SUMMARY. The goal of Chapter 25 has been to describe electric phenomena in terms of charges, forces, and fields.

Transcription

1 Summay 743 SUMMARY The goal of Chapte 5 has been to descibe electic phenomena in tems of chages, foces, and fields. Geneal Pinciples Coulomb s Law The foces between two chaged paticles q and q sepaated by distance ae F on = F on = K 0 q 0 0 q 0 These foces ae an action/eaction pai diected along the line joining the paticles. The foces ae epulsive fo two like chages, attactive fo two opposite chages. F on q q F on The net foce on a chage is the sum of the foces fom all othe chages. The unit of chage is the coulomb (C). The electostatic constant is K = 9.0 * 0 9 N m /C. Impotant Concepts The Chage Model Thee ae two kinds of chage, positive and negative. Fundamental chages ae potons and electons, with chage {e whee e =.60 * 0-9 C. Objects ae chaged by adding o emoving electons. The amount of chage is q = (N p - N e )e. An object with an equal numbe of potons and electons is neutal, meaning no net chage. Chaged objects exet electic foces on each othe. Like chages epel, opposite chages attact. The foce inceases as the chage inceases. The foce deceases as the distance inceases. Thee ae two types of mateial, insulatos and conductos. Chage emains fixed in o on an insulato. Chage moves easily though o along conductos. Chage is tansfeed by contact between objects. Chaged objects attact neutal objects. Chage polaizes metal by shifting the electon sea. Chage polaizes atoms, ceating electic dipoles. The polaization foce is always an attactive foce. Extenal chages Net foce Net foce Polaized neutal objects The Field Model Chages inteact with each othe via the electic field E u. Chage A altes the space aound it by ceating an electic field. A The field is the agent that exets a foce. The foce on chage q B is F u on B = q B E u. An electic field is identified and measued in tems of the foce on a pobe chage q: E u = F u on q/q F on B The electic field exists at all points in space. An electic field vecto shows the field only at one point, the point at the tail of the vecto. The electic field of a point chage is A B E u = q 4pP 0 n

2 744 chapte 5. Electic Chages and Foces Tems and Notation neutal chaging chage model chage, q o Q like chages opposite chages dischaging conducto insulato electon cloud fundamental unit of chage, e chage quantization ionization law of consevation of chage sea of electons ion coe cuent chage caies electostatic equilibium gounded chage polaization polaization foce electic dipole chaging by induction Coulomb s law electostatic constant, K point chage coulomb, C pemittivity constant, P 0 field electic field, E u field model souce chage electic field stength, E field diagam CONCEPTU AL QUESTIO NS. Can an insulato be chaged? If so, how would you chage an insulato? If not, why not?. Can a conducto be chaged? If so, how would you chage a conducto? If not, why not? 3. Fou lightweight balls A, B, C, and D ae suspended by theads. Ball A has been touched by a plastic od that was ubbed with wool. When the balls ae bought close togethe, without touching, the following obsevations ae made: Balls B, C, and D ae attacted to ball A. Balls B and D have no effect on each othe. Ball B is attacted to ball C. What ae the chage states (glass, plastic, o neutal) of balls A, B, C, and D? Explain. 4. Chaged plastic and glass ods hang by theads. a. An object epels the plastic od. Can you pedict what it will do to the glass od? If so, what? If not, why not? b. A diffeent object attacts the plastic od. Can you pedict what it will do to the glass od? If so, what? If not, why not? 5. A lightweight metal ball hangs by a thead. When a chaged od is held nea, the ball moves towad the od, touches the od, then quickly flies away fom the od. Explain this behavio. 6. Suppose thee exists a thid type of chage in addition to the two types we ve called glass and plastic. Call this thid type X chage. What expeiment o seies of expeiments would you use to test whethe an object has X chage? State clealy how each possible outcome of the expeiments is to be intepeted. 7. A negatively chaged electoscope has sepaated leaves. a. Suppose you bing a negatively chaged od close to the top of the electoscope, but not touching. How will the leaves espond? Use both chage diagams and wods to explain. b. How will the leaves espond if you bing a positively chaged od close to the top of the electoscope, but not touching? Use both chage diagams and wods to explain. 8. The two oppositely chaged metal sphees in FIGURE Q5.8 have equal quantities of chage. They ae bought into contact with a neutal metal od. What is the final chage state of each sphee and of the od? Use both chage diagams and wods to explain. A Metal FIGURE Q5.8 B A Touch FIGURE Q5.9 B 9. Metal sphee A in FIGURE Q5.9 has 4 units of negative chage and metal sphee B has units of positive chage. The two sphees ae bought into contact. What is the final chage state of each sphee? Explain. 0. Metal sphees A and B in FIGURE Q5.0 ae initially neutal and ae touching. A positively chaged od is bought nea A, but not touching. Is A now positive, negative, o neutal? Use both chage diagams and wods to explain. FIGURE Q5.0. If you bing you finge nea a lightweight, negatively chaged hanging ball, the ball swings ove towad you finge as shown in FIGURE Q5.. Use chage diagams and wods to explain this obsevation.. Repoduce FIGURE Q5. on you pape. Then daw a dot (o dots) on the figue to show the position (o positions) whee an electon would expeience no net foce. FIGURE Q5. 3. Chages A and B in FIGURE Q5.3 ae equal. Each chage exets a foce on the othe of magnitude F. Suppose the chage of B is inceased by a facto of 4, A B Finge FIGURE Q5. FIGURE Q5.3 but eveything else is unchanged. In tems of F, (a) what is the magnitude of the foce on A, and (b) what is the magnitude of the foce on B? 4. The electic field stength at one point nea a point chage is 000 N/C. What is the field stength if (a) the distance fom the point chage is doubled, and (b) the distance fom the point chage is halved? 5. The electic foce on a chaged paticle in an electic field is F. What will be the foce if the paticle s chage is tipled and the electic field stength is halved? F A B A? B? F

