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Physics for Scientists and Engineers with Modern Physics

Paul Tipler, Gene Mosca

Chapter 23

Electric Fields - all with Video Answers

Educators

+ 3 more educators

Chapter Questions

06:20

Problem 1

(a) Find to three significant digits the charge and the mass of an ionized hydrogen atom, represented as $\mathrm{H}^{+}$ . Suggestion: Begin by looking up the mass of a neutral atom on the periodic table of the elements. (b) Find the charge and the mass of $\mathrm{Na}^{+},$ a singly ionized sodium atom. (c) Find the charge and the average mass of a chloride ion Cl" that joins with the Na' to make one molecule of table salt. (d) Find the charge and the mass of $\mathrm{Ca}^{++}=\mathrm{Ca}^{2+},$ a doubly ionized calcium atom. (e) You can model the center of an ammonia molecule as an $\mathrm{N}^{3-}$ ion. Find its charge and mass. (f) The plasma in a hot star contains quadruply ionized nitrogen atoms, $\mathrm{N}^{4}+$ . Find their charge and mass. (g) Find the charge and the mass of the nucleus of a nitrogen atom. (h) Find the charge and the mass of the molecular ion $\mathrm{H}_{2} \mathrm{O}^{-}$ .

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
06:57

Problem 2

(a) Calculate the number of electrons in a small, electrically neutral silver pin that has a mass of 10.0 $\mathrm{g}$ . Silver has 47 electrons per atom, and its molar mass is 107.87 $\mathrm{g} / \mathrm{mol}$ . (b) Electrons are added to the pin until the net negative charge is 1.00 $\mathrm{mC}$ . How many electrons are added for every $10^{9}$ electrons already present?

Mihajlo Grcic
Mihajlo Grcic
Numerade Educator
07:39

Problem 3

The Nobel laureate Richard Feynman 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-of-magnitude calculation to substantiate this assertion.

Mihajlo Grcic
Mihajlo Grcic
Numerade Educator
02:07

Problem 4

Two protons in an atomic nucleus are typically separated by a distance of $2 \times 10^{-15} \mathrm{m}$ . The electric repulsion force between the protons is huge, but the attractive nuclear
force is even stronger and keeps the nucleus from bursting apart. What is the magnitude of the electric force between two protons separated by $2.00 \times 10^{-15} \mathrm{m}^{2}$

James Erikson
James Erikson
Numerade Educator
06:13

Problem 5

(a) Two protons in a molecule are separated by $3.80 \times$ $10^{-10} \mathrm{m}$ . Find the electric force exerted by one proton on the other. (b) How does the magnitude of this force compare to the magnitude of the gravitational force between the two protons? (c) What If? What must be the charge-
to-mass ratio of a particle if the magnitude of the gravitational force between two of these particles equals the magnitude of electric force between them?

Katie Mcalpine
Katie Mcalpine
Numerade Educator
06:46

Problem 6

Two small silver spheres, each with a mass of 10.0 $\mathrm{g}$ , are separated by 1.00 $\mathrm{m}$ . Calculate the fraction of the electrons in one sphere that must be transferred to the other in
order to produce an attractive force of $1.00 \times 10^{4} \mathrm{N}$ (about 1 ton) between the spheres. (The number of electrons per atom of silver is $47,$ and the number of atoms per gram is Avogadro's number divided by the molar mass of silver, $107.87 \mathrm{g} / \mathrm{mol} . )$

Vishal Gupta
Vishal Gupta
Numerade Educator
06:21

Problem 7

Three point charges are located at the corners of an equilateral triangle as shown in Figure $\mathrm{P} 23.7$ . Calculate the resultant electric force on the $7.00-\mu \mathrm{C}$ charge.

Mihajlo Grcic
Mihajlo Grcic
Numerade Educator
04:26

Problem 8

Suppose that 1.00 $\mathrm{g}$ of hydrogen is separated into electrons and protons. Suppose also that the protons are placed at the Earth's north pole and the electrons are placed at the south pole. What is the resulting compressional force on the Earth?

