Summary 703

SUMMARY The goal of Chapter 21 has been to calculate and use the electric potential and electric potential energy.

GENERAL PRINCIPLES

Elect{ic Potential and Potential Energy The electric potential V is created by charges and ex ists at every point surrounding those charges.

When a charge q is brought near these charges, it acquires an electric potential energy

These charges crt:atc Lhe electric potential.

V = - IOOV \ , (±) • <;;I e

EfJ EfJ V=300V V=200V

at a point where the other charges have created an electric potential V.

Energy is conserved for a charged particle in an electric potent ial:

or I!.K ~ -q l!.V

IMPORTANT CONCEPTS

For a conductor in electrostatic equilibrium

Any excess charge is on the surface.

The electric fi eld inside is zero.

The exterior electric field is perpendicular to the surface.

\ The field strength is largest at shalll corners.

The surface is an equipotential.

The entire conductor is at the same potential and so the surface is an equipote ntial .

APPLICATIONS

Capacitors and dielectrics

The potential difference .6. Vc between two conductors charged to :!: Q is proportional to the charge:

I!.Vc ~ QIC

where C is the capacitance of the two conductors.

A parallel-plate capacitor with plates of area A and separation d has a capacitance

C = KEoAId

When a dielectric is inserted between the plates of a capaci tor, its capac it ance is increased by a factor K, the dielectric constant of the material.

The energy stored in a capacitor is Uc = tC( .6. Vc)2.

This energy is stored in the electric field, which has energy density liE = tKEu E 2

Sources of Potential Potential differences .6. V are created by a separation of charge. Two important sources of potential difference are

A balte1)', which uses chemical means 10 separate charge and produce a potential difference.

The opposite charges on the plates of a capacito/; which create a potential difference between the plates.

The e lec tri c potential of a point charge q is V = K ~ r

Connecting potential and field

kif ~ £ poinL~ "downhill:' in the

if i ~ everywhere : "'_ :: = _ ::: !.-:---" . ' f=

d irec tion o f decrealoi ng V.

perpend icular 10 -- - - _... ............ !he field sl rc ngl~ IS

the equ ipotent ial Di rection of ........ " .... ~}nVe~eIY propom o naJ to surfaces. decreasing " the dl ~lance d between

potcntial ' thc equipotc nt ial surfaces.

Graphical represe ntations of the potential

v

~, Potential graph Equipotential surfaces

Equipotential mall Elevation gfallh

Parallel-plate capacitor

For a capac itor charged to .6. Vc the potential at distance x from the negalive plate is

x V ~d l!.Vc

The electri c field ins ide is

E = I!.Vc1d

Units Electric potential: I V = 1 J/C

Electric field: I VIm = I N/C

-

-

-

o

, , ,

, , , ,

,

,

,

ii ,

, , ,

,

,

Energy: 1 electron volt = I eV = 1.60 X lO- t9 J is the kinetic energy gained by an electron upon accelerat ing through a potential difference of 1 V.

+ + + + + + + + + +

x

704 CHAPTER 21 Electric Potential

tMP)TM For homewo~k assig~ed on MasteringPhysics, go to

;.:;./ www.mastenngphyslcs.com

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

QUESTIONS

Conceptual Questions

I. By moving a 10 nC charge from point Po. to point B, you deter­mine that the electric potential at B is 150 V. What would be the potential at B if a 20 nC charge were moved from A to B?

2. Charge q is fired through a small hole in the pos iti ve plate of a capacitor, as shown in Figure Q21.2. a. If q is a positi ve charge ,

does it speed up or slow down inside the capaci­tor? Answer thi s question twice: (i) Us ing the con­cept of force. (ii ) Using the concept of energy.

b. Repeat part a if q is a neg- FIGURE 021 .2

ali ve charge. 3. Why is the potential energy of two opposite charges a negat ive

number? (Note: Saying that the formula gives a negative num­ber is not an explanation .)

4. An electron (q = -e) completes haIr of a circular orb it of radius ,. arou nd a nucleus wi th Q = +3e, as shown in Figure Q21.4. a. How much work is done on the

electron as it moves from i to f? Give FIGURE 021 .4

either a numerical value or an expression from which you could calculate the value if YOll knew the radius. Justify your answer.

b. By how much does the electric potential energy change as the e lectron moves from i to f?

c. Is the electron's speed at f greater than, less than, or equal to its speed at i?

d. Are your answers to parts a and c consistent with each other? 5. An electron moves along the tra ­

jectory from i to f in Figure Q2 1.5. a. Does the electric potential

energy increase. decrease, or stay the same? Explain.

b. Is the electron' s speed at f FIGURE 021 .5

greater than, less than, or equal to its speed at i? Explain. 6. The graph in Figure Q2 1.6 V (V)

shows the electric potenti al 20 along the x-axis. Draw a graph of the potential energy

'0 of a 0.10 C charged parti cle in th is region of space, pro­viding a numeri cal scale on

o ¥~-~~-".f--~-~x (01) o 2 3

the energy axis. FIGURE 021 .6

Problems labeled INT integrate significant material from earlier

chapters; BID are of biological or medical interest.