3 Execises and Poblems 745 EXE R CISES AND PROBLEMS Poblems labeled integate mateial fom ealie chaptes. Section 5.4 Coulomb s Law Execises Section 5. Developing a Chage Model Section 5. Chage. A plastic od is chaged to - nc by ubbing. a. Have electons been added to the od o potons emoved? Explain. b. How many electons have been added o potons emoved?. A glass od is chaged to +8.0 nc by ubbing. a. Have electons been emoved fom the od o potons added? Explain. b. How many electons have been emoved o potons added? 3. A glass od that has been chaged to + nc touches a metal sphee. Aftewad, the od s chage is +8.0 nc. a. What kind of chaged paticle was tansfeed between the od and the sphee, and in which diection? That is, did it move fom the od to the sphee o fom the sphee to the od? b. How many chaged paticles wee tansfeed? 4. A plastic od that has been chaged to -5 nc touches a metal sphee. Aftewad, the od s chage is -0 nc. a. What kind of chaged paticle was tansfeed between the od and the sphee, and in which diection? That is, did it move fom the od to the sphee o fom the sphee to the od? b. How many chaged paticles wee tansfeed? 5. What is the total chage of all the potons in.0 mol of He gas? 6. What is the total chage of all the electons in.0 L of liquid wate? Section 5.3 Insulatos and Conductos 7. Figue 5.8 showed how an electoscope becomes negatively chaged. The leaves will also epel each othe if you touch the electoscope with a positively chaged glass od. Use a seies of chage diagams to explain what happens and why the leaves epel each othe. 8. A plastic balloon that has been ubbed with wool will stick to a wall. a. Can you conclude that the wall is chaged? If not, why not? If so, whee does the chage come fom? b. Daw a seies of chage diagams showing how the balloon is held to the wall. 9. Two neutal metal sphees on wood stands ae touching. A negatively chaged od is held diectly above the top of the left sphee, not quite touching it. While the od is thee, the ight sphee is moved so that the sphees no longe touch. Then the od is withdawn. Aftewad, what is the chage state of each sphee? Use chage diagams to explain you answe. 0. You have two neutal metal sphees on wood stands. Devise a pocedue fo chaging the sphees so that they will have like chages of exactly equal magnitude. Use chage diagams to explain you pocedue.. You have two neutal metal sphees on wood stands. Devise a pocedue fo chaging the sphees so that they will have opposite chages of exactly equal magnitude. Use chage diagams to explain you pocedue.. Two.0 kg masses ae.0 m apat (cente to cente) on a fictionless table. Each has +0 mc of chage. a. What is the magnitude of the electic foce on one of the masses? b. What is the initial acceleation of this mass if it is eleased and allowed to move? 3. Two small plastic sphees each have a mass of.0 g and a chage of nc. They ae placed.0 cm apat (cente to cente). a. What is the magnitude of the electic foce on each sphee? b. By what facto is the electic foce on a sphee lage than its weight? 4. A small glass bead has been chaged to +0 nc. A metal ball beaing.0 cm above the bead feels a 0.08 N downwad electic foce. What is the chage on the ball beaing? 5. Two potons ae.0 fm apat. a. What is the magnitude of the electic foce on one poton due to the othe poton? b. What is the magnitude of the gavitational foce on one poton due to the othe poton? c. What is the atio of the electic foce to the gavitational foce? 6. What is the net electic foce on chage A in FIGURE EX5.6?.0 nc.0 nc 4.0 nc A B C.0 cm FIGURE EX5.6.0 cm 7. What is the net electic foce on chage B in FIGURE EX5.7? 8. Object A, which has been chaged to +4.0 nc, is at the oigin. Object B, which has been chaged to -8.0 nc, is at (x, y) = (0.0 cm,.0 cm). Detemine the electic foce on each object. Wite each foce vecto in component fom. 9. A small plastic bead has been chaged to -5 nc. What ae the magnitude and diection of the acceleation of (a) a poton and (b) an electon that is.0 cm fom the cente of the bead? Section 5.5 The Field Model FIGURE EX5.7.0 nc.0 nc.0 nc 0. What ae the stength and diection of the electic field.0 mm fom (a) a poton and (b) an electon?. The electic field at a point in space is E u = (400 in + 00 jn) N/C. a. What is the electic foce on a poton at this point? Give you answe in component fom. b. What is the electic foce on an electon at this point? Give you answe in component fom. c. What is the magnitude of the poton s acceleation? d. What is the magnitude of the electon s acceleation? A B C.0 cm.0 cm

4 746 chapte 5. Electic Chages and Foces. What magnitude chage ceates a.0 N/C electic field at a point.0 m away? 3. What ae the stength and diection of the electic field 4.0 cm fom a small plastic bead that has been chaged to -8.0 nc? 4. The electic field.0 cm fom a small object points away fom the object with a stength of 70,000 N/C. What is the object s chage? 5. What ae the stength and diection of an electic field that will balance the weight of a.0 g plastic sphee that has been chaged to -3.0 nc? 6. A + nc chage is located at the oigin. a. What ae the electic fields at the positions (x, y) = (5.0 cm, 0 cm), ( -5.0 cm, 5.0 cm), and (-5.0 cm,-5.0 cm)? Wite each electic field vecto in component fom. b. Daw a field diagam showing the electic field vectos at these points. 7. A nc chage is located at (x, y) = (.0 cm, 0 cm). What ae the electic fields at the positions (x, y) = (5.0 cm, 0 cm), (-5.0 cm, 0 cm), and (0 cm, 5.0 cm)? Wite each electic field vecto in component fom. 35. What is the foce F u on the -0 nc chage in FIGURE P5.35? Give you answe as a magnitude and an angle measued cw o ccw (specify which) fom the +x@axis. 5 nc 5.0 nc 3.0 cm FIGURE P cm 0 nc FIGURE P What is the foce F u on the -0 nc chage in FIGURE P5.36? Give you answe as a magnitude and an angle measued cw o ccw (specify which) fom the +x@axis. 37. What is the foce F u on the 5.0 nc chage in FIGURE P5.37? Give you answe as a magnitude and an angle measued cw o ccw (specify which) fom the +x@axis. 3.0 cm 0 nc 0 nc 0 nc 3.0 cm 3.0 cm.0 cm 8.0 nc 0 nc 0 nc Poblems 4.0 cm 4.0 cm 8. Pennies today ae coppe-coveed zinc, but olde pennies ae 3. g of solid coppe. What ae the total positive chage and total negative chage in a solid coppe penny that is electically neutal? 9. A.0 g plastic bead chaged to -4.0 nc and a 4.0 g glass bead chaged to +8.0 nc ae.0 cm apat (cente to cente). What ae the acceleations of (a) the plastic bead and (b) the glass bead? 30. The nucleus of a 5 Xe atom (an isotope of the element xenon with mass 5 u) is 6.0 fm in diamete. It has 54 potons and chage q = +54e. a. What is the electic foce on a poton.0 fm fom the suface of the nucleus? b. What is the poton s acceleation? Hint: Teat the spheical nucleus as a point chage. 3. Two.0 g sphees ae chaged equally and placed.0 cm apat. When eleased, they begin to acceleate at 50 m/s. What is the magnitude of the chage on each sphee? 3. Objects A and B ae both positively chaged. Both have a mass of 00 g, but A has twice the chage of B. When A and B ae placed 0 cm apat, B expeiences an electic foce of 0.45 N. a. What is the chage on A? b. If the objects ae eleased, what is the initial acceleation of A? 33. What is the foce F u on the.0 nc chage in FIGURE P5.33? Give you answe as a magnitude and a diection..0 nc.0 nc.0 cm.0 cm cm.0 nc FIGURE P nc.0 cm.0 cm cm.0 nc.0 nc FIGURE P What is the foce F u on the.0 nc chage in FIGURE P5.34? Give you answe as a magnitude and a diection. 5.0 nc FIGURE P nc 5.0 nc FIGURE P What is the foce F u on the 5.0 nc chage in FIGURE P5.38? Give you answe as a magnitude and an angle measued cw o ccw (specify which) fom the +x@axis. 39. What is the foce F u on the.0 nc chage in the middle of FIGURE P5.39 due to the fou othe chages? Give you answe in component fom. FIGURE P nc.0 nc.0 cm.0 nc.0 nc.0 nc.0 cm 40. What is the foce F u on the.0 nc chage at the bottom in FIGURE P5.40? Give you answe in component fom. 6.0 nc.0 nc.0 nc 5.0 cm 45.0 nc FIGURE P cm 6.0 nc.0 nc.0 nc.0 nc FIGURE P What is the foce F u on the.0 nc chage at the bottom in FIGURE P5.4? Give you answe in component fom. 5.0 cm cm

