University Physics with Modern Physics

Hugh D. Young

Chapter 21

Electric Charge and Electric Field - all with Video Answers

Educators

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Chapter Questions

Problem 1

Excess electrons are placed on a small lead sphere with mass 8.00 g so that its net charge is $-3.20 \times 10^{-9} \mathrm{C}$ (a) Find the M 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 $\mathrm{g} / \mathrm{mol}$ .

Merlin Bacon

Merlin Bacon

Numerade Educator

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$\mu$ or less. How much charge flows between the ground and the cloud in this time? How many electrons flow during this time?

Prashant Bana

Numerade Educator

Problem 3

BIO Estimate how many electrons there are in your body. Make any assumptions you feel are necessary, but clearly state what they are. (Hint: Most of the atoms in your body have equal numbers of electrons, protons, and neutrons.) What is the combined charge of all these electrons?

Kayla Gephart

Numerade Educator

Problem 4

Particles in a Gold Ring. You have a pure (24 karat) gold ring with mass 17.7 $\mathrm{g}$ . Gold has atomic mass of 197 $\mathrm{g} / \mathrm{mol}$ and an atomic number of $79 .$ (a) How many protons are in the ring, and what is their total positive charge? (b) If the ring carries no net charge, how many electrons are in it?

Kayla Gephart

Numerade Educator

Problem 5

BI0 Signal Propagation in Neurons. 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?

Kayla Gephart

Numerade Educator

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 $4.57 \times 10^{-21} \mathrm{N} ?$

Kayla Gephart

Numerade Educator

Problem 7

An average human weighs about 650 $\mathrm{N} .$ If two such generic humans each carried 1.0 coulomb of excess charge, one positive and one negative, how far apart would they have to be for the electric attraction hetween them to equal their $650-\mathrm{N}$ weight?

Chris Murray

Numerade Educator

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

Problem 9

Two small plastic spheres are given positive electrical 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?

Matthew Miranda

Matthew Miranda

Numerade Educator

Problem 10

What If We Were Not Neutral? A 75 -kg person holds out his arms so that his hands are 1.7 $\mathrm{m}$ apart. Typically, a person's hand makes up about 1.0$\%$ of his or her body weight. For round numbers, we shall assume that all the weight of each hand is due to the calcium in the bones, and we shall treat the hands as point charges. One mole of Ca contains $40.18 \mathrm{g},$ and each atom has 20 protons and 20 electrons. Suppose that only 1.0$\%$ of the positive charges in each hand were unbalanced by negative charge. (a) How many Ca atoms does each hand contain? (b) How many coulombs of unbalanced charge does each hand contain? (c) What force would the person's arms have to exert on his hands to prevent them from flying off? Does it seem likely that his arms are capable of exerting such a force?

Dading Chen

Numerade Educator

Problem 11

Two very small $8.55-$ g spheres, 15.0 $\mathrm{cm}$ apart from center to center, are charged by adding equal numbers of electrons to each of them. Disregarding all other forces, how many electrons
would you have to add to each sphere so that the two spheres will accelerate at 25.0 $\mathrm{g}$ when released? Which way will they accelerate?

Steven Li

Numerade Educator

Problem 12

Just How Strong Is the Electric Force? Suppose you had two small boxes, each containing 1.0 $\mathrm{g}$ of protons. (a) If one were placed on the moon by an astronaut and the other were left on the earth, and if they were connected by a very light (and very long!) string, what would be the tension in the string? Express your answer in newtons and in pounds. Do you need to take into account the
gravitational forces of the earth and moon on the protons? Why? (b) What gravitational force would each box of protons exert on the other box?

Kayla Gephart

Numerade Educator

Problem 13

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.

Janielle Madlansacay

Janielle Madlansacay

Numerade Educator

Problem 14

A negative charge of $-0.550 \mu C$ exerts an upward $0.200-\mathrm{N}$ force on an unknown charge 0.300 $\mathrm{m}$ directly below it. (a) What is the unknown charge (magnitude and sign)? (b) What are the magnitude and direction of the force that the unknown charge exerts on the $-0.550-\mu \mathrm{C}$ charge?

Ryan Hood

Numerade Educator

Problem 15

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

Problem 16

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.

Matthew Miranda

Matthew Miranda

Numerade Educator

Problem 17

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

Rachel Wellington

Rachel Wellington

University of Georgia

Problem 18

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

Problem 19

Three point charges are arranged along the $x$ -axis. Charge $q_{1}=+3.00 \mu C$ is at the origin, and charge $q_{2}=-5.00 \mu C$ 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

Numerade Educator

Problem 20

Repeat Exercise 21.19 for $q_{3}=+8.00 \mu \mathrm{C}$

Narayan Hari

Numerade Educator

Problem 21

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 $q_{3}=+5.00 \mathrm{nClocated}$ at $y=-0.400 \mathrm{m} ?$

Nicholas Mogoi

Numerade Educator

Problem 22

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?

