Chapter 1 Review

Samuel J. Ling; Jeff Sanny; William Moebs; Daryl Janzen

Chapter 1 Review

Key Terms

charging by induction
process by which an electrically charged object brought near a neutral object creates a charge separation in that object

conduction electron
electron that is free to move away from its atomic orbit

conductor
material that allows electrons to move separately from their atomic orbits; object with properties that allow charges to move about freely within it

continuous charge distribution
total source charge composed of so large a number of elementary charges that it must be treated as continuous, rather than discrete

coulomb
SI unit of electric charge

Coulomb force
another term for the electrostatic force

Coulomb’s law
mathematical equation calculating the electrostatic force vector between two charged particles

dipole
two equal and opposite charges that are fixed close to each other

dipole moment
property of a dipole; it characterizes the combination of distance between the opposite charges, and the magnitude of the charges

electric charge
physical property of an object that causes it to be attracted toward or repelled from another charged object; each charged object generates and is influenced by a force called an electric force

electric field
physical phenomenon created by a charge; it “transmits” a force between a two charges

electric force
noncontact force observed between electrically charged objects

electron
particle surrounding the nucleus of an atom and carrying the smallest unit of negative charge

electrostatic attraction
phenomenon of two objects with opposite charges attracting each other

electrostatic force
amount and direction of attraction or repulsion between two charged bodies; the assumption is that the source charges remain motionless

electrostatic repulsion
phenomenon of two objects with like charges repelling each other

electrostatics
study of charged objects which are not in motion

field line
smooth, usually curved line that indicates the direction of the electric field

field line density
number of field lines per square meter passing through an imaginary area; its purpose is to indicate the field strength at different points in space

induced dipole
typically an atom, or a spherically symmetric molecule; a dipole created due to opposite forces displacing the positive and negative charges

infinite plane
flat sheet in which the dimensions making up the area are much, much greater than its thickness, and also much, much greater than the distance at which the field is to be calculated; its field is constant

infinite straight wire
straight wire whose length is much, much greater than either of its other dimensions, and also much, much greater than the distance at which the field is to be calculated

insulator
material that holds electrons securely within their atomic orbits

ion
atom or molecule with more or fewer electrons than protons

law of conservation of charge
net electric charge of a closed system is constant

linear charge density
amount of charge in an element of a charge distribution that is essentially one-dimensional (the width and height are much, much smaller than its length); its units are $\mathrm{C/m}$

neutron
neutral particle in the nucleus of an atom, with (nearly) the same mass as a proton

permanent dipole
typically a molecule; a dipole created by the arrangement of the charged particles from which the dipole is created

permittivity of vacuum
also called the permittivity of free space, and constant describing the strength of the electric force in a vacuum

polarization
slight shifting of positive and negative charges to opposite sides of an object

principle of superposition
useful fact that we can simply add up all of the forces due to charges acting on an object

proton
particle in the nucleus of an atom and carrying a positive charge equal in magnitude to the amount of negative charge carried by an electron

static electricity
buildup of electric charge on the surface of an object; the arrangement of the charge remains constant (“static”)

superposition
concept that states that the net electric field of multiple source charges is the vector sum of the field of each source charge calculated individually

surface charge density
amount of charge in an element of a two-dimensional charge distribution (the thickness is small); its units are $\mathrm{C/m}^2$

volume charge density
amount of charge in an element of a three-dimensional charge distribution; its units are $\mathrm{C/m}^3$