5 Execises and Poblems A +.0 nc chage is at the oigin and a -4.0 nc chage is at x =.0 cm. a. At what x-coodinate could you place a poton so that it would expeience no net foce? b. Would the net foce be zeo fo an electon placed at the same position? Explain. 43. The net foce on the.0 nc chage in FIGURE P5.43 is zeo. What is q?.0 cm.0 nc 3.0 cm FIGURE P5.43 q.0 nc.0 nc 3.0 cm 4.0 cm q 5.0 nc q 0 cm FIGURE P cm 44. Chage q in FIGURE P5.44 is in static equilibium. What is q? 45. A positive point chage Q is located at x = a and a negative point chage -Q is at x = -a. A positive chage q can be placed anywhee on the y-axis. Find an expession fo (F net ) x, the x-component of the net foce on q. 46. A positive point chage Q is located at x = a and a negative point chage -Q is at x = -a. A positive chage q can be placed anywhee on the x-axis. Find an expession fo (F net ) x, the x-component of the net foce on q, when (a) 0 x 0 6 a and (b) 0 x 0 7 a. 47. FIGURE P5.47 shows fou chages at the cones of a squae of side L. What is the magnitude of the net foce on q? Q L 4Q L q Q q FIGURE P5.47 FIGURE P5.48 L q Q aq 48. FIGURE P5.48 shows thee chages and the net foce on chage -q. Chage Q is some multiple a of q. What is a? 49. Two positive point chages q and 4q ae at x = 0 and x = L, espectively, and fee to move. A thid chage is placed so that the entie thee-chage system is in static equilibium. What ae the magnitude, sign, and x-coodinate of the thid chage? 50. Suppose the magnitude of the poton chage diffes fom the magnitude of the electon chage by a mee pat in 0 9. a. What would be the foce between two.0-mm-diamete coppe sphees.0 cm apat? Assume that each coppe atom has an equal numbe of electons and potons. b. Would this amount of foce be detectable? What can you conclude fom the fact that no such foces ae obseved? 5. In a simple model of the hydogen atom, the electon moves in a cicula obit of adius nm aound a stationay poton. How many evolutions pe second does the electon make? 45 L 5. You have two small,.0 g balls that have been given equal but opposite chages, but you don t know the magnitude of the chage. To find out, you place the balls distance d apat on a slippey hoizontal suface, elease them, and use a motion detecto to measue the initial acceleation of one of the balls towad the othe. Afte epeating this fo seveal diffeent sepaation distances, you data ae as follows: Distance (cm) Acceleation (m/s ) Use an appopiate gaph of the data to detemine the magnitude of the chage. 53. A 0.0 g honeybee acquies a chage of +3 pc while flying. BIO a. The eath s electic field nea the suface is typically (00 N/C, downwad). What is the atio of the electic foce on the bee to the bee s weight? b. What electic field (stength and diection) would allow the bee to hang suspended in the ai? 54. As a science poject, you ve invented an electon pump that moves electons fom one object to anothe. To demonstate you invention, you bolt a small metal plate to the ceiling, connect the pump between the metal plate and youself, and stat pumping electons fom the metal plate to you. How many electons must be moved fom the metal plate to you in ode fo you to hang suspended in the ai.0 m below the ceiling? You mass is 60 kg. Hint: Assume that both you and the plate can be modeled as point chages. 55. You have a lightweight sping whose unstetched length is 4.0 cm. Fist, you attach one end of the sping to the ceiling and hang a.0 g mass fom it. This stetches the sping to a length of 5.0 cm. You then attach two small plastic beads to the opposite ends of the sping, lay the sping on a fictionless table, and give each plastic bead the same chage. This stetches the sping to a length of 4.5 cm. What is the magnitude of the chage (in nc) on each bead? 56. An electic dipole consists of two opposite chages {q sepaated by a small distance s. The poduct p = qs is called the dipole moment. FIGURE P5.56 shows an electic dipole pependicula to an electic field E u. Find an expession in tems of p and E fo the magnitude of the toque that the electic field exets on the dipole. FIGURE P5.56 q 57. You sometimes ceate a spak when you touch a dooknob afte shuffling you feet on a capet. Why? The ai always has a few fee electons that have been kicked out of atoms by cosmic ays. If an electic field is pesent, a fee electon is acceleated until it collides with an ai molecule. It will tansfe its kinetic enegy to the molecule, then acceleate, then collide, then acceleate, collide, and so on. If the electon s kinetic enegy just befoe a collision is.0 * 0-8 J o moe, it has sufficient enegy to kick an electon out of the molecule it hits. Whee thee was one fee electon, now thee ae two! Each of these can then s q Electic dipole E