Narayan Hari

Numerade Educator

Problem 23

BIO Base Pairing in DNA, I. 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 E21.23 shows the thymine-adenine bond. 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. (b) Calculate the force on the electron in the hydrogen atom, which is 0.0529 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.

Ryan Hood

Numerade Educator

Problem 24

BIO Base Pairing in DNA, II. Refer to Exercise 21.23 . Figure E 21.24 shows the bonding of the cytosine and guanine molecules. 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?

Jacob Schulze

Numerade Educator

Problem 25

LA 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$\mu$ in the field, assuming it starts from rest.

Rachel Wellington

Rachel Wellington

University of Georgia

Problem 26

A particle has charge $-3.00 \mathrm{nC}$ . (a) Find the magnitude and direction of the electric field due to this particle at a point 0.250 m directly above it. (b) At what distance from this particle does its electric field have a magnitude of 12.0 $\mathrm{N} / \mathrm{C} ?$

Vishal Gupta

Numerade Educator

Problem 27

CP 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)?

Matthew Miranda

Matthew Miranda

Numerade Educator

Problem 28

CP 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$\mu$ 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,

Kayla Gephart

Numerade Educator

Problem 29

(a) What must the charge (sign and magnitude) of a 1.45 -g particle be for it to remain stationary when placed in a downward-directed 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?

Jeremy Hurley

Numerade Educator

Problem 30

A point charge is placed at each corner of a square with side length $a$ . The charges all have the
same magnitude $q .$ Two of the charges are positive and two are negative, as shown in Fig. E21.30. 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 $a$ ?

Dading Chen

Numerade Educator

Problem 31

Two point charges are separated by 25.0 $\mathrm{cm}$ (Fig. E21.31). 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

Numerade Educator

Problem 32

Electric Field of the Earth. The earth has a net electric charge that causes a field at points near its surface equal to 150 $\mathrm{N} / \mathrm{C}$ and directed in toward the center of the earth. (a) What
magnitude and sign of charge would a $60-\mathrm{kg}$ human have to acquire to overcome his or her weight by the force exerted by the earth's electric field? (b) What would be the force of repulsion between two people each with the charge calculated in part (a) and separated by a distance of 100 $\mathrm{m} ?$ Is use of the earth's electric field a feasible means of flight? Why or why not?

Ze-Han Lee

Numerade Educator

Problem 33

CP An electron is projected with an initial speed $v_{0}=1.60 \times$ $10^{6} \mathrm{m} / \mathrm{s} \quad$ into the uniform field between the parallel plates in Fig. E21.33. 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 in Fig. $\mathrm{E} 21.33$ the electron is replaced by a proton with the same initial speed $v_{0} .$ Would the proton hit one of the plates? If the proton would not hit one of the plates, 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

University of Washington

Problem 34

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 $\vec{\boldsymbol{E}}_{1}$ and $\vec{\boldsymbol{E}}_{2}$ 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

Numerade Educator

Problem 35

Cp In Exercise $21.33,$ what is the speed of the electron
as it emerges from the field?

Janielle Madlansacay

Janielle Madlansacay

Numerade Educator

Problem 36

(a) Calculate the magnitude and direction (relative to the $+x$ -axis) of the electric field in Example $21.6 .$ (b) $\mathrm{A}-2.5$ -nC point charge is placed at point $P$ in Fig. $21.19 .$ Find the magnitude and direction of (i) the force that the $-8.0-$ nC charge at the origin exerts on this charge and (ii) the force that this charge exerts on the $-8.0-\mathrm{nC}$ charge at the origin.

Ze-Han Lee

Numerade Educator

Problem 37

If two electrons are each $1.50 \times$ $10^{-10} \mathrm{m}$ from a proton, as shown in Fig.
$\mathrm{E} 21.37,$ find the magnitude and direction of the net electric force they will exert on the proton.

DD

Dylan Delcol

Numerade Educator

Problem 38

CP 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 $1.50 \times 10^{-6}$ s. (a) Find the magnitude of the electric field. (b) Find the speed of the proton when it strikes the negatively charged plate.

