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Physics for Scientists and Engineers with Modern Physics

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

Chapter 22

Electric Fields - all with Video Answers

Educators

+ 7 more educators

Chapter Questions

12:11

Problem 1

Find to three significant digits the charge and the mass of the following particles. Suggestion: Begin by looking up the mass of a neutral atom on the periodic table of the elements in Appendix C. (a) an ionized hydrogen atom, represented as $\mathrm{H}^{+}$ (b) a singly ionized sodium atom, $\mathrm{Na}^{+}$ (c) a chloride ion $\mathrm{Cl}^{-}$
(d) a doubly ionized calcium atom, $\mathrm{Ca}^{++}=\mathrm{Ca}^{2+}$ (e) the center of an ammonia molecule, modeled as an $\mathrm{N}^{3-}$ ion (f) quadruply ionized nitrogen atoms, $\mathrm{N}^{4+},$ found in plasma in a hot star
(g) the nucleus of a nitrogen atom
(h) the molecular ion $\mathrm{H}_{2} \mathrm{O}^{-}$

Brandy Heflin
Brandy Heflin
Numerade Educator
03:12

Problem 2

(a) Find the magnitude of the electric force between a $\mathrm{Na}^{+}$ ion and a Cl $^{-}$ ion separated by $0.50 \mathrm{nm}$.
(b) Would the
answer change if the sodium ion were replaced by Lit and the chloride ion by Br"? Explain.

Jacob Schulze
Jacob Schulze
Numerade Educator
01:20

Problem 3

In a thundercloud, there may be electric charges of $+40.0 \mathrm{C}$ near the top of the cloud and $-40.0 \mathrm{C}$ near the bottom of the cloud. These charges are separated by $2.00 \mathrm{~km}$. What is the electric force on the top charge?

Nishant Kumar
Nishant Kumar
Numerade Educator
06:52

Problem 4

Nobel laureate Richard Feynman $(1918-1988)$ once said that if two persons stood at arm's length from each other and each person had $1 \%$ more electrons than protons, the force of repulsion between them would be enough to lift a "weight" equal to that of the entire Earth. Carry out an order-ofmagnitude calculation to substantiate this assertion.

Jacob Shpiece
Jacob Shpiece
Numerade Educator
01:39

Problem 5

A 7.50-nC point charge is located 1.80 m from a 4.20 -nC point charge. (a) Find the magnitude of the electric force that one particle exerts on the other. (b) Is the force attractive or repulsive?

Supratim Pal
Supratim Pal
Numerade Educator
03:45

Problem 6

This afternoon, you have a physics symposium class, and you are the presenter. You will be presenting a topic to physics majors and faculty, You have been so busy that you have not had time to prepare and you don't even have an idea for a topic. You are frantically reading your physics textbook looking for an idea. In your reading, you have learned that the Earth carries a charge on its surface of about $10^{5} \mathrm{C}$, which results in electric fields in the atmosphere. This gets you very excited about a new theory. Suppose the Moon also carries a charge on the order of $10^{5} \mathrm{C},$ with the opposite sign! Maybe the orbit of the Moon around the Earth is due to electrical attraction between the Moon and the Earth! There's an idea for your symposium presentation! You quickly jot down a few notes and run off to your symposium. While you are speaking, you notice one of the professors doing some calculations on a scrap of paper. Uh-oh! He has just raised his hand with a question. Why are you embarrassed?

Jacob Shpiece
Jacob Shpiece
Numerade Educator
03:17

Problem 7

Two small beads having positive charges $q_{1}=3 q$ and $q_{2}=q$ are fixed at the opposite ends of a horizontal insulating rod of length $d=1.50 \mathrm{~m}$. The bead with charge $q_{1}$ is at the origin. As shown in Figure $\mathrm{P} 22.7$, a third small, charged bead is free to slide on the rod. (a) At what position $x$ is the third bead in equilibrium? (b) Can the equilibrium be stable?

Susan Hallstrom
Susan Hallstrom
Numerade Educator
03:32

Problem 8

Two small beads having charges $q_{1}$ and $q_{2}$ of the same sign are fixed at the opposite ends of a horizontal insulating rod of length $d$. The bead with charge $q_{1}$ is at the origin. As shown in Figure $\mathrm{P} 22.8,$ a third small, charged bead is free to slide on the rod. (a) At what position $x$ is the third bead in equilibrium?
(b) Can the equilibrium be stable?

