Review & Summary. Questions

Transcription

1 QUESTIONS 87 Reew & Summar Magnetc Fel A magnetc fel s efne n terms of the force F : actng on a test partcle wth charge q mong through the fel wth eloct : The SI unt for : : : F : q : :. s the tesla (T): T N/(A m) 4 gauss. (8-) The Hall Effect When a conuctng strp carrng a current s place n a unform magnetc fel :, some charge carrers (wth charge e) bul up on one se of the conuctor, creatng a potental fference V across the strp. The polartes of the ses ncate the sgn of the charge carrers. A Charge Partcle Crculatng n a Magnetc Fel A charge partcle wth mass m an charge magntue q mong wth eloct : perpencular to a unform magnetc fel : wll trael n a crcle.applng Newton s secon law to the crcular moton els q m, (8-5) r from whch we fn the raus r of the crcle to be r m (8-6) q. The frequenc of reoluton f, the angular frequenc, an the pero of the moton T are gen b f p T q pm. : (8-9, 8-8, 8-7) Magnetc Force on a Current-Carrng Wre A straght wre carrng a current n a unform magnetc fel eperences a sewas force F : L : :. (8-6) The force actng on a current element L : n a magnetc fel s (8-8) The recton of the length ector L : or L : s that of the current. Torque on a Current-Carrng Col A col (of area A an N turns, carrng current ) n a unform magnetc fel : wll : eperence a torque gen b F : L : :. t : m : :. (8-37) Here m : s the magnetc pole moment of the col, wth magntue m NA an recton gen b the rght-han rule. Orentaton Energ of a Magnetc Dpole The orentaton energ of a magnetc pole n a magnetc fel s U(u) m : :. (8-38) If an eternal agent rotates a magnetc pole from an ntal orentaton u to some other orentaton u f an the pole s statonar both ntall an fnall, the work W a one on the pole b the agent s W a U U f U. (8-39) Questons Fgure 8- shows three stuatons n whch a postel charge partcle moes at eloct : through a unform magnetc fel : an eperences a magnetc force F :. In each stuaton, etermne whether the orentatons of the ectors are phscall reasonable. F F F (a) (b) (c) Fgure 8- Queston. Fgure 8-3 shows a wre that carres current to the rght through a unform magnetc fel. It also shows four choces for the recton of that fel. (a) Rank the choces accorng to the magntue of the electrc potental fference that woul be set up across the wth of the wre, greatest frst. (b) For whch choce s the top se of the wre at hgher potental than the bottom se of the wre? 3 Fgure 8-4 shows a metallc, rectangular sol that s to moe at a certan spee through the unform magnetc fel :. The mensons of the sol are multples of, as shown. You hae s choces for the recton of the eloct: parallel to,, or z n e- 3 4 Wre Choces for Fgure 8-3 Queston. ther the poste or negate recton. (a) Rank the s choces accorng to the potental fference set up across the sol, greatest frst. (b) For whch choce s the front face at lower potental? 4 Fgure 8-5 shows the path of a z 3 partcle through s regons of unform magnetc fel, where the path s ether Fgure 8-4 Queston 3. a half-crcle or a quarter-crcle. Upon leang the last regon, the partcle traels between two charge, parallel plates an s eflecte towar the plate of hgher potental. What s the recton of the magnetc fel n each of the s regons? a b c f Fgure 8-5 Queston 4. e

