Summary 667

SUMMARY The goal of Chapter 20 has been to develop a basic understanding of electric phenomena in terms of charges, forces, and fields.

GENERAL PRINCIPLES

Charge Coulomb's Law There are two kinds of charges, called positive and negative.

The forces between two charged particles ql and q2 separated by di stance r are

Atoms consist of a nucleus contain ing positi vely charged protons sunu unded by a cloud of negatively charged electrons.

The fundamental charge e is

where K = 8.99X 109 N· m2/C2 is the electrostatic constant. These forces are an action/reaction pair directed along the line joi ning the particles.

the magn itude of the charge on anelectronorproton: e= 1.60 X 10 I~ C.

The forces are repulsive for two like ch<rrges. allractive for two oppos ite ch<rrges.

Matter with equal amounts of positive and negative charge is neutral.

The net force on a charge is the vector sum of the forces from all other charges.

Charge is conserved; it can't be created or destroyed. The unit of charge is the coulomb (C).

IMPORTANT CONCEPTS

The Electric Field Visuallzlng the electric field

Charges interact with each other via the electric fie ld E. The electric field exists at all points in space.

An electric field vector shows the field onl y at one point. Charge A alters the space

around it by creating an electri c field.

The field is the agent that exerts a force on charge B.

An electric tield is identified and measured in terms of the force on a probe charge q. The unit of the electric field is N/C.

The electric fi eld is a vector. The elec tric field from multiple charges is the vector sum of the fields from the individual charges.

APPLICATIONS

There are two types of material , insulators and conductors.

Charge remains fixed on an insulator.

Charge moves easil y through conductors.

Charge is transferred by contact between objects.

A dipole has no net charge, but has a field because the two charges are separated.

A dipole will rotate to align with an electric fi eld.

_ F onq E ~-

q

E lOtal =

EI + £2 + .

the point at the tail of the vec tor.

A field diagram shows field vectors at several points.

Electric field lines:

are always paraliel to the field vectors.

are close where the field is strong, far apart where the field is weak.

go from positive to negati ve charges.

- (1 -

Electric fields : important cases Conductors in electric fields

The electric field inside The electri c field of a point charge is

E ~ (KI;I, [ away from q if q > 0]) r ~ towardqlfq < O

The electric field inside a parallel·plate capacitor is uniform:

- (Q f .. . ) E ~ --, rom pOSItive to negative f oA

where €o = 8.85 X 10 12 C2/N ' m2 is the permittivity constant.

a conductor in electrostatic equi­librium is zero.

Any excess charge is on the surface.

The electric fi eld is perpendicular to the surface.

The density of charge and the electric field are highest near a pointed end.

,

,

I

,

668 CHAPTER 20 Electric Fields and Forces

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Problem difficulty is labeled as I (straightforward) to 11111 (challenging).

0,0 •

QUESTIONS

Conceptual Ouestions

I. What is alike abollt charges when we say "two like charges"? Do they look, smell , or taste the same?

2. Four lightweight balls A, B, C, and Dare sllspended by threads. Ball A has been touched by a plastic rod that was rubbed with wool. When the balls are brought close together, withom·touch­ing, the following observations are made: • Balls B, C, and 0 are allracted to ball A. • Balls Band 0 have no effect on each other. • Ball B is attracted to ball C. What are the charge states (posi tive, negative, or neutral) of balls A, S, C, and D? Explain.

3. Plastic and glass rods that have been charged by rubbing with wool and silk, respectively. hang by threads. a. An object repels the plastic rod. Can you predict what it will

do to the glass rod? If so, what? If not, why not? Explain. b. A different object attracts the plast ic rod. Can you predict

what it will do to the glass rod? If so, what? If not, why not? Explain.

4. a. Can an insulator be charged? If so, how would you charge an insulator? If not. why not?

b. Can a conductor be charged? if so, how would you charge a conductor? If not, why not?

5. When you take clothes out of the drier right after it stops, the clothes often stick to your hands and arms. Is your body charged? If so, how did it acqu ire a charge? If not, why does thi s happen ?

6. A lightweight metal ball hangs by a thread. When a charged rod is held near, the ball moves toward the rod , touches the rod, then quickJy "fl ies away" from the rod. Explain this behavior.