Key Equations

Coulomb’s law	$\vec{\mathbf{F}}_{12}(r)=\frac{1}{4\pi\epsilon_0}\frac{q_1q_2}{r_{12}^2}\hat{\mathbf{r}}_{12}$
Superposition of electric forces	$\vec{\mathbf{F}}(r)=\frac{1}{4\pi\epsilon_0}Q\sum_{i=1}^{N}\frac{q_i}{r_{i}^2}\hat{\mathbf{r}}_{i}$
Electric force due to an electric field	$\vec{\mathbf{F}}=Q\vec{\mathbf{E}}$
Electric field at point $P$	$\vec{\mathbf{E}}(P)=\frac{1}{4\pi\epsilon_0}\sum_{i=1}^{N}\frac{q_i}{r_{i}^2}\hat{\mathbf{r}}_{i}$
Field of an infinite wire	$\vec{\mathbf{E}}(z)=\frac{1}{4\pi\epsilon_0}\frac{2\lambda}{z}\hat{\mathbf{k}}$
Field of an infinite plane	$\vec{\mathbf{E}}=\frac{\sigma}{2\epsilon_0}\hat{\mathbf{k}}$
Dipole moment	$\vec{\mathbf{p}}=q\vec{\mathbf{d}}$
Torque on dipole in external $E$ -field	$\vec{\mathbf{\tau}}=\vec{\mathbf{p}}\times\vec{\mathbf{E}}$

Summary

1.1 Electric Charge

There are only two types of charge, which we call positive and negative. Like charges repel, unlike charges attract, and the force between charges decreases with the square of the distance.
The vast majority of positive charge in nature is carried by protons, whereas the vast majority of negative charge is carried by electrons. The electric charge of one electron is equal in magnitude and opposite in sign to the charge of one proton.
An ion is an atom or molecule that has nonzero total charge due to having unequal numbers of electrons and protons.
The SI unit for charge is the coulomb ( $\mathrm{C}$ ), with protons and electrons having charges of opposite sign but equal magnitude; the magnitude of this basic charge is $e=1.602\times10^{-19}~\mathrm{C}$
Both positive and negative charges exist in neutral objects and can be separated by bringing the two objects into physical contact; rubbing the objects together can remove electrons from the bonds in one object and place them on the other object, increasing the charge separation.
For macroscopic objects, negatively charged means an excess of electrons and positively charged means a depletion of electrons.
The law of conservation of charge states that the net charge of a closed system is constant.

1.2 Conductors, Insulators, and Charging by Induction

A conductor is a substance that allows charge to flow freely through its atomic structure.
An insulator holds charge fixed in place.
Polarization is the separation of positive and negative charges in a neutral object. Polarized objects have their positive and negative charges concentrated in different areas, giving them a charge distribution.

1.3 Coulomb’s Law

Coulomb’s law gives the magnitude of the force between point charges. It is
$\vec{\mathbf{F}}_{12}(r)=\frac{1}{4\pi\epsilon_0}\frac{q_1q_2}{r_{12}^2}\hat{\mathbf{r}}_{12}$

where $q_1$ and $q_2$ are two point charges separated by a distance $r$ . This Coulomb force is extremely basic, since most charges are due to point-like particles. It is responsible for all electrostatic effects and underlies most macroscopic forces.

1.4 Electric Field

The electric field is an alteration of space caused by the presence of an electric charge. The electric field mediates the electric force between a source charge and a test charge.
The electric field, like the electric force, obeys the superposition principle
The field is a vector; by definition, it points away from positive charges and toward negative charges.

1.5 Calculating Electric Fields of Charge Distributions

A very large number of charges can be treated as a continuous charge distribution, where the calculation of the field requires integration. Common cases are:
- one-dimensional (like a wire); uses a line charge density $\lambda$
- two-dimensional (metal plate); uses surface charge density $\sigma$
- three-dimensional (metal sphere); uses volume charge density $\rho$
The “source charge” is a differential amount of charge $dq$ . Calculating $dq$ depends on the type of source charge distribution:
$dq=\lambda dl;~~dq=\sigma dA;~~dq=\rho dV.$
Symmetry of the charge distribution is usually key.
Important special cases are the field of an “infinite” wire and the field of an “infinite” plane.

1.6 Electric Field Lines

Electric field diagrams assist in visualizing the field of a source charge.
The magnitude of the field is proportional to the field line density.
Field vectors are everywhere tangent to field lines.