Anand Jangid

Numerade Educator

Problem 39

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

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 40

$\mathrm{A}+8.75-\mu \mathrm{C}$ point charge is glued down on a horizontal frictionless table. It is tied to a $-6.50-\mu C$ point charge by a light, nonconducting $2.50-\mathrm{cm}$ wire. 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. E21.40. (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

Problem 41

(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

Numerade Educator

Problem 42

Two point charges $Q$ and $+q$ (where $q$ is positive) produce the net electric field shown at point $P$ in Fig. $E 21.42 .$ 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.

Manne Andergronde

Manne Andergronde

Numerade Educator

Problem 43

Two positive point charges $q$ are placed on the $x$ -axis, one al $x=a$ and one at $x=-a$ . (a) Find the magnitude and direction of the electric field at $x=0 .$ (b) Derive an expression for the electric field at points on the $x$ -axis. Use your result to graph the $x$ -component of the electric field as a function of $x,$ for values of $x$ between $-4 a$ and $+4 a .$

Kayla Gephart

Numerade Educator

Problem 44

The two charges $q_{1}$ and $q_{2}$ shown in Fig. E21.44 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.

Prashant Bana

Numerade Educator

Problem 45

$A+2.00-n C \quad$ point charge is at the origin, and a second $-5.00$ -n $\mathrm{C}$ 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).

Dading Chen

Numerade Educator

Problem 46

Repeat Exercise $21.44,$ but now let $q_{1}=-4.00 \mathrm{nC}$

Janielle Madlansacay

Janielle Madlansacay

Numerade Educator

Problem 47

Three negative point charges lie along a line as shown in Fig. E21.47. 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.

Sheh Lit Chang

University of Washington

Problem 48

BI0 Electric Field of Axons. A nerve signal is transmitted through a neuron when an excess of 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. Let us look at a 0.10 -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 -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 sharks 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?

Surjit Tewari

Numerade Educator

Problem 49

In a rectangular coordinate system a positive point charge $q=6.00 \times 10^{-9} \mathrm{Cis~placed}$ 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) $x=0.300 \mathrm{m}, y=0 ;(\mathrm{c}) x=0.150 \mathrm{m}, y=-0.400 \mathrm{m} ;(\mathrm{d}) x=0$ $y=0.200 \mathrm{m}$

Sheh Lit Chang

University of Washington

Problem 50

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

Numerade Educator

Problem 51

Repeat Exercise 21.49 for the case where the point charge at $x=+0.150 \mathrm{m}, y=0$ is positive and the other is negative, each with magnitude $6.00 \times 10^{-9} \mathrm{C}$ .

Janielle Madlansacay

Janielle Madlansacay

Numerade Educator

Problem 52

A very long, straight wire has charge per unit length $1.50 \times 10^{-10} \mathrm{C} / \mathrm{m} .$ At what distance from the wire is the electric-field magnitude equal to 2.50 $\mathrm{N} / \mathrm{C}$?

Kayla Gephart

Numerade Educator

Problem 53

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, as shown in 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 $q=-2.50 \mu C$ is placed at the point $P$ described in part (a). What are the magnitude and direction of the force exerted by the charge $q$ on the ring?

Kayla Gephart

Numerade Educator

Problem 54

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

Numerade Educator

Problem 55

A charge of $-6.50 \mathrm{nC}$ is spread uniformly over the surface of one face of a nonconducting disk of radius 1.25 $\mathrm{cm}$ . (a) Find the magnitude and direction of the electric field this disk produces at a point $P$ on the axis of the disk a distance of 2.00 $\mathrm{cm}$ from its center. (b) Suppose that the charge were all pushed away from the center and distributed uniformly on the outer rim of the disk. Find the magnitude and direction of the electric field at point $P$ . (c) If the charge is all brought to the center of the disk, find the magnitude and direction of the electric field at point $P .$ (d) Why is the field in part (a) stronger than the field in part (b)? Why is the field in part (c) the strongest of the three fields?

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 56

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 a uniform electric field $\vec{\boldsymbol{E}}$ 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 $\vec{\boldsymbol{E}}$ 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

Numerade Educator

Problem 57

Point charges $q_{1}=-4.5 \mathrm{nC}$ and $q_{2}=+4.5 \mathrm{nC}$ are separated by 3.1 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
$7.2 \times 10^{-9} \mathrm{N} \cdot \mathrm{m} ?$

Kayla Gephart

Numerade Educator

Problem 58

The dipole moment of the water molecule $\left(\mathrm{H}_{2} \mathrm{O}\right)$ is $6.17 \times 10^{-30} \mathrm{C} \cdot \mathrm{m} .$ Consider a water molecule located at the origin whose dipole moment $\vec{p}$ points in the $+x$ -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.