7. As shown in Figure Q20.7, metal A B

sphere A has 4 unit s of negative charge and metal sphere B has 2 units of positi ve charge. The two Touch

spheres are brought into contact. FIGURE 020.7

What is the final charge state of each sphere? Explain.

8. Figure Q20.8 shows a positively charged rod held near, but not touch­ing, a neutral metal sphere. a. Add plusses and minuses to the

figure to show the charge di strib­ution on the sphere.

b. Does the sphere experience a net force? If so, in which direction? Explain.

10 FIGURE 020.8

Problems labeled INT integrate significant material from earlier

chapters; BID are of biological or medical interest.

9. A plastic balloon that has been rubbed with wool wiU stick to a wall. a. Can you conclude that the wall is charged? If not , why not?

IJ so, where does the charge come from? b. Draw a charge diagram showi ng how the balloon is held to

the wall. 10. You are given two metal spheres on portable insulating stands, a

glass rod , and a piece of silk. Explain how to give the spheres exactly equal but opposite charges.

II. A honeybee acquires a positive e lectric charge as it flies BID through the air. This charge causes pollen grains to be attracted

to the bee. Explain, using words and diagrams, how a neutral , conducting pollen grain will be attracted to a positi ve ly charged bee.

12. A metal rod A and a metal sphere B, on in sulating stands, touch each other as shown in Figure Q20.12. They are origi­nally neutral. A positi ve ly charged rod is brought near (but not touching) the far end of A. While the charged rod is st ill close, A and B are separated. The charged rod is then withdrawn. Is the sphere then positive ly charged, negatively charged, or neu­tral ? Explain.

~ C=' ~A ==::::J(s) FIGURE 020.12 ~ ')( I y

13. Each part of Figure Q20.1 3 shows two points near two charges. Compare the electric fi e ld strengths EI and £2 at these two points. Is EI > E2, E1 = E2, or EI < E2?

(, ) (h)

0 • 0 • 0 • • • , 2 2

(e) (d) ,. • 0 •

2

0 • 0

(e) (I)

0 • ED • e • • , 2 2

FIGURE 020 .13

c 14. Ion tophores is is a non invas ive BID process that transports drugs

thro ugh the skin without need les. In the photo, the red elec trode is pos iti ve and the b lac k e lec trode is negati ve. The e lectri c ri eld between the electrodes will drive the negati vely c harged molecules of an anest heti c through the skin . Should the drug be placed at the red OJ the black electrode? Ex pla in .

15. Rank in order, from largest 10

smallest, not ing any ti es , the d ec· tr ic fie ld strengths E I to £~ a l po in ts 1 to 4 in Figure Q20. 1S.

16. A 10 nC charge sits at a poinl in space whe re the mag ni tude of the electr ic fi eld is 1200 N/C. What

17. BIO 18.

FIGURE Q20.15 wiLl the magn itude of the fi e ld be if the 10 nC charge is replaced by a 20 nC charge? When DNA breaks in to fragments in a cell , electrostatic forces may actually inhi b it repair. Ex plai n why thi s happe ns. A ho ll ow soda straw is uniformly c harged, as shown in Fi gu re Q20. 18. What is the elec tric field at the center (ins ide) of the straw? Ex plain .

+ +

Inside straw

++++++++ ++++++++

+ + + + + + + +

FIGURE 020.18 FIGURE 020.19

19. A small pos iti ve charge q experi ences a force of mag nitude FI when placed at po in t I in Figure Q20. l9. In terms of F1:

a. What is the mag ni tude of the force on charge q at po int 3? b. What is the magn itude o f the fo rce o n a charge 3q at point I? c. What is the magn itude of the force on a charge 2q at point 2? d. What is the magn itude of the force on a charge - 2q at

po int 2? 20. A ty pica l commerc ial a irplane is s truck by lig ht ni ng about

once pe r year. W hen thi s happe ns. the ex te rnal metal ski n o f the ai rplane mig ht be bu rned, bu t the people and equipme nt ins ide the a ircraft exper ie nce no ill e ffects. Ex pla in why thi s is so.