1.7 Electric Dipoles

If a permanent dipole is placed in an external electric field, it results in a torque that aligns it with the external field.
If a nonpolar atom (or molecule) is placed in an external field, it gains an induced dipole that is aligned with the external field.
The net field is the vector sum of the external field plus the field of the dipole (physical or induced).
The strength of the polarization is described by the dipole moment of the dipole, $\vec{\mathbf{p}}=q\vec{\mathbf{d}}$ .

Answers to Check Your Understanding

1.1 The force would point outward.

1.2 The net force would point $58^{\circ}$ below the $-x$ -axis.

1.3 $\vec{\mathbf{E}}=\frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{r}$

1.4 We will no longer be able to take advantage of symmetry. Instead, we will need to calculate each of the two components of the electric field with their own integral.

1.5 The point charge would be $Q=\sigma ab$ where $a$ and $b$ are the sides of the rectangle but otherwise identical.

1.6 The electric field would be zero in between, and have magnitude $\frac{\sigma}{\epsilon_0}$ everywhere else.

Conceptual Questions

1.1 Electric Charge

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

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

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?

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

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

1.2 Conductors, Insulators, and Charging by Induction

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.

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.

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

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

A sphere is shown suspended by a thread from the ceiling. A negatively charged rod is brought near the sphere.

10. While walking on a rug, a person frequently becomes charged because of the rubbing between his shoes and the rug. 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?

11. Compare charging by conduction to charging by induction.

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.

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

14. Why do electrostatic experiments work so poorly in humid weather?

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

16. Can induction be used to produce charge on an insulator?

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.

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

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?

1.3 Coulomb’s Law

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

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?

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

1.4 Electric Field

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

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

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

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?

1.5 Calculating Electric Fields of Charge Distributions

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

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

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.

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?

1.6 Electric Field Lines

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?

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

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

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?

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

1.7 Electric Dipoles

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?

Problems

1.1 Electric Charge

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 $0.500~\mu\mathrm{C}$ ?

38. If $1.80\times10^{20}$ electrons move through a pocket calculator during a full day’s operation, how many coulombs of charge moved through it?

39. To start a car engine, the car battery moves $3.75\times10^{21}$ electrons through the starter motor. How many coulombs of charge were moved?

40. A certain lightning bolt moves $40.0~\mathrm{C}$ of charge. How many fundamental units of charge is this?

41. A $2.5$ – $\mathrm{g}$ copper penny is given a charge of $-2.0\times10^{-9}~\mathrm{C}$ . (a) How many excess electrons are on the penny? (b) By what percent do the excess electrons change the mass of the penny?

42. A $2.5$ – $\mathrm{g}$ copper penny is given a charge of $4.0\times10^{-9}~\mathrm{C}$ . (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?

1.2 Conductors, Insulators, and Charging by Induction

43. Suppose a speck of dust in an electrostatic precipitator has $1.0000\times10^{12}$ 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?

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?

45. A $50.0$ – $\mathrm{g}$ 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 $63.5~\mathrm{u}$ .)

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

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?

1.3 Coulomb’s Law

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.

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 $+3~\mu\mathrm{C}$ charge (c) half a meter above the $+12~\mu\mathrm{C}$ charge in a direction perpendicular to the line joining the two fixed charges.

50. In a salt crystal, the distance between adjacent sodium and chloride ions is $2.82\times10^{-10}~\mathrm{m}$ What is the force of attraction between the two singly charged ions?

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

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?

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 $q_3=10~\mu\mathrm{C}$ placed midway between $q_1$ and $q_2$ ?

54. Where must $q_3$ of the preceding problem be placed so that the net force on it is zero?

55. Two small balls, each of mass $5.0~\mathrm{g}$ , 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?

Two small balls are attached to threads which are in turn tied to the same point on the ceiling. The threads hang at an angle of 5.0 degrees to either side of the vertical. Each ball has a charge Q.

56. Point charges $Q_1=2.0~\mu\mathrm{C}$ and $Q_2=4.0~\mu\mathrm{C}$ are located at $\vec{\mathbf{r}}_1=(4.0\hat{\mathbf{i}}-2.0\hat{\mathbf{j}}+5.0\hat{\mathbf{k}})~\mathrm{m}$ and $\vec{\mathbf{r}}_2=(8.0\hat{\mathbf{i}}+5.0\hat{\mathbf{j}}-9.0\hat{\mathbf{k}})~\mathrm{m}$ . What is the force of $Q_2$ on $Q_1$ ?

