Essential University Physics Global Edition

Richard Wolfson

Chapter 20

Electric Charge, Force, and Field - all with Video Answers

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Chapter Questions

Problem 1

Conceptual Example $20.1$ shows that the gravitational force between an electron and a proton is about $10^{-40}$ times weaker than the electric force between them. Since matter consists largely of electrons and protons, why is the gravitational force important at all?

Zulfiqar Ali

Zulfiqar Ali

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Problem 2

A free neutron is unstable and soon decays to other particles, one of them a proton. Must there be others? If so, what electric properties must it or they have?

Zulfiqar Ali

Numerade Educator

Problem 3

Where in Fig. $20.5$ could you put a third charge so it would experience no net force? Would it be in stable or unstable equilibrium?

Zulfiqar Ali

Numerade Educator

Problem 4

Equation $20.3$ gives the electric field of a point charge. Does the direction of (a) $\hat{r}$ or (b) $\vec{E}$ depend on whether the charge is positive or negative?

Zulfiqar Ali

Numerade Educator

Problem 5

Is the electric force on a charged particle always in the direction of the field? Explain.

Zulfiqar Ali

Numerade Educator

Problem 6

Why does a dipole, which has no net charge, produce an electric field?

Zulfiqar Ali

Numerade Educator

Problem 7

The ring in Example $20.6$ carries total charge $Q$, and the point $P$ is the same distance $r=\sqrt{x^{2}+a^{2}}$ from all parts of the ring. So why isn't the electric field of the ring just $k Q / r^{2}$ ?

Zulfiqar Ali

Numerade Educator

Problem 8

A spherical balloon is initially uncharged. If you spread positive charge uniformly over the balloon's surface, would it expand or contract? What would happen if you spread negative charge instead?

Zulfiqar Ali

Numerade Educator

Problem 9

Why should there be a force between two dipoles, which each have zero net charge?

Zulfiqar Ali

Numerade Educator

Problem 10

Dipoles $A$ and $B$ are both located in the field of a point charge $Q$, as shown in Fig. 20.27. Does either experience a net torque? A net force? If each dipole is released from rest, qualitatively describe its subsequent motion.

Zulfiqar Ali

Numerade Educator

Problem 11

Suppose the electron and proton charges differed by one part in one billion. Estimate the net charge on your body, assuming it contains equal numbers of electrons and protons.

Zulfiqar Ali

Numerade Educator

Problem 12

A typical lightning flash delivers about $25 \mathrm{C}$ of negative charge from cloud to ground. How many electrons are involved?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 13

Protons and neutrons are made from combinations of the two most common quarks, the $u$ quark (charge $+\frac{2}{3} e$ ) and the $d$ quark (charge $-\frac{1}{3} e$ ). How could three of these quarks combine to make (a) a proton and (b) a neutron?

Vishal Gupta

Numerade Educator

Problem 14

Earth carries a net charge of about $-5 \times 10^{5} \mathrm{C}$. How many more electrons are there than protons on Earth?

Zulfiqar Ali

Numerade Educator

Problem 15

As they fly, honeybees may acquire electric charges of about $180 \mathrm{pC}$. Electric forces between charged honeybees and spider webs can make the bees more vulnerable to capture by spiders. How many electrons would a honeybee have to lose to acquire a charge of $+180 \mathrm{pC}$ ?

Zulfiqar Ali

Numerade Educator

Problem 16

The electron and proton in a hydrogen atom are $52.9 \mathrm{pm}$ apart. Find the magnitude of the electric force between them.

Zulfiqar Ali

Numerade Educator

Problem 17

An electron at Earth's surface experiences a gravitational force $m_{\mathrm{e}} g$. How far away can a proton be and still produce the same force on the electron? (Your answer should show why gravity is unimportant on the molecular scale!)

Zulfiqar Ali

Numerade Educator

Problem 18

You break a piece of Styrofoam packing material, and it releases lots of little spheres whose electric charge makes them stick annoyingly to you. If two of the spheres carry equal charges and repel with a force of $20 \mathrm{mN}$ when they're $17 \mathrm{~mm}$ apart, what's the magnitude of the charge on each?