2 1. Microbes such as bac teria have small pos itive charges when in BIO solu tion. Publ ic heal th agenc ies are ex plor ing a new way to

measure the presence o f small num bers o f microbes in dri nk ing water by us ing electric forces to concentrate the microbes. Water is sent between the two o ppos itely charged electrodes of a parallel-plate capacitor. Any microbes in the water w iU collect on o ne of the electrodes. a . On which e lectrode will the microbes collec t? b. How could the microbes be eas il y removed fro m the elec­

trodes fo r anaJ ys is? 22. a. Is there a poin t between a 10 nC charge and a 20 nC charge at

which the electr ic fie ld is zero? If so, which charge is thi s po int closer to? If not, why not?

b . Repeat part a for the case of a 10 nC charge and a - 20 nC charge.

Questions 669

Multiple-Choice Questions

23. I Two lightweight. elec trica lly neutral cond ucting baUs hang from threads. Choose the d iagram in Figure Q20.23 that shows how the baUs hang after:

24.

25.

26.

a. Both are to uched by a negati vely charged rod. b. Ball 1 is touched by a negat ive ly charged rod and ball 2 is

touched by a pos iti vely charged rod. c. 8 0th are touched by a negati vely charged rod but ba ll 2

picks up more charge than ball I. d. O nl y ball 1 is touched by a negative ly charged rod. Note that parts a through d are indepe ndent; these are not actions taken in sequence.

A. B. c. D. E.

2 2 I 2 2 2

FIGURE 020.23

I AU the charges in Fig ure Q 20.24 have the same magn itude. In which case does the electr ic field at the dot have the largest magni tude?

A. ~ • B. @ •

c. • D. • FIGURE 020.24

I All the charges in Fig ure Q20.25 have the same magn itude. In which case does the e lectric fi eld at the dot have the largest magnitude?

A. @ • B. <!l •

c. • D. • FIGURE 020.25

I Al l the charges in Fig ure Q20.26 have the same magn itude. In which case does the electric fi eld at the dol have the largest magni tude?

A. @ • B. @ •

c. @ • D. • FIGURE 020.26

27. I A glass bead c harged to +3.5 nC exe rts an 8.0 X 10- 4 N repu ls ive electri c force o n a plas ti c bead 2.9 cm away. What is the charge on the plasti c bead? A. + 2. 1 nC B. +7.4 nC C. +2 1 "C D. +740 "C

28. I A +7.5 nC point charge and a - 2.0 nC point charge are 3.0cm apart. What is the elec tr ic field strength at the midpo int between the two charges? A. 3.3 X 10] NrC B. 5.7 x 10] Nrc c. 2.2 x 105 Nrc D. 3.8 x 105 Nrc

670 CHAPTER 20 Electric Fields and Forces

29. II Three point charges are arranged as shown in Figure Q20.29. Which arrow best re presents the direction of the e lectric field vector at the position of the dot?

- I nC ~.

A. C.

D.

E.

G) +3nC

FIGURE Q20 .29 - 12 nC

PROBLEMS

Section 20.1 Charges and Forces

Section 20.2 Charges, Atoms, and Molecules

I . III A glass rod is c harged to +5 .0 nC by rubb ing.

/

a. Have electrons been removed from the rod or protons added? Explain.

b. How many electrons have been re moved or protons added? 2. III A plast ic rod is charged to - 20 nC by rubb ing.

a. Have e lec trons been added to the rod or protons removed? Explain.

b. How many electrons have been added or protons removed? 3. III Suppose you have 1.0 mol of O2 gas. How many coulombs of

INT positi ve charge are con tained in the ato mic nuclei of lhjs gas? 4. II A plastic rod that has been charged to - I 5.0 nC touches a

meta l sphere. Afterward, the rod's charge is - 10.0 nCo a. What kind of charged particle was transferred between the rod

and the sphere, and in which direction? That is, did it move

from the rod to the sphere or from the sphere to the rod? b. How many charged particles were transferred?

5. A glass rod that has been c harged to + 12.0 nC to uc hes a metal sphere. Afterward, the rod' s charge is +8.0 nC a. What kind of charged particle was transfel1-ed between the rod

and the sphere, and in which direction? That is, did it move from the rod to the sphere or from the sphere to the rod?

b. How many charged part icles were transferred? 6. III Two identi ca l meta l spheres A and B are connected by a

metal rod. Both are ini tially neutral. 1.0 X 10 12 electrons are added to sphere A. the n the connec ting rod is removed. Afte r· ward, what are the charge of A and the charge of B?

7. II Two iden tical metal spheres A and B are connected by a plas· tic rod. Both are initi aJly neutral. 1.0 X 10 12 e lectrons are added to sphere A, then the con nect ing rod is removed. Afte rward, what are the charge of A and the charge of B?