2 88 CHAPTER 8 MAGNETIC FIELDS 5 In Moule 8-, we scusse a charge partcle mong through crosse fels wth the forces F : an F : E n opposton. We foun that the partcle moes n a straght lne (that s, nether force omnates the moton) f ts spee s gen b Eq. 8-7 ( E/). Whch of the two forces omnates f the spee of the partcle s (a) E/ an (b) E/? 6 Fgure 8-6 shows crosse unform electrc an magnetc fels E : an : an, at a certan nstant, the eloct ectors of the charge partcles lste n Table 8-3. (The ectors are not rawn to scale.) The spees gen n the table are ether less than or greater than E/ (see Queston 5). Whch partcles wll moe out of the page towar ou after the nstant shown n Fg. 8-6? E Table 8-3 Queston 6 Fgure 8-6 Queston 6. Partcle Charge Spee Partcle Charge Spee Less 6 Greater Greater 7 Less 3 Less 8 Greater 4 Greater 9 Less 5 Less Greater 7 Fgure 8-7 shows the path of an electron that passes through two regons contanng unform magnetc fels of magntues an. Its path n each regon s a half-crcle. (a) Whch fel s stronger? (b) What s the recton of each fel? (c) Is the tme spent b the electron n the : regon greater than, Fgure 8-7 Queston 7. less than, or the same as the tme spent n the : regon? 8 Fgure 8-8 shows the path of an electron n a regon of unform magnetc fel. The path conssts of two straght sectons, each between a par of unforml charge plates, an two half-crcles. Whch plate s at the hgher electrc potental n (a) Fgure 8-8 Queston 8. the top par of plates an (b) the bottom par? (c) What s the recton of the magnetc fel? 9 (a) In Checkpont 5, f the pole moment m : s rotate from orentaton to orentaton b an eternal agent, s the work one on the pole b the agent poste, negate, or zero? (b) Rank the work one on the pole b the agent for these three rotatons, greatest frst: :, : 4, : 3. Partcle rounabout. Fgure 8-9 shows paths through a regon of unform magnetc fel. One path s a straght lne; the rest are half-crcles. Table 8-4 ges the masses, charges, an spees of partcles that take these paths through the fel n the rectons shown. Whch path n the fgure correspons to whch partcle n the table? (The recton of the magnetc fel can be etermne b means of one of the paths, whch s unque.) b c Table 8-4 Queston e k j a g Fgure 8-9 Queston. Partcle Mass Charge Spee m q m q 3 m/ q 4 3m 3q 3 5 m q 6 m q 7 m 4q 8 m q 9 m q 3 m q 8 3m 3 In Fg. 8-3, a charge partcle enters a unform magnetc fel : wth spee, moes through a halfcrcle n tme T, an then leaes the fel. (a) Is the charge poste or negate? (b) Is the fnal spee of the partcle greater than, less than, Fgure 8-3 Queston. or equal to? (c) If the ntal spee ha been.5, woul the tme spent n fel : hae been greater than, less than, or equal to T? () Woul the path hae been a half-crcle, more than a half-crcle, or less than a half-crcle? Fgure 8-3 ges snapshots for three stuatons n whch a postel charge partcle passes through a unform magnetc fel :. The eloctes : of the partcle ffer n orentaton n the three snapshots but not n magntue. Rank the stuatons accorng to (a) the pero, (b) the frequenc, an (c) the ptch of the partcle s moton, greatest frst. () () Fgure 8-3 Queston. h (3) f

3 PROLEMS 89 Problems SSM Tutorng problem aalable (at nstructor s screton) n WlePLUS an WebAssgn Worke-out soluton aalable n Stuent Solutons Manual WWW Worke-out soluton s at Number of ots ncates leel of problem ffcult ILW Interacte soluton s at Atonal nformaton aalable n The Flng Crcus of Phscs an at flngcrcusofphscs.com Moule 8- Magnetc Fels an the Defnton of SSM ILW A proton traelng at 3. wth respect to the recton of a magnetc fel of strength.6 mt eperences a magnetc force of N. Calculate (a) the proton s spee an (b) ts knetc energ n electron-olts. A partcle of mass g an charge 8 mc moes through a unform magnetc fel, n a regon where the free-fall acceleraton s 9.8ĵ m/s. The eloct of the partcle s a constant î km/s, whch s perpencular to the magnetc fel. What, then, s the magnetc fel? 3 An electron that has an nstantaneous eloct of : (. 6 m/s)î (3. 6 m/s)ĵ s mong through the unform magnetc fel : (.3 T)î (.5 T)ĵ. (a) Fn the force on the electron ue to the magnetc fel. (b) Repeat our calculaton for a proton hang the same eloct. 4 An alpha partcle traels at a eloct : of magntue 55 m/s through a unform magnetc fel : of magntue.45 T. (An alpha partcle has a charge of 3. 