Supratim Pal
Supratim Pal
Numerade Educator
03:22

Problem 9

Review. In the Bohr theory of the hydrogen atom, an electron moves in a circular orbit about a proton, where the radius of the orbit is $5.29 \times 10^{-11} \mathrm{~m}$. (a) Find the magnitude of the electric force exerted on each particle. (b) If this force causes the centripetal acceleration of the electron, what is the speed of the electron?

Jacob Shpiece
Jacob Shpiece
Numerade Educator
06:55

Problem 10

Three point charges lie along a straight line as shown in Figure $\mathrm{P} 22.10,$ where $q_{1}=6.00 \mu \mathrm{C}, q_{2}=1.50 \mu \mathrm{C},$ and $q_{9}=$
$-2.00 \mu \mathrm{C} .$ The separation distances are $d_{1}=3.00 \mathrm{~cm}$ and $d_{2}=2.00 \mathrm{~cm} .$ Calculate the magnitude and direction of the net electric force on
(a) $q_{1},$ (b) $q_{2},$ and
(c) $q_{y}$

Jacob Schulze
Jacob Schulze
Numerade Educator
08:23

Problem 11

A point charge $+2 Q$ is at the origin and a point charge $-Q$ is located along the $x$ axis at $x=d$ as in Figure P22.11. Find a symbolic expression for the net force on a third point charge $+Q$ located along the $y$ axis at $y=d$

Dading Chen
Dading Chen
Numerade Educator
09:31

Problem 12

Particle A of charge $3.00 \times$ $10^{-4} \mathrm{C}$ is at the origin, particle $\mathrm{B}$ of charge $-6.00 \times 10^{-4} \mathrm{C}$ is at $(4.00 \mathrm{~m}, 0),$ and particle C of charge $1.00 \times 10^{-4} \mathrm{C}$ is at $(0,3.00 \mathrm{~m})$. We wish to find the net electric force on $\mathrm{C}$. (a) What is the $x$ component of the electric force exerted by A on C? (b) What is the $y$ component of the force exerted by A on C? (c) Find the magnitude of the force exerted by $\mathrm{B}$ on $\mathrm{C}$. (d) Calculate the $x$ component of the force exerted by $\mathrm{B}$ on $\mathrm{C}$. (e) Calculate the $y$ component of the force exerted by $\mathrm{B}$ on $\mathrm{C}$. (f) Sum the two $x$ components from parts (a) and (d) to obtain the resultant $x$ component of the electric force acting on $\mathrm{C}$. (g) Similarly, find the $y$ component of the resultant force vector acting on
C. (h) Find the magnitude and direction of the resultant electric force acting on $\mathrm{C}$.

Jacob Schulze
Jacob Schulze
Numerade Educator
14:21

Problem 13

Review. Two identical particles, each having charge $+q$, are fixed in space and separated by a distance $d$. A third particle with charge $-Q$ is free to move and lies initially at rest on the perpendicular bisector of the two fixed charges a distance $x$ from the midpoint between those charges (Fig. P22.13). (a) Show that if $x$ is small compared with $d$, the motion of $-Q$ is simple harmonic along the perpendicular bisector.
(b) Determine the period of that motion. (c) How fast will the charge $-Q$ be moving when it is at the midpoint between the two fixed charges if initially it is released at a distance $a<<d$ from the midpoint?

Dading Chen
Dading Chen
Numerade Educator
04:14

Problem 14

Why is the following situation impossible? Two identical dust particles of mass $1.00 \mu \mathrm{g}$ are floating in empty space, far from any external sources of large gravitational or electric fields, and at rest with respect to each other. Both particles carry electric charges that are identical in magnitude and sign. The gravitational and electric forces between the particles happen to have the same magnitude, so each particle experiences zero net force and the distance between the particles remains constant.

Dading Chen
Dading Chen
Numerade Educator
03:01

Problem 15

What are the magnitude and direction of the electric field that will balance the weight of (a) an electron and
(b) a proton?

