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University Physics with Modern Physics

Roger A. Freedman, Hugh D. Young

Chapter 21

Electric Charge and Electric Field - all with Video Answers

Educators

+ 8 more educators

Chapter Questions

02:20

Problem 1

Excess electrons are placed on a small lead sphere with mass $8.00 \mathrm{~g}$ so that its net charge is $-3.20 \times 10^{-9} \mathrm{C}$. (a) Find the number of excess electrons on the sphere. (b) How many excess electrons are there per lead atom? The atomic number of lead is $82,$ and its atomic mass is $207 \mathrm{~g} / \mathrm{mol} .$

Narayan Hari
Narayan Hari
Numerade Educator
01:53

Problem 2

Lightning occurs when there is a flow of electric charge (principally electrons) between the ground and a thundercloud. The maximum rate of charge flow in a lightning bolt is about $20,000 \mathrm{C} / \mathrm{s} ;$ this lasts for $100 \mu \mathrm{s}$ or less. How much charge flows between the ground and the cloud in this time? How many electrons flow during this time?

Prashant Bana
Prashant Bana
Numerade Educator
01:19

Problem 3

If a proton and an electron are released when they are $2.0 \times 10^{-10} \mathrm{~m}$ apart (a typical atomic distance), find the initial acceleration of each particle.

Narayan Hari
Narayan Hari
Numerade Educator
06:11

Problem 4

In Example $21.4,$ what is the net force (magnitude and direction) on charge $q_{1}$ exerted by the other two charges?

Mohamed Mustafa
Mohamed Mustafa
Numerade Educator
01:07

Problem 5

Neurons are components of the nervous system of the body that transmit signals as electrical impulses travel along their length. These impulses propagate when charge suddenly rushes into and then out of a part of the neuron called an axon. Measurements have shown that, during the inflow part of this cycle, approximately $5.6 \times 10^{11} \mathrm{Na}^{+}$ (sodium ions) per meter, each with charge $+e,$ enter the axon. How many coulombs of charge enter a $1.5 \mathrm{~cm}$ length of the axon during this process?

Narayan Hari
Narayan Hari
Numerade Educator
16:20

Problem 6

Two small spheres spaced $20.0 \mathrm{~cm}$ apart have equal charge. How many excess electrons must be present on each sphere if the magnitude of the force of repulsion between them is $3.33 \times 10^{-21} \mathrm{~N} ?$

DL
Daniel Lebrun
George Mason University
01:04

Problem 7

An average human weighs about $650 \mathrm{~N}$. If each of two average humans could carry $1.0 \mathrm{C}$ of excess charge, one positive and one negative, how far apart would they have to be for the electric attraction between them to equal their $650 \mathrm{~N}$ weight?

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
26:52

Problem 8

Two small aluminum spheres, each having mass $0.0250 \mathrm{~kg}$, are separated by $80.0 \mathrm{~cm}$. (a) How many electrons does each sphere contain? (The atomic mass of aluminum is $26.982 \mathrm{~g} / \mathrm{mol}$, and its atomic number is $13 .$ ) (b) How many electrons would have to be removed from one sphere and added to the other to cause an attractive force between the spheres of magnitude $1.00 \times 10^{4} \mathrm{~N}$ (roughly 1 ton)? Assume that the spheres may be treated as point charges. (c) What fraction of all the electrons in each sphere does this represent?

DL
Daniel Lebrun
George Mason University
01:11

Problem 9

Two small plastic spheres are given positive electric charges. When they are $15.0 \mathrm{~cm}$ apart, the repulsive force between them has magnitude $0.220 \mathrm{~N}$. What is the charge on each sphere (a) if the two charges are equal and (b) if one sphere has four times the charge of the other?

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
01:43

Problem 10

Two point charges are placed on the $x$ -axis as follows: Charge $q_{1}=+4.00 \mathrm{nC}$ is located at $x=0.200 \mathrm{~m},$ and charge $q_{2}=+5.00 \mathrm{nC}$ is at $x=-0.300 \mathrm{~m} .$ What are the magnitude and direction of the total force exerted by these two charges on a negative point charge $q_{3}=-6.00 \mathrm{nC}$ that is placed at the origin?

Ze-Han Lee
Ze-Han Lee
Numerade Educator
06:50

Problem 11

In an experiment in space, one proton is held fixed and another proton is released from rest a distance of $2.50 \mathrm{~mm}$ away. (a) What is the initial acceleration of the proton after it is released? (b) Sketch qualitative (no numbers!) acceleration-time and velocity-time graphs of the released proton's motion.

Rachel Wellington
Rachel Wellington
University of Georgia
11:24

Problem 12

A negative charge of $-0.550 \mu \mathrm{C}$ exerts an upward $0.600 \mathrm{~N}$ force on an unknown charge that is located $0.300 \mathrm{~m}$ directly below the first charge. What are (a) the value of the unknown charge (magnitude and sign); (b) the magnitude and direction of the force that the unknown charge exerts on the $-0.550 \mu \mathrm{C}$ charge?

DL
Daniel Lebrun
George Mason University
05:23

Problem 13

Three point charges are arranged on a line. Charge $q_{3}=+5.00 \mathrm{nC}$ and is at the origin. Charge $q_{2}=-3.00 \mathrm{nC}$ and is at $x=+4.00 \mathrm{~cm} .$ Charge $q_{1}$ is at $x=+2.00 \mathrm{~cm} .$ What is $q_{1}$ (magnitude and sign) if the net force on $q_{3}$ is zero?

Rachel Wellington
Rachel Wellington
University of Georgia
28:30

Problem 14

In Example $21.4,$ suppose the point charge on the $y$ -axis at $y=-0.30 \mathrm{~m}$ has negative charge $-2.0 \mu \mathrm{C},$ and the other charges remain the same. Find the magnitude and direction of the net force on $Q$. How does your answer differ from that in Example $21.4 ?$ Explain the differences.

DL
Daniel Lebrun
George Mason University
05:40

Problem 15

In Example 21.3 , calculate the net force on charge $q_{1}$.

Rachel Wellington
Rachel Wellington
University of Georgia
03:28

Problem 16

Three point charges are arranged along the $x$ -axis. Charge $q_{1}=+3.00 \mu \mathrm{C}$ is at the origin, and charge $q_{2}=-5.00 \mu \mathrm{C}$ is at $x=0.200 \mathrm{~m} .$ Charge $q_{3}=-8.00 \mu \mathrm{C} .$ Where is $q_{3}$ located if the net force on $q_{1}$ is $7.00 \mathrm{~N}$ in the $-x$ -direction?

Narayan Hari
Narayan Hari
Numerade Educator
04:06

Problem 17

Two point charges are located on the $y$ -axis as follows: charge $q_{1}=-1.50 \mathrm{nC}$ at $y=-0.600 \mathrm{~m},$ and charge $q_{2}=+3.20 \mathrm{nC}$ at the origin $(y=0) .$ What is the total force (magnitude and direction) exerted by these two charges on a third charge $q_{3}=+5.00 \mathrm{nC}$ located at $y=-0.400 \mathrm{~m} ?$

Ajay Singhal
Ajay Singhal
Numerade Educator
07:08

Problem 18

The two sides of the DNA double helix are connected by pairs of bases (adenine, thymine, cytosine, and guanine). Because of the geometric shape of these molecules, adenine bonds with thymine and cytosine bonds with guanine. Figure $\mathbf{E} 21.18$ shows the bonding of thymine and adenine. Each charge shown is $\pm e,$ and the $\mathrm{H}-\mathrm{N}$ distance is $0.110 \mathrm{nm}$. (a) Calculate the net force that thymine exerts on adenine. Is it attractive or repulsive? To keep the calculations fairly simple, yet reasonable, consider only the forces due to the $\mathrm{O}-\mathrm{H}-\mathrm{N}$ and the $\mathrm{N}-\mathrm{H}-\mathrm{N}$ combinations, assuming that these two combinations are parallel to each other. Remember, however, that in the $\mathrm{O}-\mathrm{H}-\mathrm{N}$ set, the $\mathrm{O}^{-}$ exerts a force on both the $\mathrm{H}^{+}$ and the $\mathrm{N}^{-},$ and likewise along the $\mathrm{N}-\mathrm{H}-\mathrm{N}$ set. $(\mathrm{b})$ Calculate the force on the electron in the hydrogen atom, which is $0.0529 \mathrm{nm}$ from the proton. Then compare the strength of the bonding force of the electron in hydrogen with the bonding force of the adenine- - thymine molecules.