Kayla Gephart

Numerade Educator

Problem 59

Torque on a Dipole. An electric dipole with dipole moment $\vec{p}$ is in a uniform 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 displacement 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.

Kayla Gephart

Numerade Educator

Problem 60

Consider the electric dipole of Example 21.14 . (a) Derive an expression for the magnitude of the electric field produced by the dipole at a point on the $x$ -axis in Fig. $21.33 .$ What is the direction of this electric field? (b) How does the electric field at points on the $x$ -axis depend on $x$ when $x$ is very large?

Dading Chen

Numerade Educator

Problem 61

Three charges are at the corners of an isosceles triangle as shown in Fig. E21.61. The $\pm 5.00-\mu C$ charges form a dipole. (a) Find the force (magnitude and direction) the $-10.00-\mu C$ charge exerts on the dipole. (b) For an axis perpendicular to the line connecting the $\quad \pm 5.00-\mu C$ charges at the mid-point of this line, find the torque (magnitude and direction) exerted on the dipole by the $-10.00-\mu \mathrm{C}$ charge.

Maria Gabriela Cota Moreira

Maria Gabriela Cota Moreira

Numerade Educator

Problem 62

A dipole consisting of charges $\pm e, 220 \mathrm{nm}$ apart, is placed between two very large (essentially infinite) sheets carrying equal but opposite charge densities of 125$\mu \mathrm{C} / \mathrm{m}^{2}$ . (a) What is the maximum potential energy this dipole can have due to the sheets,
and how should it be oriented relative to the sheets to attain this value? (b) What is the maximum torque the sheets can exert on the dipole, and how should it be oriented relative to the sheets to attain this value? (c) What net force do the two sheets exert on the dipole?

Sheh Lit Chang

University of Washington

Problem 63

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.

Banhishikha Sinha

Banhishikha Sinha

Numerade Educator

Problem 64

Two charges, one of 2.50$\mu \mathrm{C}$ and the other of $-3.50 \mu \mathrm{C},$ are placed on the $x$ -axis, one at the origin and the other at $x=0.600 \mathrm{m},$ as shown in Fig. $\mathrm{P} 21.64 .$ Find the position on the $x$ -axis where the net force on a small charge $+q$ would be zero.

William Dunkerton

William Dunkerton

Numerade Educator

Problem 65

Three point charges are arranged along the $x$ -axis. Charge $q_{1}=-4.50 \mathrm{nC}$ is located at $x=0.200 \mathrm{m},$ and charge $q_{2}=+2.50 \mathrm{nC}$ is at $x=-0.300 \mathrm{m} .$ A positive point charge $q_{3}$ is located at the origin. (a) What must the value of $q_{3}$ be for the net
force on this point charge to have magnitude 4.00$\mu \mathrm{N} ?$ (b) What is the direction of the net force on $q_{3} ?(\mathrm{c})$ Where along the $x$ -axis can $q_{3}$ be placed and the net force on it be zero, other than the trivial answers of $x=+\infty$ and $x=-\infty ?$

Vishal Gupta

Numerade Educator

Problem 66

A charge $q_{1}=+5.00 \mathrm{nC}$ is placed at the origin of an $x y$ -coordinate system, and a charge $q_{2}=-2.00 \mathrm{nC}$ is placed on the positive $x$ -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.

Maria Gabriela Cota Moreira

Maria Gabriela Cota Moreira

Numerade Educator

Problem 67

CP Two positive point charges $Q$ are held fixed on the $x$ -axis at $x=a$ and $x=-a .$ A third positive point charge $q,$ with mass $m,$ is placed on the $x$ -axis away from the origin at a coordinate $x$ such that $|x| << a$ . The charge $q,$ which is free to move along the $x$ -axis, is then released. (a) Find the frequency of oscillation of the charge $q .$ (Hint: Review the definition of simple harmonic motion in Section $14.2 .$ Use the binomial expansion $(1+z)^{n}=1+n z+n(n-1) z^{2} / 2+\cdots,$ valid for the case $|z|<1 .$ (b) Suppose instead that the charge $q$ were placed on the $y$ -axis at a coordinate $y$ such that $|y| < < a$ , and then released. If this charge is free to move anywhere in the $x y$ -plane, what will happen to it? Explain your answer.

Janielle Madlansacay

Janielle Madlansacay

Numerade Educator

Problem 68

CP Two identical spheres with mass $m$ are hung from silk threads of length $L,$ as shown in Fig. $P 21.68 .$ Each sphere has 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 . )

Maria Gabriela Cota Moreira

Maria Gabriela Cota Moreira

Numerade Educator

Problem 69

CP Two small spheres with mass $m=15.0$ g are hung by silk threads of length $L=1.20 \mathrm{m}$ from a common point (Fig. P21.68). 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 self- consistent answer is obtained.)