8. II If two identical conducti ng spheres are in contact, any excess charge will be evenly d istributed between the two. Three identi· cal metal spheres are labeled A, S , and C Initi all y, A has c harge q, B has charge -qI2, and C is uncharged . a. What is the final c harge on each sphere if C is touched to 8 ,

removed, and then touched to A? b. Starting again froll1the ini tial conditions, what is the charge

on each sphere if C is touched to A, removed, and then touched to 8?

30. II A rod has positive charge +q at one end and negati ve charge -q at the other, forming a dipo le. The d ipo le is placed in a nonuniform electric field represented by the field lines in Figure Q20.30. Which arrow best indicates the d irect ion of the net electric force on the d ipole?

FIGURE 020 .30

Section 20.3 Coulomb's Law

9. II Two 1.0 kg masses are 1.0 m apart on a fr ict ion less table. IN! Each has + 1.0 !,-C of charge.

a. What is the magnitude of the electric force on one of the masses?

b. Whal is the ini tial acce lerati on of each mass if they are released and allowed to move?

10. III Two small plastic spheres each have a mass o f 2 .0 g and a charge of -50.0 nC They are placed 2.0 cm apart. a. What is the magn itude of the electric fo rce between the

spheres? b. By what factor is the e lectric force on a sphere larger than

its we ight? 11. III A small plast ic sphe re wi th a charge of -5 .0 nC is near

another small plastic sphere with a charge of - 12 nC If the sphe res repel one another with a force of mag nitude 8.2 X .10- -1 N, what is the di stance betwee n the spheres?

12. III A small meta l bead, labeled A, has a c harge of 25 nCo It is touched to metal bead B, initi a ll y neut ra l, so that the two beads share the 25 nC charge, but not necessaril y equally. When the two beads are then placed 5.0 c m apart, the force between them is 5.4 X 10 -I N. What are the c harges qA and qs

on the beads? 13. 1111 A small g lass bead has been charged to +20 nC A tiny ball

bearing 1.0 e m above the bead feels a 0.018 N dow nward elec· tric force. What is the charge on the baJi bearing?

14. I What are the magni tude and d irection of the electric force on charge A in Figure P20. 14?

I.One - 1.OnC 4.0nC

AG) B c0 FIGURE P20 .14 1.0em 1.0em

15 . III in Figure P20. 15, charge Ch experiences no net e lectric force. What is ql?

FIGURE P20 .1S 10cm 10 em

16. I Objec t A. which has been charged to + 10 nCo is at the origin. Object B, which has been charged to - 20 nC, is at (x, y) = (0.0 cm, 2.0 cm). What are the magnitude and direction of the electric force on each object?

17. I A small glass bead has been charged to +20 nCo What are the INT magnitude and direction of the acceleration of (a) a proton and

(b) an elcctron that is 1.0 cm from the center of the bead?

Section 20.4 The Concept of the Electric Field

18. II What magnitude charge crea tes a 1.0 N/C electric field at a poin t 1.0 m away?

19. II What are the strength and direction of the electric field 2.0 cm from a small glass bead that has been charged to +6.0 nC?

20. II A 30 nC charged particle and a 50 nC charged particle are near each other. There are no other charges nearby. The electri c force on the 30 nC particle is 0.035 N. The 50 nC particle is then moved very far away. Afterward , what is the magnitude of the electric field at its original position?

2 1. J What are the strength and direction of the electric field 1.0 mm from (a) a proton and (b) an e lectron?

22. j A + 10 nC charge is located at the origin. a. What are the strength s o f the e lectr ic fields at the

positions (x, y) = (5.0 cm, 0.0 cm), (- 5.0 cm, 5.0 cm), and (- 5.0 em, - 5.0 cm)?

b. Draw a fi eld diagram showing the elec tri c field vectors at these points.

23. I A - 10 nC charge is located at the origin. a. What are the streng ths o f the electric fi e lds at the posi­

lions (x, y) = (0.0 em, 5.0 em), (- 5.0 em, - 5.0 em), and (- 5.0 cm, 5.0 cm)?

b. Draw a field diagram showing the elec tri c fi e ld vectors at these points.

24. III What are the strength and direction of the electri c fi eld al the position indicated by the dot in Figure P20.24? Specify the direct ion as an angle above or below horizon tal.