Gopesh Vishwakarma

Gopesh Vishwakarma

Numerade Educator

Problem 19

A charge $q$ is at the point $x=5 \mathrm{~m}, y=0 \mathrm{~m}$. Write expressions for the unit vectors you would use in Coulomb's law if you were finding the force that $q$ exerts on other charges located at (a) $x=5 \mathrm{~m}, y=2.5 \mathrm{~m}$; (b) the origin; and (c) $x=7 \mathrm{~m}, y=3.5 \mathrm{~m}$. You're not given the sign of $q$. Why doesn't this matter?

Gopesh Vishwakarma

Gopesh Vishwakarma

Numerade Educator

Problem 20

A proton is at the origin and an electron is at the point $x=0.41 \mathrm{~nm}, y=0.36 \mathrm{~nm}$. Find the electric force on the proton.

Zulfiqar Ali

Numerade Educator

Problem 21

An electron experiences an electric force of $0.61 \mathrm{nN}$. What's the field strength at its location?

Zulfiqar Ali

Numerade Educator

Problem 22

Find the magnitude of the electric force on a $6.0-\mu \mathrm{C}$ charge in a $50-\mathrm{N} / \mathrm{C}$ electric field.

Gopesh Vishwakarma

Gopesh Vishwakarma

Numerade Educator

Problem 23

A $75-\mathrm{nC}$ charge experiences a $144-\mathrm{mN}$ force in a certain electric field. Find (a) the field strength and (b) the force that a $35-\mu \mathrm{C}$ charge would experience in the same field.

Gopesh Vishwakarma

Gopesh Vishwakarma

Numerade Educator

Problem 24

The electric field inside a cell membrane is $8.0 \mathrm{MN} / \mathrm{C}$. What's the force on a singly charged ion in this field?

Prabhu Ramji

Numerade Educator

Problem 25

$\mathrm{A}-3.0-\mu \mathrm{C}$ charge experiences a $9.0 \hat{\imath}-\mathrm{N}$ electric force in a certain electric field. What force would a proton experience in the same field?

Gopesh Vishwakarma

Gopesh Vishwakarma

Numerade Educator

Problem 26

The electron in a hydrogen atom is $52.9 \mathrm{pm}$ from the proton. At this distance, what's the strength of the electric field due to the proton?

Zulfiqar Ali

Numerade Educator

Problem 27

In Fig. 20.28, point $P$ is midway between the two charges. Find the electric field in the plane of the page (a) $5.0 \mathrm{~cm}$ the the left of $P$, (b) $5.0 \mathrm{~cm}$ directly above $P$, and (c) at $P$.

Zulfiqar Ali

Numerade Educator

Problem 28

The water molecule's dipole moment is $6.17 \times 10^{-30} \mathrm{C} \cdot \mathrm{m}$. What would be the separation distance if the molecule consisted of charges $\pm e ?$ (The effective charge is actually less because $\mathrm{H}$ and $\mathrm{O}$ atoms share the electrons.)

Gopesh Vishwakarma

Gopesh Vishwakarma

Numerade Educator

Problem 29

The electric field $22 \mathrm{~cm}$ from a long wire carrying a uniform line charge density is $1.9 \mathrm{kN} / \mathrm{C}$. What's the field strength $38 \mathrm{~cm}$ from the wire?

Zulfiqar Ali

Numerade Educator

Problem 30

Find the line charge density on a long wire if the electric field $39 \mathrm{~cm}$ from the wire has magnitude $210 \mathrm{kN} / \mathrm{C}$ and points toward the wire.

Gopesh Vishwakarma

Gopesh Vishwakarma

Numerade Educator

Problem 31

Find the magnitude of the electric field due to a charged ring of radius $a$ and total charge $Q$ on the ring axis at distance $a$ from the ring's center.

Vanessa Lamar

Numerade Educator

Problem 32

In his famous 1909 experiment that demonstrated quantization of electric charge, R. A. Millikan suspended small oil drops in an electric field. With field strength $20 \mathrm{MN} / \mathrm{C}$, what mass drop can be suspended when the drop carries 10 elementary charges?