Jacob Shpiece
Jacob Shpiece
Numerade Educator
04:56

Problem 16

Consider $n$ equal positively charged particles each of magnitude $Q / n$ placed symmetrically around a circle of radius $a$. Calculate the magnitude of the electric field at a point a distance $x$ from the center of the circle and on the line passing through the center and perpendicular to the plane of the circle.

Jacob Schulze
Jacob Schulze
Numerade Educator
08:53

Problem 17

Two equal positively charged particles are at opposite corners of a trapezoid as shown in Figure P22.17. Find symbolic expressions for the total electric field at (a) the point $P$ and (b) the point $P^{\prime}$

Dading Chen
Dading Chen
Numerade Educator
06:48

Problem 18

Two charged particles are located on the $x$ axis. The first is a charge $+Q$ at $x=-a .$ The second is an unknown charge located at $x=+3 a$. The net electric field these charges produce at the origin has a magnitude of $2 k_{e} Q / a^{2} .$ Explain how many values are possible for the unknown charge and find the possible values.

Dading Chen
Dading Chen
Numerade Educator
05:19

Problem 19

Three point charges are located on a circular arc as shown in Figure P22.19. (a) What is the total electric field at $P$, the center of the arc? (b) Find the clectric force that would be exerted on a -5.00 -nC point charge placed at $P$.

Jacob Schulze
Jacob Schulze
Numerade Educator
06:58

Problem 20

Two $2.00-\mu \mathrm{C}$ point charges are located on the $x$ axis. One is at $x=1.00 \mathrm{~m},$ and the other is at $x=-1.00 \mathrm{~m}$
(a) Determine the electric field on the $y$ axis at $y=0.500 \mathrm{~m}$
(b) Calculate the electric force on a $-3.00-\mu \mathrm{C}$ charge placed on the $y$ axis at $y=0.500 \mathrm{~m}$.

Meghan Miholics
Meghan Miholics
Numerade Educator
05:12

Problem 21

Three point charges are arranged as shown in Figure $\mathrm{P} 22.21 .$ (a) Find the vector electric field that the $6.00-n C$ and $-3.00-n C$ charges together create at the origin.
(b) Find the vector force on the $5.00-\mathrm{nC}$ charge.

Jacob Schulze
Jacob Schulze
Numerade Educator
03:10

Problem 22

Consider the electric dipole shown in Figure $\mathrm{P} 22.22$. Show that the electric field at a distant point on the $+x$ axis is $E_{x}=4 k_{e} q a / x^{3}$C

Prashant Bana
Prashant Bana
Numerade Educator
09:56

Problem 23

Three equal positive charges $\underline{q}$ are at the corners of an equilateral triangle of side $a$ as shown in Figure $\mathrm{P} 22.23$. Assume the three charges together create an electric field. (a) Sketch the field lines in the plane of the charges. (b) Find the location of one point (other than $\infty$ ) where the electric field is zero. What are (c) the magnitude and (d) the direction of the electric field at $P$ due to the two charges at the base?

Jacob Schulze
Jacob Schulze
Numerade Educator
04:30

Problem 24

A proton accelerates from rest in a uniform electric field of $640 \mathrm{~N} / \mathrm{C}$. At one later moment, its speed is $1.20 \mathrm{Mm} / \mathrm{s}$ (nonrelativistic because $v$ is much less than the speed of light).
(a) Find the acceleration of the proton. (b) Over what time interval does the proton reach this speed? (c) How far does it move in this time interval? (d) What is its kinetic energy at the end of this interval?

Dading Chen
Dading Chen
Numerade Educator
05:45

Problem 25

A proton moves at $4.50 \times 10^{5} \mathrm{~m} / \mathrm{s}$ in the horizontal direction. It enters a uniform vertical electric field with a magnitude of $9.60 \times 10^{3} \mathrm{~N} / \mathrm{C}$. Ignoring any gravitational effects, find (a) the time interval required for the proton to travel $5.00 \mathrm{~cm}$ horizontally, (b) its vertical displacement during the time interval in which it travels $5.00 \mathrm{~cm}$ horizontally, and (c) the horizontal and vertical components of its velocity after it has traveled $5.00 \mathrm{~cm}$ horizontally.