. 1.0nC . ,ooC 5.0em 5.0 e111

• • 5.0em 5.0 em

5.0 em 5.0C111

. ,OoC - - 1.0 nC

FIGURE P20.24 FIGURE P20.25

25. What are the strength and direction of the electric fi eld at the position indicatcd by the dot in Figure P20.25? Specify the direction as an angle above or below hori zontal.

Section 20.5 Applications of the Electric Field

26. II What are the strength and direction of an e lectri c fi eld that INT will balance the weight of a 1.0 g plastic sphere that has been

charged to - 3.0 nC? 27. I What are the strength and direction of an electri c field that INT will balance the weight of (a) a proton and (b) an electron? 28. II A 0.10 g plastic bead is charged by the addition of 1.0 X 1010

INT excess elec trons. What elec tri c fi eld E (strength and direction) will cause the bead to hang suspended in the air?

Problems 671

29. II A parallel-plate capac itor is construc ted of two square plates, size L X L, separated by distance d. The plates are given charge ± Q. What is the ratio ErlE j of the final electri c fi eld strength Er to the initial e lectric fi eld strength Ej if: a. Q is doubled? b. L is doubled? c. d is doubled?

30. II A parallel-plate capac itor is formed from two 4.0 cm X 4.0 cm electrodes spaced 2.0 mm apart. The e lectri c field strength inside the capac itor is 1.0 X lOb N/C. What is the charge (in nC) on each electrode?

3 1. I Two identical closely spaced c ircular disks form a parallel­plate capacitor. Transferring 1.5 X 109 electrons from one di sk to the other causes the electric fi eld strength between them to be 1.0 X 105 N/C. What are the diameters of the disks?

Section 20.6 Conductors and Electric Fields

32. III Storm clouds may build up large negative charges near their bottom edges. The earth is a good conductor, so the charge on the cloud attracts an equal and opposite charge on the earth under the cloud. The electric field strength near the earth depends on the shape of the earth's surface, as we can explain with a simple model. The top metal plate in Figure 20.32 has uniformly distributed negative charge. The bollom metal plate, which has a high point, has an equal and opposite charge that is free to move. a. Sketch the two plates and the region between them, show­

ing the distribution of positive charge on the bottom plate. b. Complete your diagram by sketching electric field lines

between the two plates . .Be sure to note the direction of the fi eld. Where is the field strongest?

c. Explain why it is more dangerous to be on lOp of a hill or mountain during a lightning storm than on level ground.

o t + + ++++++

FIGURE P20.32 FIGURE P20.33

33. III A neutral conducting sphere is between two parallel charged plates, as shown in Figure P20.33. Sketch the electric field lines in the region between the plates. Be sure to include the effect of the conduct ing sphere.

Section 20.7 Forces and Torques in Electric Fields

34. III Two small plastic spheres, one charged to 17 nC and the INT other to - 17 nC, are con nec ted by a 25-mm-Iong insulating

rod. Suppose thi s dipole is placed in a uniform electri c field with strength 7.4 X \05 N/C. What is the maximum possible torque on the dipole?

35. II A prote in molecule in an electrophoresis gel has a negati ve BID charge. The exact charge depends on the pH of the solution, but

30 excess electrons is lypical. What is the magn itude of the e lec tri c force on a protein with thi s charge in a 1500 N/C elec­tric field?

672 CHAPTER 20 Electric Fields and Forces

36. II Large electric field s in cell membranes cause ions to BID move through the ce ll waH, as we will explore in Chapter 23.

The fi e ld st rength in a typical me mbrane is 1.0 X 107 N/C. What is the magnitude of the force on a calcium ion with charge +e?

37. II I Molecules of carbon mon- + 3.4 x 10- 21 C - 3.4 X 10- 11 C IN! ox ide are permanent electric

dipoles due to unequa l sharing of electrons between the carbon

@-@ 0.11 nm

and oxygen atoms. Figure P20.37 FIGURE P20 .37

shows the di stance and charges. Suppose a carbon monoxide molecule with a hori zon tal axi s is in a vertical electric fi e ld of sU'ength 15,000 N/C. a. What is the magnitude of the net force on the molecule? b. What is the magnitude of the torque on the molecule?

