathrm{C} \hat{\mathbf{j}} ?$

Mayukh Banik

Numerade Educator

Problem 107

A water molecule consists of two hydrogen atoms bonded with one oxygen atom. The bond angle between the two hydrogen atoms is $104^{\circ}$ (see below). Calculate the net dipole moment of a water molecule that is placed in a uniform, horizontal electric field of magnitude $2.3 \times 10^{-8} \mathrm{N} / \mathrm{C} .$ (You are missing some information for solving this problem; you will need to determine what information you need, and look it up.)

Mayukh Banik

Numerade Educator

Problem 108

Point charges $q_{1}=2.0 \mu \mathrm{C}$ and $q_{1}=4.0 \mu \mathrm{C}$ are located at $r_{1}=(4.0 \hat{\mathbf{i}}-2.0 \hat{\mathbf{j}}+2.0 \hat{\mathbf{k}}) \mathrm{m}$ and $r_{2}=(8.0 \hat{\mathbf{i}}+5.0 \hat{\mathbf{j}}-9.0 \hat{\mathbf{k}}) \mathrm{m}$. What is the force of $q_{2}$ on $q_{1} ?$

Katie Mcalpine

Numerade Educator

Problem 109

What is the force on the $5.0-\mu \mathrm{C}$ charge shown below?

Mayukh Banik

Numerade Educator

Problem 110

What is the force on the $2.0-\mu \mathrm{C}$ charge placed at the center of the square shown below?

Mayukh Banik

Numerade Educator

Problem 111

Four charged particles are positioned at the comers of a parallelogram as shown below. If $q=5.0 \mu \mathrm{C}$ and $Q=8.0 \mu \mathrm{C}, \quad$ what is the net force on $q ?$

Mayukh Banik

Numerade Educator

Problem 112

A charge $Q$ is fixed at the origin and a second charge $q$ moves along the $x$ -axis, as shown below. How much work is done on $q$ by the electric force when $q$ moves from $x_{1}$ to $x_{2} ?$

Katie Mcalpine

Numerade Educator

Problem 113

A charge $q=-2.0 \mu \mathrm{C}$ is released from rest when it is $2.0 \mathrm{m}$ from a fixed charge $Q=6.0 \mu \mathrm{C}$. What is the kinetic energy of $q$ when it is $1.0 \mathrm{m}$ from $Q ?$

Katie Mcalpine

Numerade Educator

Problem 114

What is the electric field at the midpoint $M$ of the hypotenuse of the triangle shown below?

Katie Mcalpine

Numerade Educator

Problem 115

Find the electric field at $P$ for the charge configurations shown below.

Mayukh Banik

Numerade Educator

Problem 116

(a) What is the electric field at the lower-right-hand comer of the square shown below? (b) What is the force on a charge $q$ placed at that point?

Katie Mcalpine

Numerade Educator

Problem 117

Point charges are placed at the four corners of a rectangle as shown below: $q_{1}=2.0 \times 10^{-6} \mathrm{C}$, $q_{2}=-2.0 \times 10^{-6} \mathrm{C}$, $q_{3}=4.0 \times 10^{-6} \mathrm{C}$, and $q_{4}=1.0 \times 10^{-6} \mathrm{C}$. What is the electric field at $P ?$

Mayukh Banik

Numerade Educator

Problem 118

Three charges are positioned at the corners of a parallelogram as shown below. (a) If $Q=8.0 \mu \mathrm{C},$ what is the electric field at the unoccupied corner? (b) What is the force on a $5.0-\mu \mathrm{C}$ charge placed at this corner?

Mayukh Banik

Numerade Educator

Problem 119

A positive charge $q$ is released from rest at the origin of a rectangular coordinate system and moves under the influence of the electric field $\overrightarrow{\mathbf{E}}=E_{0}(1+x / a) \hat{\mathbf{i}}$. What is the kinetic energy of $q$ when it passes through $x=3 a ?$

Katie Mcalpine

Numerade Educator

Problem 120

A particle of charge $-q$ and mass $m$ is placed at the center of a uniformaly charged ring of total charge $Q$ and radius $R .$ The particle is displaced a small distance along the axis perpendicular to the plane of the ring and released. Assuming that the particle is constrained to move along the axis, show that the particle oscillates in simple harmonic motion with a frequency $f=\frac{1}{2 \pi} \sqrt{\frac{q Q}{4 \pi \varepsilon_{0} m R^{3}}}$

Mayukh Banik

Numerade Educator

Problem 121

Charge is distributed uniformly along the entire $y$ -axis with a density $\lambda_{y}$ and along the positive $x$ -axis
from $x=a$ to $x=b$ with a density $\lambda_{x} .$ What is the force between the two distributions?

Mayukh Banik

Numerade Educator

Problem 122

The circular arc shown below carries a charge per unit length $\lambda=\lambda_{0} \cos \theta, \quad$ where $\theta$ is measured from the X-axis. What is the electric field at the origin?

Katie Mcalpine

Numerade Educator

Problem 123

Calculate the electric field due to a uniformly charged rod of length $L$, aligned with the $x$ -axis with one end at the origin; at a point $P$ on the z-axis.

Mayukh Banik

Numerade Educator

Problem 124

The charge per unit length on the thin rod shown below is $\lambda .$ What is the electric force on the point charge q? Solve this problem by first considering the electric force $d \space\overrightarrow{\mathbf{F}} \quad$ on $q$ due to a small segment $d x$ of the rod, which contains charge $\lambda d x$. Then, find the net force by integrating $d \space\overrightarrow{\mathbf{F}}$ over the length of the rod.

Mayukh Banik

Numerade Educator

Problem 125

The charge per unit length on the thin rod shown here is $\lambda .$ What is the electric force on the point charge $q$ ? (See the preceding problem.)

Mayukh Banik

Numerade Educator

Problem 126

The charge per unit length on the thin semicircular wire shown below is $\lambda$. What is the electric force on the point charge $q$ ? (See the preceding problems.)

Mayukh Banik

Numerade Educator