9 C an a mass of kg.) The angle between : an : s 5. What s the magntue of (a) the force F : actng on the partcle ue to the fel an (b) the acceleraton of the partcle ue to F :? (c) Does the spee of the partcle ncrease, ecrease, or reman the same? 5 An electron moes through a unform magnetc fel gen b : î (3. )ĵ. At a partcular nstant, the electron has eloct : (.î 4.ĵ) m/s an the magnetc force actng on t s (6.4 9 N)kˆ. Fn. 6 A proton moes through a unform magnetc fel gen b : (î ĵ 3kˆ ) mt. At tme t, the proton has a eloct gen b : î ĵ (. km/s)kˆ an the magnetc force on the proton s F : (4. 7 N)î (. 7 N)ĵ. At that nstant, what are (a) an (b)? Moule 8- Crosse Fels: Dscoer of the Electron 7 An electron has an ntal eloct of (.ĵ 5.kˆ ) km/s an a constant acceleraton of (. m/s )î n a regon n whch unform electrc an magnetc fels are present. If : (4 mt)î, fn the electrc fel E :. 8 An electrc fel of.5 kv/m an a perpencular magnetc fel of.4 T act on a mong electron to prouce no net force. What s the electron s spee? 9 ILW In Fg. 8-3, an electron accelerate from rest through potental fference V. kv enters the gap between two parallel plates hang separaton. mm an potental fference V Fgure 8-3 Problem 9. V : V V.The lower plate s at the lower potental. Neglect frngng an assume that the electron s eloct ector s perpencular to the electrc fel ector between the plates. In unt-ector notaton, what unform magnetc fel allows the electron to trael n a straght lne n the gap? A proton traels through unform magnetc an electrc fels. The magnetc fel s :.5î mt. At one nstant the eloct of the proton s : ĵ m/s. At that nstant an n unt-ector notaton, what s the net force actng on the proton f the electrc fel s (a) 4.kˆ V/m, (b) 4.kˆ V/m, an (c) 4.î V/m? An on source s proucng 6 L ons, whch hae charge e an mass kg. The ons are accelerate b a potental fference of kv an pass horzontall nto a regon n whch there s a unform ertcal magnetc fel of magntue. T. Calculate the strength of the smallest electrc fel, to be set up oer the same regon, that wll allow the 6 L ons to pass through uneflecte. At tme t, an electron s sent along the poste recton of an as, through both an electrc fel E : an a magnetc fel :, wth E : recte parallel to the as. Fgure 8-33 ges the component s F net, of the net force on the electron ue to the two fels, as a functon of the electron s spee at tme t.the scale of the eloct as s set b (m/s) s. m/s. The an z components of the net force are zero at t. Fgure 8-33 Problem. Assumng, fn (a) the magntue E an (b) n unt-ector notaton. : Moule 8-3 Crosse Fels: The Hall Effect 3 A strp of copper 5 mm thck an 4.5 mm we s place n a unform magnetc fel : of magntue.65 T, wth : perpencular to the strp. A current 3 A s then sent through the strp such that a Hall potental fference V appears across the wth of the strp. Calculate V. (The number of charge carrers per unt olume for copper s electrons/m 3.) 4 A metal strp 6.5 cm long,.85 cm we, an.76 mm thck moes wth constant eloct : through a unform magnetc fel. mt recte perpencular to the strp, as shown n Fg A potental fference of 3.9 mv s measure between ponts an across the strp. Calculate the spee. 5 A conuctng rectangular sol of mensons 5. m, Fgure 8-34 Problem m, an z. m moes wth a constant eloct : (. m/s)î through a unform magnetc fel F net, ( 9 N)