Sheh Lit Chang
Sheh Lit Chang
University of Washington
05:28

Problem 19

Refer to Exercise 21.18 . Figure E21.19 shows the bonding of cytosine and guanine. The $\mathrm{O}-\mathrm{H}$ and $\mathrm{H}-\mathrm{N}$ distances are each $0.110 \mathrm{nm} .$ In this case, assume that the bonding is due only to the forces along the $\mathrm{O}-\mathrm{H}-\mathrm{O}, \mathrm{N}-\mathrm{H}-\mathrm{N},$ and $\mathrm{O}-\mathrm{H}-\mathrm{N}$
combinations, and assume also that these three combinations are parallel to each other. Calculate the net force that cytosine exerts on guanine due to the preceding three combinations. Is this force attractive or repulsive?

Salamat Ali
Salamat Ali
Numerade Educator
02:32

Problem 20

A uniform electric field exists in the region between two oppositely charged plane parallel plates. A proton is released from rest at the surface of the positively charged plate and strikes the surface of the opposite plate, $1.60 \mathrm{~cm}$ distant from the first, in a time interval of $3.20 \times 10^{-6} \mathrm{~s} .$ (a) Find the magnitude of the electric field. (b) Find the speed of the proton when it strikes the negatively charged plate.

Ze-Han Lee
Ze-Han Lee
Numerade Educator
04:15

Problem 21

A proton is placed in a uniform electric field of $2.75 \times 10^{3} \mathrm{~N} / \mathrm{C} .$ Calculate (a) the magnitude of the electric force felt by the proton; (b) the proton's acceleration; (c) the proton's speed after $1.00 \mu \mathrm{s}$ in the field, assuming it starts from rest.

Rachel Wellington
Rachel Wellington
University of Georgia
02:39

Problem 22

A particle has charge $-5.00 \mathrm{nC}$. (a) Find the magnitude and direction of the electric field due to this particle at a point $0.250 \mathrm{~m}$ directly above it. (b) At what distance from this particle does its electric field have a magnitude of $12.0 \mathrm{~N} / \mathrm{C} ?$

Prashant Bana
Prashant Bana
Numerade Educator
09:40

Problem 23

A proton is traveling horizontally to the right at $4.50 \times 10^{6} \mathrm{~m} / \mathrm{s}$. (a) Find the magnitude and direction of the weakest electric field that can bring the proton uniformly to rest over a distance of $3.20 \mathrm{~cm}$. (b) How much time does it take the proton to stop after entering the field? (c) What minimum field (magnitude and direction) would be needed to stop an electron under the conditions of part (a)?

Rachel Wellington
Rachel Wellington
University of Georgia
09:40

Problem 23

A proton is traveling horizontally to the right at $4.50 \times 10^{6} \mathrm{~m} / \mathrm{s}$. (a) Find the magnitude and direction of the weakest electric field that can bring the proton uniformly to rest over a distance of $3.20 \mathrm{~cm} .$ (b) How much time does it take the proton to stop after entering the field? (c) What minimum field (magnitude and direction) would be needed to stop an electron under the conditions of part (a)?

Rachel Wellington
Rachel Wellington
University of Georgia
02:50

Problem 24

An electron is released from rest in a uniform electric field. The electron accelerates vertically upward, traveling $4.50 \mathrm{~m}$ in the first $3.00 \mu \mathrm{s}$ after it is released. (a) What are the magnitude and direction of the electric field? (b) Are we justified in ignoring the effects of gravity? Justify your answer quantitatively.

Ze-Han Lee
Ze-Han Lee
Numerade Educator
01:19

Problem 25

(a) What must the charge (sign and magnitude) of a $1.45 \mathrm{~g}$ particle be for it to remain stationary when placed in a downwarddirected electric field of magnitude $650 \mathrm{~N} / \mathrm{C} ?$ (b) What is the magnitude of an electric field in which the electric force on a proton is equal in magnitude to its weight?

Narayan Hari
Narayan Hari
Numerade Educator
11:33

Problem 26

A $+8.75 \mu \mathrm{C}$ point charge is glued down on a horizontal frictionless table. It is tied to a $-6.50 \mu \mathrm{C}$ point charge by a light, nonconducting $2.50 \mathrm{~cm}$ wire. $\mathrm{A}$ uniform electric field of magnitude $1.85 \times 10^{8} \mathrm{~N} / \mathrm{C}$ is directed parallel to the wire, as shown in Fig. E21.26. (a) Find the tension in the wire. (b) What would the tension be if both charges were negative?

Maria Gabriela Cota Moreira
Maria Gabriela Cota Moreira
Numerade Educator
12:31

Problem 27

An electron is projected with an initial speed $v_{0}=$ $1.60 \times 10^{6} \mathrm{~m} / \mathrm{s}$ into the uniform field between two parallel plates (Fig. E21.27). Assume that the field between the plates is uniform and directed vertically downward and that the field outside the plates is zero. The electron enters the field at a point midway between the plates. (a) If the electron just misses the upper plate as it emerges from the field, find the magnitude of the electric field. (b) Suppose that the electron in Fig. E21.27 is replaced by a proton with the same initial speed $v_{0} .$ Would the proton hit one of the plates? If not, what would be the magnitude and direction of its vertical displacement as it exits the region between the plates? (c) Compare the paths traveled by the electron and the proton, and explain the differences. (d) Discuss whether it is reasonable to ignore the effects of gravity for each particle.

Sheh Lit Chang
Sheh Lit Chang
University of Washington
06:01

Problem 28

In Exercise 21.27 , what is the speed of the electron as it emerges from the field?

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator
03:43

Problem 29

A point charge is at the origin. With this point charge as the source point, what is the unit vector $\hat{r}$ in the direction of the field point
(a) at $x=0, y=-1.35 \mathrm{~m}$
(b) at $x=12.0 \mathrm{~cm}, y=12.0 \mathrm{~cm} ;$ (c) at $x=-1.10 \mathrm{~m}, y=2.60 \mathrm{~m} ?$ Express your results in terms of the unit vectors $\hat{\imath}$ and $\hat{\jmath}$

Jayashree Behera
Jayashree Behera
Numerade Educator
03:55

Problem 30

(a) An electron is moving east in a uniform electric field of $1.50 \mathrm{~N} / \mathrm{C}$ directed to the west. At point $A,$ the velocity of the electron is $4.50 \times 10^{5} \mathrm{~m} / \mathrm{s}$ toward the east. What is the speed of the electron when it reaches point $B, 0.375 \mathrm{~m}$ east of point $A ?$ (b) A proton is moving in the uniform electric field of part (a). At point $A,$ the velocity of the proton is $1.90 \times 10^{4} \mathrm{~m} / \mathrm{s},$ east. What is the speed of the proton at point $B ?$

Ryan Hood
Ryan Hood
Numerade Educator
01:28

Problem 31

A uniform line of charge with length $20.0 \mathrm{~cm}$ is along the $x$ -axis, with its midpoint at $x=0 .$ Its charge per length is $+4.80 \mathrm{nC} / \mathrm{m}$ A small sphere with charge $-2.00 \mu \mathrm{C}$ is located at $x=0, y=5.00 \mathrm{~cm}$ What are the magnitude and direction of the force that the charged sphere exerts on the line of charge?

Penny Riley
Penny Riley
Numerade Educator
06:52

Problem 32

Two point charges $Q$ and $+q$ (where $q$ is positive) produce the net electric field shown at point $P$ in Fig. $\mathrm{E} 21.32 .$ The field points parallel to the line connecting the two charges. (a) What can you conclude about the sign and magnitude of $Q ?$ Explain your reasoning. (b) If the lower charge were negative instead, would it be possible for the field to have the direction shown in the figure? Explain your reasoning.