6 748 chapte 5. Electic Chages and Foces acceleate, hit a molecule, and kick out anothe electon. Then thee will be fou fee electons. In othe wods, as FIGURE P5.57 shows, a sufficiently stong electic field causes a chain eaction of electon poduction. This is called a beakdown of the ai. The cuent of moving electons is what gives you the shock, and a spak is geneated when the electons ecombine with the positive ions and give off excess enegy as a bust of light. a. The aveage distance an electon tavels between collisions is.0 mm. What acceleation must an electon have to gain.0 * 0-8 J of kinetic enegy in this distance? b. What foce must act on an electon to give it the acceleation found in pat a? c. What stength electic field will exet this much foce on an electon? This is the beakdown field stength. Note: The measued beakdown field stength is a little less than you calculated value because ou model of the pocess is a bit too simple. Even so, you calculated value is close. d. Suppose a fee electon in ai is.0 cm away fom a point chage. What minimum chage must this point chage have to cause a beakdown of the ai and ceate a spak? Atoms Oiginal electon Beakdown of the ai FIGURE P5.57 Electon knocked fom fist atom mm Electons 58. Two 5.0 g point chages on.0-m-long theads epel each othe afte being chaged to +00 nc, as shown in FIGURE P5.58. What is the angle u? You can assume that u is a small angle..0 m u 5.0 g 5.0 g 00 nc 00 nc FIGURE P5.58 u.0 m.0 m g 3.0 g q q FIGURE P m 59. Two 3.0 g point chages on.0-m-long theads epel each othe afte being equally chaged, as shown in FIGURE P5.59. What is the chage q? 60. What ae the electic fields at points,, and 3 in FIGURE P5.60? Give you answe in component fom. 6. What ae the electic fields at points and in FIGURE P5.6? Give you answe as a magnitude and diection..0 cm.0 nc.0 cm FIGURE P5.6.0 cm 6. What ae the electic fields at points,, and 3 in FIGURE P5.6? Give you answe in component fom. 63. A -0.0 nc chage is located at position (x, y) = (.0 cm,.0 cm). At what (x, y) position(s) is the electic field a. - 5,000 in N/C? b. (6,000 in - 80,500jn) N/C? c. (8,800 in +,600jn) N/C? 64. A 0.0 nc chage is located at position (x, y) = (.0 cm,.0 cm). At what (x, y) position(s) is the electic field a. - 5,000 in N/C? b. (6,000 in + 80,500jn) N/C? c. (,600 in - 8,800jn) N/C? 65. Thee.0 nc chages ae placed as shown in FIGURE P5.65. Each of these chages ceates an electic field E u q.0 nc at a point 3.0 cm in font of the middle chage. a. What ae the thee fields E u, E u, and E u 3 ceated by the thee chages? Wite you answe fo each as a vecto in component fom..0 cm 5.0 nc.0 cm FIGURE P5.6 b. Do you think that electic fields obey a pinciple of supeposition? That is, is thee a net field at this point given by E u net = E u + E u + E u 3? Use what you leaned in this chapte and peviously in ou study of foces to ague why this is o is not tue. c. If it is tue, what is E u net? 66. An electic field E u = 00,000 in N/C causes the 5.0 g point chage in FIGURE P5.66 to hang at a 0 angle. What is the chage on the ball? 0 q.0 nc q 3.0 nc FIGURE P cm.0 cm.0 cm u 3.0 cm 3 q 5.0 g 5 nc.0 g 3.0 cm FIGURE P5.66 FIGURE P5.67 FIGURE P nc 4.0 cm An electic field E u = 00,000 in N/C causes the point chage in FIGURE P5.67 to hang at an angle. What is u?

7 Execises and Poblems 749 In Poblems 68 though 7 you ae given the equation(s) used to solve a poblem. Fo each of these, a. Wite a ealistic poblem fo which this is the coect equation(s). b. Finish the solution of the poblem. 68. (9.0 * 0 9 N m /C ) * N * (.60 * 0-9 C) =.5 * 0 6 N/C (.0 * 0-6 m) 69. (9.0 * 09 N m /C )q (0.050 m) = 0.00 N 70. (9.0 * 09 N m /C )(5 * 0-9 C) 7. = 54,000 N/C a F x = * (9.0 * 09 N m /C )(.0 * 0-9 C)q (0.00 m)/sin 30 * cos 30 = 5.0 * 0-5 N a F y = 0 N Challenge Poblems 7. A.0-mm-diamete coppe ball is chaged to +50 nc. What faction of its electons have been emoved? 73. Thee 3.0 g balls ae tied to 80-cm-long theads and hung fom a single fixed point. Each of the balls is given the same chage q. At equilibium, the thee balls fom an equilateal tiangle in a hoizontal plane with 0 cm sides. What is q? 74. The identical small sphees shown in FIGURE CP5.74 ae chaged to +00 nc and -00 nc. They hang as shown in a 00,000 N/C electic field. What is the mass of each sphee? 0 50 cm 00 nc 00 nc FIGURE CP cm 0 nc.0 nc FIGURE CP The foce on the -.0 nc chage is as shown in FIGURE CP5.75. What is the magnitude of this foce? 76. In Section 5.3 we claimed that a chaged object exets a net attactive foce on an electic dipole. Let s investigate this. FIGURE CP5.76 shows a pemanent electic dipole consisting of chages +q and -q sepaated by the fixed distance s. Chage +Q is distance fom the cente of the dipole. We ll assume, as is usually the case in pactice, that s V. a. Wite an expession fo the net foce exeted on the dipole by chage +Q. b. Is this foce towad +Q o away fom +Q? Explain. c. Use the binomial appoximation ( + x) -n - nx if x V to show that you expession fom pat a can be witten F net = KqQs/ 3. d. How can an electic foce have an invese-cube dependence? Doesn t Coulomb s law say that the electic foce depends on the invese squae of the distance? Explain. FIGURE CP5.76 s q q 60 F 5.0 cm Q 30 q STOP TO THINK ANSWERS Stop to Think 5.: b. Chaged objects ae attacted to neutal objects, so an attactive foce is inconclusive. Repulsion is the only sue test. Stop to Think 5.: q e ( 3e) + q a (e) + q d (0) + q b (e) + q c (e). Stop to Think 5.3: a. The negative plastic od will polaize the electoscope by pushing electons down towad the leaves. This will patially neutalize the positive chage the leaves had acquied fom the glass od. Stop to Think 5.4: b. The two foces ae an action/eaction pai, opposite in diection but equal in magnitude. Stop to Think 5.5: c. Thee s an electic field at all points, whethe an E u vecto is shown o not. The electic field at the dot is to the ight. But an electon is a negative chage, so the foce of the electic field on the electon is to the left. Stop to Think 5.6: E b + E a + E d + E c.