General Problems

38. 11111 A 2.0-mm-diameter copper ball is c harged to + 50 nC. What INT fraction of its elec tro ns have been re moved? The dens ity of

copper is 8900 kg/m3.

39. I Pennies today are copper-covered zinc, but o lder pennies are INT 3.1 g of so lid copper. What are the total positi ve charge and

total negative charge in a so lid copper penny that is electri cally neutral ? The density of copper is 8900 kg/m 3.

40. II Two protons are 2.0 fm apart. (I fm = I femtometer = INT I X 10- 15 m.)

a . What is the magnitude of the electr ic force on one proton due to the other proton?

b. What is the magnitude of the gravitational force on one pro­ton due to the other proton?

c. What is the ratio of the elec tri c force to the gravitat ional force?

41 . 1111 The nucleus of a 125 Xe atom (an isotope of the element xenon INT with mass 125 u) is 6.0 fm in diameter. It has 54 protons and

charge q = +54e. ( I fm = 1 femtometer = 1 X 10- 15 m.) a . What is the electr ic force on a proton 2.0 fm from the sur­

face of the nucleus? b. What is the proton 's acceleration? Hint: Treat the spherical nucleus as a point charge.

42. III Two equall y charged, 1.00 g spheres are placed with 2.00 cm INT between the ir centers. When released, each beg ins to accelerate

at 225 m/s2. What is the magnitude of the c harge on each sphere?

43. II Objects A and B are both positively charged. Both have a mass of 100 g, but A has twice the charge of B. When A and B are placed with 10 cm between their centers, B experiences an electric force of 0.45 N. a. How large is the force on A? b. What are the charges qA and qA?

44. 11111 An electric dipole is formed from ± 1.0 nC point c harges spaced 2.0 mm apart. The dipole is centered at the o ri g in, orie nted along the y-axi s . What is the e lec tric field s tre ngth at the points (a) (x, y) ~ ( 10 mm, a mm) and (b) (.t, y) ~ (0 mm, 10 mm)?

45. III What are the strength and direction of the e lectr ic field at the position indicated by the dot in Figure P20.45? Spec ify the direction as an ang le above or be low horizontal.

46.

47.

• e l.O,C • S.O cm S.O cm

10cm IOcm

. 1.0nC - - I.OnC

FIGURE P20 .45 FIGURE P20 .46

11111 Whal are the strength and direction of the electric fi eld at the position indicated by the dot in Figure P20.46? Specify the direct ion as an angle above or below hori zontal. II What is the force on the 1.0 nC charge in Figure P20.47? Gi ve your answer as a magnitude and a direction.

1.0 nC 1.0nC

~ ~ , , 1.0 cm " \ 1.0em 1.0cm ,' \ 1.Oem , , , , ,

'''-(;)' 60'/'

2.0 ,C @ --\ ---- f --0 2.O'c 1.0em

,"- (;). (;)/' 2.0nc @ -_L ___ f __ '- - 2.0nC

I.Ocm

FIGURE P20 .47 FIGURE P20 .48

48. II What is the force on the 1.0 nC charge in Figure P20.48? Gi ve your answer as a magnitude and a direction.

49. I What is the magnitude of the force on the 1.0 nC charge in the middle of Figure P20.49 due to the four other charges?

50.

5 1.

- 2.0 nC - 2.0 nC - 2.0 nC 2.0 nC 1.0em 1.0cm

€I ,

,.. : 1.0 cm : 1.0 em

1.0nC 1.0 nC

-(t) - 2.0 nC - 2.0 nC - 2.0 nC 2.0 nC

FIGURE P20 .49 FIGURE P20 .50

III What are the magnitude and direction of the force on the t .O nC charge in the middle of Figure P20.50 due to the four other charges? II What are the magnitude and direct ion of the force on the 1.0 nC c harge at the bottom of Figure P20.51 ?

- 6.0 nC

2.0 nC 2.0nC

@ B

FIGURE P20 .51

52. 11111 A 5.0 nC point charge sits at x = O. At the same time, a 4500 N/C uniform elec tric field (created by di stan t source charges) points in the positi ve x-direc tion . At what point along the x-ax is, if any, would (a) a proton and (b) an electron experi ­ence no net force?