Zulfiqar Ali

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Problem 33

How strong an electric field is needed to accelerate electrons in an X-ray tube from rest to one-tenth the speed of light in a distance of $4.7 \mathrm{~cm}$ ?

Gopesh Vishwakarma

Gopesh Vishwakarma

Numerade Educator

Problem 34

A proton moving to the right at $3.3 \times 10^{5} \mathrm{~m} / \mathrm{s}$ enters a region where a $60-\mathrm{kN} / \mathrm{C}$ electric field points to the left. (a) How far will the proton get before it momentarily stops? (b) Describe its subsequent motion.

Gopesh Vishwakarma

Gopesh Vishwakarma

Numerade Educator

Problem 35

An electrostatic analyzer like that of Example $20.8$ has $b=7.5 \mathrm{~cm}$. What value of $E_{0}$ will enable the device to select protons moving at $84 \mathrm{~m} / \mathrm{s}$ ?

Zulfiqar Ali

Numerade Educator

Problem 36

Charges on raindrops vary widely in both magnitude and sign. Consider a case where the two drops on the $x$-axis in Example $20.2$ are $2.18 \mathrm{~mm}$ apart and have charge $q=645 \mathrm{nC}$, while the third drop is $12.3 \mathrm{~mm}$ up the $y$-axis and has charge $Q=-1.87 \mu \mathrm{C}$. Find the electric force on the upper drop.

Gopesh Vishwakarma

Gopesh Vishwakarma

Numerade Educator

Problem 37

Suppose that all three raindrops in Example $20.2$ have equal charges and that their positions form an equilateral triangle with side $3.36 \mathrm{~mm}$. If the electric force on the upper charge is $96.2 \hat{\jmath} \mathrm{N}$, (a) what's the magnitude of the charge? (b) Can you determine the sign of the charge from the information given?

Anand Jangid

Numerade Educator

Problem 38

(a) Repeat Example $20.2$ to find the force on $Q$, now taking the right-hand charge on the $x$-axis to be $-q$. (b) For $y \gg a$, how does the force you found in (a) depend on the distance $y$ ?

Gopesh Vishwakarma

Gopesh Vishwakarma

Numerade Educator

Problem 39

(a) Use calculus to show that the maximum force in the situation of Example $20.2$ occurs when $y=a / \sqrt{2}$, and (b) find the magnitude of that maximum force.

Gopesh Vishwakarma

Gopesh Vishwakarma

Numerade Educator

Problem 40

A $1.00-\mathrm{km}$ length of power line carries a total charge of $264 \mathrm{mC}$ distributed uniformly over its length. Find the magnitude of the electric field $54.3 \mathrm{~cm}$ from the axis of the power line, and not near either end (staying away from the ends means you can approximate the field as that of an infinitely long wire).

Gopesh Vishwakarma

Gopesh Vishwakarma

Numerade Educator

Problem 41

A uniformly charged wire is $2.18 \mathrm{~m}$ long and $0.15 \mathrm{~mm}$ in diameter. You measure the electric field $1.20 \mathrm{~cm}$ from the wire's axis, not near either end, and you find it to be 455 $\mathrm{kN} / \mathrm{C}$, pointing toward the wire. Find the total charge on the wire.

Gopesh Vishwakarma

Gopesh Vishwakarma

Numerade Educator

Problem 42

A thin rod of length $L$ lies on the $x$-axis with its center at the origin, as shown in Fig. $20.29$. The rod carries charge $Q$ distributed uniformly over its length. (a) Modify the calculation of Example $20.7$ the electric field at point $A$ in Fig. $20.29$, located on the positive $y$-axis an arbitrary distance $y$ from the origin (but with $y$ large enough to put point $A$ outside the thin rod). (b) Show that your result reduces to the field of a point charge $Q$ when $y \gg L$.

Gopesh Vishwakarma

Gopesh Vishwakarma

Numerade Educator

Problem 43

A thin rod of length $L$ lies on the $x$-axis with its center at the origin, as shown in Fig. 20.29. The rod carries charge $Q$ distributed uniformly over its length. (a) Find an expression for the electric field at point $B$ in Fig. $20.29$, located on the positive $x$-axis a distance $x$ from the origin, where $x>L / 2$, so that point $B$ is beyond the right end of the rod. (b) Show that your result reduces to the field of a point charge $Q$ when $x \gg L$.