Dading Chen
Dading Chen
Numerade Educator
14:35

Problem 26

Protons are projected with an initial speed $v_{i}=$ $9.55 \mathrm{~km} / \mathrm{s}$ from a field-free region through a plane and into a region where a uniform electric field $\overrightarrow{\mathbf{E}}=-720 \hat{\mathbf{j}} \mathrm{N} / \mathrm{C}$ is present above the plane as shown
in Figure $\mathrm{P} 22.26 .$ The initial velocity vector of the protons makes an angle $\theta$ with the plane. The protons are to hit a target that lies at a horizontal distance of $R=1.27 \mathrm{~mm}$ from the point where the protons cross the plane and enter the electric field. We wish to find the angle $\theta$ at which the protons must pass through the plane to strike the target. (a) What analysis model describes the horizontal motion of the protons above the plane? (b) What analysis model describes the vertical motion of the protons above the plane? (c) Argue that Fquation 4.20 would be applicable to the protons in this situation. (d) Use Equation 4.20 to write an expression for $R$ in terms of $v_{i}$ $E$, the charge and mass of the proton, and the angle $\theta$. (e) Find the two possible values of the angle $\theta$. (f) Find the time interval during which the proton is above the plane in Figure $\mathrm{P} 22.26$ for each of the two possible values of $\theta$

Dading Chen
Dading Chen
Numerade Educator
11:09

Problem 27

You are still fascinated by the process of inkjet printing, as described in the opening storyline for this chapter. You convince your father to take you to his manufacturing facility to see the machines that print expiration dates on eggs. You strike up a conversation with the technician operating the machine. He tells you that the ink drops are created using a piezoelectric crystal, acoustic waves, and the Plateau-Rayleigh instability, which creates uniform drops of mass $m=1.25 \times 10^{-8}$ g. While you don't understand the fancy words, you do recognize mass! The technician also tells you that the drops are charged to a controllable value of $q$ and then projected vertically downward between parallel deflecting plates at a constant terminal speed of $18.5 \mathrm{~m} / \mathrm{s} .$ The plates are $\ell=2.25 \mathrm{~cm}$ long and have a uniform electric field of magnitude $E=6.35 \times 10^{4} \mathrm{~N} / \mathrm{C}$ between them. Noting your interest in the process, the technician asks you, "If the position on the egg at which the drop is to be deposited requires that its deflection at the bottom end of the plates be $0.17 \mathrm{~mm},$ what is the required charge on the drop?" You quickly get to work to find the answer.

Stylianos Gregoriou
Stylianos Gregoriou
Numerade Educator
07:58

Problem 28

You are working on a research project in which you must control the direction of travel of electrons using deflection plates. You have devised the apparatus shown in Figure $\bar{P} 22.28 .$ The plates are of length $\ell=0.500 \mathrm{~m}$ and are separated by a distance $d=3.00 \mathrm{~cm} .$ Electrons are fired at $v_{i}=5.00 \times 10^{6} \mathrm{~m} / \mathrm{s}$ into a uniform electric field from the left edge of the lower, positive plate, aimed directly at the right edge of the upper, negative plate. Therefore, if there is no electric field between the plates, the electrons will follow the broken line in the figure. With an electric field existing between the plates, the electrons will follow a curved path, bending downward. You need to determine (a) the range of angles over which the electron can leave the apparatus and
(b) the electric field required to give the maximum possible deviation angle.

Rashmi Sinha
Rashmi Sinha
Numerade Educator
02:50

Problem 29

Consider an infinite number of identical particles, each with charge $q$, placed along the $x$ axis at distances $a, 2 a, 3 a$, $4 a, \ldots$ from the origin. What is the electric field at the origin due to this distribution? Suggestion: Use
$$
1+\frac{1}{2^{2}}+\frac{1}{3^{2}}+\frac{1}{4^{2}}+\cdots=\frac{\pi^{2}}{6}
$$

Jacob Shpiece
Jacob Shpiece
Numerade Educator
01:56

Problem 30

A particle with charge $-3.00 \mathrm{nC}$ is at the origin, and a particle with negative charge of magnitude $Q$ is at $x=50.0 \mathrm{~cm} . \mathrm{A}$ third particle with a positive charge is in equilibrium at $x=20.9 \mathrm{~cm} .$ What is $Q ?$