Sheh Lit Chang
Sheh Lit Chang
University of Washington
04:39

Problem 33

A very long line of charge with charge per unit length $+8.00 \mu \mathrm{C} / \mathrm{m}$ is on the $x$ -axis and its midpoint is at $x=0 .$ A second very long line of charge with charge per length $-4.00 \mu \mathrm{C} / \mathrm{m}$ is parallel to the $x$ -axis at $y=10.0 \mathrm{~cm}$ and its midpoint is also at $x=0 .$ At what point on the $y$ -axis is the resultant electric field of the two lines of charge equal to zero?

Vishal Gupta
Vishal Gupta
Numerade Educator
09:20

Problem 34

The two charges $q_{1}$ and $q_{2}$ shown in Fig. E21.34 have equal magnitudes. What is the direction of the net electric field due to these two charges at points $A$ (midway between the charges), $B,$ and $C$ if (a) both charges are negative, (b) both charges are positive, (c) $q_{1}$ is positive and $q_{2}$ is negative.

Susan Hallstrom
Susan Hallstrom
Numerade Educator
16:28

Problem 35

A +2.00 nC point charge is at the origin, and a second $-5.00 \mathrm{nC}$ point charge is on the $x$ -axis at $x=0.800 \mathrm{~m}$. (a) Find the electric field (magnitude and direction) at each of the following points on the $x$ -axis: (i) $x=0.200 \mathrm{~m} ;$ (ii) $x=1.20 \mathrm{~m} ;$ (iii) $x=-0.200 \mathrm{~m}$. (b) Find the net electric force that the two charges would exert on an electron placed at each point in part (a).

Vishal Gupta
Vishal Gupta
Numerade Educator
04:20

Problem 36

Repeat Exercise $21.35,$ but now let the charge at the origin be $-4.00 \mathrm{nC}$

Ze-Han Lee
Ze-Han Lee
Numerade Educator
04:23

Problem 37

Three negative point charges lie along a line as shown in Fig. $\mathrm{E} 21.37 .$ Find the magnitude and direction of the electric field this combination of charges produces at point $P,$ which lies $6.00 \mathrm{~cm}$ from the $-2.00 \mu \mathrm{C}$ charge measured perpendicular to the line connecting the three charges.

Supratim Pal
Supratim Pal
Numerade Educator
04:08

Problem 38

A point charge is placed at each corner of a square with side length a. All charges have magnitude $q$. Two of the charges are positive and two are negative (Fig. E21.38). What is the direction of the net electric field at the center of the square due to the four charges, and what is its magnitude in terms of $q$ and $a$ ?

Sheh Lit Chang
Sheh Lit Chang
University of Washington
04:30

Problem 39

Two point charges are separated by 25.0 cm (Fig. E21.39). Find the net electric field these charges produce at (a) point $A$ and (b) point $B$. (c) What would be the magnitude and direction of the electric force this combination of charges would produce on a proton at $A ?$

Kayla Gephart
Kayla Gephart
Numerade Educator
03:27

Problem 40

Point charge $q_{1}=-5.00 \mathrm{nC}$ is at the origin and point charge $q_{2}=+3.00 \mathrm{nC}$ is on the $x$ -axis at $x=3.00 \mathrm{~cm} .$ Point $P$ is on the $y$ -axis at $y=4.00 \mathrm{~cm} .$ (a) Calculate the electric fields $\vec{E}_{1}$ and $\vec{E}_{2}$ at point $P$ due to the charges $q_{1}$ and $q_{2}$. Express your results in terms of unit vectors (see Example 21.6 ). (b) Use the results of part (a) to obtain the resultant field at $P$, expressed in unit vector form.

Ze-Han Lee
Ze-Han Lee
Numerade Educator
04:50

Problem 41

If two electrons are each $1.50 \times 10^{-10} \mathrm{~m}$ from a proton (Fig. E21.41), find the magnitude and direction of the net electric force they will exert on the proton.

Sheh Lit Chang
Sheh Lit Chang
University of Washington
05:37

Problem 42

A nerve signal is transmitted through a neuron when an excess of $\mathrm{Na}^{+}$ ions suddenly enters the axon, a long cylindrical part of the neuron. Axons are approximately $10.0 \mu \mathrm{m}$ in diameter, and measurements show that about $5.6 \times 10^{11} \mathrm{Na}^{+}$ ions per meter (each of charge $+e$ ) enter during this process. Although the axon is a long cylinder, the charge does not all enter everywhere at the same time. A plausible model would be a series of point charges moving along the axon. Consider a $0.10 \mathrm{~mm}$ length of the axon and model it as a point charge. (a) If the charge that enters each meter of the axon gets distributed uniformly along it, how many coulombs of charge enter a $0.10 \mathrm{~mm}$ length of the axon?
(b) What electric field (magnitude and direction) does the sudden influx of charge produce at the surface of the body if the axon is $5.00 \mathrm{~cm}$ below the skin? (c) Certain shark can respond to electric fields as weak as $1.0 \mu \mathrm{N} / \mathrm{C}$. How far from this segment of axon could a shark be and still detect its electric field?

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator
05:52

Problem 43

In a rectangular coordinate system a positive point charge $q=6.00 \times 10^{-9} \mathrm{C}$ is placed at the point $x=+0.150 \mathrm{~m}, y=0,$ and an identical point charge is placed at $x=-0.150 \mathrm{~m}, y=0 .$ Find the $x$ - and $y$ -components, the magnitude, and the direction of the electric field at the following points:
(a) the origin;
(b) $x=0.300 \mathrm{~m}, y=0$
(c) $x=0.150 \mathrm{~m}, y=-0.400 \mathrm{~m}$
(d) $x=0, y=0.200 \mathrm{~m}$

Salamat Ali
Salamat Ali
Numerade Educator
03:29

Problem 44

A point charge $q_{1}=-4.00 \mathrm{nC}$ is at the point $x=0.600 \mathrm{~m}, y=0.800 \mathrm{~m},$ and a second point charge $q_{2}=+6.00 \mathrm{nC}$ is at the point $x=0.600 \mathrm{~m}, y=0 .$ Calculate the magnitude and direction of the net electric field at the origin due to these two point charges.

Ze-Han Lee
Ze-Han Lee
Numerade Educator
05:06

Problem 45

Three parallel sheets of charge, large enough to be treated as infinite sheets, are perpendicular to the $x$ -axis. Sheet $A$ has surface charge density $\sigma_{A}=+8.00 \mathrm{nC} / \mathrm{m}^{2}$. Sheet $B$ is $4.00 \mathrm{~cm}$ to the right of sheet $A$ and has surface charge density $\sigma_{B}=-4.00 \mathrm{nC} / \mathrm{m}^{2} .$ Sheet $C$ is $4.00 \mathrm{~cm}$ to the right of sheet $B,$ so is $8.00 \mathrm{~cm}$ to the right of sheet $A,$ and has surface charge density $\sigma_{C}=+6.00 \mathrm{nC} / \mathrm{m}^{2}$. What are the magnitude and direction of the resultant electric field at a point that is midway between sheets $B$ and $C,$ or $2.00 \mathrm{~cm}$ from each of these two sheets?

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator
02:13

Problem 46

Point charge $q_{1}=-5.00 \mathrm{nC}$ is on the $x$ -axis at $x=-0.400 \mathrm{~m}$. Point $P$ is on the $x$ -axis at $x=+0.200 \mathrm{~m}$. Point charge $q_{2}$ is at the origin. What are the sign and magnitude of $q_{2}$ if the resultant electric field at point $P$ is zero?

Prashant Bana
Prashant Bana
Numerade Educator
09:54

Problem 47

Point charge $A$ is on the $x$ -axis at $x=-3.00 \mathrm{~cm}$. At $x=1.00 \mathrm{~cm}$ on the $x$ -axis its electric field is $2700 \mathrm{~N} / \mathrm{C}$. Point charge $B$ is also on the $x$ -axis, at $x=5.00 \mathrm{~cm}$. The absolute magnitude of charge $B$ is twice that of $A .$ Find the magnitude and direction of the total electric field at the origin if (a) both $A$ and $B$ are positive; (b) both are negative; (c) $A$ is positive and $B$ is negative; (d) $A$ is negative and $B$ is positive.