4 83 CHAPTER 8 MAGNETIC FIELDS : (3. mt)ĵ (Fg. 8-35). What are the resultng (a) electrc fel wthn the sol, n unt-ector notaton, an (b) potental fference across the sol? 6 Fgure 8-35 shows a metallc block, wth ts faces parallel to coornate aes.the block s n a unform magnetc fel of magntue. T. One ege length of the block s 5 cm; the block s not rawn to z scale. The block s moe at 3. m/s z parallel to each as, n turn, an the resultng potental fference V that Fgure 8-35 Problems 5 appears across the block s measure. Wth the moton parallel to the an 6. as, V mv; wth the moton parallel to the z as, V 8 mv; wth the moton parallel to the as, V. What are the block lengths (a), (b), an (c) z? Moule 8-4 A Crculatng Charge Partcle 7 An alpha partcle can be prouce n certan raoacte ecas of nucle an conssts of two protons an two neutrons. The partcle has a charge of q e an a mass of 4. u, where u s the atomc mass unt, wth u.66 7 kg. Suppose an alpha partcle traels n a crcular path of raus 4.5 cm n a unform magnetc fel wth. T. Calculate (a) ts spee, (b) ts pero of reoluton, (c) ts knetc energ, an () the potental fference through whch t woul hae to be accelerate to achee ths energ. 8 In Fg. 8-36, a partcle moes along a crcle n a regon of unform magnetc fel of magntue 4. mt. The partcle s ether a proton or an electron (ou must ece whch). It eperences a magnetc force of magntue 3. 5 N. What are (a) the partcle s spee, (b) the raus of the crcle, an (c) the pero of the moton? 9 A certan partcle s sent nto a unform magnetc fel, wth the partcle s eloct ector perpencular to the recton of the fel. Fgure 8-37 ges the pero T of the partcle s moton ersus the nerse of the fel magntue. The ertcal as scale s set b T s 4. ns, an the horzontal as scale s set b s 5. T. What s the rato m/q of the partcle s mass to the magntue of ts charge? T (ns) T s (T ) Fgure 8-37 Problem 9. Fgure 8-36 Problem 8. An electron s accelerate from rest through potental fference V an then enters a regon of unform magnetc fel, where t s unergoes unform crcular moton. r s Fgure 8-38 ges the raus r of that moton ersus V /. The ertcal as scale s set b r s 3. mm, an the horzontal as scale s set b V / s 4. V /. What s the magntue of the magnetc fel? V / SSM An electron of knetc energ. kev crcles n a plane per- V / (V / ) s pencular to a unform magnetc Fgure 8-38 Problem. fel. The orbt raus s 5. cm. Fn (a) the electron s spee, (b) the magnetc fel magntue, (c) the crclng frequenc, an () the pero of the moton. In a nuclear eperment a proton wth knetc energ. MeV moes n a crcular path n a unform magnetc fel.what energ must (a) an alpha partcle (q e, m 4. u) an (b) a euteron (q e, m. u) hae f the are to crculate n the same crcular path? 3 What unform magnetc fel, apple perpencular to a beam of electrons mong at.3 6 m/s, s requre to make the electrons trael n a crcular arc of raus.35 m? 4 An electron s accelerate from rest b a potental fference of 35 V. It then enters a unform magnetc fel of magntue mt wth ts eloct perpencular to the fel. Calculate (a) the spee of the electron an (b) the raus of ts path n the magnetc fel. 5 (a) Fn the frequenc of reoluton of an electron wth an energ of ev n a unform magnetc fel of magntue 35. mt. (b) Calculate the raus of the path of ths electron f ts eloct s perpencular to the magnetc fel. 6 In Fg. 8-39, a charge partcle moes nto a regon of unform magnetc fel :, goes through half a crcle, an then ets that regon. The partcle s ether a proton or an electron (ou must ece whch). Fgure 8-39 Problem 6. It spens 3 ns n the regon. (a) What s the magntue of :? (b) If the partcle s sent back through the magnetc fel (along the same ntal path) but wth. tmes ts preous knetc energ, how much tme oes t spen n the fel urng ths trp? 7 A mass spectrometer (Fg. 8-) s use to separate uranum ons of mass kg an charge 3. 9 C from relate speces. The ons are accelerate through a potental fference of kv an then pass nto a unform magnetc fel, where the are bent n a path of raus. m. After traelng through 8 an passng through a slt of wth. mm an heght. cm, the are collecte n a cup. (a) What s the magntue of the (perpencular) magnetc fel n the separator? If the machne s use to separate out mg of materal per hour, calculate (b) the current of the esre ons n the machne an (c) the thermal energ prouce n the cup n. h. 8 A partcle unergoes unform crcular moton of raus 6. mm n a unform magnetc fel. The magnetc force on the partcle has a magntue of.6 7 N. What s the knetc energ of the partcle? 9 An electron follows a helcal path n a unform magnetc fel of magntue.3 T.The ptch of the path s 6. mm, an the r (mm)