Mihajlo Grcic
Mihajlo Grcic
Numerade Educator
06:58

Problem 9

Two identical conducting small spheres are placed with their centers 0.300 $\mathrm{m}$ apart. One is given a charge of 12.0 $\mathrm{nC}$ and the other a charge of $-18.0 \mathrm{nC}$ . (a) Find the electric force exerted by one sphere on the other. (b) What If? The spheres are connected by a conducting wire. Find the electric force between the two after they have come to equilibrium.

Mihajlo Grcic
Mihajlo Grcic
Numerade Educator
06:39

Problem 10

Two small beads having positive charges 3$q$ and $q$ are fixed at the opposite ends of a horizontal, insulating rod, extending from the origin to the point, $x=d$ . As shown in Figure $\mathrm{P} 23.10$ , a third small charged bead is free to slide on the rod. At what position is the third bead in equilibrium? Can it be in stable equilibrium?

Vishal Gupta
Vishal Gupta
Numerade Educator
05:56

Problem 11

Review problem. 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 $0.529 \times 10^{-10} \mathrm{m} .$ (a) Find the electric force between the two. (b) If this force causes the centripetal acceleration of the electron, what is the speed of the electron?

Mihajlo Grcic
Mihajlo Grcic
Numerade Educator
17:26

Problem 12

Review problem. Two identical particles, each having charge $+q,$ are fixed in space and separated by a distance $d$ . A third point 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 the two fixed charges (Fig. P23.12). (a) Show that if $x$ is small compared with $d,$ the motion of $-Q$ will be simple harmonic
along the perpendicular bisector. Determine the period of that motion. (b) 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?

Shareef Jackson
Shareef Jackson
Numerade Educator
04:12

Problem 13

What are the magnitude and direction of the electric field that will balance the weight of (a) an electron and (b) a proton? (Use the data in Table $23.1 .$ )

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
02:31

Problem 14

An object having a net charge of 24.0$\mu \mathrm{C}$ is placed in a uni- form electric field of 610 $\mathrm{N} / \mathrm{C}$ directed vertically. What is the mass of this object if it "floats" in the field?

Mihajlo Grcic
Mihajlo Grcic
Numerade Educator
04:50

Problem 15

In Figure $\mathrm{P} 23.15$ , determine the point (other than infinity) at which the electric field is zero.

Ren Jie Tuieng
Ren Jie Tuieng
Numerade Educator
07:41

Problem 16

An airplane is flying through a thundercloud at a height of 2000 $\mathrm{m}$ . (This is a very dangerous thing to do because of updrafts, turbulence, and the possibility of electric discharge.) If a charge concentration of $+40.0 \mathrm{C}$ is above the plane at a height of 3000 $\mathrm{m}$ within the cloud and a charge concentration of $-40.0 \mathrm{C}$ is at height $1000 \mathrm{m},$ what is the electric field at the aircraft?

Mihajlo Grcic
Mihajlo Grcic
Numerade Educator
06:37

Problem 17

Two point charges 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$k_{e} Q / a^{2}$ . What are the two possible values of the unknown charge?

Mihajlo Grcic
Mihajlo Grcic
Numerade Educator
11:05

Problem 18

Three charges are at the corners of an equilateral triangle as shown in Figure $\mathrm{P} 23.7 .$ (a) Calculate the electric ficld at the position of the $2.00-\mu \mathrm{C}$ charge due to the $7.00-\mu \mathrm{C}$ and $-4.00-\mu \mathrm{C}$ charges. (b) Use your answer to part (a) to determine the force on the $2.00-\mu \mathrm{C}$ charge.

Mihajlo Grcic
Mihajlo Grcic
Numerade Educator
03:38

Problem 19

Three point charges are arranged as shown in Figure P23.19. (a) Find the vector electric field that the $6.00-\mathrm{nC}$ and $-3.00-\mathrm{nC}$ charges together create at the origin. (b) Find the vector force on the $5.00-\mathrm{nC}$ charge.

Khoobchandra Agrawal
Khoobchandra Agrawal
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 $\mathrm{y}=0.500 \mathrm{m} .$

Meghan Miholics
Meghan Miholics
Numerade Educator
12:00

Problem 21

Four point charges are at the corners of a square of side a as shown in Figure $\mathrm{P} 23.21$ . (a) Determine the magnitude and direction of the electric field at the location of charge $q$ . (b) What is the resultant force on $q$ ?