Meghan Miholics

Meghan Miholics

Numerade Educator

Problem 70

CP Two identical spheres are each attached to silk threads of length $L=0.500 \mathrm{m}$ and hung from a common point (Fig. P21.68). Each sphere has mass $m=8.00 \mathrm{g} .$ The radius of each sphere is very small compared to the distance between the spheres, so they may be treated as point charges. One sphere is given positive charge $q_{1},$ and the other a different positive charge $q_{2} ;$ this causes the spheres to separate so that when the spheres are in equilibrium, each thread makes an angle $\theta=20.0^{\circ}$ with the vertical. (a) Draw a free-body diagram for each sphere when in equilibrium, and label all the forces that act on each sphere. (b) Determine the magnitude of the electrostatic force that acts on each sphere, and determine the tension in each thread. (c) Based on the information you have been given, what can you say about the magnitudes of $q_{1}$ and $q_{2} ?$ Explain your answers. (d) A small wire is now connected between the spheres, allowing charge to be transferred from one sphere to the other until the two spheres have equal charges; the wire is then removed. Each thread now makes an angle of $30.0^{\circ}$ with the vertical. Determine the original charges. (Hint: The total charge on the pair of spheres is conserved.)

Ze-Han Lee

Numerade Educator

Problem 71

Sodium chloride ( NaCl, ordinary table salt) is made up of positive sodium ions $\left(\mathrm{Na}^{+}\right)$ and negative chloride ions $\left(\mathrm{Cl}^{-}\right) .$ (a) If a point charge with the same charge and mass as all the $\mathrm{Na}^{+}$ ions in 0.100 mol of $\mathrm{NaCl}$ is 2.00 $\mathrm{cm}$ from a point charge with the same charge and mass as all the $\mathrm{Cl}^{-}$ ions, what is the its initial acceleration? (See Appendix D for atomic masses.)
(c) Does it seem reasonable that the ions in NaCl could be separated in this way? Why or why not? (In fact, when sodium chloride dissolves in water, it breaks up into $\mathrm{Na}^{+}$ and $\mathrm{Cl}^{-}$ ions. However, in this situation there are additional electric forces
exerted by the water molecules on the ions.)

Janielle Madlansacay

Janielle Madlansacay

Numerade Educator

Problem 72

$\mathrm{A}-5.00$ -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 $+x$ -x-direction, (b) 45.0 $\mathrm{N} / \mathrm{C}$ in the - $x$ -direction?

Maria Gabriela Cota Moreira

Maria Gabriela Cota Moreira

Numerade Educator

Problem 73

CP A small 12.3 -g plastic ball is tied to a very light 28.6 -cm string that is attached to the vertical wall of a room (Fig. P21.73). 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 i remains suspended, with the string making a angle of $17.4^{\circ}$ with the wall. Find the magnitude and direction of the electric field in the room.

DS

Drew Strella

Numerade Educator

Problem 74

CP At $t=0$ a very small object with mass 0.400 $\mathrm{mg}$ and charge $+9.00 \mu \mathrm{C}$ is traveling at 125 $\mathrm{m} / \mathrm{s}$ in the $-x$ -direction. The charge is moving in a uniform electric field that is in the +y-direction and that has magnitude $E=895 \mathrm{N} / \mathrm{C}$ .
The gravitational force on the particle can be neglected. How far is the particle from the origin at $t=7.00 \mathrm{ms} ?$

Maria Gabriela Cota Moreira

Maria Gabriela Cota Moreira

Numerade Educator

Problem 75

Two particles having charges $q_{1}=0.500 \mathrm{nC}$ 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?

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 76

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 each of these charges (Fig. $\mathrm{P} 21.76$ ) 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 $q_{2}$ .

Ze-Han Lee

Numerade Educator

Problem 77

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

Numerade Educator

Problem 78

Three point charges are placed on the $y$ -axis: a charge $q$ at $y=a,$ a charge $-2 q$ at the origin, and a charge $q$ at $y=-a$ . Such an arrangement is called an electric quadrupole. (a) Find the
magnitude and direction of the electric field at points on the positive $x$ -axis. (b) Use the binomial expansion to find an approximate expression for the electric field valid for $x \gg a$ . Contrast this
behavior to that of the electric field of a point charge and that of the electric field of a dipole.

Corinna Pena

Numerade Educator

Problem 79

cp Strength of the Electric Force. Imagine two 1.0 -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 electrical 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?