7. As shown in Figure Q21.7, two protons are launched with the same speed from point I inside a paralle l-plate capacitor. One proton moves along the path from I to 2, the other from I to 3. Poims 2 and 3 are the same distance from the pos itive plate.

21 / 03

,~ 1+ + + + + ++ + + + + + + + ++ 1

FIGURE 021.7

a. Is 6. UI : 2. the change in potential energy along the path 1: 2, larger than , smalJenhan, or equal to 6. UI : 3? Explain.

b. Is the proton 's speed \'2 at point 2 larger than, smaller than, or equal to the proton's speed \') at point 3? Expla in.

8. Figure Q2 1.8 shows two points inside a capacitor. Let V = 0 V at the negat ive plate. a. What is the ratio V21V1 of the e lectri c potential at these two

points? Explain . b. What is the ratio £2IEJ of the elec tri c field strength at these

two points? Explain.

I mm + + + + + + +

2 + • • + + + + + + +

FIGURE 021 .8 3mm +

9. A capaci tor with plates separated by di stance d is charged to a potential difference 6.Ve . All wires and batteries are di scon­nected, then the two plates are pulled apart (with insulated han­dles) to a new separation of di stance 2d. a. Does the capacitor charge Q change as the separation

increases? If so, by what factor? If not , why not? b. Does the elec tric fi e ld strength £ change as the separation

increases? If so, by what factor? If not , why not? c. Does the potential difference 6. Vc change as the separation

increases? If so, by what factor? If nol, why not? 10. Rank in order, from most positive to most negati ve, the electric

potentials VI to Vs at points 1 to 5 in Figure Q21.10. Explain.

,. , , ,

o 2,

... --.. - ... 3

FIGURE 021 .10

, , , '4 , ., 1 nun

2 o

3 mm

FIGURE 021 .11

II . Figure Q21.11 shows two points near a positive point charge. a. What is the ratio V I/V2 of the electri c potenti als at these two

points? Ex plain . b. What is the rat io E/£2 of the electric rie ld strengths at these

two points? Ex plain . 12. Each pan of Figure 021.12 shows three points in the vic inity of

two point charges. The charges have equal mag nitudes. Rank in order, from largest to smallest, the potentials VI, V2, and VJ .

(a) (b)

• ~ ® ® 2 3 2 3

«) I · (d ) I ·

® ,. ® ® 2 ·

3· 3 ·

FIGURE 021 .12

13. a. Suppose that E = 6 throughout some reg ion of space. Can you conclude that V = 0 V in thi s region? Explain.

b. Suppose that V = 0 V throughout some reg ion of space. Can you concl ude that E = 6 in this region? Explai n.

14. Rank in order, from largest to smallest, the e lectri c fi eld strengths £1 . E2, E). and E4 at the four labeled points in Figure Q21.14. Explain.

IOV , , OV

-10 V

20V - 20 V ;--, , I , _, , -" 30 V "" ,' I ,"-30 V ",

I .. - - .. ' \ I! , ... - - " , I' , , ,1/ I , "

I I' -.. ,\ III I I ' - • \ \

I 2 . 40V I I I ! I I 1-40V , 4 I

I '''-'''II ~ \''''-'I I \ ' "" ,I, '" ... I "-;',',',,'-- , , ,

FIGURE 021.14

~ E

FIGURE 021 .15

15. Figure Q2 1.1 5 shows an elec tric fi eld diagram. Dotted lines I and 2 are two surfaces in space, not physical objects. a . Is the e lectric potenti al at point a higher than , lower than , or

equal to the e lectric potential at point b? Explain . b. Rank in order, from largest to smallest, the potenti al differ­

ences 6. Vab, 6. Vcd ' and 6. VCf' Explain. c. Is surface I an equipotential surface? What about surface 2?

Explain why or why not. 16. Figure Q21.16 shows a

negat ively charged electro­scope. The go ld leaf stands away from the rigid meta l post. Is the electric poten-

17.

tial of the leaf higher than, less than , or equal to the potential of the post? Explain . FIGURE 021 .16

- Leaf

Posl

Rank in order, from largest to smallest, the energies CUd l to CUd4 stored in each of the capac itors in Figure Q21.17. Explain .

Questions 705

FIGURE 021 .17

18. A paralle l-plate capac itor with plate separation d is connected to a battery that has potential difference 6. V lxlI" Without break­ing any of the connect ions, insu lat ing handles are used to increase the plate separation to 2d. a. Does the potential difference 6. Vc change as the separation

increases? If so, by what factor? If not, why not? b. Does the capac itance change? If so, by what factor? If not,

why not? c. Does the capac itor charge Q change? If so, by what factor?