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.

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 $0.060~\mathrm{N}$ . What is the original charge on each sphere?

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

Two charges are shown, placed on a horizontal line and separated by 2.0 meters. The charge on the left is a positive 1.0 micro Coulomb charge. The charge on the right is a negative 2.0 micro Coulomb charge. Point P is 1.0 to the right of the negative charge.

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

Charges are shown at the vertices of an equilateral triangle with sides length a. The bottom of the triangle is on the x axis of an x y coordinate system, and the bottom left vertex is at the origin. The charge at the origin is positive q. The charge at the bottom right hand corner is also positive q. The charge at the top vertex is negative two q.

61. Two fixed particles, each of charge $5.0\times10^{-6}~\mathrm{C}$ are $24~\mathrm{cm}$ apart. What force do they exert on a third particle of charge $-2.5\times10^{-6}~\mathrm{C}$ that is $13~\mathrm{cm}$ from each of them?

62. The charges $q_1=2.0\times10^{-7}~\mathrm{C}$ , $q_2=-4.0\times10^{-7}~\mathrm{C}$ , and $q_3=-1.0\times10^{-7}~\mathrm{C}$ are placed at the corners of the triangle shown below. What is the force on $q_1$ ?

Charges are shown at the vertices of a right triangle. The bottom of the triangle is length 4 meters, the vertical side on the left is length 3 meters, and the hypotenuse is length 5 meters. The charge at the top is q sub one and positive, the charge at the bottom left is q sub 3 and negative and the charge at the bottom right is q sub 2 and negative.

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

Charges are shown at the corners of a square with sides length a. All of the charges are positive and all are magnitude q.

64. Point charges $q_1=10~\mu\mathrm{C}$ and $q_2=-30~\mu\mathrm{C}$ are fixed at $\vec{\mathbf{r}}_1=(3.0\hat{\mathbf{i}}-4.0\hat{\mathbf{j}})~\mathrm{m}$ and $\vec{\mathbf{r}}_1=(9.0\hat{\mathbf{i}}+6.0\hat{\mathbf{j}})~\mathrm{m}$ . What is the force of $q_2$ on $q_1$ ?

1.4 Electric Field

65. A particle of charge $2.0\times10^{-8}~\mathrm{C}$ experiences an upward force of magnitude $4.0\times10^{-6}~\mathrm{N}$ 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 $q=-1.0\times10^{-8}~\mathrm{C}$ is placed there, what is the force on it?

66. On a typical clear day, the atmospheric electric field points downward and has a magnitude of approximately $100~\mathrm{N/C}$ . Compare the gravitational and electric forces on a small dust particle of mass $2.0\times10^{-15}~\mathrm{g}$ that carries a single electron charge. What is the acceleration (both magnitude and direction) of the dust particle?

67. Consider an electron that is $10^{-10}~\mathrm{m}$ from an alpha particle ( $q=3.2\times10^{-19}~\mathrm{C}$ ). (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?

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 $\sin^{2}\theta\tan\theta=\frac{q^2}{16\pi\epsilon_0gl^2m}$ .

Two small balls are attached to threads of length l which are in turn tied to the same point on the ceiling. The threads hang at an angle of theta to either side of the vertical. Each ball has a charge q and mass m.

69. What is the electric field at a point where the force on a charge $q=-2.0\times10^{-6}~\mathrm{C}$ is $\left(4.0\hat{\mathbf{i}}-6.0\hat{\mathbf{j}}\right)\times10^{-6}~\mathrm{N}$ ?

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?

71. The electric field in a particular thundercloud is $2.0\times10^{5}~\mathrm{N/C}$ What is the acceleration of an electron in this field?