Gopesh Vishwakarma

Gopesh Vishwakarma

Numerade Educator

Problem 44

Two charges, of which one has a magnitude three times as large as the other's, are located $14.5 \mathrm{~cm}$ apart and experience an attractive force of $156 \mathrm{~N}$. (a) What's the magnitude of the larger charge? (b) Can you determine the sign of the larger charge?

Gopesh Vishwakarma

Gopesh Vishwakarma

Numerade Educator

Problem 45

A proton is on the $x$-axis at $x=1.3 \mathrm{~nm}$. An electron is on the $y$-axis at $y=0.86 \mathrm{~nm}$. Find the net force the two exert on a helium nucleus (charge $+2 e$ ) at the origin.

Gopesh Vishwakarma

Gopesh Vishwakarma

Numerade Educator

Problem 46

A charge $3 q$ is at the origin, and a charge $-2 q$ is on the positive $x$-axis at $x=a$. Where would you place a third charge so it would experience no net electric force?

Zulfiqar Ali

Numerade Educator

Problem 47

A negative charge $-q$ lies midway between two positive charges $+Q$. What must $Q$ be such that the electric force on all three charges is zero?

Zulfiqar Ali

Numerade Educator

Problem 48

In Fig. 20.30, take $q_{1}=68 \mu \mathrm{C}, q_{2}=$ $-34 \mu \mathrm{C}$, and $q_{3}=15 \mu \mathrm{C}$. Find the electric force on $q_{3}$.

Gopesh Vishwakarma

Gopesh Vishwakarma

Numerade Educator

Problem 49

In Fig. 20.30, take $q_{1}=21 \mu \mathrm{C}$ and $q_{2}=18 \mu \mathrm{C}$. If the force on $q_{1}$ points in the $-x$-direction, find (a) $q_{3}$ and (b) the magnitude of the force on $q_{1}$.

Zulfiqar Ali

Numerade Educator

Problem 50

DNA fragments introduced into an electrophoresis apparatus (see Application, page 390 ) generally carry negative charges equivalent to two extra electrons per base pair of nucleotides in the fragment. The table below shows the forces on several DNA fragments in an electrophoresis apparatus, as a function of the number of base pairs. Plot these data, establish a best-fit line, and use the resulting slope to determine the strength of the electric field in the electrophoresis apparatus.
$$
\begin{array}{|l|l|l|l|l|l|l|}
\hline \text { Base pairs } & 400 & 800 & 1200 & 2000 & 3000 & 5000 \\
\hline \text { Force }(\mathrm{pN}) & 0.235 & 0.472 & 0.724 & 1.15 & 1.65 & 2.87 \\
\hline
\end{array}
$$

Mayukh Banik

Numerade Educator

Problem 51

A proton is at the origin and an ion at $x=8.0 \mathrm{~nm}$. If the electric field is zero at $x=-4.0 \mathrm{~nm}$, what's the ion's charge?

Gopesh Vishwakarma

Gopesh Vishwakarma

Numerade Educator

Problem 52

Four equal charges $Q$ are at the corners of a square of side $a$. Find an expression for the magnitude of the force on each charge.

Gopesh Vishwakarma

Gopesh Vishwakarma

Numerade Educator

Problem 53

A dipole lies on the $y$-axis and consists of anelectron at $y=0.60 \mathrm{~nm}$ and a proton at $y=-0.60 \mathrm{~nm}$. Find the electric field (a) midway between the two charges; (b) at the point $x=2.0 \mathrm{~nm}, y=0 \mathrm{~nm}$; and (c) at the point $x=-20 \mathrm{~nm}, y=0 \mathrm{~nm}$.

Gopesh Vishwakarma

Gopesh Vishwakarma

Numerade Educator

Problem 54

Show that the field on the $x$-axis for the dipole of Example $20.5$ is given by Equation $20.6 \mathrm{~b}$, for $|x| \gg a$.