If not, why not? 19. The gap between the capacitor plates shown in Figure Q21.19 is

partially filled with a dielectric. The capac itor was charged by a 9 V battery, then di scon nec ted from the battery. Rank in order, from smallest to largest, the electri c fi eld strengths E I , E2 , and EJ at the points labe led in the fi gure, as well as the fie ld strength E4 between the plates if the dielectr ic is removed. Explain.

+ + + + + + + + • • -+ 3 -+ + + + +

FIGURE 021 .19 +

Multiple-Choice Questions

20. I A 1.0 nC posi ti ve point charge is located at point A in Figure Q21.20. The electri c poten­ti al at point B is A. 9.0V B. 9.0 sin 30° V C. 9.0cos300 V D. 9.0 tan 30° V

2 1. I For the capac itor shown in Figure Q2 1.2 1, the potent ial di fference 6. Vab between points a and b is A. 6V B. 6sin30° V C. 6cos300 V D. 6/s in 300 V

•

k-om B

A 300

(!)- - - -

FIGURE 021.20

E. 6/cos 300 V FIGURE 021 .21

22. I The electri c potential is 300 V at x = 0 cm, is - 100 V at x = 5 cm, and varies linearl y with x. If a positi ve charge is re leased from res t at x = 2.5 cm, and is subject only to electr ic forces, the charge wi ll A. Move to the right. B. Move to the left. C. Stay atx = 2.5 cm. D. Not enough informat ion to tell.

706 CHAPTER 21 Electric Potential

Questions 23 through 27 refer to Figure Q21.23, which shows equ ipoten­tial lines in a region of space. The equipotent ial lines are spaced by the same difference in poten­tial , and several of the pOientials are given. 23. I What is the

potential at point c? A. - 400 V B. -350 V C. - IOOV D. 350 V

o

, , , ,.

FI G URE 021 .23

ov

E. 400V

, , loqv" ..

b -- . - --

24. I At which point, a, b, or c, is the magnitude of the electric field the greatest?

25. I What is the approximate magnitude of the eJectric field at point c? A. 100 Vim D. 1500 Vim

B.300V/m C. 800 Vim E. 3000 Vim

26. I The direction of the e lectr ic field at point b is closest to which direction? A. Right B. Up C. Left D. Down

27. II A + 10 nC c harge is moved from point c to point a. How much work is required in order to do thi s? A. 3.5 X 10-6 J B. 4.0 X lO-6 J C. 3.5 X 10-3 J D. 4.0 X 10- 3 J E. 3.5 J

PROBLEMS

Section 21.1 Electric Potential E nergy and E lectric Potential

Section 21.2 Sources of Electric Potential

I . III Moving a charge from point A, where the potential is 300 V, to po int B, where the potential is ISO V, takes 4.5 X 10-1 J of work. What is the value of the charge?

2. 1111 The graph in Figure P21.2 shows the electric potential energy as a function of separation for two point charges. If one charge is +0.44 nC, what is the other charge?

012 3 45678910 o r (em)

- 2

- 4

- 6

FIGURE P21 .2

3. II I It takes 3.0,uJ of work to move a 15 nC c harge from point A to B. It takes -5.0,uJ of work to move the charge from C to B. What is the potential difference Vc - VA?

4. I A 20 nC charge is moved from a point where V = 150 V to a point where V = - 50 V. How much work is done by the force that moves the c harge?

28. I A bug zapper consists of two metal plates connected to a high- vo ltage power supply. The vo ltage between the plates is set to g ive an e lectric field s lightly less than I X 106 VIm. When a bug nies between the two plates, it increases the field enough to initiate a spark that inc inerates the bug. If a bug zap­per has a 4000 V power supply, what is the approximate separa­tion between the plates? A. 0.05 em B. 0.5 em C. 5 em D. 50cm

29. I An atom of helium and one of argon are s ingly ion ized-one elec tron is removed from each. The two ions are then acceler­ated from rest by the elecuic field between two plates with a potential difference of ISO V. After accelerating from one plate to the other, A. The helium ion has more kinetic energy. 8. The argon ion has more kinetic energy. C. Both ions have the same kinetic energy. D. There is not e nough information to say which ion has more

kinetic energy. 30. II The dipole moment of the heart is shown Bl0 at a particular instant in Figure Q2 1.30.

Which of the foUowing potentiaJ differences will have the largest positive val ue? A. VI - V2

8 . VI - V3 C. V2 - VI

D. V) - VI FIGURE 021.30

5. I At one point in space , the electric pote ntial energy of a 15 nC charge is 45 ,uJ. a. What is the electric potential at this point? b. If a 25 nC charge were placed at this point, what would its

electric potential e nergy be?

Section 21.3 Electric Potential and Conservation of Energy

6. I An electron has been accelerated from rest through a poten­tial difference of 1000 V. a. What is its kinetic energy, in elec tron volts? b. What is its kinetic energy, in joules? c. What is its speed?