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator
02:07

Problem 48

A very long, straight wire has charge per unit length $3.20 \times 10^{-10} \mathrm{C} / \mathrm{m} .$ At what distance from the wire is the electric-field magnitude equal to $2.50 \mathrm{~N} / \mathrm{C} ?$

Vishal Gupta
Vishal Gupta
Numerade Educator
03:21

Problem 49

A ring-shaped conductor with radius $a=2.50 \mathrm{~cm}$ has a total positive charge $Q=+0.125 \mathrm{nC}$ uniformly distributed around it (see Fig. 21.23 ). The center of the ring is at the origin of coordinates $O$. (a) What is the electric field (magnitude and direction) at point $P,$ which is on the $x$ -axis at $x=40.0 \mathrm{~cm} ?$ (b) A point charge $q=-2.50 \mu \mathrm{C}$ is placed at $P .$ What are the magnitude and direction of the force exerted by the charge $q$ on the ring?

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator
03:06

Problem 50

A straight, nonconducting plastic wire $8.50 \mathrm{~cm}$ long carries a charge density of $+175 \mathrm{nC} / \mathrm{m}$ distributed uniformly along its length. It is lying on a horizontal tabletop. (a) Find the magnitude and direction of the electric field this wire produces at a point $6.00 \mathrm{~cm}$ directly above its midpoint. (b) If the wire is now bent into a circle lying flat on the table, find the magnitude and direction of the electric field it produces at a point $6.00 \mathrm{~cm}$ directly above its center.

Ze-Han Lee
Ze-Han Lee
Numerade Educator
01:07

Problem 51

Point charges $q_{1}=-4.5 \mathrm{nC}$ and $q_{2}=+4.5 \mathrm{nC}$ are separated by $3.1 \mathrm{~mm}$, forming an electric dipole. (a) Find the electric dipole moment (magnitude and direction).
(b) The charges are in a uniform electric field whose direction makes an angle of $36.9^{\circ}$ with the line connecting the charges. What is the magnitude of this field if the torque exerted on the dipole has magnitude $7.2 \times 10^{-9} \mathrm{~N} \cdot \mathrm{m} ?$

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
01:55

Problem 52

The ammonia molecule $\left(\mathrm{NH}_{3}\right)$ has a dipole moment of $5.0 \times 10^{-30} \mathrm{C} \cdot \mathrm{m} .$ Ammonia molecules in the gas phase are placed in a uniform electric field $\vec{E}$ with magnitude $1.6 \times 10^{6} \mathrm{~N} / \mathrm{C}$. (a) What is the change in electric potential energy when the dipole moment of a molecule changes its orientation with respect to $\vec{E}$ from parallel to perpendicular? (b) At what absolute temperature $T$ is the average translational kinetic energy $\frac{3}{2} k T$ of a molecule equal to the change in potential energy calculated in part (a)? (Note: Above this temperature, thermal agitation prevents the dipoles from aligning with the electric field.)

Kayla Gephart
Kayla Gephart
Numerade Educator
05:08

Problem 53

An electric dipole with dipole moment $\vec{p}$ is in a uniform external electric field $E$. (a) Find the orientations of the dipole for which the torque on the dipole is zero. (b) Which of the orientations in part (a) is stable, and which is unstable? (Hint: Consider a small rotation away from the equilibrium position and see what happens.) (c) Show that for the stable orientation in part (b), the dipole's own electric field tends to oppose the external field.

Jayashree Behera
Jayashree Behera
Numerade Educator
02:57

Problem 54

The dipole moment of the water molecule $\left(\mathrm{H}_{2} \mathrm{O}\right)$ is $6.17 \times 10^{-36} \mathrm{C} \cdot \mathrm{m} .$ Consider a water molecule located at the origin whose dipole moment $\vec{p}$ points in the $+x$ -direction. A chlorine ion $\left(\mathrm{Cl}^{-}\right),$ of charge $-1.60 \times 10^{-19} \mathrm{C},$ is located at $x=3.00 \times 10^{-9} \mathrm{~m}$
Find the magnitude and direction of the electric force that the water molecule exerts on the chlorine ion. Is this force attractive or repulsive? Assume that $x$ is much larger than the separation $d$ between the charges in the dipole, so that the approximate expression for the electric field along the dipole axis derived in Example 21.14 can be used.

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator
11:39

Problem 55

Three charges are at the corners of an isosceles triangle as shown in Fig. E21.55. The $\pm 5.00 \mu \mathrm{C}$ charges form a dipole.
(a) Find the force (magnitude and direction) the $-10.00 \mu \mathrm{C}$ charge exerts on the dipole. (b) For an axis perpendicular to the line connecting the $\pm 5.00 \mu \mathrm{C}$ charges at the midpoint of this line, find the torque (magnitude and direction) exerted on the dipole by the $-10.00 \mu \mathrm{C}$ charge.

Brandy Heflin
Brandy Heflin
Numerade Educator
06:24

Problem 56

An electric dipole with a dipole moment of magnitude $p$ is placed at various orientations in an electric field $\vec{E}$ that is directed to the left. (a) What orientation of the dipole will result in maximum torque directed into the page? What then is the electric potential energy?
(b) What orientation of the dipole will give zero torque and maximum electric potential energy? What type of equilibrium is this: stable, unstable, or neutral?

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator
07:04

Problem 57

Four identical charges $Q$ are placed at the corners of a square of side $L$. (a) In a free-body diagram, show all of the forces that act on one of the charges. (b) Find the magnitude and direction of the total force exerted on one charge by the other three charges.

Jayashree Behera
Jayashree Behera
Numerade Educator
04:54

Problem 58

Two charges are placed on the $x$ -axis: one, of $2.50 \mu \mathrm{C}$, at the origin and the other, of $-3.50 \mu \mathrm{C},$ at $x=0.600 \mathrm{~m}$ (Fig. $\mathrm{P} 21.58$ ). Find the position on the $x$ -axis where the net force on a small charge $+q$ would be zero.

Vishal Gupta
Vishal Gupta
Numerade Educator
08:08

Problem 59

A charge $q_{1}=+5.00 \mathrm{nC}$ is placed at the origin of an $x y-$$x$ -axis at $x=4.00 \mathrm{~cm} .$ (a) If a third charge $q_{3}=+6.00 \mathrm{nC}$ is now placed at the point $x=4.00 \mathrm{~cm}, y=3.00 \mathrm{~cm},$ find the $x$ - and $y$ -components of the total force exerted on this charge by the other two. (b) Find the magnitude and direction of this force.

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator
05:41

Problem 60

Two identical spheres with mass $m$ are hung from silk threads of length $L$ (Fig. $\mathbf{P} 2 \mathbf{1 . 6 0}$ ). The spheres have the same charge, so $q_{1}=q_{2}=q .$ The radius of each sphere is very small compared to the distance between the spheres, so they may be treated as point charges. Show that if the angle $\theta$ is small, the equilibrium separation $d$ between the spheres is $d=\left(q^{2} L / 2 \pi \epsilon_{0} m g\right)^{1 / 3} .$ (Hint: If $\theta$ is small, then $\tan \theta \cong \sin \theta .)$

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator
06:43

Problem 61

Two small spheres with mass $m=15.0 \mathrm{~g}$ are hung by silk threads of length $L=1.20 \mathrm{~m}$ from a common point (Fig. $\mathrm{P} 21.60$ ). When the spheres are given equal quantities of negative charge, so that $q_{1}=q_{2}=q,$ each thread hangs at $\theta=25.0^{\circ}$ from the vertical. (a) Draw a diagram showing the forces on each sphere. Treat the spheres as point charges. (b) Find the magnitude of $q$. (c) Both threads are now shortened to length $L=0.600 \mathrm{~m},$ while the charges $q_{1}$ and $q_{2}$ remain unchanged. What new angle will each thread make with the vertical? (Hint: This part of the problem can be solved numerically by using trial values for $\theta$ and adjusting the values of $\theta$ until a selfconsistent answer is obtained.)