Zulfiqar Ali

Numerade Educator

Problem 55

You're $1.44 \mathrm{~m}$ from a charge distribution that is well under $1 \mathrm{~cm}$ in size. You measure an electric field strength of 296 N/C due to this distribution. You then move to a distance of $2.16 \mathrm{~m}$ from the distribution, where you measure a field strength of $87.7 \mathrm{~N} / \mathrm{C}$. What's the net charge of the distribution? Hint: Don't try to calculate the charge. Determine instead how the field decreases with distance, and from that infer the charge.

Zulfiqar Ali

Numerade Educator

Problem 56

Three identical charges $q$ form an equilateral triangle of side $a$, with two charges on the $x$-axis and one on the positive $y$-axis. (a) Find an expression for the electric field at points on the $y$-axis above the uppermost charge. (b) Show that your result reduces to the field of a point charge $3 q$ for $y \gg a$.

Laszlo Zalavari

Laszlo Zalavari

Numerade Educator

Problem 57

Two identical small metal spheres initially carry charges $q_{1}$ and $q_{2}$. When they're $1.0 \mathrm{~m}$ apart, they experience a $2.5-\mathrm{N}$ attractive force. Then they're brought together so charge moves from one to the other until they have the same net charge. They're again placed $1.0 \mathrm{~m}$ apart, and now they repel with a $2.5-\mathrm{N}$ force.
What were the original charges $q_{1}$ and $q_{2}$ ?

Zulfiqar Ali

Numerade Educator

Problem 58

Two $32.0-\mu \mathrm{C}$ charges are attached to opposite ends of a spring with spring constant $k=135 \mathrm{~N} / \mathrm{m}$ and equilibrium length $49.3 \mathrm{~cm}$. By how much does the spring stretch? Hint: You'll need to use a computer or advanced calculator to solve the cubic equation that arises in this problem.

Zulfiqar Ali

Numerade Educator

Problem 59

A positive charge $Q$ is located at the origin, and another charge $q$ is at $x=a$, where $a>0$. Given that the electric field is zero at $x=2 a$, find an expression for $q$ in terms of $Q$.

Gopesh Vishwakarma

Gopesh Vishwakarma

Numerade Educator

Problem 60

An electron is moving in a circular path around a long, uniformly charged wire carrying $1.4 \mathrm{nC} / \mathrm{m}$. What's the electron's speed?

Ankur S

Numerade Educator

Problem 61

Find the line charge density on a long wire if a $6.5-\mu \mathrm{g}$ particle carrying $2.2 \mathrm{nC}$ describes a circular orbit about the wire with speed $270 \mathrm{~m} / \mathrm{s}$.

Anand Jangid

Numerade Educator

Problem 62

A dipole with dipole moment $1.6 \mathrm{nC} \cdot \mathrm{m}$ is oriented at $30^{\circ}$ to a $5.0-\mathrm{MN} / \mathrm{C}$ electric field. Find (a) the magnitude of the torque on the dipole and (b) the work required to rotate the dipole until it's antiparallel to the field.

Gopesh Vishwakarma

Gopesh Vishwakarma

Numerade Educator

Problem 63

You have a job examining patent applications. You're presented with the device in Fig. 20.31, which its inventor claims will separate isotopes of a particular element. Atoms are first stripped completely of their electrons, then accelerated from rest through an electric field chosen to give the desired isotope exactly the right speed to pass through the electrostatic analyzer (see Example 20.8). Will the device work?

Rashmi Sinha

Numerade Educator

Problem 64

A $5.0-\mu \mathrm{m}$ strand of DNA carries charge $+e$ per $\mathrm{nm}$ of length. Treating it as a charged line, what's the electric field strength $21 \mathrm{~nm}$ from the DNA, not near either end?

Gopesh Vishwakarma

Gopesh Vishwakarma

Numerade Educator

Problem 65

Heating in a microwave oven occurs as water molecules rotate back and forth to align their dipole moments with a rapidly changing electric field. Given water's dipole moment of $6.17 \times 10^{30} \mathrm{C} \cdot \mathrm{m}$, what's the energy change when a water molecule, with its dipole moment initially opposite a $2.95-\mathrm{kN} / \mathrm{C}$ electric field, swings to align with the field?