5 PROLEMS 83 magntue of the magnetc force on the electron s. 5 N. 37 Estmate the total path length traele b a euteron n a What s the electron s spee? cclotron of raus 53 cm an operatng frequenc MHz urng the (entre) acceleraton process. Assume that the acceleratng 3 In Fg. 8-4, an electron wth an ntal knetc energ of 4. kev enters regon potental between the ees s 8 kv. at tme t. That regon contans a Regon 38 In a certan cclotron a proton moes n a crcle of raus unform magnetc fel recte nto the page, wth magntue. T. The electron goes through a half-crcle an then ets Regon.5 m. The magntue of the magnetc fel s. T. (a) What s the oscllator frequenc? (b) What s the knetc energ of the proton, n electron-olts? regon, heae towar regon across a gap of 5. cm. There s an electrc potental fference V V across the 39 SSM A horzontal power lne carres a current of 5 A Moule 8-6 Magnetc Force on a Current-Carrng Wre Fgure 8-4 gap, wth a polart such that the electron s Problem 3. spee ncreases unforml as t traerses the gap. Regon contans a unform magnetc fel recte out of the page, wth magntue. T. The electron goes through a halfcrcle an then leaes regon. At what tme t oes t leae? 3 A partcular tpe of funamental partcle ecas b transformng nto an electron e an a postron e. Suppose the ecang partcle s at rest n a unform magnetc fel : of magntue 3.53 mt an the e an e moe awa from the eca pont n paths lng n a plane perpencular to :. How long after the eca o the e an e colle? 3 A source njects an electron of spee.5 7 m/s nto a unform magnetc fel of magntue. 3 T. The eloct of the electron makes an angle u wth the recton of the magnetc fel. Fn the stance from the pont of njecton at whch the electron net crosses the fel lne that passes through the njecton pont. 33 SSM WWW A postron wth knetc energ. kev s projecte nto a unform magnetc fel : of magntue. T, wth ts eloct ector makng an angle of 89. wth :. Fn (a) the pero, (b) the ptch p, an (c) the raus r of ts helcal path. 34 An electron follows a helcal path n a unform magnetc fel gen b : (î 5ĵ 3kˆ ) mt. At tme t, the electron s eloct s gen b : (î 3ĵ 5kˆ ) m/s. (a) What s the angle f between : an :? The electron s eloct changes wth tme. Do (b) ts spee an (c) the angle f change wth tme? () What s the raus of the helcal path? Moule 8-5 Cclotrons an Snchrotrons 35 A proton crculates n a cclotron, begnnng appromatel at rest at the center. Wheneer t passes through the gap between ees, the electrc potental fference between the ees s V. (a) how much oes ts knetc energ ncrease wth each passage through the gap? (b) What s ts knetc energ as t completes passes through the gap? Let r be the raus of the proton s crcular path as t completes those passes an enters a ee, an let r be ts net raus, as t enters a ee the net tme. (c) what percentage oes the raus ncrease when t changes from r to r? That s, what s percentage ncrease r r r %? 36 A cclotron wth ee raus 53. cm s operate at an oscllator frequenc of. MHz to accelerate protons. (a) What magntue of magnetc fel s requre to achee resonance? (b) At that fel magntue, what s the knetc energ of a proton emergng from the cclotron? Suppose, nstea, that.57 T. (c) What oscllator frequenc s requre to achee resonance now? () At that frequenc, what s the knetc energ of an emergng proton? from south to north. Earth s magnetc fel (6. mt) s recte towar the north an nclne ownwar at 7. to the horzontal. Fn the (a) magntue an (b) recton of the magnetc force on