Maria Gabriela Cota Moreira

Maria Gabriela Cota Moreira

Numerade Educator

Problem 80

Electric Force Within the Nucleus. Typical dimensions of atomic nuclei are of the order of $10^{-15} \mathrm{m}(1 \mathrm{fm})$ . (a) If two protons in a nucleus are 2.0 fm apart, find the magnitude of
the electric force each one exerts on the other. Express the answer in newtons and in pounds. Would this force be large enough for a person to feel? (b) since the protons repel each other so strongly, why don't they shoot out of the nucleus?

Kayla Gephart

Numerade Educator

Problem 81

If Atoms Were Not Neutral... Because the charges on the electron and proton have the same absolute value, atoms are electrically neutral. Suppose this were not precisely true, and the absolute value of the charge of the electron were less than the charge of the proton by 0.00100$\% .$ (a) Estimate what the net charge of this textbook would be under these circumstances. Make any assumptions you feel are justified, but state clearly what they are. (Hint: Most of the atoms in this textbook have equal numbers of electrons, protons, and neutrons.) (b) What would be the magnitude of the electric force between two textbooks placed 5.0 $\mathrm{m}$ apart? Would this force be attractive or repulsive? Estimate what the acceleration of each book would be if the books were 5.0 $\mathrm{m}$ apart and there were no non-electric forces on them. (c) Discuss how the fact that ordinary matter is stable shows that the absolute values of the charges on the electron and
proton must be identical to a very high level of accuracy.

Rashmi Sinha

Numerade Educator

Problem 82

CP 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 $50.0^{\circ}($ Fig. .21 .82$)$ . (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?

Narayan Hari

Numerade Educator

Problem 83

CP 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?

Maria Gabriela Cota Moreira

Maria Gabriela Cota Moreira

Numerade Educator

Problem 84

CP A small sphere with mass 9.00$\mu g$ and charge $-4.30 \mu C$ is moving in a circular orbit around a stationary sphere that has charge $+7.50 \mu \mathrm{C}$ . If the speed of the small sphere is
$5.90 \times 10^{3} \mathrm{m} / \mathrm{s},$ what is the radius of its orbit? Treat the spheres as point charges and ignore gravity.

Maria Gabriela Cota Moreira

Maria Gabriela Cota Moreira

Numerade Educator

Problem 85

Two small copper spheres each have radius 1.00 $\mathrm{mm} .$ (a) How many atoms does each sphere contain? (b) Assume that each copper atom contains 29 protons and 29 electrons. We know that electrons and protons have charges of exactly the same magnitude, but let's explore the effect of small differences (see also Problem 21.81 ). If the charge of a proton is $+e$ and the magnitude of the charge of an electron is 0.100$\%$ smaller, what is the net charge of the each sphere and what force would one sphere exert on the other if they were separated by 1.00 $\mathrm{m} ?$

Dading Chen

Numerade Educator

Problem 86

CP Operation of an Inkjet Printer. In an inkjet printer, letters are built up by squirting drops of ink at the paper from a rapidly moving nozzle. The ink drops, which have a mass of $1.4 \times 10^{-8}$ g each, leave the nozzle and travel toward the paper at $20 \mathrm{m} / \mathrm{s},$ passing through a charging unit that gives each drop a positive charge $q$ by removing some electrons from it. The drops then pass between parallel deflecting plates 2.0 $\mathrm{cm}$ long where there is a uniform vertical electric field with magnitude $8.0 \times 10^{4} \mathrm{N} / \mathrm{C}$ . 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?

Prashant Bana

Numerade Educator

Problem 87

CP 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. You can 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 $\alpha=30.0^{\circ} .$

Ajay Singhal

Numerade Educator

Problem 88

A negative point charge $q_{1}=-4.00 \mathrm{nC}$ is on the $x$ -axis at $x=0.60 \mathrm{m} .$ A second point charge $q_{2}$ is on the $x$ -axis at $x=-1.20 \mathrm{m} .$ What must the sign and magnitude of $q_{2}$ be for the net electric field at the origin to be (a) 50.0 $\mathrm{N} / \mathrm{C}$ in the $+x$ -direction and $(\mathrm{b}) 50.0 \mathrm{N} / \mathrm{C}$ in the $-x$ -x-direction?

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 89

CALC 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. P21.89). (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.