Mihajlo Grcic
Mihajlo Grcic
Numerade Educator
07:08

Problem 22

Consider the electric dipole shown in Figure $\mathrm{P} 23.22 .$ Show that the electric field at a distant point on the $+x$ axis is $E_{x}=4 k_{e} q a / x^{3}$ .

Mihajlo Grcic
Mihajlo Grcic
Numerade Educator
07:30

Problem 23

Consider $n$ equal positive point charges each of magnitude $Q / n$ placed symmetrically around a circle of radius $R .$ (a) Calculate the magnitude of the electric field at a point a distance $x$ on the line passing through the center of the circle and perpendicular to the plane of the circle. (b) Explain why this result is identical to that of the calculation done in Example 23.8 .

Mihajlo Grcic
Mihajlo Grcic
Numerade Educator
05:23

Problem 24

Consider an infinite number of identical charges (each of 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 the fact that
$$
1+\frac{1}{2^{2}}+\frac{1}{3^{2}}+\frac{1}{4^{2}}+\cdots=\frac{\pi^{2}}{6}
$$

Mihajlo Grcic
Mihajlo Grcic
Numerade Educator
11:04

Problem 25

A rod 14.0 $\mathrm{cm}$ long is uniformly charged and has a total charge of $-22.0 \mu \mathrm{C}$ . Determine the magnitude and direction of the electric field along the axis of the rod at a point 36.0 $\mathrm{cm}$ from its center.

Mihajlo Grcic
Mihajlo Grcic
Numerade Educator
06:56

Problem 26

A continuous line of charge lies along the $x$ axis, extending from $x=+x_{0}$ to positive infinity. The line carries charge with a uniform linear charge density $\lambda_{0}$ . What are the magnitude and direction of the electric field at the origin?

Mihajlo Grcic
Mihajlo Grcic
Numerade Educator
05:43

Problem 27

A uniformly charged ring of radius 10.0 $\mathrm{cm}$ has a total charge of 75.0$\mu \mathrm{C}$ . Find the electric field on the axis of the ring at (a) $1.00 \mathrm{cm},$ (b) $5.00 \mathrm{cm},(\mathrm{c}) 30.0 \mathrm{cm},$ and (d) 100 $\mathrm{cm}$ from the center of the ring.

Dading Chen
Dading Chen
Numerade Educator
04:46

Problem 28

A line of charge starts at $x=+x_{0}$ and extends to positive infinity. The linear charge density is $\lambda=\lambda_{0} x_{0} / x$ Determine the electric field at the origin.

Mihajlo Grcic
Mihajlo Grcic
Numerade Educator
06:21

Problem 29

Show that the maximum magnitude $E_{\max }$ of the electric field along the axis of a uniformly charged ring occurs at $x=a / \sqrt{2}$ (see Fig. 23.18 ) and has the value $Q /\left(6 \sqrt{3} \pi \epsilon_{0} a^{2}\right) .$

Dading Chen
Dading Chen
Numerade Educator
06:21

Problem 30

A uniformly charged disk of radius 35.0 $\mathrm{cm}$ carries charge with a density of $7.90 \times 10^{-3} \mathrm{C} / \mathrm{m}^{2}$ . Calculate the electric field on the axis of the disk at (a) $5.00 \mathrm{cm},$ (b) $10.0 \mathrm{cm},$ (c) $50.0 \mathrm{cm},$ and $(\mathrm{d}) 200 \mathrm{cm}$ from the center of the disk.

Dading Chen
Dading Chen
Numerade Educator
11:29

Problem 31

Example 23.9 derives the exact expression for the electric field at a point on the axis of a uniformly charged disk. Consider a disk, of radius $R=3.00 \mathrm{cm},$ having a uniformly distributed charge of $+5.20 \mu \mathrm{C}$ (a) Using the result of Example $23.9,$ compute the electric field at a point on the axis and 3.00 $\mathrm{mm}$ from the center. What If? Compare this answer with the field computed from the near-field approximation $E=\sigma / 2 \epsilon_{0}$ . (b) Using the result of Example 23.9 , compute the electric field at a point on the axis and 30.0 $\mathrm{cm}$ from the center of the disk. What If? Compare this with the electric field obtained by treating the disk as a $+5.20-\mu \mathrm{C}$ point charge at a distance of $30.0 \mathrm{cm} .$