8 Summay 773 SUMMARY The goal of Chapte 6 has been to lean how to calculate and use the electic field. Geneal Pinciples Souces of E u Electic fields ae ceated by chages. Two majo tools fo calculating E u ae The field of a point chage: E u = q 4pP 0 n The pinciple of supeposition Consequences of E u The electic field exets a foce on a chaged paticle: F u = qe u The foce causes acceleation: a u = (q/m)e u Tajectoies of chaged paticles ae calculated with kinematics. F F Multiple point chages Use supeposition: E u = E u + E u + E u 3 + # # # Continuous distibution of chage Divide the chage into segments Q fo which you aleady know the field. The electic field exets a toque on a dipole: t = pe sin u The toque tends to align the dipoles with the field. Find the field of each Q. Find E u by summing the fields of all Q. The summation usually becomes an integal. A citical step is eplacing Q with an expession involving a chage density (l o h) and an integation coodinate. In a nonunifom electic field, a dipole has a net foce in the diection of inceasing field stength. F Applications Electic dipole q p The electic dipole moment is p u = (qs, fom negative to positive) Field on axis: E u = u p 4pP 0 3 Field in bisecting plane: E u = - 4pP 0 s u p Infinite line of chage with linea chage density l 3 E u = 4pP 0 l, pependicula to line q Infinite plane of chage with suface chage density h E u = h P 0, pependicula to plane Sphee of chage Same as a point chage Q fo 7 R R Paallel-plate capacito The electic field inside an ideal capacito is a unifom electic field: E u = h P 0, fom positive to negative A eal capacito has a weak finge field aound it.

9 774 chapte 6. The Electic Field Tems and Notation dipole moment, p u electic field line linea chage density, l suface chage density, h unifomly chaged line of chage electode plane of chage sphee of chage paallel-plate capacito finge field unifom electic field chage-to-mass atio, q/m CONCEPTUAL QUES TIONS (a) (b). You ve been assigned the task of detemining the magnitude and diection of the electic field at a point in space. Give a step-bystep pocedue of how you will do so. List any objects you will use, any measuements you will make, and any calculations you will need to pefom. Make sue that you measuements do not distub the chages that ae ceating the field.. Repoduce FIGURE Q6. on you pape. Fo each pat, daw a dot o dots on the figue to show any position o positions (othe than infinity) whee E u = 0 u. FIGURE Q6. 3. Rank in ode, fom lagest to smallest, the electic field stengths E to E 4 at points to 4 in FIGURE Q6.3. Explain. FIGURE Q6.3 FIGURE Q A small segment of wie in FIGURE Q6.4 contains 0 nc of chage. a. The segment is shunk to one-thid of its oiginal length. What is the atio l f /l i, whee l i and l f ae the initial and final linea chage densities? b. A poton is vey fa fom the wie. What is the atio F f /F i of the electic foce on the poton afte the segment is shunk to the foce befoe the segment was shunk? c. Suppose the oiginal segment of wie is stetched to 0 times its oiginal length. How much chage must be added to the wie to keep the linea chage density unchanged? 5. An electon expeiences a foce of magnitude F when it is cm fom a vey long, chaged wie with linea chage density l. If the chage density is doubled, at what distance fom the wie will a poton expeience a foce of the same magnitude F? 6. FIGURE Q6.6 shows a hollow soda staw that has been unifomly chaged with positive chage. What is the electic field at the cente (inside) of the staw? Explain. 3 4 FIGURE Q6.6 Inside staw 7. The iegulaly shaped aea of chage in FIGURE Q6.7 has suface chage density h i. Each dimension (x and y) of the aea is educed by a facto of a. What is the atio h f /h i, whee h f is the final suface chage density? b. An electon is vey fa fom the aea. FIGURE Q6.7 What is the atio F f /F i of the electic foce on the electon afte the aea is educed to the foce befoe the aea was educed? 8. A cicula disk has suface chage density 8 nc/cm. What will the suface chage density be if the adius of the disk is doubled? 9. A sphee of adius R has chage Q. The electic field stength at distance 7 R is E i. What is the atio E f /E i of the final to initial electic field stengths if (a) Q is halved, (b) R is halved, and (c) is halved (but is still 7 R)? Each pat changes only one quantity; the othe quantities have thei initial values. 0. The ball in FIGURE Q6.0 is suspended fom a lage, unifomly chaged positive plate. It swings with peiod T. If the ball is dischaged, will the peiod incease, decease, o stay the same? Explain. FIGURE Q6.0 Mass m Chage q FIGURE Q6.. Rank in ode, fom lagest to smallest, the electic field stengths E to E 5 at the five points in FIGURE Q6.. Explain.. A paallel-plate capacito consists of two squae plates, size L * L, sepaated by distance d. The plates ae given chage {Q. What is the atio E f /E i of the final to initial electic field stengths if (a) Q is doubled, (b) L is doubled, and (c) d is doubled? Each pat changes only one quantity; the othe quantities have thei initial values. 3. A small object is eleased at point 3 in the cente of the capacito in FIGURE Q6.. Fo each situation, does the object move to the ight, to the left, o emain in place? If it moves, does it acceleate o move at constant speed? a. A positive object is eleased fom est. b. A neutal but polaizable object is eleased fom est. c. A negative object is eleased fom est

10 Execises and Poblems A poton and an electon ae eleased fom est in the cente of a capacito. a. Is the foce atio F p /F e geate than, less than, o equal to? Explain. b. Is the acceleation atio a p /a e geate than, less than, o equal to? Explain. 5. Thee chages ae placed at the cones of the tiangle in FIGURE Q6.5. The ++ chage has twice the quantity of chage of the two chages; the net chage is zeo. Is the tiangle in equilibium? If so, explain why. If not, daw the equilibium oientation. FIGURE Q6.5 EXE R CIS E S A N D PROBLEMS Poblems labeled Execises integate mateial fom ealie chaptes. Section 6. The Electic Field of Multiple Point Chages. What ae the stength and diection of the electic field at the position indicated by the dot in FIGURE EX6.? Specify the diection as an angle above o below hoizontal. 3.0 nc 3.0 nc FIGURE EX6.. What ae the stength and diection of the electic field at the position indicated by the dot in FIGURE EX6.? Specify the diection as an angle above o below hoizontal. 3. What ae the stength and diection of the electic field at the position indicated by the dot in FIGURE EX6.3? Specify the diection as an angle above o below hoizontal. 0 cm 5.0 cm 3.0 nc 5.0 cm 3.0 nc FIGURE EX cm 5.0 cm FIGURE EX6. 4. What ae the stength and diection of the electic field at the position indicated by the dot in FIGURE EX6.4? Specify the diection as an angle above o below hoizontal. 5. An electic dipole is fomed fom {.0 nc chages spaced.0 mm apat. The dipole is at the oigin, oiented along the x-axis. What is the electic field stength at the points (a) (x, y) = (0 cm, 0 cm) and (b) (x, y) = (0 cm, 0 cm)? 6. An electic dipole is fomed fom two chages, {q, spaced.0 cm apat. The dipole is at the oigin, oiented along the y-axis. 0 cm 3.0 nc 5.0 cm 3.0 nc 5.0 cm 3.0 nc 3.0 nc FIGURE EX cm 5.0 cm The electic field stength at the point (x, y) = (0 cm, 0 cm) is 360 N/C. a. What is the chage q? Give you answe in nc. b. What is the electic field stength at the point (x, y) = (0 cm, 0 cm)? Section 6.3 The Electic Field of a Continuous Chage Distibution 7. The electic field stength 0.0 cm fom a vey long chaged wie is 000 N/C. What is the electic field stength 5.0 cm fom the wie? 8. A 0-cm-long thin glass od unifomly chaged to +0 nc and a 0-cm-long thin plastic od unifomly chaged to -0 nc ae placed side by side, 4.0 cm apat. What ae the electic field stengths E to E 3 at distances.0 cm,.0 cm, and 3.0 cm fom the glass od along the line connecting the midpoints of the two ods? 9. Two 0-cm-long thin glass ods unifomly chaged to +0 nc ae placed side by side, 4.0 cm apat. What ae the electic field stengths E to E 3 at distances.0 cm,.0 cm, and 3.0 cm to the ight of the od on the left along the line connecting the midpoints of the two ods? 0. A small glass bead chaged to +6.0 nc is 4.0 cm fom a thin, unifomly chaged, 0-cm-long glass od. The bead is epelled fom the od with a foce of 840 mn. What is the total chage on the od? Section 6.4 The Electic Fields of Rings, Disks, Planes, and Sphees. Two 0-cm-diamete chaged ings face each othe, 0 cm apat. The left ing is chaged to -0 nc and the ight ing is chaged to +0 nc. a. What is the electic field E u, both magnitude and diection, at the midpoint between the two ings? b. What is the foce F u on a -.0 nc chage placed at the midpoint?. Two 0-cm-diamete chaged ings face each othe, 0 cm apat. Both ings ae chaged to +0 nc. What is the electic field stength at (a) the midpoint between the two ings and (b) the cente of the left ing? 3. Two 0-cm-diamete chaged disks face each othe, 0 cm apat. The left disk is chaged to -50 nc and the ight disk is chaged to +50 nc. a. What is the electic field E u, both magnitude and diection, at the midpoint between the two disks? b. What is the foce F u on a -.0 nc chage placed at the midpoint?