53. II The net fo rce on the 1.0 nC charge in Figure P20.53 is zero. What isq?

FIGURE P20.53

[ 200m

(±} 2.0nC

3.0el11

q

o 1,0nC (±) 4.0 em

2.0nC @

3.0(:111

54. II Two parti cles have positive charges q and Q. A thi rd charged parti cle is placed halfway between the m. What mllstthis parti ­cle's charge be so that the net force on charge Q is zero?

55. 11111 Figure P20.55 shows four charges at _Q

the corners of a square of side L. Assume q

e

56. INT

q and Q are pos itive. a. Draw a diagram showing the three

forces on charge q due to the other charges. Gi ve your vectors the cor­rect relati ve lengths.

b. Find an express ion for the mag ni tude

,

® L 4Q

of the net force on q. FIGURE P20.55 Suppose the magn itude of the proto n

- Q

charge differs from that of the e lectron charge by a mere I part in 109.

a. What would be the force be tween two 2.0-mm-di ameter copper spheres 1.0 cm apart? Assume that each copper atom has an equal number of elec trons and protons. The density of copper is 8900 kg/m3

.

h. Wo uld thi s amount of force be detectab le? What can YOll conclude from the fact that no such forces are observed?

57. III In a simple model of the hydrogen atom, the elec tron INT moves in a c ircular orbit of rad ius 0.053 nm around a station­

ary proton . How many revo lu tions per second does the elec­tron make? Hint: What must be true for a force that causes c irc ular motion?

58. III A 0. 10 g honeybee acquires a charge of +23 pC whi le fl ying. BIO a. The e leClric fi eld near the surface of the earth is typically

100 N/C, directed downward. What is (he rat io of the e lec­tric force on the bee to the bee 's we ight?

b. W hat e lectri c fie ld strength and direction would allow the bee to hang suspended in the air?

59. 11111 A + 10 nC charge is located at (x. y) = (0 cm, 10 cm) and a -5.0 nC charge is located (x, y) = (5.0 cm, 0 cm). Where would a - 10 nC charge need to be located in order that the electri c fie ld at the origin be zero?

60. III Two 2.0-cm-diameter disks face each o ther, 1.0 mm apart. INT They are charged to ± 10 nCo

a. What is the electri c fi eld strength between the disks? b. A proton is shot from the negati ve disk toward the pos iti ve

disk. What launch speed must the proton have to j ust barely reac h the pos iti ve di sk?

61. 1111 The electron gun in a television tube uses a uniform electri c INT field to accelerate electrons from rest to 5.0 X 107 m/s in a dis­

tance of 1.2 cm. What is the electric fi eld strength?

62. 1111 A 0.020 g plasti c bead hangs from a INT lightwe ight thread. Another bead is

fi xed in pos ition beneath the point where the thread is ti ed. If both beads have charge q, the moveable bead swings out to the pos ition shown in Figure P20.62. What is q?

63. 11 11 You have a lightweight spring whose INT unstretched length is 4.0 cm. You' re

Problems 673

5.0el11

FIGURE P20.62

curious [0 see if you can lise thi s spri ng to measure charge. First, you attach one end of the spri ng to the ce iling and hang a 1.0 g mass from it. This stretches the spring to a length of 5.0 cm. You then attach two small plast ic beads to the opposite ends of the spri ng, lay the spring on a friction less tab le, and give each plasti c bead the same charge. T his stretches the spring to a length of 4.5 cm. What is the mag nitude of the charge (in nC) on each bead?

64. 11111 Two 3.0 g spheres on 1.0-m-long threads repel each other INT after being equally charged, as shown in Figure P20.64. What is

the charge q?

3.0 g 3.0 g

FIGURE P20.64 q q

65. 11111 An e lec tri c field if = ( 100,000 N/C. ri ght) causes the 5.0 g INT ball in Figure P20.65 to hang al a 200 angle. What is the charge

on the bal.l ?

E -, -­,

q 5.0 g

T------ -

FIGURE P20.65

o E ---:25 nC + 2.0g

,------+ -

FIGURE P20.66

66. 1111 An electric fie ld E = (200,000 N/C, ri ght) causes the 2.0 g INT ball in Figure P20.66 to hang at an angle. What is (J? 67. II A small charged bead has a mass of 1.0 g. It is held in a uni­INT form electri c fi eld E = (200,000 N/C, up) .When the bead is

re leased, it acce lerates upward with an acce leralion of 20 m/s2•

What is the charge on the bead? 68. II A bead with a mass of 0.050 g and a charge of 15 nC is free 10

INT slide on a ve rti ca l rod. At the base of the rod is a fi xed 10 nC charge. In equilibrium, at what he ight above the fi xed charge does the bead rest?