Dading Chen
Dading Chen
Numerade Educator
10:49

Problem 31

A small block of mass $m$ and charge $Q$ is placed on an insulated, frictionless, inclined plane of angle $\theta$ as in Figure P22.31. An electric field is applied parallel to the incline. (a) Find an expression for the magnitude of the electric field that enables the block to remain at rest.
(b) If $m=5.40 \mathrm{~g}$, $Q=-7.00 \mu \mathrm{C},$ and $\theta=25.0^{\circ}$
determine the magnitude and the direction of the elec-
tric ficld that enables the
block to remain at rest on the incline.

Stylianos Gregoriou
Stylianos Gregoriou
Numerade Educator
05:29

Problem 32

A small sphere of charge $q_{1}=0.800 \mu \mathrm{C}$ hangs from
the end of a spring as in Figure P22.32a. When another small sphere of charge $q_{2}=$ $-0.600 \mu \mathrm{C}$ is held beneath the first sphere as in Figure P22.32b, the spring stretches by $d=3.50 \mathrm{~cm}$ from its original length and reaches a new equilibrium position with a separation between the charges of $r=5.00 \mathrm{~cm}$. What is the force constant of the spring?

Susan Hallstrom
Susan Hallstrom
Numerade Educator
05:08

Problem 33

A charged cork ball of mass $1.00 \mathrm{~g}$ is suspended on a light string in the presence of a uniform electric field as shown in Figure $\mathrm{P} 22.33 .$ When $\overrightarrow{\mathbf{E}}=$ $(3.00 \hat{\mathbf{i}}+5.00 \hat{\mathbf{j}}) \times 10^{5} \mathrm{~N} / \mathrm{C},$ the
ball is in equilibrium at $\theta=$ $37.0^{\circ}$. Find
(a) the charge on the ball and (b) the tension in the string.

Dading Chen
Dading Chen
Numerade Educator
14:05

Problem 34

A charged cork ball of mass $m$ is suspended on a light string in the presence of a uniform electric field as shown in Figure $\mathrm{P} 22.33 .$ When $\overrightarrow{\mathbf{E}}=A \hat{\mathbf{i}}+B \hat{\mathbf{j}},$ where $A$ and $B$ are positive quantities, the ball is in equilibrium at the angle $\theta .$ Find (a) the charge on the ball and (b) the tension in the string.

Stylianos Gregoriou
Stylianos Gregoriou
Numerade Educator
11:47

Problem 35

Three charged particles are aligned along the $x$ axis as shown in Figure $\mathrm{P} 22.35 .$ Find the electric field at (a) the position $(2.00 \mathrm{~m}, 0)$ and $(\mathrm{b})$ the position $(0,2.00 \mathrm{~m})$

Melissa Munoz
Melissa Munoz
Numerade Educator
09:38

Problem 36

Two point charges $q_{A}=-12.0 \mu \mathrm{C}$ and $q_{\mathrm{B}}=45.0 \mu \mathrm{C}$ and a third particle with unknown charge $q_{\mathrm{c}}$ are located on the $x$ axis. The particle $q_{\mathrm{A}}$ is at the origin, and $q_{\mathrm{B}}$ is at $x=15.0 \mathrm{~cm}$. The third particle is to be placed so that each particle is in equilibrium under the action of the electric forces exerted by the other two particles. (a) Is this situation possible? If so, is it possible in more than one way? Explain. Find (b) the required location and (c) the magnitude and the sign of the charge of the third particle.

Dading Chen
Dading Chen
Numerade Educator
03:14

Problem 37

Two small spheres hang in equilibrium at the bottom ends of threads, $40.0 \mathrm{~cm}$ long, that have their top ends tied to the same fixed point. One sphere has mass $2.40 \mathrm{~g}$ and charge $+300 \mathrm{nC}$. The other sphere has the same mass and charge $+200 \mathrm{nC}$. Find the distance between the centers of the spheres.