Mihajlo Grcic
Mihajlo Grcic
Numerade Educator
09:20

Problem 32

The electric field along the axis of a uniformly charged disk of radius $R$ and total charge $Q$ was calculated in Example $23.9 .$ Show that the electric field at distances $x$ that are large compared with $R$ approaches that of a point charge $Q=\sigma \pi R^{2}$ (Suggestion: First show that $x /\left(x^{2}+R^{2}\right)^{1 / 2}=$ $\left(1+R^{2} / x^{2}\right)^{-1 / 2}$ and use the binomial expansion
$(1+\delta)^{n} \approx 1+n \delta$ when $\delta<1 . )$

Mihajlo Grcic
Mihajlo Grcic
Numerade Educator
10:11

Problem 33

A uniformly charged insulating rod of length 14.0 $\mathrm{cm}$ is bent into the shape of a semicircle as shown in Figure $\mathrm{P} 23.33$ . The rod has a total charge of $-7.50 \mu \mathrm{C}$ . Find the
magnitude and direction of the electric field at $O,$ the center of the semicircle.

Ren Jie Tuieng
Ren Jie Tuieng
Numerade Educator
17:28

Problem 34

(a) Consider a uniformly charged thin-walled right circular cylindrical shell having total charge Q , radius R, and height h. Determine the electric field at a point a distance d from the right side of the cylinder as shown in Figure P23.34. (Suggestion: Use the result of Example 23.8 and treat the cylinder as a collection of ring charges.) (b) What If? Consider now a solid cylinder with the same dimensions and carrying the same charge, uniformly distributed through its volume. Use the result of Example 23.9 to find
the field it creates at the same point.

Mihajlo Grcic
Mihajlo Grcic
Numerade Educator
13:36

Problem 35

A thin rod of length $\ell$ and uniform charge per unit length $\lambda$ lies along the $x$ axis, as shown in Figure $\mathrm{P} 23.35 .$ (a) Show that the electric field at $P,$ a distance $y$ from the rod along
its perpendicular bisector, has no $x$ component and is given by $E=2 k_{e} \lambda \sin \theta_{0} / y .$ (b) What If? Using your result to part $(\mathrm{a}),$ show that the field of a rod of infinite length is
$E=2 k_{e} \lambda / \mathrm{y}$ . (Suggestion: First calculate the field at $P$ due to an element of length $d x,$ which has a charge $\lambda$ dx. Then change variables from $x$ to $\theta,$ using the relationships $x=$ $y \tan \theta$ and $d x=y \sec ^{2} \theta d \theta,$ and integrate over $\theta . )$

Mihajlo Grcic
Mihajlo Grcic
Numerade Educator
04:38

Problem 36

Three solid plastic cylinders all have radius 2.50 $\mathrm{cm}$ and length $6.00 \mathrm{cm} .$ One (a) carries charge with uniform density 15.0 $\mathrm{nC} / \mathrm{m}^{2}$ everywhere on its surface. Another (b) carries charge with the same uniform density on its curved lateral surface only. The third (c) carries charge with uniform density 500 $\mathrm{nC} / \mathrm{m}^{3}$ throughout the plastic. Find the charge of each cylinder.

Mihajlo Grcic
Mihajlo Grcic
Numerade Educator
08:15

Problem 37

Eight solid plastic cubes, each 3.00 $\mathrm{cm}$ on each edge, are glued together to form each one of the objects (i, ii, in, and iv) shown in Figure $\mathrm{P} 23.37$ . (a) Assuming each object carries charge with uniform density 400 $\mathrm{nC} / \mathrm{m}^{3}$ throughout its volume, find the charge of each object. (b) Assuming each object carries charge with uniform density 15.0 $\mathrm{nC} / \mathrm{m}^{2}$ everywhere on its exposed surface, find the charge on each object. (c) Assuming charge is placed only on the edges where perpendicular surfaces meet, with uniform density $80.0 \mathrm{pC} / \mathrm{m},$ find the charge of each object.