69. II A small bead with a pos iti ve charge q is free 10 slide on a hor­INT izontal wire of length 4.0 cm. At the left end of the wire is a

fi xed charge q, and at the right end is a fi xed charge 4q. How fa r from the left end of the wire does the bead come to rest?

674 CHAPTER 20 Electric Fields and Forces

In Problems 70 and 7 1 you are given the equation used to so lve a problem. For each of these,

a. Write a realistic problem for which this is the correct equat ion. b. Finish the solution of the problem.

(9.0 X 10' N . m' /C' ) X N X (1.60 X 10- 19 C) 70. III ~---~c:---:-::--;c-=----~

( I.O X 10 6 m) 2

= 1.5 X 10' N/C

71. III (9.0 X 10' N . m' /C' )q'

(0.0 15 m)' 0.020 N

Passage Problems

Flow Cytometry BID

Flow cytometry, illustrated In Figure P20.72, is a technique used to sort cells by type. The ce ll s are placed in a conducting sal ine solution which is then forced from a nozzle. The st ream breaks up into smaLl droplets, each containing one cell. A metal collar su rrounds the stream ri ght at the point where the droplets separate from the st ream. Charging the co ll ar polarizes the conduct ing liquid, causing the droplels to be· come charged as they break off from the st ream. A lase r beam probes the solut ion just upslfeam from the charging collar, looking for the pres· ence of certa in types of cells. ALI droplets containing one particular type of cell are given the same charge by the charging collar. Droplets with other desired types of cells receive a different charge, and droplets with no

NO,J.Io- D Uq9id ______ ~ C, II,

''','' ~~7 Lascr ~ ; ,

probe "" ~ Charging -© ~ collar

FIGURE P20.72

Stop to Think 20.1: A. The electroscope is originally given a posi· tive charge. The charge spreads out , and the leaves repel each other. When a rod Wilh a negative charge is brought near, some of the posi· live charge is attracted to the top of the eleclroscope. away from the leaves. There is less charge on the leaves, and so they move closer together.

Stop to Think 20.2: q,,( + 3e) > qA( + Ie) > qo(O) > qll( -Ie) > qd- 2e).

Stop to Think 20.3: B. The two forces are an actionireaclion pair, opposite in direction but eqllal in magn itude.

desired cell receive no charge. The charged droplets then pass between two parallel charged electrodes where they receive a hori· zontal force that directs them into different collection tubes, depending on their charge.

72. I If the charging collar has a positive charge, the net charge on a droplet separating from the stream will be A. Positive. B. Negative. C. Neutral. D. The charge will depend on the type of cell.

73. I Which of the following describes the charges on the droplets that end up in the five tubes, moving from left to right? A. +2q. +q. O.-q, - 2q B. +q, +2q,0,-2q, - q C. - q,-2q, 0, +2q, +q D. - 2q,-q, 0,+q,+2q

74. I Because the drop lets are conductors, a droplet's positi ve and negati ve charges will separate while the droplet is in the region between the deflection plates. Suppose a neutra l droplet passes between the plates. The droplet ' s dipole moment will point A. Up. B. Down. C. Left. D. Right.

75. I Another way to son the droplets would be to give each droplet the same charge, then vary the electric field between the deflection plates. For the apparatus as sketched, thi s technique will not work because A. Several droplets are between the plates at one time, and

they would all feel the same force. B. The ce ll s in the solution have net charges that would affect

the droplet charge. C. A droplet with a net charge would always experience a net

force between the plates. D. The droplets would all repel each other, and thi s force

would dominate the deflecting force.

Stop to Think 20.4: En > EA > ED > Eco The field is propor· tional to the charge, and inverse ly proportional 10 the square of the distance.

Stop to Think 20.5 : C. Electric field lines sfarl on positi ve charges. Very near to each of the positive charges, the field lines should look like the fi eld lines of a single positive charge.

Stop to Think 20.6: FA = F 8 = F c = F D = FE" The field inside a capacitor is the same at all points. Because the field is uniform, the force on the proton will be the same at all points. The electric field ex ists at aU points whether or not a vec tor is shown at that point.