7. I A proton has been accelerated from rest through a potential difference of - 1000 V. a. What is its kinetic energy, in electron volts? b. What is its kinetic energy, in joules? c. What is its speed?

8. 11 111 What potential difference is needed to accelerate a I-Ie+ ion (charge +e, mass 4 u) from rest to a speed of 1.0 X lOb m/s?

9. II An electron with an initial speed of 500,000 mls is brought to rest by an electric field. a. Did the electron move inlO a region of higher po tentiaJ or

lower potential? b. What was the potentiaJ difference that stopped the electron? c. What was the initial kinetic e nergy of the electron, in elec­

tron volts?

10. III A pro to n with an initi al speed of 800,000 mls is brought to rest by an electric field. a. Did the proton move into a region of higher poten ti al or

lower po ten ti aJ? b. What was the poten tial difference that stopped the proton? c . What was the initial kinetic energy of the proton, in elecU"on

volts?

Section 21.4 Calculating the Electric Potential

11. II The electric potential at a point that is halfway between two ident ical charged particles is 300 V. What is the potential at a point that is 25% ofthc way from one particlc to the other?

12. II A 2.0 cm X 2.0 cm parallel-p late capacitor has a 2.0 mm spac ing. The elcctric ficld stre ngth ins ide the capac itor is 1.0 X 10' Vim. a. What is the potenti al difference across the capac itor? b. How much charge is on each plate?

L3. III Two 2.00 cm X 2.00 cm plates that form a parallel-plate capaci tor are charged to ± 0.708 nCo What are the electric field sU"ength inside and the potential difference across the capacitor if the spacing betwcen the plates is (a) 1.00 mm and (b) 2.00 nun?

14. J a. In Figure P2 1. 14, which capaci tor plate, le ft or right, is the positive plate?

b. What is the electric field strength inside the capacitor? c. What is the potential energy of a proton at the midpoint of

the capacitor?

ov 3.0mm

300 v

, , , ,

FIGURE P21.14 looV 200V

IS. A +25 nC charge is at the origin.

16.

a. What are the radii of the 1000 V, 2000 V, 3000 V, and 4000 V equipotential surfaces?

b. Draw an cquipotenti al map in charge and these four surfaces.

a. What is the electric potential at points A, B, and C in FigureP21.16?

b. What is the potent ial energy of an electron at each of these points?

c. What are the potent ial differ­ences .6.VAB and .6.VBC?

the xy-plane showing thc

" 2%Ocm "" I ,,~ - \

, " .. 10em A I B ;

,2.0nC" , - -.-,

c FIGURE P21 .16

17. III A I.O-mm-diameter ball bearing has 2.0 X 109 excess elec­trons. What is the ball bearing's potential?

18. II What is the elec tri c poten tial at the point indicated with the dot in Figure P2 1.1 8?

2.0 nC 2.0 nC @- ------ (!)

I 3.0em I

, ,

(!)--------. FIGURE P21.18 2.0 nC

Problems 707

19. II a. What is the potential difference between the terminals of an ord inary AA or AAA battery? (If you'rc not surc, find one and look at the labe l.)

b. An AA battery is con nected to a parallel-plate capac itor hav ing 4.0-cm-diameter plates spaced 2 mill apart. How much charge does the battery move from one plate to the other?

Section 21.5 Connecting Potential and Field

20. II a. In Figure P21.20, which point, A or S, has a higher elec­tri c potential?

b. What is the potential difference between A and B?

E [~~V/Ill ~ --.,...; B . --

7.0 (Ill

FIGURE P21.20

E = 1200 VIm

~h / M ' FIGURE P21 .21

2 1. III In Figure P2 1.2 1 , the elec tri c potential at point A is - 300 V. What is the potential at point B, whic h is 5.0 e m to the ri ght of A?

22. II What is the potenti al diffe re nce between Xi = 10 cm and Xf= 30 cm in thc uniform e lectric fie ld Ex = 1000 VIm?

23. I What are the magni tude and direction of the electri c field at the dot in Figurc P21.23?

I.Oem 1.0em ------,

o V 100 V 200 v FIGURE P21 .23

y

1.0 ern

1.0 em

•

45'

- 200 V

FIGURE P21 .24

200 V

ov

24. I What are the magnitude and direction of the electric field at the dot in Figurc P21.24?

Section 2 t.6 The Electrocardiogram

25 . I Onc standard location fo r a pair BID of electrodes during an EKG is

shown in Figure P21.25. The poten­ti al difference 6. V ) I = V) - V I is recorded. For each of the three instants a, b, and c during the heart's cycle shown in Figure 2 1.29. will a V31 be pos iti ve or negat ive? Explain.

26. I Thrce e lccU"odes, 1- 3, are at­BID tached to a patient as shown in

Figure P21 .26. During ventri cu lar depolarization (see Figure 2 1.29), across which pair of elec trodes is the magn itude of the potenti al dif­ference likely to be the smallest? Explain.