Sheh Lit Chang

University of Washington

Problem 90

CALC 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. P21.90). (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 $x>a, F_{x} \cong-Q q / 4 \pi \epsilon_{0} x^{2}$ 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

Problem 91

A charged line like that shown in Fig. 21.24 extends from $y=2.50 \mathrm{cm}$ to $y=-2.50 \mathrm{cm} .$ The total charge distributed uniformly along the line is $-7.00 \mathrm{nC}$ . (a) Find the electric field (magnitude and direction) on the $x$ -axis at $x=10.0 \mathrm{cm} .$ (b) Is the
magnitude of the electric field you calculated in part (a) larger or smaller than the electric field 10.0 $\mathrm{cm}$ from a point charge that has the same total charge as this finite line of charge? In terms of the approximation used to derive $E=Q / 4 \pi \epsilon_{0} x^{2}$ for a point charge from Eq. $(21.9)$ , explain why this is so. (c) At what distance $x$ does the result for the finite line of charge differ by 1.0$\%$ from that for the point charge?

Ajay Singhal

Numerade Educator

Problem 92

CP A Parallel Universe. Imagine a parallel universe in which the electric force has the same properties as in our universe but there is no gravity. In this parallel universe, the sun carries charge $Q,$ the earth carries charge $-Q,$ and the electric attraction between them keeps the earth in orbit. The earth in the parallel universe has the same mass, the same orbital radius, and the same orbital period as in our universe. Calculate the value of $Q$ . (Consult Appendix F as needed.)

Maria Gabriela Cota Moreira

Maria Gabriela Cota Moreira

Numerade Educator

Problem 93

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 $x>>R,$ 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 $E=O / 4 \pi \epsilon_{0} x^{2}$ 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 $x=10.0 \mathrm{cm} ?$

Linda Winkler

Numerade Educator

Problem 94

BIO Electrophoresis. Electrophoresis is a process used by biologists to separate different biological molecules (such as proteins) from each other according to their ratio of charge to size. The materials to be separated are in a viscous solution that produces a drag force $F_{\mathrm{D}}$ proportional to the size and speed of the molecule. We can express this relation- ship as $F_{\mathrm{D}}=K R v,$ where $R$ is the radius of the molecule (modeled as being spherical), $v$ is its its speed, and $K$ is a constant that depends on the viscosity of the solution. The solution is placed in an external electric field $E$ so that the electric force on a particle of charge $q$ is $F=q E$ . (a) Show that when the electric field is adjusted so that the two forces (electric and viscous drag) just balance, the ratio of $q$ to $R$ is $K v / E$ . (b) Show that if we leave the electric field on for a time $T,$ the distance
$x$ that the molecule moves during that time is $x=(E T / k)(q / R)$ . (c) Suppose you have a sample containing three different biological molecules for which the molecular ratio $q / R$ for material 2 is
twice that of material 1 and the ratio for material 3 is three times that of material 1. Show that the distances migrated by these molecules after the same amount of time are $x_{2}=2 x_{1}$ and $x_{3}=3 x_{1}$ . In other words, material 2 travels twice as far as material $1,$ and material 3 travels three times as far as material $1 .$ Therefore, we have separated these molecules according to their ratio of charge to size. In practice, this process can be carried out in a special gel or paper, along which the biological molecules migrate. (Fig. P21.94). The process can be rather slow, requiring several hours for separations of just a centimeter or so.

Jacob Schulze

Numerade Educator

Problem 95

CALC Positive charge $+Q$ is distributed uniformly along the $+x$ -axis from $x=0$ to $x=a .$ Negative charge $-Q$ is distributed uniformly along the $-x$ -axis from $x=0$ to $x=-a$ . (a) A positive point charge $q$ lies on the positive $y$ -axis, a distance $y$ from the origin. Find the force (magnitude and direction) that the positive and negative charge distributions together exert on $q .$ Show that this force is proportional to $y^{-3}$ for $y>>$ a. (b) Suppose instead that the positive point charge $q$ lies on the positive $x$ -axis, a distance $x>a$ from the origin. Find the force (magnitude and direction) that the charge distribution exerts on $q .$ Show that this force is proportional to $x^{-3}$ for $x>>a$ .

Janielle Madlansacay

Janielle Madlansacay

Numerade Educator

Problem 96

CP 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.

Maria Gabriela Cota Moreira

Maria Gabriela Cota Moreira

Numerade Educator

Problem 97

CALC 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.

Maria Gabriela Cota Moreira

Maria Gabriela Cota Moreira

Numerade Educator

Problem 98

CALC 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. P21.98). What are the magnitude and direction of the net produced by this distribution of charge?