Keshav Singh
Keshav Singh
Numerade Educator
01:20

Problem 38

A positively charged disk has a uniform charge per unit area as described in Example 23.9 . Sketch the electric field lines in a plane perpendicular to the plane of the disk passing through its center.

Banhishikha Sinha
Banhishikha Sinha
Numerade Educator
00:54

Problem 39

A negatively charged rod of finite length carries charge with a uniform charge per unit length. Sketch the electric field lines in a plane containing the rod.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
02:17

Problem 40

Figure $\mathrm{P} 23.40$ shows the electric field lines for two point charges separated by a small distance. (a) Determine the ratio $q_{1} / q_{2} .$ (b) What are the signs of $q_{1}$ and $q_{2} ?$

Deepak Kohli
Deepak Kohli
Numerade Educator
09:06

Problem 41

Three equal positive charges $q$ are at the corners of an equilateral triangle of side $a$ as shown in Figure $\mathrm{P} 23.41 .$ (a) Assume that the three charges together create an electric field. Sketch the field lines in the plane of the charges. Find the location of a point (other than of) where the electric
field is zero. (b) What are the magnitude and direction of the electric field at $P$ due to the two charges at the base?

Mihajlo Grcic
Mihajlo Grcic
Numerade Educator
07:41

Problem 42

An electron and a proton are each placed at rest in an electric field of 520 $\mathrm{N} / \mathrm{C}$ . Calculate the speed of each particle 48.0 $\mathrm{ns}$ after being released.

Mihajlo Grcic
Mihajlo Grcic
Numerade Educator
06:45

Problem 43

A proton accelerates from rest in a uniform electric field of 640 $\mathrm{N} / \mathrm{C}$ . At some later time, its speed is $1.20 \times 10^{6} \mathrm{m} / \mathrm{s}$
(nonrelativistic, because $v$ is much less than the speed of light). (a) Find the acceleration of the proton. (b) How long does it take the proton to reach this speed? (c) How far has it moved in this time? (d) What is its kinetic energy at this time?

Mihajlo Grcic
Mihajlo Grcic
Numerade Educator
03:32

Problem 44

A proton is projected in the positive $x$ direction into a region of a uniform electric field $\mathbf{E}=-6.00 \times 10^{5} \hat{\mathbf{i}} \mathrm{N} / \mathrm{C}$ at $t=0 .$ The proton travels 7.00 $\mathrm{cm}$ before coming to rest. Determine (a) the acceleration of the proton, (b) its initial speed, and (c) the time at which the proton comes to rest.

Vishal Gupta
Vishal Gupta
Numerade Educator
02:49

Problem 45

The electrons in a particle beam each have a kinetic energy $K$ . What are the magnitude and direction of the electric field that will stop these electrons in a distance $d$ ?

Ren Jie Tuieng
Ren Jie Tuieng
Numerade Educator
08:22

Problem 46

A positively charged bead having a mass of 1.00 g falls from rest in a vacuum from a height of 5.00 $\mathrm{m}$ in a uniform vertical electric field with a magnitude of $1.00 \times$ $10^{4} \mathrm{N} / \mathrm{C}$ . The bead hits the ground at a speed of 21.0 $\mathrm{m} / \mathrm{s}$ . Determine (a) the direction of the electric field (up or down), and (b) the charge on the bead.

Mihajlo Grcic
Mihajlo Grcic
Numerade Educator
07:26

Problem 47

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 $(\mathrm{a})$ 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 $(\mathrm{c})$ the horizontal and vertical components of its vlocity after it has traveled 5.00 $\mathrm{cm}$ horizontally.

Mihajlo Grcic
Mihajlo Grcic
Numerade Educator
10:23

Problem 48

Two horizontal metal plates, each $100 \mathrm{~mm}$ square, are aligned $10.0 \mathrm{~mm}$ apart, with one above the other. They are given equal-magnitude charges of opposite sign so that a uniform downward electric field of $2000 \mathrm{~N} / \mathrm{C}$ exists in the region between them. A particle of mass $2.00 \times 10^{-16} \mathrm{~kg}$ and with a positive charge of $1.00 \times 10^{-6} \mathrm{C}$ leaves the center of the bottom negative plate with an initial speed of $1.00 \times 10^{5} \mathrm{~m} / \mathrm{s}$ at an angle of $37.0^{\circ}$ above the horizontal. Describe the trajectory of the particle. Which plate does it strike? Where does it strike, relative to its starting point?