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator
05:47

Problem 62

A small sphere with charge $q=5.00 \mu \mathrm{C}$ and mass $0.500 \mathrm{~g}$ is traveling horizontally toward the east at a height of $60.0 \mathrm{~cm}$ above the ground. The sphere has a speed of $2.00 \mathrm{~m} / \mathrm{s}$ as it enters a region of uniform electric field with magnitude $E$. What is $E$ if the sphere has a speed of $5.00 \mathrm{~m} / \mathrm{s}$ just before it strikes the ground?

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator
02:53

Problem 63

A small $12.3 \mathrm{~g}$ plastic ball is tied to a very light $28.6 \mathrm{~cm}$ string that is attached to the vertical wall of a room (Fig. $\mathbf{P} 2 \mathbf{1 . 6 3}$ ). A uniform horizontal electric field exists in this room. When the ball has been given an excess charge of $-1.11 \mu \mathrm{C},$ you observe that it remains suspended, with the string making an angle of $17.4^{\circ}$ with the wall. Find the magnitude and direction of the electric field in the room.

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator
01:07

Problem 64

A small sphere with positive charge $q$ and mass $m$ is released from rest in a uniform electric field $\vec{E}$ that is directed vertically upward. The magnitude of the field is large enough for the sphere to travel upward when it is released. How long does it take the sphere to travel upward a distance $d$ after being released from rest? Give your answer in terms of $q, m, d, E,$ and the acceleration due to gravity, $g$.

Narayan Hari
Narayan Hari
Numerade Educator
03:46

Problem 65

If we rub a balloon on our hair, the balloon sticks to a wall or ceiling. This is because the rubbing transfers electrons from our hair to the balloon, giving it a net negative charge. When the balloon is placed near the ceiling, the extra electrons in it repel nearby electrons in the ceiling, creating a separation of charge on the ceiling, with positive charge closer to the balloon. Model the interaction as two point-like charges of equal magnitude and opposite signs, separated by a distance of $500 \mu \mathrm{m}$. Neglect the more distant negative charges on the ceiling. (a) A typical balloon has a mass of $4 \mathrm{~g}$. Estimate the minimum magnitude of charge the balloon requires to stay attached to the ceiling. (b) since a balloon sticks handily to the ceiling after being rubbed, assume that it has attained 10 times the estimated minimum charge. Estimate the number of electrons that were transferred to the balloon by the process of rubbing.

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator
08:24

Problem 66

An American penny is $97.5 \%$ zinc and $2.5 \%$ copper and has a mass of $2.5 \mathrm{~g}$. (a) Use the approximation that a penny is pure zinc, which has an atomic mass of $65.38 \mathrm{~g} / \mathrm{mol},$ to estimate the number of electrons in a penny. (Each zinc atom has 30 electrons.) (b) Estimate the net charge on all of the electrons in one penny. (c) The net positive charge on all of the protons in a penny has the same magnitude as the charge on the electrons. Estimate the force on either of two objects with this net magnitude of charge if the objects are separated by $2 \mathrm{~cm}$. (d) Estimate the number of leaves on an oak tree that is 60 feet tall. (e) Imagine a forest filled with such trees, arranged in a square lattice, each $10 \mathrm{~m}$ distant from its neighbors. Estimate how large such a forest would need to be to include as many leaves as there are electrons in one penny. (f) How does that area compare to the surface area of the earth?

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator
01:19

Problem 67

Two particles having charges $q_{1}=0.500 \mathrm{nC} \quad$ and$q_{2}=8.00 \mathrm{nC}$ are separated by a distance of $1.20 \mathrm{~m}$. At what point along the line connecting the two charges is the total electric field due to the two charges equal to zero?

Narayan Hari
Narayan Hari
Numerade Educator
03:39

Problem 68

A -3.00 nC point charge is on the $x$ -axis at $x=1.20 \mathrm{~m}$. A second point charge, $Q,$ is on the $x$ -axis at $-0.600 \mathrm{~m}$. What must be the sign and magnitude of $Q$ for the resultant electric field at the origin to be (a) $45.0 \mathrm{~N} / \mathrm{C}$ in the $+x$ -direction, (b) $45.0 \mathrm{~N} / \mathrm{C}$ in the $-x$ -direction?

Ajay Singhal
Ajay Singhal
Numerade Educator
03:03

Problem 69

A charge $+Q$ is located at the origin, and a charge $+4 Q$ is at distance $d$ away on the $x$ -axis. Where should a third charge, $q$, be placed, and what should be its sign and magnitude, so that all three charges will be in equilibrium?

Narayan Hari
Narayan Hari
Numerade Educator
10:34

Problem 70

A charge of $-3.00 \mathrm{nC}$ is placed at the origin of an $x y-$coordinate system, and a charge of $2.00 \mathrm{nC}$ is placed on the $y$ -axis at $y=4.00 \mathrm{~cm} .$ (a) If a third charge, of $5.00 \mathrm{nC}$, is now placed at the point $x=3.00 \mathrm{~cm}, y=4.00 \mathrm{~cm},$ find the $x-$ and $y-$ components of the total force exerted on this charge by the other two charges. (b) Find the magnitude and direction of this force.

Meghan Miholics
Meghan Miholics
Numerade Educator
View

Problem 71

Three identical point charges $q$ are placed at each of three corners of a square of side $L$. Find the magnitude and direction of the net force on a point charge $-3 q$ placed (a) at the center of the square and (b) at the vacant corner of the square. In each case, draw a free-body diagram showing the forces exerted on the $-3 q$ charge by each of the other three charges.

Yaqub Khan
Yaqub Khan
Numerade Educator
11:40

Problem 72

Two point charges $q_{1}$ and $q_{2}$ are held in place $4.50 \mathrm{~cm}$ apart. Another point charge $Q=-1.75 \mu \mathrm{C},$ of mass $5.00 \mathrm{~g},$ is initially located $3.00 \mathrm{~cm}$ from both of these charges (Fig. $\mathbf{P} 2 \mathbf{1 . 7 2}$ ) and released from rest. You observe that the initial acceleration of $Q$ is $324 \mathrm{~m} / \mathrm{s}^{2}$ upward, parallel to the line connecting the two point charges. Find $q_{1}$ and $q_{2}$.

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator
09:17

Problem 73

Imagine two $1.0 \mathrm{~g}$ bags of protons, one at the earth's north pole and the other at the south pole.
(a) How many protons are in each bag?
(b) Calculate the gravitational attraction and the electric repulsion that each bag exerts on the other.
(c) Are the forces in part (b) large enough for you to feel if you were holding one of the bags?

Jayashree Behera
Jayashree Behera
Numerade Educator
05:45

Problem 74

Two tiny spheres of mass $6.80 \mathrm{mg}$ carry charges of equal magnitude, $72.0 \mathrm{nC}$, but opposite sign. They are tied to the same ceiling hook by light strings of length $0.530 \mathrm{~m}$. When a horizontal uniform electric field $E$ that is directed to the left is turned on, the spheres hang at rest with the angle $\theta$ between the strings equal to $58.0^{\circ}$ (Fig. P21.74). (a) Which ball (the one on the right or the one on the left) has positive charge? (b) What is the magnitude $E$ of the field?

Ze-Han Lee
Ze-Han Lee
Numerade Educator
03:33

Problem 75

Consider a model of a hydrogen atom in which an electron is in a circular orbit of radius $r=5.29 \times 10^{-11} \mathrm{~m}$ around a stationary proton. What is the speed of the electron in its orbit?