11 776 chapte 6. The Electic Field 4. Two 0-cm-diamete chaged disks face each othe, 0 cm apat. Both disks ae chaged to +50 nc. What is the electic field stength at (a) the midpoint between the two disks and (b) a point on the axis 5.0 cm fom one disk? 5. The electic field stength.0 cm fom a 0-cm-diamete metal ball is 50,000 N/C. What is the chage (in nc) on the ball? 6. A 0 cm * 0 cm hoizontal metal electode is unifomly chaged to +80 nc. What is the electic field stength.0 mm above the cente of the electode? Section 6.5 The Paallel-Plate Capacito 7. Two cicula disks spaced 0.50 mm apat fom a paallel-plate capacito. Tansfeing 3.0 * 0 9 electons fom one disk to the othe causes the electic field stength to be.0 * 0 5 N/C. What ae the diametes of the disks? 8. A paallel-plate capacito is fomed fom two 6.0-cm-diamete electodes spaced.0 mm apat. The electic field stength inside the capacito is.0 * 0 6 N/C. What is the chage (in nc) on each electode? 9. Ai beaks down when the electic field stength eaches 3.0 * 0 6 N/C, causing a spak. A paallel-plate capacito is made fom two 4.0 cm * 4.0 cm disks. How many electons must be tansfeed fom one disk to the othe to ceate a spak between the disks? Section 6.6 Motion of a Chaged Paticle in an Electic Field 0. A 0.0 g glass bead is chaged by the emoval of.0 * 0 0 electons. What electic field E u (stength and diection) will cause the bead to hang suspended in the ai?. Two.0-cm-diamete disks face each othe,.0 mm apat. They ae chaged to {0 nc. a. What is the electic field stength between the disks? b. A poton is shot fom the negative disk towad the positive disk. What launch speed must the poton have to just baely each the positive disk?. An electon in a unifom electic field inceases its speed fom.0 * 0 7 m/s to 4.0 * 0 7 m/s ove a distance of. cm. What is the electic field stength? 3. The suface chage density on an infinite chaged plane is -.0 * 0-6 C/m. A poton is shot staight away fom the plane at.0 * 0 6 m/s. How fa does the poton tavel befoe eaching its tuning point? 4. A.0-mm-diamete oil doplet (density 900 kg/m 3 ) is negatively chaged with the addition of 5 exta electons. It is eleased fom est.0 mm fom a vey wide plane of positive chage, afte which it acceleates towad the plane and collides with a speed of 3.5 m/s. What is the suface chage density of the plane? Section 6.7 Motion of a Dipole in an Electic Field 5. The pemanent electic dipole moment of the wate molecule (H O) is 6. * 0-30 C m. What is the maximum possible toque on a wate molecule in a 5.0 * 0 8 N/C electic field? 6. A point chage Q is distance fom the cente of a dipole consisting of chages {q sepaated by distance s. The chage is located in the plane that bisects the dipole. At this instant, what ae (a) the foce (magnitude and diection) and (b) the magnitude of the toque on the dipole? You can assume W s. 7. An ammonia molecule (NH 3 ) has a pemanent electic dipole moment 5.0 * 0-30 C m. A poton is.0 nm fom the molecule in the plane that bisects the dipole. What is the electic foce of the molecule on the poton? Poblems 8. What ae the stength and diection of the electic field at the position indicated by the dot in FIGURE P6.8? Give you answe (a) in component fom and (b) as a magnitude and angle measued cw o ccw (specify which) fom the positive x-axis..0 cm 5.0 nc 0 nc 9. What ae the stength and diection of the electic field at the position indicated by the dot in FIGURE P6.9? Give you answe (a) in component fom and (b) as a magnitude and angle measued cw o ccw (specify which) fom the positive x-axis. 30. What ae the stength and diection of the electic field at the position indicated by the dot in FIGURE P6.30? Give you answe (a) in component fom and (b) as a magnitude and angle measued cw o ccw (specify which) fom the positive x-axis. 0 nc 3.0 cm 4.0 cm 5.0 nc FIGURE P6.8 FIGURE P cm 0 nc 5.0 nc Q 4Q FIGURE P FIGURE P6.3 shows thee chages at the cones of a squae. Wite the electic field at point P in component fom. 3. Chages -q and +q in FIGURE P6.3 ae located at x = {a. Detemine the electic field at points to 4. Wite each field in component fom. FIGURE P6.3 a y a 3 x a q q a a a 33. Two positive chages q ae on the y-axis at y = { s. a. Find an expession fo the electic field stength at distance x on the axis that bisects the two chages. b. Fo q =.0 nc and s = 6.0 mm, evaluate E at x = 0,, 4, 6, and 0 mm nc.0 cm FIGURE P6.9 L 4.0 cm L 0 nc 0 nc P Q