Mihajlo Grcic
Mihajlo Grcic
Numerade Educator
09:01

Problem 49

Protons are projected with an initial speed $v_{i}=9.55 \times$ $10^{3} \mathrm{m} / \mathrm{s}$ into a region where a uniform electric field $\mathbf{E}=-720 \hat{\mathbf{j}} \mathrm{N} / \mathrm{C}$ is present, as shown in Figure $\mathrm{P} 23.49$ . The protons are to hit a target that lies at a horizontal distance of 1.27 $\mathrm{mm}$ from the point where the protons cross the plane and enter the electric field in Figure $\mathrm{P} 23.49$ . Find (a) the two projection angles $\theta$ that will result in a hit and
(b) the total time of flight (the time interval during which the proton is above the plane in Figure $\mathrm{P} 23.49$ ) for each trajectory.

Mihajlo Grcic
Mihajlo Grcic
Numerade Educator
11:13

Problem 50

Two known charges, $-12.0 \mu \mathrm{C}$ and 45.0$\mu \mathrm{C}$ , and an unknown charge are located on the $x$ axis. The charge $-12.0 \mu \mathrm{C}$ is at the origin, and the charge 45.0$\mu \mathrm{C}$ is at $x=$ 15.0 $\mathrm{cm}$ . The unknown charge is to be placed so that each charge is in equilibrium under the action of the electric forces exerted by the other two charges. Is this situation possible? Is it possible in more than one way? Find the required location, magnitude, and sign of the unknown charge.

Mihajlo Grcic
Mihajlo Grcic
Numerade Educator
13:17

Problem 51

A uniform electric field of magnitude 640 $\mathrm{N} / \mathrm{C}$ exists between two parallel plates that are 4.00 $\mathrm{cm}$ apart. A proton is released from the positive plate at the same instant that
an electron is released from the negative plate. (a) Determine the distance from the positive plate at which the two pass each other. (Ignore the electrical attraction between the proton and electron.) (b) What If? Repeat part (a) for a sodium ion $\left(\mathrm{Na}^{+}\right)$ and a chloride ion $\left(\mathrm{Cl}^{-}\right)$ .

Dading Chen
Dading Chen
Numerade Educator
07:50

Problem 52

Three point charges are aligned along the $x$ axis as shown in Figure $\mathrm{P} 23.52$ . Find the electric field at (a) the position $(2.00,0)$ and $(\mathrm{b})$ the position $(0,2.00) .$

Keshav Singh
Keshav Singh
Numerade Educator
04:23

Problem 53

A researcher studying the properties of ions in the upper atmosphere wishes to construct an apparatus with the following characteristics: Using an electric field, a beam of ions, each having charge $q$ , mass $m,$ and initial velocity vi, is turned through an angle of $90^{\circ}$ as each ion undergoes displacement $R \hat{\mathbf{j}}$ . The ions enter a chamber as shown in Figure $\mathrm{P} 23.58$ , and leave through the exit port with the same speed they had when they entered the chamber. The electric field acting on the ions is to have constant magnitude. (a) Suppose the electric field is produced by two concentric cylindrical electrodes not shown in the diagram, and hence is radial. What magnitude should the field have? What If? (b) If the field is produced by two flat plates and is uniform in direction, what value should the field have in this case?

Rashmi Sinha
Rashmi Sinha
Numerade Educator
03:37

Problem 54

A small, 2.00 g plastic ball is suspended by a 20.0 -m-long string in a uniform electric field as shown in Figure $\mathrm{P} 23.54$ . If the ball is in equilibrium when the string makes a $15.0^{\circ}$ angle with the vertical, what is the net charge on the ball?

Mihajlo Grcic
Mihajlo Grcic
Numerade Educator
05:08

Problem 55

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 $\mathrm{P} 23.55$ . When $\mathbf{E}=(3.00 \hat{\mathbf{i}}+5.00 \hat{\mathbf{j}}) \times$ $10^{5} \mathrm{N} / \mathrm{C},$ 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
10:58

Problem 56

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 $\mathrm{P} 23.55$ . When $\mathbf{E}=(A \hat{\mathbf{i}}+B \hat{\mathbf{j}}) \mathrm{N} / \mathrm{C},$ where $A$ and $B$ are positive numbers, the ball is in equilibrium at the angle $\theta .$ Find $(a)$ the charge on the ball and $(b)$ the tension in the string.