Maria Gabriela Cota Moreira

Maria Gabriela Cota Moreira

Numerade Educator

Problem 99

Two 1.20 -m nonconducting wires meet at a right angle. One segment carries $+2.50 \mu C$ of charge distributed uniformly along its length, and the other carries $-2.50 \mu \mathrm{C}$ distributed uniformly along it, as shown in Fig. $\mathrm{P} 21.99$ . (a) Find the magnitude and direction of the electric field these wires produce at point $P$ , which is 60.0 $\mathrm{cm}$ from each wire. (b) If
an electron is released at $P,$ what are the magnitude and direction of the net force that these wires exert on it?

Narayan Hari

Numerade Educator

Problem 100

Two very large parallel sheets are 5.00 $\mathrm{cm}$ apart. Sheet $A$ carries a uniform surface charge density of $-9.50 \mu \mathrm{C} / \mathrm{m}^{2},$ 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 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 $B$ .

Maria Gabriela Cota Moreira

Maria Gabriela Cota Moreira

Numerade Educator

Problem 101

Repeat Problem 21.100 for the case where sheet $B$ is positive.

Janielle Madlansacay

Janielle Madlansacay

Numerade Educator

Problem 102

Two very large horizontal sheets are 4.25 $\mathrm{cm}$ apart and carry equal but opposite uniform surface charge densities of magnitude $\sigma .$ You want to use these sheets to hold stationary in the
region between them an oil droplet of mass 324$\mu$ that carries an excess of five electrons. Assuming that the drop is in vacuum, (a) which way should the electric field between the plates point, and (b) what should $\sigma$ be?

Maria Gabriela Cota Moreira

Maria Gabriela Cota Moreira

Numerade Educator

Problem 103

An infinite sheet with positive charge per unit area $\sigma$ lies in the $x y$ -plane. A second infinite sheet with negative charge per unit area $-\sigma$ lies in the $y z$ -plane. Find the net electric field at
all points that do not lie in either of these planes. Express your answer in terms of the unit vectors $\hat{\imath}, \hat{j},$ and $\hat{k} .$

Maria Gabriela Cota Moreira

Maria Gabriela Cota Moreira

Numerade Educator

Problem 104

CP A thin disk with a circular hole at its center, called an annulus, has inner radius $R_{1}$
and outer radius $R_{2}$ (Fig. P21. 104 ). 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 $\vec{E} .$ Consider points both above and below the annulus in Fig. P21. $104 .$ (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.)

Sheh Lit Chang

University of Washington

Problem 105

Three charges are placed as shown in Fig. P21.105. The magnitude of $q_{1}$ is 2.00$\mu C$ , but its sign and the value of the charge $q_{2}$ are not $2.00 \mu C,$ but its sign and the value of the charge $q_{2}$ are not known. Charge $q_{3}$ is $+4.00 \mu C$ , and the net force $\vec{F}$ on $q_{3}$ is entirely in the neaative $x$ -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 $\vec{\boldsymbol{F}},$ deduce the signs of the charges $q_{1}$ and $q_{2},$ (c) Calculate the magnitude of $q_{2} .(\mathrm{d})$ Determine $F,$ the magnitude of the net force on $q_{3}$

Keshav Singh

Numerade Educator

Problem 106

Two charges are placed as shown in Fig. P21. 106. The magnitude of $q_{1}$ is $3.00 \mu C,$ but its sign and the value of the charge $q_{2}$ are not $3.00 \mu C,$ but its sign and the value of the charge $q_{2}$ are not known. The direction of the net electric field $\vec{E}$ at point $P$ is entirely in the negative $y$ -direction. (a) Considering the different possible signs of $q_{1}$ and $q_{2},$ there are four possible diagrams that could represent the electric fields $\vec{E}_{1}$ and $\vec{E}_{2}$ produced by $q_{1}$ and $q_{2} .$ Sketch the four possible electric-field configurations. (b) Using the sketches from part (a) and the direction of $\vec{E}$ , deduce the signs of $q_{1}$ and $q_{2}$ . (c) Determine the magnitude of $\vec{E}$ .

Penny Riley

Numerade Educator

Problem 107

CALC Two thin rods of length $L$ lie along the $x$ -axis, one between $x=a / 2$ and $x=a / 2+L$ and the other between $x=-a / 2$ and $x=-a / 2-L .$ 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 $a>>L,$ the magnitude of this force reduces to $F=Q^{2} / 4 \pi \epsilon_{0} a^{2} .$ (Hint: Use the expansion $\ln (1+z)=z-$ $z^{2} / 2+z^{3} / 3-\cdots,$ valid for $|z|<<1 .$ Carry all expansions to at
least order $L^{2} / a^{2} .$ ) Interpret this result.

Sheh Lit Chang

University of Washington