~L '\ ~·r FIGURE P21 .25

FIGURE P21 .26

708 CHAPTER 21 Electric Potential

Section 21.7 Capacitance and Capacitors

27. III Two 2.0 em X 2.0 em square aluminum electrodes, spaced 0.50 mm apan , are connected to a 100 V battery. a. What is the capacitance? b. What is the charge on the pos iti ve elec trode?

28. III An uncharged capac itor is connec ted to the termi nals of a 3.0 V battery, and 6.0 j.LC flows to the positive plate. The 3.0 V battery is then disconnected and replaced with a 5.0 V ballery, with the positi ve and negat ive terminals connected in the same manner as before. How much addi tional charge flows to the positive plate?

29. III You need to construct a 100 pF capacitor for a sc ience pro­ject. You plan to cut two L X L metal squares and place spacers between them. The thinnest spacers you have are 0.20 mm thick. What is the proper value of L?

30. I A switch that con nec ts a battery to a 10 j.LF capaci tor is closed. Several seconds later you find that the capac itor plates are charged to ± 30 j.Lc. What is the battery voltage?

3 1. I What is the vo ltage of a battery that will charge a 2.0 j.LF capacilor to ± 48 j.LC?

32. I Two elec trodes con nected to a 9 .0 V battery are charged to ± 45 nCo What is the capacitance of the electrodes?

33. I Initially, the switch in Figure P21.33 is open and the capac i­tor is uncharged. How much charge nows through the switch after the switch is closed?

FIGURE P21 .33

Section 21.8 Dielectrics and Capacitors

34. II A 1.2 nF parallel-plate capacitor has an air gap between its plates. It s capacitance increases by 3.0 nF when the gap is fiJled by a dielectri c. What is the dielectric constant of that dielectric?

35. III A sc ience-fair radio uses a homemade capacitor made of two 35 cm X 35 cm sheets of aluminum foil separated by a 0.2S-mm-thick sheet of paper. What is its capacitance?

36. III A 25 pF parallel-plate capacitor with an air gap between the plates is connec ted to a 100 V battery. A Teflon slab is then insefled between the plates and complete ly fill s the gap. What is the change in the charge on the positive plate when the Teflon is inserled?

37. II Two 2.0-cm-diameter electrodes with a O. IO-mm-thick sheet of Tenon between them are allached to a 9.0 V battery. Without di sconnect ing the battery, the Teflon is removed. What are the charge, potential difference, and electric field (a) before and (b) after the Tefl on is removed?

38. 11111 A capacitor with its plates separated by paper stores 4.4 nC of charge when it is connected to a paflicular baltery. An otherwise identical capacitor, but with its plates separated by Pyrex glass, is connected to the same battery. How much charge does that capaci tor store?

Section 21.9 Energy and Capacitors

39. III To what potent ial should you charge a 1.0 j.LF capaci tor to store 1.0 J of energy?

40. II A pair of 10 j.LF capac itors in a high-power laser are charged to 1.7 kY. a. What charge is stored in each capaci tor? b. How much energy is stored in each capacitor?

4 1. I Capac itor 2 has half the capac itance and twice the potenti al difference as capac itor I. What is the ratio (Udl/(Uch?

42. 1111 Two uncharged metal spheres, spaced 15.0 cm apart, have a capac itance of 24.0 pF. How Illuch work would it take to move J 2.0 nC of charge from one sphere to the other?

43. 1111 50 pJ of energy is stored in a 2.0 cm X 2.0 cm X 2.0 cm region of uniform electric field. What is the electric field strength?

General Problems

44. II A 2.0-cm-diameter paraHel-plate capacitor with a spac ing of 0.50 mm is charged to 200 V. What are (a) the total energy stored in the elec tric field and (b) the energy density?

45. III What is the change in e lectric potential energy or a 3.0 nC point charge when it is moved from point A to point B in Figure P21.45?

o A

5.0 em AVD = 50V

25.0 nC

® o B

1.5 em

FIGURE P21 .45 FIGURE P21.46

46. II What is the potential difference !:J. V34 in Figure P2 1 A6? 47. II A-50 nC charged panicle is in a uniform e lec tric field INT E = ( 10 V 1m, cast). An external force moves the particle 1.0 m

nonh, then 5.0 m east, then 2.0 m south, and fin all y 3.0 m west. The particle begins and ends its mot ion with zero veloc ity. a. How much work is done on it by the ex ternal force? b. What is the potential difference between the particle 's fi na l

and initial positions? 48. II At a di stance r from a po int charge, the electr ic potenti a l INT is 3000 V and the magni tude of the electric field is

2.0 X 10' Vim. a. What is the di stance r? b. What are the electric potent ial and the magn itude or the

elec tric fie ld at di stance rl2 from the charge? 49. What is the electric poten tial energy of the electron 111

Figure P21A9? The protons are fixed and can' t move.