Jayashree Behera
Jayashree Behera
Numerade Educator
03:29

Problem 76

A disk with radius $R$ and uniform positive charge density $\sigma$ lies horizontally on a tabletop. A small plastic sphere with mass $M$ and positive charge $Q$ hovers motionless above the center of the disk, suspended by the Coulomb repulsion due to the charged disk.
(a) What is the magnitude of the net upward force on the sphere as a function of the height $z$ above the disk? (b) At what height $h$ does the sphere hover? Express your answer in terms of the dimensionless constant $v \equiv 2 \epsilon_{0} M g /(Q \sigma) .$ (c) If $M=100 \mathrm{~g}, Q=1 \mu \mathrm{C}, R=5 \mathrm{~cm},$ and $\sigma=10 \mathrm{nC} / \mathrm{cm}^{2},$ what is $h ?$

Prashant Bana
Prashant Bana
Numerade Educator
06:37

Problem 77

A proton is projected into a uniform electric field that points vertically upward and has magnitude $E$. The initial velocity of the proton has a magnitude $v_{0}$ and is directed at an angle $\alpha$ below the horizontal. (a) Find the maximum distance $h_{\max }$ that the proton descends vertically below its initial elevation. Ignore gravitational forces. (b) After what horizontal distance $d$ does the proton return to its original elevation? (c) Sketch the trajectory of the proton. (d) Find the numerical values of $h_{\max }$ and $d$ if $E=500 \mathrm{~N} / \mathrm{C}, v_{0}=4.00 \times 10^{5} \mathrm{~m} / \mathrm{s},$ and $\alpha=30.0^{\circ}$

Salamat Ali
Salamat Ali
Numerade Educator
06:40

Problem 78

A small object with mass $m,$ charge $q,$ and initial speed $v_{0}=5.00 \times 10^{3} \mathrm{~m} / \mathrm{s}$ is projected into a uniform electric field between two parallel metal plates of length $26.0 \mathrm{~cm}$ (Fig. $\mathrm{P} 21.78$ ). The electric field between the plates is directed downward and has magnitude $E=800 \mathrm{~N} / \mathrm{C}$. Assume that the field is zero outside the region between the plates. The separation between the plates is large enough for the object to pass between the plates without hitting the lower plate. After passing through the field region, the object is deflected downward a vertical distance $d=1.25 \mathrm{~cm}$ from its original direction of motion and reaches a collecting plate that is $56.0 \mathrm{~cm}$ from the edge of the parallel plates. Ignore gravity and air resistance. Calculate the object's charge-to-mass ratio, $q / m$.

Sheh Lit Chang
Sheh Lit Chang
University of Washington
15:06

Problem 79

Positive charge $Q$ is distributed uniformly along the $x$ axis from $x=0$ to $x=a$. A positive point charge $q$ is located on the positive $x$ -axis at $x=a+r,$ a distance $r$ to the right of the end of $Q$ (Fig. P21.79). (a) Calculate the $x$ - and $y$ -components of the electric field produced by the charge distribution $Q$ at points on the positive $x$ -axis where $x>a$. (b) Calculate the force (magnitude and direction) that the charge distribution $Q$ exerts on $q$. (c) Show that if $r \gg a$, the magnitude of the force in part (b) is approximately $Q q / 4 \pi \epsilon_{0} r^{2}$. Explain why this result is obtained.

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator
25:35

Problem 80

Positive charge $Q$ is distributed uniformly along the positive $y$ -axis between $y=0$ and $y=a$ A negative point charge $-q$ lies on the positive $x$ -axis, a distance $x$ from the origin (Fig. P21.80). (a) Calculate the $x$ - and $y$ -components of the electric field produced by the charge distribution $Q$ at points on the positive $x$ -axis. (b) Calculate the $x$ - and $y$ -components of the force that the charge distribution $Q$ exerts on $q$ (c) Show that if $x \gg a, F_{x} \cong-Q q / 4 \pi \epsilon_{0} x^{2}$ and $F_{y} \cong+Q q a / 8 \pi \epsilon_{0} x^{3} .$ Explain why this result is obtained.

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator
05:58

Problem 81

A uniformly charged disk like the disk in Fig. 21.25 has radius $2.50 \mathrm{~cm}$ and carries a total charge of $7.0 \times 10^{-12} \mathrm{C}$. (a) Find the electric field (magnitude and direction) on the $x$ -axis at $x=20.0 \mathrm{~cm}$. (b) Show that for $x \gg R$, Eq. (21.11) becomes $E=Q / 4 \pi \epsilon_{0} x^{2}$, where $Q$ is the total charge on the disk. (c) Is the magnitude of the electric field you calculated in part (a) larger or smaller than the electric field $20.0 \mathrm{~cm}$ from a point charge that has the same total charge as this disk? In terms of the approximation used in part (b) to derive $E=Q / 4 \pi \epsilon_{0} x^{2}$ for a point charge from Eq. ( 21.11 ), explain why this is so. (d) What is the percent difference between the electric fields produced by the finite disk and by a point charge with the same charge at $x=20.0 \mathrm{~cm}$ and at $x=10.0 \mathrm{~cm} ?$

Salamat Ali
Salamat Ali
Numerade Educator
01:55

Problem 82

A small sphere with mass $m$ carries a positive charge $q$ and is attached to one end of a silk fiber of length $L .$ The other end of the fiber is attached to a large vertical insulating sheet that has a positive surface charge density $\sigma$. Show that when the sphere is in equilibrium, the fiber makes an angle equal to arctan $\left(q \sigma / 2 m g \epsilon_{0}\right)$ with the vertical sheet.

Narayan Hari
Narayan Hari
Numerade Educator
06:00

Problem 83

Negative charge $-Q$ is distributed uniformly around a quarter-circle of radius $a$ that lies in the first quadrant, with the center of curvature at the origin. Find the $x$ - and $y$ -components of the net electric field at the origin.

Salamat Ali
Salamat Ali
Numerade Educator
06:15

Problem 84

A semicircle of radius $a$ is in the first and second quadrants, with the center of curvature at the origin. Positive charge $+Q$ is distributed uniformly around the left half of the semicircle, and negative charge $-Q$ is distributed uniformly around the right half of the semicircle (Fig. P21.84). What are the magnitude and direction of the net electric field at the origin produced by this distribution of charge?

Sheh Lit Chang
Sheh Lit Chang
University of Washington
26:50

Problem 85

Two $1.20 \mathrm{~m}$ nonconducting rods meet at a right angle. One rod carries $+2.50 \mu \mathrm{C}$ of charge distributed uniformly along its length, and the other carries $-2.50 \mu \mathrm{C}$ distributed uniformly along it (Fig. $\mathbf{P} 2 \mathbf{1 . 8 5}$ ). (a) Find the magnitude and direction of the electric field these rods produce at point $P,$ which is $60.0 \mathrm{~cm}$ from each rod. (b) If an electron is released at $P$, what are the magnitude and direction of the net force that these rods exert on it?

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator
02:59

Problem 86

Two very large parallel sheets are $5.00 \mathrm{~cm}$ apart. Sheet $A$ carries a uniform surface charge density of $-8.80 \mu \mathrm{C} / \mathrm{m}^{2},$ and sheet $B,$ which is to the right of $A,$ carries a uniform charge density of $-11.6 \mu \mathrm{C} / \mathrm{m}^{2} .$ Assume that the sheets are large enough to be treated as infinite. Find the magnitude and direction of the net electric field these sheets produce at a point (a) $4.00 \mathrm{~cm}$ to the right of sheet $A ;$ (b) $4.00 \mathrm{~cm}$ to the left of sheet $A ;$ (c) $4.00 \mathrm{~cm}$ to the right of sheet $B$.

Ze-Han Lee
Ze-Han Lee
Numerade Educator
16:26

Problem 87

A thin disk with a circular hole at its center, called an $a n-$ nulus, has inner radius $R_{1}$ and outer radius $R_{2}$ (Fig. $\mathbf{P 2 1 . 8 7}$ ). The disk has a uniform positive surface charge density $\sigma$ on its surface. (a) Determine the total electric charge on the annulus. (b) The annulus lies in the $y z$ plane, with its center at the origin. For an arbitrary point on the $x$ -axis (the axis of the annulus), find the magnitude and direction of the electric field $\overrightarrow{\boldsymbol{E}}$. Consider points both above and below the annulus. (c) Show that at points on the $x$ -axis that are sufficiently close to the origin, the magnitude of the electric field is approximately proportional to the distance between the center of the annulus and the point. How close is "sufficiently close"?
(d) A point particle with mass $m$ and negative charge $-q$ is free to move along the $x$ -axis (but cannot move off the axis). The particle is originally placed at rest at $x=0.01 R_{1}$ and released. Find the frequency of oscillation of the particle. (Hint: Review Section 14.2. The annulus is held stationary.)