12 Execises and Poblems Deive Equation 6. fo the field E u dipole in the plane that bisects an electic dipole. 35. Thee chages ae on the y-axis. Chages -q ae at y = {d and chage +q is at y = 0. a. Find an expession fo the electic field E u at a point on the x-axis. b. Veify that you answe to pat a has the expected behavio as x becomes vey small and vey lage. 36. FIGURE P6.36 is a coss section of two infinite lines of chage that extend out of the page. Both have linea chage density l. Find an expession fo the electic field stength E at height y above the midpoint between the lines. y d Lines of chage coming out of the page FIGURE P6.36 FIGURE P FIGURE P6.37 is a coss section of two infinite lines of chage that extend out of the page. The linea chage densities ae {l. Find an expession fo the electic field stength E at height y above the midpoint between the lines. 38. Two infinite lines of chage, each with linea chage density l, lie along the x- and y-axes, cossing at the oigin. What is the electic field stength at position (x, y)? 39. The electic field 5.0 cm fom a vey long chaged wie is (000 N/C, towad the wie). What is the chage (in nc) on a.0-cm-long segment of the wie? 40. FIGURE P6.40 shows a thin od of length L with total chage Q. a. Find an expession fo the electic field stength at point P on the axis of the od at distance fom the cente. b. Veify that you expession has the expected behavio if W L. c. Evaluate E at = 3.0 cm if L = 5.0 cm and Q = 3.0 nc. L FIGURE P FIGURE P6.4 shows a thin od of length L with total chage Q. Find an expession fo the electic field E u at point P. Give you answe in component fom. 4. Show that the on-axis electic field of a ing of chage has the expected behavio when z V R and when z W R. 43. A ing of adius R has total chage Q. a. At what distance along the z-axis is the electic field stength a maximum? b. What is the electic field stength at this point? 44. Chage Q is unifomly distibuted along a thin, flexible od of length L. The od is then bent into the semicicle shown in FIGURE P6.44. P L y d Lines of chage coming out of the page y x FIGURE P6.4 P x a. Find an expession fo the electic field E u at the cente of the semicicle. Hint: A small piece of ac length s spans a small angle u = s/r, whee R is the adius. b. Evaluate the field stength if L = 0 cm and Q = 30 nc. L FIGURE P6.44 Cente 45. A plastic od with linea chage density l is bent into the quate cicle shown in FIGURE P6.45. We want to find the electic field at the oigin. a. Wite expessions fo the x- and y-components of the electic field at the oigin due to a small piece of chage at angle u. b. Wite, but do not evaluate, definite integals fo the x- and y-components of the net electic field at the oigin. c. Evaluate the integals and wite E u net in component fom. 46. You ve hung two vey lage sheets of plastic facing each othe with distance d between them, as shown in FIGURE P6.46. By ubbing them with wool and silk, you ve managed to give one sheet a unifom suface chage density h = -h 0 and the othe a unifom suface chage density h = +3h 0. What ae the electic field vectos at points,, and 3? 47. Two.0-cm-diamete insulating sphees have a 6.0 cm space between them. One sphee is chaged to +0 nc, the othe to -5 nc. What is the electic field stength at the midpoint between the two sphees? 48. Two paallel plates.0 cm apat ae equally and oppositely chaged. An electon is eleased fom est at the suface of the negative plate and simultaneously a poton is eleased fom est at the suface of the positive plate. How fa fom the negative plate is the point at which the electon and poton pass each othe? 49. A paallel-plate capacito has.0 cm *.0 cm electodes with suface chage densities {.0 * 0-6 C/m. A poton taveling paallel to the electodes at.0 * 0 6 m/s entes the cente of the gap between them. By what distance has the poton been deflected sideways when it eaches the fa edge of the capacito? Assume the field is unifom inside the capacito and zeo outside the capacito. 50. An electon is launched at a 45 angle and a speed of 5.0 * 0 6 m/s fom the positive plate of the paallel-plate capacito shown in FIGURE P6.50. The electon lands 4.0 cm away. a. What is the electic field stength inside the capacito? b. What is the smallest possible spacing between the plates? R FIGURE P6.45 d 3 h h 0 h 3h 0 FIGURE P6.46 u FIGURE P6.50 y v cm x

13 778 chapte 6. The Electic Field 5. The two paallel plates in FIGURE P6.5 ae.0 cm apat and the electic field stength between them is.0 * 0 4 N/C. An electon is launched at a 45 angle fom the positive plate. What is the maximum initial speed v 0 the electon can have without hitting the negative plate? v 0 45 FIGURE P6.5.0 cm 5. A poblem of pactical inteest is to make a beam of electons tun a 90 cone. This can be done with the paallel-plate capacito shown in FIGURE P6.5. An electon with kinetic enegy 3.0 * 0-7 J entes though a small hole in the bottom plate of the capacito. a. Should the bottom plate be chaged positive o negative elative to the top plate if you want the electon to tun to the ight? Explain. b. What stength electic field is needed if the electon is to emege fom an exit hole.0 cm away fom the entance hole, taveling at ight angles to its oiginal diection? Hint: The difficulty of this poblem depends on how you choose you coodinate system. c. What minimum sepaation d min must the capacito plates have? 53. The combustion of fossil fuels poduces micon-sized paticles of soot, one of the majo components of ai pollution. The teminal speeds of these paticles ae extemely small, so they emain suspended in ai fo vey long peiods of time. Futhemoe, vey small paticles almost always acquie small amounts of chage fom cosmic ays and vaious atmospheic effects, so thei motion is influenced not only by gavity but also by the eath s weak electic field. Conside a small spheical paticle of adius, density, and chage q. A small sphee moving with speed v expeiences a dag foce F dag = 6phv, whee h is the viscosity of the ai. (This diffes fom the dag foce you leaned in Chapte 6 because thee we consideed macoscopic athe than micoscopic objects.) a. A paticle falling at its teminal speed v tem is in dynamic equilibium with no net foce. Wite Newton s fist law fo this paticle falling in the pesence of a downwad electic field of stength E, then solve to find an expession fo v tem. b. Soot is pimaily cabon, and cabon in the fom of gaphite has a density of 00 kg/m 3. In the absence of an electic field, what is the teminal speed in mm/s of a.0-mmdiamete gaphite paticle? The viscosity of ai at 0C is.8 * 0-5 kg/m s. c. The eath s electic field is typically (50 N/C, downwad). In this field, what is the teminal speed in mm/s of a.0-mmdiamete gaphite paticle that has acquied 50 exta electons? 54. A.0-mm-diamete glass sphee has a chage of +.0 nc. What speed does an electon need to obit the sphee.0 mm above the suface? 55. In a classical model of the hydogen atom, the electon obits the poton in a cicula obit of adius nm. What is the d FIGURE P6.5 Electons obital fequency? The poton is so much moe massive than the electon that you can assume the poton is at est. 56. In a classical model of the hydogen atom, the electon obits a stationay poton in a cicula obit. What is the adius of the obit fo which the obital fequency is.0 * 0 s -? 57. An electic field can induce an electic dipole in a neutal atom o molecule by pushing the positive and negative chages in opposite diections. The dipole moment of an induced dipole is diectly popotional to the electic field. That is, p u = ae u, whee a is called the polaizability of the molecule. A bigge field stetches the molecule fathe and causes a lage dipole moment. a. What ae the units of a? b. An ion with chage q is distance fom a molecule with polaizability a. Find an expession fo the foce F u ion on dipole. 58. Show that an infinite line of chage with linea chage density l exets an attactive foce on an electic dipole with magnitude F = lp/4pp 0. Assume that is much lage than the chage sepaation in the dipole. In Poblems 59 though 6 you ae given the equation(s) used to solve a poblem. Fo each of these a. Wite a ealistic poblem fo which this is the coect equation(s). b. Finish the solution of the poblem. 59. (9.0 * 0 9 N m /C ) (.0 * 0-9 C) s (0.05 m) 3 = 50 N/C 60. (9.0 * 0 9 N m /C ) (.0 * 0-7 C/m) 6. h P 0 c - z z + R d = h P * 0 m/s = (.60 * 0-9 C)E (.67 * 0-7 kg) Q E = (8.85 * 0 - C /N m )(0.00 m) Challenge Poblems = 5,000 N/C 63. You physics assignment is to figue out a way to use electicity to launch a small 6.0-cm-long plastic dink stie. You decide that you ll chage the little plastic od by ubbing it with fu, then hold it nea a long, chaged wie, as shown in FIGURE CP6.63. When you let go, the electic foce of the wie on the plastic od will shoot it away. Suppose you can unifomly chage the plastic stie to 0 nc and that the linea chage density of the long wie is.0 * 0-7 C/m. What is the net electic foce on the plastic stie if the end closest to the wie is.0 cm away? FIGURE CP6.63 l C/m.0 cm Plastic stie 6.0 cm