72. A small piece of cork whose mass is $2.0~\mathrm{g}$ is given a charge of $5.0\times10^{-7}~\mathrm{C}$ . What electric field is needed to place the cork in equilibrium under the combined electric and gravitational forces?

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

74. What is the electric field of a proton at the first Bohr orbit for hydrogen ( $r=5.29\times10^{-11}~\mathrm{m}$ )? What is the force on the electron in that orbit?

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?

76. Two point charges, $q_1=2.0\times10^{-7}~\mathrm{C}$ and $q_2=-6.0\times10^{-8}~\mathrm{C}$ 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?

77. Point charges $q_1=50~\mu\mathrm{C}$ and $q_1=-25~\mu\mathrm{C}$ are placed $1.0~\mathrm{m}$ 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?

78. Can you arrange the two point charges $q_1=-2.0\times10^{-6}~\mathrm{C}$ and $q_2=4.0\times10^{-6}~\mathrm{C}$ along the $x$ -axis so that $E=0$ at the origin?

79. Point charges $q_1=q_2=4.0\times10^{-6}~\mathrm{C}$ 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 $x=0$ , $y=3.0~\mathrm{m}$

1.5 Calculating Electric Fields of Charge Distributions

80. A thin conducting plate $1.0~\mathrm{m}$ on the side is given a charge of $-2.0\times10^{-6}~\mathrm{C}$ . An electron is placed $1.0~\mathrm{cm}$ above the centre of the plate. What is the acceleration of the electron?

81. Calculate the magnitude and direction of the electric field 2.0 m from a long wire that is charged uniformly at $\lambda=4.0\times10^{-6}~\mathrm{C/m}$

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?

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 $d\vec{\mathbf{E}}$ at $P$ due to a small segment $dx$ of the rod, which contains charge $dq=\lambda dx$ . Then find the net field by integrating $d\vec{\mathbf{E}}$ over the length of the rod.)

A horizontal rod of length L is shown. The rod has total charge q. Point P is a distance a to the right of the right end of the rod.

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

A semicircular arc of radius r is shown. The arc has total charge q. Point P is at the center of the circle of which the arc is a part.

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 $8.0~\mu\mathrm{C},$ 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?

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?

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 $P_2$ ?
A horizontal rod of length L is shown. The rod has total charge q. Point P 1 is a distance a over 2 above the midpoint of the rod, so that the horizontal distance from P 1 to each end of the rod is L over 2. Point P 2 is a distance a to the right of the right end of the rod.

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 $y=b$ ?

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 (a) $\vec{\mathbf{r}}=a\hat{\mathbf{i}}+b\hat{\mathbf{j}}$ and (b) $\vec{\mathbf{r}}=c\hat{\mathbf{k}}$

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

An arc that is part of a circle of radius R and with center P is shown. The arc extends from an angle theta to the left of vertical to an angle theta to the right of vertical.

91. A proton moves in the electric field $\vec{\mathbf{E}}=100\hat{\mathbf{i}}~\mathrm{N/C}$ . (a) What are the force on and the acceleration of the proton? (b) Do the same calculation for an electron moving in this field.

92. An electron and a proton, each starting from rest, are accelerated by the same uniform electric field of $200~\mathrm{N/C}$ . Determine the distance and time for each particle to acquire a kinetic energy of $3.2\times10^{-16}~\mathrm{J}$

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?

94. A proton enters the uniform electric field produced by the two charged plates shown below. The magnitude of the electric field is $4.0\times10^{5}~\mathrm{N/C},$ and the speed of the proton when it enters is $1.5\times10^7~\mathrm{m/s}.$ What distance $d$ has the proton been deflected downward when it leaves the plates?

Two oppositely charged horizontal plates are parallel to each other. The upper plate is positive and the lower is negative. The plates are 12.0 centimeters long. The path of a positive proton is shown passing from left to right between the plates. It enters moving horizontally and deflects down toward the negative plate, emerging a distance d below the straight line trajectory.