Zulfiqar Ali

Numerade Educator

Problem 66

A dipole with charges $\pm q$ and separation $2 a$ is located a distance $x$ from a point charge $+Q$, oriented as shown in Fig. $20.32$. Find expressions for the magnitude of (a) the net torque and (b) the net force on the dipole, both in the limit $x \gg a$. (c) What's the direction of the net force?

Mayukh Banik

Numerade Educator

Problem 67

You're taking physical chemistry, and your professor is discussing molecular dipole moments. Water, he says, has a dipole moment of " $1.85$ debyes," while carbon monoxide's dipole moment is only "0.12 debye." Your physics professor wants these moments expressed in SI. She tells you that the atomic separation in these two covalent compounds is about the same, and asks what that indicates about the way shared charge is distributed. What do you tell her?

Zulfiqar Ali

Numerade Educator

Problem 68

The electric field on the axis of a uniformly charged ring has magnitude $340 \mathrm{kN} / \mathrm{C}$ at a point $10 \mathrm{~cm}$ from the ring center. The magnitude $25 \mathrm{~cm}$ from the center is $110 \mathrm{kN} / \mathrm{C}$; in both cases the field points away from the ring. Find (a) the ring's radius and (b) its charge.

Vishal Gupta

Numerade Educator

Problem 69

An electric quadrupole consists of two oppositely directed dipoles in close proximity. (a) Calculate the field of the quadrupole shown in Fig. $20.33$ for points to the right of $x=a$ and (b) show that for $x>a$ the quadrupole field falls off as $1 / x^{4}$.

Zulfiqar Ali

Numerade Educator

Problem 70

Four charges lie at the corners of a square of side $a$, with the center of the square at the origin. The two charges with $y=a / 2$ have magnitude $Q$ and are positive. The two charges with $y=-a / 2$ also have magnitude $Q$ but are negative. (a) Find an expression for the magnitude of the electric field for points on the $y$-axis with $y>a / 2$. (b) Show that, for $y>a$, your result exhibits the $1 / y^{3}$ falloff you would expect for an electric dipole. (c) Compare the result of (b) with Equation $20.6 \mathrm{~b}$ and write an expression for the magnitude of the dipole moment of this four-charge distribution. Hint: Be careful with your approximation in (b)! If you get 0 for your answer, then you've gone too far. You can neglect $a^{2}$ when compared with $y^{2}$, but you can't neglect $a$ when compared with $y$ or you'll be throwing out the charge separation that makes this distribution resemble a dipole at large distances.

Carson Merrill

Numerade Educator

Problem 71

A straight wire $12 \mathrm{~m}$ long carries $28 \mu \mathrm{C}$ distributed uniformly over its length. (a) What's the line charge density on the wire? Find the electric field strength (b) $20 \mathrm{~cm}$ from the wire axis, not near either end, and (c) $450 \mathrm{~m}$ from the wire. Make suitable approximations in both cases.

Zulfiqar Ali

Numerade Educator

Problem 72

Two thin rods, each of length $a$, lie along the $x$-axis as shown in Fig. 20.34. The left-hand rod, which extends from $x=-a$ to the origin, carries negative charge $-Q$ distributed uniformly over its length. The right-hand rod extends from the origin to $x=a$ and carries positive $Q$ distributed uniformly over its length. (a) Find an expression for the magnitude of the electric field on the $x$-axis for $x>a$ (that is, beyond the right end of the right-hand rod). (b) Show that, for $x \gg a$, the field exhibits the $1 / x^{3}$ falloff of an electric dipole. (c) Compare the result of (b) with Equation $20.6 \mathrm{~b}$ and write an expression for the magnitude of the dipole moment of this two-rod structure.