14 Execises and Poblems Thee 0-cm-long ods fom an equilateal tiangle in a plane. Two of the ods ae chaged to +0 nc, the thid to -0 nc. What is the electic field stength at the cente of the tiangle? 65. A od of length L lies along the y-axis with its cente at the oigin. The od has a nonunifom linea chage density l = a 0 y 0, whee a is a constant with the units C/m. a. Daw a gaph of l vesus y ove the length of the od. b. Detemine the constant a in tems of L and the od s total chage Q. Hint: This equies an integation. Think about how to handle the absolute value sign. c. Find the electic field stength of the od at distance x on the x-axis. 66. a. An infinitely long sheet of chage of width L lies in the xyplane between x = -L / and x = L /. The suface chage density is h. Deive an expession fo the electic field E u at height z above the centeline of the sheet. b. Veify that you expession has the expected behavio if z V L and if z W L. c. Daw a gaph of field stength E vesus z. 67. a. An infinitely long sheet of chage of width L lies in the xyplane between x = -L / and x = L /. The suface chage density is h. Deive an expession fo the electic field E u along the x-axis fo points outside the sheet (x 7 L /). b. Veify that you expession has the expected behavio if x W L. Hint: ln( + u) u if u V. c. Daw a gaph of field stength E vesus x fo x 7 L /. 68. One type of ink-jet pinte, called an electostatic ink-jet pinte, foms the lettes by using deflecting electodes to stee chaged ink dops up and down vetically as the ink jet sweeps hoizontally acoss the page. The ink jet foms 30@mm@diamete dops of ink, chages them by spaying 800,000 electons on the suface, and shoots them towad the page at a speed of 0 m/s. Along the way, the dops pass though two hoizontal, paallel electodes that ae 6.0 mm long, 4.0 mm wide, and spaced.0 mm apat. The distance fom the cente of the electodes to the pape is.0 cm. To fom the tallest lettes, which have a height of 6.0 mm, the dops need to be deflected upwad (o downwad) by 3.0 mm. What electic field stength is needed between the electodes to achieve this deflection? Ink, which consists of dye paticles suspended in alcohol, has a density of 800 kg/m A poton obits a long chaged wie, making.0 * 0 6 evolutions pe second. The adius of the obit is.0 cm. What is the wie s linea chage density? 70. A positon is an elementay paticle identical to an electon except that its chage is +e. An electon and a positon can otate about thei cente of mass as if they wee a dumbbell connected by a massless od. What is the obital fequency fo an electon and a positon.0 nm apat? 7. You have a summe inten position with a company that designs and builds nanomachines. An enginee with the company is designing a micoscopic oscillato to help keep time, and you ve been assigned to help him analyze the design. He wants to place a negative chage at the cente of a vey small, positively chaged metal ing. His claim is that the negative chage will undego simple hamonic motion at a fequency detemined by the amount of chage on the ing. a. Conside a negative chage nea the cente of a positively chaged ing centeed on the z-axis. Show that thee is a estoing foce on the chage if it moves along the z-axis but stays close to the cente of the ing. That is, show thee s a foce that ties to keep the chage at z = 0. b. Show that fo small oscillations, with amplitude V R, a paticle of mass m with chage - q undegoes simple hamonic motion with fequency f = qq p A 4pP 0 mr 3 R and Q ae the adius and chage of the ing. c. Evaluate the oscillation fequency fo an electon at the cente of a.0@mm@diamete ing chaged to.0 * 0-3 C. STOP TO THINK ANSWERS Stop to Think 6.: c. Fom symmety, the fields of the positive chages cancel. The net field is that of the negative chage, which is towad the chage. Stop to Think 6.: H c H b H a. All pieces of a unifomly chaged suface have the same suface chage density. Stop to Think 6.3: b, e, and h. b and e both incease the linea chage density l. Stop to Think 6.4: E a E b E c E d E e. The field stength of a chaged plane is the same at all distances fom the plane. An electic field diagam shows the electic field vectos at only a few points; the field exists at all points. Stop to Think 6.5: F a F b F c F d F e. The field stength inside a capacito is the same at all points, hence the foce on a chage is the same at all points. The electic field exists at all points whethe o not a vecto is shown at that point. Stop to Think 6.6: c. Paabolic tajectoies equie constant acceleation and thus a unifom electic field. The poton has an initial velocity component to the left, but it s being pushed back to the ight.