95. Shown below is a small sphere of mass $0.25~\mathrm{g}$ that carries a charge of $9.0\times10^{-10}~\mathrm{C}.$ 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 $30\times10^{-6}~\mathrm{C/m}^2.$ What is the angle that the string makes with the vertical?

A small sphere is attached to the lower end of a string. The other end of the string is attached to a large vertical conducting plate that has a uniform positive charge density. The string makes an angle of theta with the vertical.

96. Two infinite rods, each carrying a uniform charge density $\lambda,$ are parallel to one another and perpendicular to the plane of the page. (See below.) What is the electrical field at $P_1$ ? At $P_2$ ?

An end view of the arrangement in the problem is shown. Two rods are parallel to one another and perpendicular to the plane of the page. They are separated by a horizontal distance of a. Pint P 1 is a distance of a over 2 above the midpoint between the rods, and so also a distance of a over 2 horizontally from each rod. Point P 2 is a distance of a to the right of the rightmost rod.

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 $O$ ?

A uniform distribution of positive charges is shown on an x y coordinate system. The charges are distributed along a 90 degree arc of a circle of radius r in the first quadrant, centered on the origin. The distribution continues along the positive x and y axes from r to infinity.

98. From a distance of $10\mathrm{cm}$ , a proton is projected with a speed of $v=4.0\times10^{6}~\mathrm{m/s}$ directly at a large, positively charged plate whose charge density is $\sigma=2.0\times10^{-5}~\mathrm{C/m}^2$ . (See below.) (a) Does the proton reach the plate? (b) If not, how far from the plate does it turn around?

A positive charge is shown at a distance of 10 centimeters and moving to the right with a speed of 4.0 times 10 to the 6 meters per second, directly toward a large, positively and uniformly charged vertical plate.

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 v0 to make this calculation? (c) What is the minimum value of $v_0$ such that $-q$ escapes from $Q$ ?

1.6 Electric Field Lines

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

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 $+20~\mu\mathrm{C}$ and $-20~\mu\mathrm{C}$ 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.

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

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.

Four charges are shown at the corners of a square. At the top left is positive 10 nano Coulombs. At the top right is negative 10 nano Coulombs. At the bottom left is negative 10 nano Coulombs. At the bottom right is positive 10 nano Coulombs.

Draw the electric field of the charge distribution.

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 $E$ .

1.7 Electric Dipoles

105. Consider the equal and opposite charges shown below. (a) Show that at all points on the $x$ -axis for which $|x|\gg a$ , $E\approx Qa/2\pi\epsilon_0x^3$ . (b) Show that at all points on the $y$ -axis for which $|y|\gg a$ , $E\approx Qa/\pi\epsilon_0y^3$ .
Two charges are shown on the y axis of an x y coordinate system. Charge +Q is a distance a above the origin, and charge −Q is a distance a below the origin.

Two charges are shown on the y axis of an x y coordinate system. Charge +Q is a distance a above the origin, and charge −Q is a distance a below the origin.

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 $4.0\times10^5~\mathrm{N/C}\hat{\mathbf{i}}$ ? (c) What is the torque on this dipole with an electric field of $-4.0\times10^5~\mathrm{N/C}\hat{\mathbf{i}}$ ? (d) What is the torque on this dipole with an electric field of $\pm4.0\times10^5~\mathrm{N/C}\hat{\mathbf{j}}$ ?

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 $2.3\times10^{-8}~\mathrm{N/C}$ . (You are missing some information for solving this problem; you will need to determine what information you need, and look it up.)

A schematic representation of the outer electron cloud of a neutral water molecule is shown. Three atoms are at the vertices of a triangle. The hydrogen atom has positive q charge and the oxygen atom has minus two q charge, and the angle between the line joining each hydrogen atom with the oxygen atom is one hundred and four degrees. The cloud density is shown as being greater at the oxygen atom.