Linda Winkler
Linda Winkler
Numerade Educator
14:41

Problem 88

Inkjet printers can be described as either continuous or drop-on-demand. In a continuous inkjet printer, letters are built up by squirting drops of ink at the paper from a rapidly moving nozzle. You are part of an engineering group working on the design of such a printer. Each ink drop will have a mass of $1.4 \times 10^{-8} \mathrm{~g}$. The drops will leave the nozzle and travel toward the paper at $50 \mathrm{~m} / \mathrm{s}$ in a horizontal direction, passing through a charging unit that gives each drop a positive charge $q$ by removing some electrons from it. The drops will then pass between parallel deflecting plates, $2.0 \mathrm{~cm}$ long, where there is a uniform vertical electric field with magnitude $8.0 \times 10^{4} \mathrm{~N} / \mathrm{C}$. Your team is working on the design of the charging unit that places the charge on the drops. (a) If a drop is to be deflected $0.30 \mathrm{~mm}$ by the time it reaches the end of the deflection plates, what magnitude of charge must be given to the drop? How many electrons must be removed from the drop to give it this charge? (b) If the unit that produces the stream of drops is redesigned so that it produces drops with a speed of $25 \mathrm{~m} / \mathrm{s},$ what $q$ value is needed to achieve the same $0.30 \mathrm{~mm}$ deflection?

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator
06:10

Problem 89

Two small spheres, each carrying a net positive charge, are separated by $0.400 \mathrm{~m}$. You have been asked to perform measurements that will allow you to determine the charge on each sphere. You set up a coordinate system with one sphere (charge $q_{1}$ ) at the origin and the other sphere (charge $q_{2}$ ) at $x=+0.400 \mathrm{~m}$. Available to you are a third sphere with net charge $q_{3}=4.00 \times 10^{-6} \mathrm{C}$ and an apparatus that can accurately measure the location of this sphere and the net force on it. First you place the third sphere on the $x$ -axis at $x=0.200 \mathrm{~m} ;$ you measure the net force on it to be $4.50 \mathrm{~N}$ in the $+x$ -direction. Then you move the third sphere to $x=+0.600 \mathrm{~m}$ and measure the net force on it now to be $3.50 \mathrm{~N}$ in the $+x$ -direction. (a) Calculate $q_{1}$ and $q_{2}$. (b) What is the net force (magnitude and direction) on $q_{3}$ if it is placed on the $x$ -axis at $x=-0.200 \mathrm{~m} ?$ (c) At what value of $x$ (other than $x=\pm \infty$ ) could $q_{3}$ be placed so that the net force on it is zero?

Salamat Ali
Salamat Ali
Numerade Educator
07:54

Problem 90

Positive charge $Q$ is distributed uniformly around a very thin conducting ring of radius $a$, as in Fig. 21.23 . You measure the electric field $E$ at points on the ring axis, at a distance $x$ from the center of the ring, over a wide range of values of $x$. (a) Your results for the larger values of $x$ are plotted in Fig. $\mathrm{P} 21.90 \mathrm{a}$ as $E x^{2}$ versus $x$. Explain why the quantity $E x^{2}$ approaches a constant value as $x$ increases. Use Fig. P21.90a to calculate the net charge $Q$ on the ring. (b) Your results for smaller values of $x$ are plotted in Fig. $\mathrm{P} 21.90 \mathrm{~b}$ as $E / x$ versus $x$. Explain why $E / x$ approaches a constant value as $x$ approaches zero. Use Fig. $\mathrm{P} 21.90 \mathrm{~b}$ to calculate $a$.

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator
38:37

Problem 91

Consider an infinite flat sheet with positive charge density $\sigma$ in which a circular hole of radius $R$ has been cut out. The sheet lies in the $x y$ -plane with the origin at the center of the hole. The sheet is parallel to the ground, so that the positive $z$ -axis describes the "upward" direction. If a particle of mass $m$ and negative charge $-q$ sits at rest at the center of the hole and is released, the particle, constrained to the $z$ -axis, begins to fall. As it drops farther beneath the sheet, the upward electric force increases. For a sufficiently low value of $m,$ the upward electrical attraction eventually exceeds the particle's weight and the particle will slow, come to a stop, and then rise back to its original position. This sequence of events will repeat indefinitely. (a) What is the electric field at a depth $\Delta$ beneath the origin along the negative $z$ -axis? (b) What is the maximum mass $m_{\max }$ that would prevent the particle from falling indefinitely? (c) If $m<m_{\max }$, how much work is done by the electric field as the particle drops from $z=0$ to $z=-\Delta ?$ (d) In this same interval, how much work is done by gravity? (e) To what ultimate depth $\Delta_{\max }$ will the particle drop? Express your answer in terms of the dimensionless parameter $\alpha \equiv m / m_{\max }$. (Use the work-energy theorem to find the speed as a function of depth, and then solve for the depth at which the speed is zero.) (f) What is the particle's speed at depth $\Delta \leq \Delta_{\max } ?$ (g) If the sheet has charge density $1.00 \mathrm{nC} / \mathrm{cm}^{2}$, the radius of the hole is $R=10.0 \mathrm{~cm},$ and the particle has mass $25.0 \mathrm{~g}$ and charge $1.00 \mu \mathrm{C}$ what are $m_{\max }$ and $\Delta_{\max } ?$

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator
10:41

Problem 92

Two rigid insulating rods, each of length $L$ and negligible mass, are connected at their centers by a frictionless hinge. One rod is fixed horizontally atop a vertical shaft, while the other rod is free to pivot about the hinge horizontally. A small ball with charge $Q$ and mass $M$ is affixed to each end of each rod. When the rods are perpendicular, there is no net torque on the second rod. We may describe the configuration of this system by the angle $\theta$ between the two rods. The first rod lies on the $x$ -axis and the hinge sits at the origin. The second rod is rotated clockwise as seen from above in Fig. 21.92 . (a) Determine the force $\vec{F}_{1}$ exerted by the charge on the left side of the fixed rod on the charge at the upper right side of the movable rod, as a function of $\theta$. (b) Determine the force $\vec{F}_{2}$ exerted by the charge on the right side of the fixed rod on the charge at the upper right side of the movable rod, as a function of $\theta$.
(c) Determine the torque about the pivot point associated with the force $\vec{F}_{1}$. (d) Determine the torque about the pivot point associated with the force $\vec{F}_{2}$. (e) Given that the net torque exerted on the movable rod has contributions from the charges at both ends, write a formula for the net torque on the movable rod. (f) The equilibrium configuration at which the torque vanishes is $\theta=\pi / 2 .$ Deviations from equilibrium may be parameterized as $\theta=\pi / 2-\epsilon .$ Using power series expansions (see Appendix B), derive the torque for small $\epsilon,$ keeping only the lowest order term. (g) For small amplitudes, this system describes a torsional oscillator. Using the previous result, write an expression for the frequency of the small oscillations.

Alan Gavel
Alan Gavel
Numerade Educator
View

Problem 93

Two thin rods, each with length $L$ and total charge $+Q,$ are parallel and separated by a distance $a .$ The first rod has one end at the origin and its other end on the positive $y$ -axis. The second rod has its lower end on the positive $x$ -axis. (a) Explain why the $y$ -component of the net force on the second rod vanishes. (b) Determine the $x$ -component of the differential force $d F_{2}$ exerted on a small portion of the second rod, with length $d y_{2}$ and position $y_{2},$ by the first rod. (This requires integrating over differential portions of the first rod, parameterized by $\left.d y_{1} .\right)$ (c) Determine the net force $\vec{F}_{2}$ on the second rod by integrating $d F_{2 x}$ over the second rod. (d) Show that in the limit $a \gg L$ the force determined in part (c) becomes $\frac{1}{4 \pi \epsilon_{0}} \frac{Q^{2}}{a^{2}} \hat{\imath}$. (e) Determine the external work required to move the second rod from very far away to the position $x=a$, provided the first rod is held fixed at $x=0 .$ This describes the potential energy of the original configuration.
(f) Suppose $L=50.0 \mathrm{~cm}, a=10.0 \mathrm{~cm}, Q=10.0 \mu \mathrm{C},$ and $m=500 \mathrm{~g}$. If the two rods are released from the original configuration, they will fly apart and ultimately achieve a particular relative speed. What is that relative speed?