Vishal Gupta
Vishal Gupta
Numerade Educator
10:58

Problem 57

Four identical point charges $(q=+10.0 \mu \mathrm{C})$ are located on the corners of a rectangle as shown in Figure $\mathrm{P} 23.57$ . The dimensions of the rectangle are $L=60.0 \mathrm{cm}$ and $W=15.0 \mathrm{cm} .$ Calculate the magnitude and direction of the resultant electric force exerted on the charge at the lower left corner by the other three charges.

Mihajlo Grcic
Mihajlo Grcic
Numerade Educator
05:13

Problem 58

Inez is putting up decorations for her sister's quinceañera (fifteenth birthday party). She ties three light silk ribbons together to the top of a gateway and hangs a rubber bal- loon from each ribbon (Fig. P23.58). To include the effects of the gravitational and buoyant forces on it, each balloon can be modeled as a particle of mass 2.00 g, with its center 50.0 cm from the point of support. To show off the colors of the balloons, Inez rubs the whole surface of each balloon with her woolen scarf, to make them hang separately with gaps between them. The centers of the hanging balloons form a horizontal equilateral triangle with sides 30.0 cm long. What is the common charge each balloon carries?

Dading Chen
Dading Chen
Numerade Educator
04:25

Problem 59

Review problem. Two identical metallic blocks resting on a frictionless horizontal surface are connected by a light metallic spring having a spring constant $k$ as shown in Figure $\mathrm{P} 23.59 \mathrm{a}$ and an unstretched length $L_{i} .$ A total charge $Q$ is slowly placed on the system, causing the spring to stretch to an equilibrium length $L,$ as shown in Figure $\mathrm{P} 23.59 \mathrm{b}$ . Determine the value of $Q,$ assuming that all the charge resides on the blocks and modeling the blocks as point charges.

Mihajlo Grcic
Mihajlo Grcic
Numerade Educator
04:07

Problem 60

Consider a regular polygon with 29 sides. The distance from the center to each vertex is $a$ . Identical charges $q$ are placed at 28 vertices of the polygon. What charge $Q$ is placed at the center of the polygon. What is the magnitude and direction of the force experienced by the charge $Q^{2}$ (Suggestion: You may use the result of Problem 63 in Chapter $3 . )$

Vishal Gupta
Vishal Gupta
Numerade Educator
06:38

Problem 61

Identical thin rods of length 2$a$ carry equal charges $+Q$ uniformly distributed along their lengths. The rods lic along the $x$ axis with their centers separated by a distance $b>2 a$ (Fig. P23.61). Show that the magnitude of the force exerted by the left rod on the right one is given by
$$
F=\left(\frac{k_{c} Q^{2}}{4 a^{2}}\right) \ln \left(\frac{b^{2}}{b^{2}-4 a^{2}}\right)
$$

Dading Chen
Dading Chen
Numerade Educator
07:55

Problem 62

Two small spheres, each of mass 2.00 $\mathrm{g}$ , are suspended by light strings 10.0 $\mathrm{cm}$ in length (Fig. P23.62). A uniform electric field is applied in the $x$ direction. The spheres
have charges equal to $-5.00 \times 10^{-8} \mathrm{C}$ and $+5.00 \times$ $10^{-8} \mathrm{C}$ . Determine the electric field that enables the spheres to be in equilibrium at an angle $\theta=10.0^{\circ} .$

Mihajlo Grcic
Mihajlo Grcic
Numerade Educator
08:39

Problem 63

A line of positive charge is formed into a semicircle of radius $R=60.0 \mathrm{cm}$ as shown in Figure $\mathrm{P} 23.63$ . The charge per unit length along the semicircle is described by the ex- pression $\lambda=\lambda_{0} \cos \theta .$ The total charge on the semicircle is 12.0$\mu \mathrm{C}$ . Calculate the total force on a charge of 3.00$\mu \mathrm{C}$ placed at the center of curvature.