Narayan Hari
Narayan Hari
Numerade Educator
03:59

Problem 38

Four identical charged particles $(q=+10.0 \mu \mathrm{C})$ are located on the corners of a rectangle as shown in Figure 122.38 . The dimensions of the rectangle are $L=60.0 \mathrm{~cm}$ and $W=$ $15.0 \mathrm{~cm} .$ Calculate (a) the magnitude and (b) the direction of the total electric force exerted on the charge at the lower left corner by the other three charges.

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
01:57

Problem 39

Review. Two identical blocks resting on a frictionless, horizontal surface are connected by a light spring having a spring constant $k=100 \mathrm{~N} / \mathrm{m}$ and an unstretched length $L_{i}=0.400 \mathrm{~m}$ as shown in Figure $\mathrm{P} 22.39 \mathrm{a}$.
A charge $Q$ is slowly placed on each block, causing the spring to stretch to an equilibrium length $L=0.500 \mathrm{~m}$ as shown in Figure $\mathrm{P} 22.39 \mathrm{~b}$. Determine the value of $Q,$ modeling the blocks as charged particles.

Dading Chen
Dading Chen
Numerade Educator
08:10

Problem 40

Review. Two identical blocks resting on a frictionless, horizontal surface are connected by a light spring having a spring constant $k$ and an unstretched length $L_{i}$ as shown in Figure $\mathrm{P} 22.39 \mathrm{a}$. A charge $Q$ is slowly placed on each block, causing the spring to stretch to an equilibrium length $I$, as shown in Figure $\mathrm{P} 22.39 \mathrm{~b}$. Determine the value of $Q, \mathrm{mod}-$ eling the blocks as charged particles.

Stylianos Gregoriou
Stylianos Gregoriou
Numerade Educator
03:43

Problem 41

Three identical point charges, each of mass $m=$ $0.100 \mathrm{~kg}$, hang from three strings as shown in Figure $\mathrm{P} 22.41 .$ If the lengths of the left and right strings are each $L=30.0$ $\mathrm{cm}$ and the angle $\theta$ is $45.0^{\circ}$, determine the value of $q$.

Dading Chen
Dading Chen
Numerade Educator
09:14

Problem 42

Why is the following situation impossible? An electron enters a region of uniform electric field between two parallel plates. The plates are used in a cathode-ray tube to adjust the position of an electron beam on a distant fluorescent screen. The magnitude of the electric ficld between the plates is $200 \mathrm{~N} / \mathrm{C} .$ The plates are $0.200 \mathrm{~m}$ in length and are separated by $1.50 \mathrm{~cm} .$ The electron enters the region at a speed of $3.00 \times 10^{6} \mathrm{~m} / \mathrm{s},$ traveling parallel to the plane of the plates in the direction of their length. It leaves the plates heading toward its correct location on the fluorescent screen.

Stylianos Gregoriou
Stylianos Gregoriou
Numerade Educator
10:08

Problem 43

Two hard rubber spheres, each of mass $m=15.0 \mathrm{~g},$ are rubbed with fur on a dry day and are then suspended with two insulating strings of length $L=5.00 \mathrm{~cm}$ whose support points are a distance $d=3.00 \mathrm{~cm}$ from each other as shown in Figure $\mathrm{P} 22.43 .$ During the rubbing process, one sphere receives exactly twice the charge of the other. They are observed to hang at equilibrium, each at an angle of $\theta=10.0^{\circ}$ with the vertical. Find the amount of charge on each sphere.

Stylianos Gregoriou
Stylianos Gregoriou
Numerade Educator
06:38

Problem 44

Two identical beads each have a mass $m$ and charge $q$. When placed in a hemispherical bowl of radius $R$ with frictionless, nonconducting walls, the beads move, and at equilibrium, they are a distance $d$ apart (Fig. $\mathrm{P} 22.44$ ). (a) Determine the charge $q$ on each bead. (b) Determine the charge required for $d$ to become equal to $2 R$.