®

/

1 Electron

:O.50nm \

Protons L - -- - - - - - - - -- -- -e

\

1 2.0nm

: 0.50 nrn

FIGURE P21 .49 (t)

50. 1111 Two point charges 2.0 cOl apart have an electri c potential energy - 180 j.LJ. The total charge is 30 nC. What are the two charges?

5 1. II Two positive point charges are 5.0 cm apart. If the electri c INT potential energy is 72 pJ, what is the magn itude of the force

between the two charges? 52. 11111 A +3.0 nC charge is at x = 0 cm and a - 1.0 nC charge is at

x = 4 cm. At what poi nt or points on the x·ax is is the electr ic potent ial zero?

53. 11111 A -3.0 nC charge is on the x-axi s at x = - 9 cm and a +4.0 nC charge is on the x·ax is at x = 16 cm. At what poi nt or points on the y·ax is is the electric poten ti al zero?

54. II A -2.0 nC charge and a +2.0 nC charge are located on the INT x·ax is at x = - 1.0 cm andx = + 1.0 cm, respectively.

a. At what position or positions on the x·ax is is the electri c rie ld zero?

b. At what position or posi tions on the x-axis is the electric potenti al zero?

c. Draw graphs of the electri c field strength and the electr ic potential along the x-ax is.

55. III A - 10.0 nC point charge and a + 20.0 nC point charge are INT 15.0 cm apart on the x-axis.

a. What is the electric potential at the point on the x-ax is where the electric rield is zero?

b. What are the magnitude and direct ion of the e lectri c field at the point on the x·axis, between the charges, where the clee· tric potential is zero?

56. 11111 A 2.0-mm-diameter glass bead is positi vely charged. The potenti al difference between a point 2.0 mm from the bead and a point 4.0 mm from the bead is 500 V. What is the charge on the bead?

57. I In a semiclass ical model of the hydrogen atom, the electron orbits the proton at a di stance of 0.053 nm. a. What is the e lectri c potential of the proton at the position of

the electron? b. What is the e lectron's poten tial energy?

58. What is the elec tric potential at the point indicated with the dot in Figure P21.58?

2.0 nC

~ , , , , 3.0 em I \ 3.0 em

, , , , , ,

• , , G----------3.0 em

- l.OnC - l.OnC

FIGURE P21 .S8

10nC - S.OnC

~ -2.0~n;-- I

, 4.0em I

, ,

. - - - - - - . A S.OnC

FIGURE P21 .S9

59. I a. What is the electric potenti al at point A in Figure P21 .59? b. What is the potential energy of a proton at point A?

60. III A proton 's speed as it passes poin t A is 50,000 m/s. It follows the trajectory shown in Figure P2 1.60. What is the proton's speed at point B?

FIGURE P21.60

, , , , , , , , , , , ,

~ 1 I 8 1 , ' , , ' , , ,

30 Y lOY - lOY

Problems 709

6 1. II Electric outlets have a vo ltage of approximately 120 V between the two parallel slots. Est imate the elec tri c fi e ld strength between these two slots.

62. I Est imate the magnitude of the e lectric fi eld in a ce ll mem­BID brane with a thickness of 8 nm. 63. II A Na+ ion moves from ins ide a cell , where the electri c poten­BID tial is -70 mV, to outside the ce ll , where the potential is 0 V.

What is the change in the ion 's elec tric potential energy as it moves from inside to outside the ce ll ? Does its energy increase or decrease?

64. III Suppose that a molecular ion with charge - IOe is embed­BID ded withi n the 5.0-nm-thick cell membrane of a cell with

membrane potential -70 mV. What is the e lectric force on the molecule?

65 . III The e lectr ic fi e ld strength is 50,000 VIm inside a parallel­plate capacitor with a 2.0 mm spacing. A proton is released from rest at the pos itive plate. What is the prolon's speed when it reaches the negat ive plate?

66. 11111 An alpha particle (the nucleus of a helium atom, wi th charge INT +2e and a mass four times that of a proton) and an ant iproton

(which has the same mass as a proton but charge -e) are released frolll rest a great di stance apart. They are oppos ite ly charged, so each acce lerates toward the other. What are the speeds of the two particles when they are 2.5 nm apart? Hint: You' ll need to use tlllO conservat ion laws. And what does "a great di stance" suggest about the ini tial value of r?

67. III A proton is released from rest at the pos iti ve plate of a parallel:plate capac itor. It crosses the capacitor and reaches the negati ve plate with a speed of 50,000 m/s. What will be the pro­ton 's fin al speed if the experiment is repeated with do uble the amount of charge on each capacitor plate?

68. II The e lectric fi e ld streng th is 20,000 VIm inside a parallel­plate capacitor with a 1.0 mm spacing. An electron is released from rest at the negative plate. What is the e lectron 's speed when it reaches the positi ve plate?