Mihajlo Grcic
Mihajlo Grcic
Numerade Educator
04:57

Problem 64

Three charges of equal magnitude $q$ are fixed in position at the vertices of an equilateral triangle (Fig. P23.64). A fourth charge $Q$ is free to move along the positive $x$ axis

Nadir Iqbal
Nadir Iqbal
Numerade Educator
08:43

Problem 65

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

Mihajlo Grcic
Mihajlo Grcic
Numerade Educator
08:49

Problem 66

Review problem. Four identical particles, each having charge $+q$ , are fixed at the corners of a square of side $L . A$ fifth point charge $-Q$ lies a distance $z$ along the line perpendicular to the plane of the square and passing through the center of the square (Fig. P23.66). (a) Show that the force exerted by the other four charges on $-Q$ is
$$
\mathbf{F}=-\frac{4 k_{e} q Q_{z}}{\left[z^{2}+\left(L^{2} / 2\right)\right]^{3 / 2}} \hat{\mathbf{k}}
$$
Note that this force is directed toward the center of the square whether $z$ is positive $(-Q \text { above the square) or neg- }$ative $(-Q \text { below the square). (b) If } z \text { is small compared }$
with $L,$ the above expression reduces to $\mathbf{F} \approx-(\text { constant) } z \mathbf{k} \text { . }$ Why does this imply that the motion of the charge $-Q$ is simple harmonic, and what is the period of this motion if the mass of $-Q$ is $m ?$

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
09:23

Problem 67

Review problem. A 1.00 -g cork ball with charge 2.00$\mu \mathrm{C}$ is suspended vertically on a 0.500 -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 gravity be included in the calculation for part
(a)? Explain.

Mihajlo Grcic
Mihajlo Grcic
Numerade Educator
05:42

Problem 68

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 $R$ apart (Fig. P23.68). Determine the charge on each bead.

Ren Jie Tuieng
Ren Jie Tuieng
Numerade Educator
08:58

Problem 69

Eight point charges, each of magnitude $q$ , are located on the corners of a cube of edge $s$ , as shown in Figure $\mathrm{P} 23.69$ . (a) Determine the $x, y,$ and $z$ components of the resultant force exerted by the other charges on the charge located at point $A$ . (b) What are the magnitude and direction of this resultant force?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
08:57

Problem 70

Consider the charge distribution shown in Figure $\mathrm{P} 23.69$ (a) Show that the magnitude of the electric field at the center of any face of the cube has a value of 2.18 $\mathrm{k}_{\mathrm{e}} \mathrm{q} / \mathrm{s}^{2}$ . (b) What is the direction of the electric field at the center of the top face of the cube?

Meghan Miholics
Meghan Miholics
Numerade Educator
03:34

Problem 71

Review problem. A negatively charged particle $-q$ is placed at the center of a uniformly charged ring, where the ring has a total positive charge $Q$ as shown in Example 23.8 . The particle, confined to move along the $x$ axis, is displaced a small distance $x$ along the axis (where $x<a$ )and released. Show that the particle oscillates in simple harmonic motion with a frequency given by
$$
f=\frac{1}{2 \pi}\left(\frac{k_{e} q Q}{m a^{3}}\right)^{1 / 2}
$$

Keshav Singh
Keshav Singh
Numerade Educator
09:17

Problem 72

A line of charge with uniform density 35.0 $\mathrm{nC} / \mathrm{m}$ lies along the line $y=-15.0 \mathrm{cm},$ between the points with coordinates $x=0$ and $x=40.0 \mathrm{cm} .$ Find the electric field it creates at the origin.

Dading Chen
Dading Chen
Numerade Educator
05:33

Problem 73

Review problem. An electric dipole in a uniform electric field is displaced slightly from its equilibrium position, as shown in Figure $\mathrm{P} 23.73$ , where $\theta$ is small. The separation of the charges is $2 a,$ and the moment of inertia of the dipole is $I .$ Assuming the dipole is released from this position, show that its angular orientation exhibits simple harmonic motion with a frequency
$$
f=\frac{1}{2 \pi} \sqrt{\frac{2 q a E}{I}}
$$

Ren Jie Tuieng
Ren Jie Tuieng
Numerade Educator