Additional Problems

108. Point charges $q_1=2.0~\mu\mathrm{C}$ and $q_2=4.0~\mu\mathrm{C}$ are located at $\vec{\mathbf{r}}_1=\left(4.0\hat{\mathbf{i}}-2.0\hat{\mathbf{j}}+2.0\hat{\mathbf{j}}\right)~\mathrm{m}$ and $\vec{\mathbf{r}}_2=\left(4.0\hat{\mathbf{i}}-2.0\hat{\mathbf{j}}+2.0\hat{\mathbf{j}}\right)~\mathrm{m}$ . What is the force of $q_2$ on $q_1$ ?

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

The following charges are shown on an x y coordinate system: Minus 3.0 micro Coulomb on the x axis, 3.0 meters to the left of the origin. Positive 5.0 micro Coulomb at the origin. Positive 9.0 micro Coulomb on the x axis, 3.0 meters to the right of the origin. Positive 6.0 micro Coulomb on the y axis, 3.0 meters above the origin.

110. What is the force on the $2.0$ – $\mu\mathrm{C}$ charge placed at the centre of the square shown below?

Charges are shown at the corners of a square with sides length 1 meter. The top left charge is positive 5.0 micro Coulombs. The top right charge is positive 4.0 micro Coulombs. The bottom left charge is negative 4.0 micro Coulombs. The bottom right charge is positive 2.0 micro Coulombs. A fifth charge of positive 2.0 micro Coulombs is at the center of the square.

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

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 $x_2?$

A charge Q is shown at the origin and a second charge q is shown to its right, on the x axis, moving to the right. Both are positive charges. Point x 1 is between the charges. Point x 2 is to the right of both.

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 $Q$ ?

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

Charges are shown at the vertices of an isosceles right triangle whose sides are length a and those hypotenuse is length M. The right angle is the bottom right corner. The charge at the right angle is positive 2 q. Both of the other two charges are positive q.

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

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

A square with sides of length a is shown. Three charges are shown as follows: At the top left, a charge of negative 2 q. At the top right, a charge of positive q. At the lower left, a charge of positive q.

117. Point charges are placed at the four corners of a rectangle as shown below: $q_1=2.0\times10^{-6}~\mathrm{C},$ $q_2=-2.0\times10^{-6}~\mathrm{C},$ $q_3=4.0\times10^{-6}~\mathrm{C},$ and $q_4=1.0\times10^{-6}~\mathrm{C}.$ What is the electric field at $P$ ?

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?

Three charges are positioned at the corners of a parallelogram. The top and bottom of the parallelogram are horizontal and are 3.0 meters long. The sides are at a thirty degree angle to the x axis. The vertical height of the parallelogram is 1.0 meter. The charges are a positive Q in the lower left corner, positive 2 Q in the lower right corner, and negative 3 Q in the upper left corner.

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 $\vec{\mathrm{E}}=E_0(1+x/a)\hat{\mathbf{i}}.$ What is the kinetic energy of $q$ when it passes through $x=3a$ ?

120. A particle of charge $-q$ and mass $m$ is placed at the centre of a uniformly 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 $f=\frac{1}{2\pi}\sqrt{\frac{qQ}{4\pi\epsilon_0mR^3}}.$

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?

122. The circular arc shown below carries a charge per unit length $\lambda=\lambda_0\cos\theta,$ where $\theta$ is measured from the $x$ -axis. What is the electric field at the origin?

An arc that is part of a circle of radius r and with center at the origin of an x y coordinate system is shown. The arc extends from an angle theta sub zero above the x axis to an angle theta sub zero below the x axis.

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.

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 $d\vec{\mathbf{F}}$ on $q$ due to a small segment $dx$ of the rod, which contains charge $\lambda dx.$ Then, find the net force by integrating $d\vec{\mathbf{F}}$ over the length of the rod.

A rod of length l is shown. The rod lies on the horizontal axis, with its left end at the origin. A positive charge q is on the x axis, a distance a to the right of the right end of the rod.

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.)

A rod of length l is shown. The rod lies on the horizontal axis, with its center at the origin, so the ends are a distance of l over 2 to the left and right of the origin. A positive charge q is on the y axis, a distance a to above the origin.

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.)

A semicircular arc that the upper half of a circle of radius R is shown. A positive charge q is at the center of the circle.

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