Lainey Roebuck
Lainey Roebuck
Numerade Educator
10:42

Problem 94

An insulating rigid rod of length $2 a$ and negligible mass is attached at its center to a pivot at the origin and is free to rotate in the $x y$ -plane. A small ball with mass $M$ and charge $Q$ is attached to one end of the rod. A second small ball with mass $M$ and no charge is attached to the other end. A constant electric field $\vec{E}=-E \hat{\imath}$ is present in the region $y>0$ while the region $y<0$ has a vanishing electric field. Define $\vec{r}$ as the vector that points from the center of the rod to the charged end of the rod, and $\theta$ as the angle between $\vec{r}$ and the positive $x$ -axis. The rod is oriented so that $\theta=0$ and is given an infinitesimal nudge in the direction of increasing $\theta$. (a) Write an expression for the vector $\vec{r}$. (b) Determine the torque $\vec{\tau}$ about the center of the rod when $0 \leq \theta \leq \pi$.
(c) Determine the torque on the rod about its center when $\pi \leq \theta \leq 2 \pi$. (d) What is the moment of inertia $I$ of the system about the $z$ -axis? (e) The potential energy $U(\theta)$ is determined by $\tau=-d U / d \theta .$ Use this equation to write an expression for $U(\theta)$ over the range $0 \leq \theta \leq 4 \pi$ using the convention that $U(0)=0 .$ Make sure that $U(\theta)$ is continuous. (f) The angular velocity of the rod is $\omega=\omega(\theta) .$ Using $\tau=I d^{2} \theta / d t^{2}$ show that the energy $\frac{1}{2} I \omega^{2}+U(\theta)$ is conserved. (g) Using energy conservation, determine an expression for the angular velocity at the $n$ th time the positive charge crosses the negative $y$ -axis.

Alan Gavel
Alan Gavel
Numerade Educator
22:20

Problem 95

Three charges are placed as shown in Fig. P21.95. The magnitude of $q_{1}$ is $2.00 \mu \mathrm{C}$, but its sign and the value of the charge $q_{2}$ are not known. Charge $q_{3}$ is $+4.00 \mu \mathrm{C},$ and the net force $\vec{F}$ on $q_{3}$ is entirely in the negative $x$ -direction. (a) Considering the different possible signs of $q_{1}$, there are four possible force diagrams representing the forces $\vec{F}_{1}$ and $\vec{F}_{2}$ that $q_{1}$ and $q_{2}$ exert on $q_{3} .$ Sketch these four possible force configurations. (b) Using the sketches from part (a) and the direction of $\vec{F},$ deduce the signs of the charges $q_{1}$ and $q_{2}$. (c) Calculate the magnitude of $q_{2}$. (d) Determine $F$, the magnitude of the net force on $q_{3}$.

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator
10:20

Problem 96

Two charges are placed as shown in Fig. P21.96. The magnitude of $q_{1}$ is $3.00 \mu \mathrm{C},$ but its sign and the value of the charge $q_{2}$ are not known. The direction of the net electric field $\overrightarrow{\boldsymbol{E}}$ at point $P$ is entirely in the negative $y$ -direction. (a) Considering the different possible signs of $q_{1}$ and $q_{2},$ four possible diagrams could represent the electric fields $\vec{E}_{1}$ and $\vec{E}_{2}$ produced by $q_{1}$ and $q_{2} .$ Sketch the four possible electricfield configurations. (b) Using the sketches from part (a) and the direction of $\overrightarrow{\boldsymbol{E}}$, deduce the signs of $q_{1}$ and $q_{2}$. (c) Determine the magnitude of $\overrightarrow{\boldsymbol{E}}$.

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator
11:44

Problem 97

Two thin rods of length $L$ lie along the $x$ -axis, one between $x=\frac{1}{2} a$ and $x=\frac{1}{2} a+L$ and the other between $x=-\frac{1}{2} a$ and $x=-\frac{1}{2} a-L .$ Each rod has positive charge $Q$ distributed uniformly along its length. (a) Calculate the electric field produced by the second rod at points along the positive $x$ -axis. (b) Show that the magnitude of the force that one rod exerts on the other is $F=\frac{Q^{2}}{4 \pi \epsilon_{0} L^{2}} \ln \left[\frac{(a+L)^{2}}{a(a+2 L)}\right]$ (c) Show that if $a \gg L,$ the magnitude of this force reduces to $F=Q^{2} / 4 \pi \epsilon_{0} a^{2} .$ (Hint: Use the expansion $\ln (1+z)=$ $z-\frac{1}{2} z^{2}+\frac{1}{3} z^{3}-\cdots,$ valid for $|z| \ll 1 .$ Carry all expansions to at least order $\left.L^{2} / a^{2} .\right)$ Interpret this result. Flying insects such as bees may accumulate a small positive electric charge as they fly. In one experiment, the mean electric charge of 50 bees was measured to be $+(30 \pm 5)$ pC per bee. Researchers also observed the electrical properties of a plant consisting of a flower atop a long stem. The charge on the stem was measured as a positively charged bee approached, landed, and flew away. Plants are normally electrically neutral, so the measured net electric charge on the stem was zero when the bee was very far away. As the bee approached the flower, a small net positive charge was detected in the stem, even before the bee landed. Once the bee landed, the whole plant became positively charged, and this positive charge remained on the plant after the bee flew away. By creating artificial flowers with various charge values, experimenters found that bees can distinguish between charged and uncharged flowers and may use the positive electric charge left by a previous bee as a cue indicating whether a plant has already been visited (in which case, little pollen may remain).

Linda Winkler
Linda Winkler
Numerade Educator
00:55

Problem 98

Consider a bee with the mean electric charge found in the experiment. This charge represents roughly how many missing electrons?
(a) $1.9 \times 10^{8} ;$ (b) $3.0 \times 10^{8} ;$ (c) $1.9 \times 10^{18}$ (d) $3.0 \times 10^{18}$.

Ze-Han Lee
Ze-Han Lee
Numerade Educator
00:37

Problem 99

What is the best explanation for the observation that the electric charge on the stem became positive as the charged bee approached (before it landed)? (a) Because air is a good conductor, the positive charge on the bee's surface flowed through the air from bee to plant. (b) Because the earth is a reservoir of large amounts of charge, positive ions were drawn up the stem from the ground toward the charged bee. (c) The plant became electrically polarized as the charged bee approached. (d) Bees that had visited the plant earlier deposited a positive charge on the stem.

Jayashree Behera
Jayashree Behera
Numerade Educator
04:57

Problem 100

After one bee left a flower with a positive charge, that bee flew away and another bee with the same amount of positive charge flew close to the plant. Which diagram in Fig. $\mathrm{P} 21.100$ (next page) best represents the electric field lines between the bee and the flower?

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator
01:39

Problem 101

In a follow-up experiment, a charge of $+40 \mathrm{pC}$ was placed at the center of an artificial flower at the end of a $30-\mathrm{cm}$ -long stem. Bees were observed to approach no closer than $15 \mathrm{~cm}$ from the center of this flower before they flew away. This observation suggests that the smallest external electric field to which bees may be sensitive is closest to which of these values? (a) $2.4 \mathrm{~N} / \mathrm{C} ;$ (b) $16 \mathrm{~N} / \mathrm{C} ;$ (c) $2.7 \times 10^{-10} \mathrm{~N} / \mathrm{C}$ (d) $4.8 \times 10^{-10} \mathrm{~N} / \mathrm{C}$.

Jayashree Behera
Jayashree Behera
Numerade Educator