69. II In the early 1900s, Robert Mill ikan used small charged INT droplets of oil , suspended in an elect ri c fi e ld, to make the first

quantitat ive measurements of the e lectron 's charge. A 0.70-pm­diameter droplet of oil, hav ing a charge of +e, is suspended in mida ir between two horizontal plates of a parallel-p late capaci­tor. The upward electric Force on the droplet is exactly balanced by the downward force of gravity. The o il has a dens ity of 860 kg/m3, and the capac itor plates are 5.0 mm apart. What must the potential difference between the plates be to hold the droplet in equilibrium?

70. 1111 Two 2.0-cm-diameter di sks spaced 2.0 mm apart form a par­allel-plate capac itor. Tbe e lectric fi e ld between the disks is 5.0 X 10' Vim. a. What is the voltage across the capac itor? b. How much charge is on each di sk? c. An electron is launched from the negati ve plate. It strikes

the positive plate at a speed of 2.0 X 107 m/s. What was the electron's speed as it left the negative plate?

7 1. II In p rotorl-beam therap)', a high-energy beam of protons is BID fired at a tumor. The protons come to rest in the tumor, deposit­

ing their kinet ic energy and breaking apart the tumor 's DNA, thus killing its cell s. For one pat ient , it is desired that 0. 10 J of proton energy be deposited in a tumor. To create the proton beam, the protons are acce lerated from rest th rough a 10 MV potential difference. What is the total charge of the protons that must be fired at the tumor to deposit the required energy?

710 CHAPTER 21 Electric Potential

72. 1111 A 2.5-mm-diameter sphere is charged to - 4.5 nCo An e lec­tron fired directly at the sphere from far away comes to within 0.30 mm of the surface of the targe t before being reflected. a. What was the electron's initial speed? b. At what distance from the surface of the sphere is the elec­

tron 's speed half of its initial value? c. What is the acceleration of the electron at its nlrning point?

73. A proton is fired from far away toward the nucleus of an iron atom. Lron is e lement number 26, and the diameter of the nucleus is 9.0 fm. (1 fm = 10- 15 m.) What initial speed does the proton need to just reach the surface of the nucleus? Assume the nucleus remains aL rest.

74. II Two 10.0-cm-diameter electrodes 0.50 cm apart fonn a paral­lel-plate capac itor. The electrodes are attached by metal wires to the tenninals of a 15 V bauery. After a long time, the capacitor is di scon nected from the battery but is not di scharged. What are the charge on each electrode, the electric field strength inside the capacitor, and the potential difference between the electrodes a. Right after the battery is disconnected? b. After insulating handles are used to pull the e lectrodes

away from each other until they are 1.0 cm apart? 75. Two 10.0-cm-diameter electrodes 0.50 cm apart form a par-

allel -plate capaci tor. The electrodes are attached by metal wires to the terminals ofa IS V battery. What are the charge on each electrode, the electric field strength inside the capacitor, and the potential difference between the electrodes a. While the capacitor is attached to the battery? b. After insulating handles are used to pull the e lectrodes

away from each other until they are 1.0 cm apart? The elec­trodes remain connected to the battery during thi s process.

76. III Determine the magnitude and direction of the e lectric field at points I and 2 in Figure P21. 76.

FtGURE P21 .76

77. I Figure P21.77 shows a series of equipotential curves. a. Is the electric field strength at point A larger than , smalJer

than , or equal to the fi eld strength at point B? Explain. b. Is the e lectric fi eld strength at point C larger than , smaller

than , or equal to the field strength aI point D? Explain. c. Determine the electric field if at point D. Express yo ur

answer as a magnitude and direction.

y(cm)

2

~v , , "A ... _0 v , , , , , , , , , _-- - 1 V

, " "' ,' 8..{ I " e C_--- - 2V

T 'f -I I '0 '

FIG URE P21. 77 oto~~0'C~-'--------C-2 x (em)

78. II Figure P2U8 shows the I 25 50

electri c potential on a grid whose squares arc 5.0 em on a side. " 50 75 a. Reproduce this figure on c

• your paper, then draw the A

50 • 75 100 50 Y, 75 Y, and 100 Y equ i-

• potent ial surfaces. 0 25 50 75

b. Estimate the electric field (strength and direction) at points A, B, C, and D. 6 25 50

c. Draw the electric field vcc- Potential in V

tors at points A. B. C, and D FIGURE P21 .78

on your diagram.

50 50

1(>0 50

100 B • 50

100 50

I

50 50

79. The plates of a 3.0 nF parallel-plate capac itor are each 0.271112 in area. a. How far apart are the plates if there's air between them? b. If the plates are separated by a Teflon sheet, how thick is the

sheet? 80. 1111 The dielectric in a capac itor serves two purposes. It in ­

creases the capacitance, compared to an otherwise ident ica l capac itor with an air gap, and it increases the maximum poten­ti al difference the capacilor can support. If the electri c fi eld in a material is suffi ciently strong, the material will suddenly become ab le to conduct , creat ing a spa rk. The critica l field strength .. at which breakdown occurs, is 3.0 MV/m for air, but 60 MY/Ill for Teflon. a. A parallel-plate capac itor cons ists of two square plates ,

15 cm on a side, spaced 0.50 111m apart with only air between them. What is the maximum energy that can be stored by the capacitor?

b. What is the maximum energy that can be stored if the plates are separated by a 0.50-t11m-thick Teflon sheet?