Dading Chen
Dading Chen
Numerade Educator
02:56

Problem 45

Two small spheres of mass $m$ are suspended from strings of length $\ell$ that are connected at a common point. One sphere has charge $Q$ and the other charge $2 Q .$ The strings make angles $\theta_{1}$ and $\theta_{2}$ with the vertical. (a) Explain how $\theta_{1}$ and $\theta_{2}$ are related. (b) Assume $\theta_{1}$ and $\theta_{2}$ are small. Show that the distance $r$ between the spheres is approximately
$$
r \approx\left(\frac{4 k_{e} Q^{2} \ell}{m g}\right)^{1 / 3}
$$

Dading Chen
Dading Chen
Numerade Educator
11:44

Problem 46

You are working as an expert witness for an inventor. The inventor devised a system that allows an 85.0 -kg human to hover above the ground at the surface of the Earth due to the repulsive force between a charge $q$ applied to his body and the normal electric charge on the Earth. The normal charge on the Earth is such that the electric ficld is uniform near the Earth's surface, directed downward toward the surface, and is of magnitude $130 \mathrm{~N} / \mathrm{C}$ at the location of the engineer's experiments. Everything went well until the engineer tried a new experiment. He attempted to transfer the same amount of charge $q$ to each of two experimental subjects standing next to each other, so they could hover and work close together on a task. The charged, hovering experimental subjects repelled each other and were injured as they flew away in opposite directions. Both experimental subjects are now suing the inventor for their injuries. The inventor is claiming that it is not his fault if the subjects find each other repulsive. To find out whether the inventor has a good defense, determine the initial acceleration of each subject if they are working $1.00 \mathrm{~m}$ apart.

Stylianos Gregoriou
Stylianos Gregoriou
Numerade Educator
04:02

Problem 47

Review. $\mathrm{A} 1.00-\mathrm{g}$ cork ball with charge $2.00 \mu \mathrm{C}$ is suspended vertically on a $0.500-\mathrm{m}$ -long light string in the presence of a uniform, downward-directed electric field of magnitude $E=1.00 \times 10^{5} \mathrm{~N} / \mathrm{C} .$ If the ball is displaced slightly from the vertical, it oscillates like a simple pendulum. (a) Determine the period of this oscillation. (b) Should the effect of gravitation be included in the calculation for part (a)? Explain.

Ren Jie Tuieng
Ren Jie Tuieng
Numerade Educator
09:31

Problem 48

Eight charged particles, each of magnitude $q$, are located on the corners of a cube of edge $s$ as shown in Figure $\mathrm{P} 22.48$.
(a) Determine the $x, y,$ and $z$ components of the total force exerted by the other charges on the charge located at point
A. What are (b) the magnitude and (c) the direction of this total force?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
11:01

Problem 49

Two particles, each with charge $52.0 \mathrm{nC},$ are located on the $y$ axis at $y=25.0 \mathrm{~cm}$ and $y=-25.0 \mathrm{~cm} .$ (a) Find the vector electric field at a point on the $x$ axis as a function of $x$. (b) Find the field at $x=36.0 \mathrm{~cm} .$ (c) $\mathrm{At}$ what location is the field $1.00 \hat{\mathrm{i}} \mathrm{kN} / \mathrm{C}_{2}$ You may need a computer to solve this equation. (d) At what location is the field $16.0 \mathrm{i} \mathrm{kN} / \mathrm{C}$ ?

Stylianos Gregoriou
Stylianos Gregoriou
Numerade Educator
15:23

Problem 50

Review. An electric dipole in a uniform horizontal electric ficld is displaced slightly from its equilibrium position as shown in Figure $\mathrm{P} 22.50$, where $\theta$ is small. The separation of the charges is $2 a$, and each of the two particles has mass $m$.
(a) Assuming the dipole is released from this position, show that its angular orientation exhibits simple harmonic motion with a frequency
$$
f=\frac{1}{2 \pi} \sqrt{\frac{q E}{m a}}
$$
What If? (b) Suppose the masses of the two charged particles in the dipole are not the same even though each particle continues to have charge $q$. Let the masses of the particles be $m_{1}$ and $m_{2^{*}}$ Show that the frequency of the oscillation in this case is
$$
f=\frac{1}{2 \pi} \sqrt{\frac{q E\left(m_{1}+m_{2}\right)}{2 a m_{1} m_{2}}}
$$

Stylianos Gregoriou
Stylianos Gregoriou
Numerade Educator