Mayukh Banik

Numerade Educator

Problem 73

Figure $20.35$ shows a thin, uniformly charged disk of radius $R$. Imagine the disk divided into rings of varying radii $r$, as suggested in the figure. (a) Show that the area of such a ring is very nearly $2 \pi r d r$. (b) If the disk carries positive surface charge density $\sigma$, use the result of part (a) to write an expression for the charge $d q$ on an infinitesimal ring. (c) Use the result of (b) along with the result of Example $20.6$ to write the infinitesimal electric field $d E$ of this ring at any point on the disk axis, taken to be the $x$-axis. (d) Integrate over all such rings to show that the net electric field on the axis has magnitude
$$
E=2 \pi k \sigma\left(1-\frac{|x|}{\sqrt{x^{2}+R^{2}}}\right)
$$

Zulfiqar Ali

Numerade Educator

Problem 74

Use the result of Problem 73 to show that the field of an infinite, uniformly charged flat sheet is $2 \pi k \sigma$, where $\sigma$ is the surface charge density. (This result is independent of distance from the sheet.)

Mayukh Banik

Numerade Educator

Problem 75

Use the binomial theorem to show that, for $x \gg R$, the result of Problem 73 reduces to the field of a point charge whose total charge is the charge density times the disk area.

Mayukh Banik

Numerade Educator

Problem 76

A semicircular loop of radius $a$ carries positive charge $Q$ distributed uniformly. Find the electric field at the loop's center (point $P$ in Fig. 20.36). (Hint: Divide the loop into charge elements $d q$ as shown, write $d q$ in terms of the angle $d \theta$, then integrate over $\theta$.)

Mayukh Banik

Numerade Educator

Problem 77

A thin rod carries charge $Q$ distributed
uniformly over its length $L$ and is situated on the $x$-axis between $x=\pm L / 2$. (a) Find the electric field at an arbitrary point $(x, y)$. (You'll have to do separate integrals for the $x$-and $y$-components.)
(b) Show that your result reduces to that of Example Variation 42 when $x=0$ and to that of Example Variation 43 when $y=0$ and $x>L / 2$.

Mayukh Banik

Numerade Educator

Problem 78

A thin rod extends along the $x$-axis from $x=0$ to $x=L$ and carries line charge density $\lambda=\lambda_{0}(x / L)^{2}$, where $\lambda_{0}$ is a constant. Find the electric field at $x=-L$.

Rashmi Sinha

Numerade Educator

Problem 79

You're working on the design of an ink-jet printer. Ink drops of mass $m$, speed $v$, and charge $q$ will enter a region of uniform electric field $E$ between two charged plates (Fig. 20.37). The drops enter midway between the plates, and the electric field deflects them toward the correct place on the page. Find an expression for the maximum electric field for which drops can still get through without hitting either plate.

Zulfiqar Ali

Numerade Educator

Problem 80

At a distance $r$, far from the heart, the heart's electric field
a. falls off as $1 / r$.
c. falls off as $1 / r^{3}$.
b. falls off as $1 / r^{2}$.
d. falls off as $1 / r^{4}$.

Rashmi Sinha

Numerade Educator

Problem 81

At a given distance, far from the heart compared with its size, the electric field
a. is weaker along an extension of the line shown in Fig. $20.38 c$ than on a perpendicular line.
b. is stronger along a an extension of the line shown in Fig. $20.38 c$ than on a perpendicular line.
c. has the same value at positions perpendicular and parallel to the line in Fig. 20.38c.

Narendra Kumar

Numerade Educator

Problem 82

The difference between Figs. $20.38 a$ and $20.38 b$ that results in an external electric field in one case but not the other is that
a. there's no net charge in Fig. $20.38 a$ but there is a net charge in Fig. 20.38b.
b. the total charge is greater in Fig. $20.38 a$.
c. the charge is distributed in Fig. $20.38 \mathrm{~b}$ so there's more negative charge to the left and more positive charge to the right.

Nicholas Majtenyi

Nicholas Majtenyi

Numerade Educator

Problem 83

At the instant shown in Fig. $20.38 c$, there's an electric field within the heart that points approximately
a. in the direction of the dipole moment vector $\vec{p}$.
b. opposite the dipole moment vector $\vec{p}$.
c. perpendicular to the dipole moment vector $\vec{p}$.

Vishal Gupta

Numerade Educator