8 1. 11 111 The flash unit in a camera uses a special circuit to "step up" INT the 3.0 V from the batteries to 300 V, which charges a capa­

c itor. The capacitor is then discharged through a fJashlamp. The di sc harge takes 10 /-Ls, and the average power dissipated in the flash lamp is 105 W. What is the capac itance of the capac itor?

In Problems 82 through 85 you are given the equation(s) used to solve a problem. For each of these, a. Write a real ist ic problem for which thi s is the con'ect equation(s). b. Finish the so lution of the problem.

(9 .0 X 10' N . m' /C' )q ,q, 82. I 90 X 10- ' J

0.030 m

q, +q,= 40nC

83. II !( 1.67 X 10-27 kg)(2.5 X 10' mI, )' + 0 =

!( 1.67 X 10-27 kg)",' +

(9.0 X 10' N • m' /C' )(2.0 X 10 ' C)(1.60 X 10 " c)

0.0010 In

(9 .0 X 10' N' m' /C' )(3.0 X 10-9 C) 84. II +

0.030 m

(9.0 X 10' N· m' /C' )(3.0 X 10-9 C)

(0.030 m) + d

R5 . II 400nC = ( IOOY)C

1200Y

(8.85 X 10- 12 F/m)(O.IO In X 0.10 m) C = ~------~~------~

d

Passage Problems

A Lightning Strike

Storm clouds build up large negati ve charges, as desc ribed in the chapter. The charges dwell in charge centers, regions of concentrated charge. Suppose a cloud has -25 C in a I .O-km-diameter spheri­cal charge center located 10 km above the ground, as ske tched in Figure P2 1.88 . The negative charge center attracts a sim­il ar amount of positi ve charge that is spread on the ground below the cloud.

The charge cen ter and the ground function as a charged capac itor, with a potential difference of app rox imately 4 X 10k V. The large e lec tri c field be-twecn these two ;;e lectrodes" may ion ize

Ikm

-25 C

IOkm

+ + + + + + +25C

the air, lead ing to a cond ucting path FIGURE P21 .88

between the cloud and the ground. Charges will fl ow along thi s con­ducting path, causing a discharge of the capac itor-a lightning strike.

Stop to Think 21.1: B I r the charge were moved From I to 2 at a con­stant speed by a hand, the force exerted by the hand would need to be

to the left, (0 oppose the ri ghtward-d irected electric force on the charge due to the source charges. Because the force due (0 the hand would be opposite the di splacement, the hand would do Ilegarive work on the charge, dec reasing its electric potential energy.

Stop to Think 21.2: C. The proton gains speed by losing potential energy. It loses potential energy by moving in the direction of decreasing elec tric potenti al.

Stop to Think 21.3: !:J. VIJ == !:J. V23 > !:J. V12 • The poten tial depends only on the distance from the charge, not the direction. !:J. VI2 = 0 because these points are at the same distance.

Problems 711

86. I What is the approx imate magni tude of the e lectric field between the charge center and the ground?

87.

A. 4X IO' Y/m B. 4X lo'Y/m C. 4X IO' Y/m D. 4X 107Y/m I Which of the curves sketched in Figure P2 1.87 best approxi­mates the shape of an equipotential drawn halfway between the charge center and the ground?

A . ........ _ _ .--"i" B. - - - - -- c. D.

FIGURE P21 .87

88. I What is the approximate capacitance of the charge cen ter + ground system? A. 6X 10 ' F B. 2X 107F C. 4X 10'P D. 8X 10'F

89. I If 12.5 C of cha.rge is transferred from the cloud to the ground in a lightning strike, what fraction of the stored energy is di ssi­pated? A. 12% B. 25% C. 50% D. 75%

90. I If the cloud transfe rs all of it s charge to the ground via sev­era l rapid lightning flashes lasting a tota l of 1 s, what is the average power? A. I GW B. 2GW C.5GW D. 10GW

Stop to Think 21.4: C. E points ;'downhill ," so V must decrease from ri ght (0 left. E is larger on the left than on the ri ght, so the equipotenti al lines must be closer together on the left.

Stop to Think 21.5: C . Capacitance is a property of the shape and pos ition of the electrodes. It does not depend on the poten ti al d iffer­ence or charge.

Stop to Think 21.6: C . The elec tr ic fi e ld is !:J. Veld. With !:J. Vc fi xed by the batte ry, in troducing the dielectric does not change E. More charge flows from the battery to compensate for the dielectric.

Stop to Think 21.7: B. The energy is 4C( !:J. vcf. !:J. Vc is constan t, but C doubles when the distance is halved.