m of the lne ue to Earth s fel. 4 A wre.8 m long carres a current of 3. A an makes an angle of 35. wth a unform magnetc fel of magntue.5 T. Calculate the magnetc force on the wre. 4 ILW A 3. g wre of length L 6. cm s suspene b a par of fleble leas n a unform magnetc fel of magntue.44 T (Fg. 8-4). What are the (a) magntue an (b) recton (left or rght) L of the current requre to remoe Fgure 8-4 Problem 4. the tenson n the supportng leas? 4 The bent wre shown n Fg. 8-4 les n a unform magnetc fel. Each straght secton s. m long an makes an angle of u 6 wth the as, an the wre carres a current of. A. What s the net mag- θ θ netc force on the wre n unt-ector notaton f the magnetc fel s gen b (a) 4.kˆ T an (b) 4.î T? 43 A sngle-turn current loop, carrng a current of 4. A, s n the Fgure 8-4 Problem 4. shape of a rght trangle wth ses 5.,, an 3 cm. The loop s n a unform magnetc fel of magntue 75. mt whose recton s parallel to the current n the 3 cm se of the loop.what s the magntue of the magnetc force on (a) the 3 cm se, (b) the 5. cm se, an (c) the cm se? () What s the magntue of the net force on the loop? 44 Fgure 8-43 shows a wre rng of raus a.8 cm that s perpencular to the general recton a of a raall smmetrc, ergng magnetc fel. The magnetc fel at the rng s eerwhere of the same θ magntue 3.4 mt, an ts recton at the rng eerwhere Fgure 8-43 Problem 44. makes an angle u wth a normal to the plane of the rng. The twste lea wres hae no effect on the problem. Fn the magntue of the force the fel eerts on the rng f the rng carres a current 4.6 ma. 45 A wre 5. cm long carres a.5 A current n the poste recton of an as through a magnetc fel : (3. mt)ĵ (. mt)kˆ. In unt-ector notaton, what s the magnetc force on the wre?

6 83 CHAPTER 8 MAGNETIC FIELDS 46 In Fg. 8-44, a metal wre of mass m 4. mg can sle 5 In Fg. 8-47, a rectangular loop carrng current les n the plane of a unform wth neglgble frcton on two horzontal parallel rals separate b stance.56 cm. The track les n a ertcal unform magnetc fel of magntue 56.3 mt. At tme t, ece G s con- loop conssts of a sngle turn of fleble con- magnetc fel of magntue.4 T. The necte to the rals, proucng a constant current 9.3 ma n the uctng wre that s wrappe aroun a fleble mount such that the mensons of the wre an rals (een as the wre moes). At t 6. ms, what are the wre s (a) spee an (b) recton of moton (left or rght)? rectangle can be change. (The total length of the wre s not change.) As ege length s are from appromatel zero to ts m 47 A. kg copper ro rests on two horzontal rals. m apart an carres a current of 5 A from one ral to the other. The coeffcent of statc frcton between ro an rals s.6. What are the (a) magntue an (b) angle (relate to the ertcal) of the smallest magnetc fel that puts the ro on the erge of slng? 48 A long, rg conuctor, lng along an as, carres a current of 5. A n the negate recton. A magnetc fel : s present, gen b : 3.î 8. ĵ, wth n meters an : n mllteslas. Fn, n unt-ector notaton, the force on the. m segment of the conuctor that les between. m an 3. m. Moule 8-7 Torque on a Current Loop 49 SSM Fgure 8-45 shows a rectangular -turn col of wre, of mensons cm b 5. cm. It carres a current of. A an s hnge Hnge along one long se. It s mounte n lne the plane, at angle u 3 to the recton of a unform magnetc fel of magntue.5 T. In unt-ector θ notaton, what s the torque actng on the col about the hnge lne? z 5 An electron moes n a crcle Fgure 8-45 Problem 49. of raus r 5.9 m wth spee.9 6 m/s. Treat the crcular path as a current loop wth a constant current equal to the rato of the electron s charge magntue to the pero of the moton. If the crcle les n a unform magnetc fel of magntue 7. mt, what s the mamum possble magntue of the torque prouce on the loop b the fel? 5 Fgure 8-46 shows a woo clner of mass m.5 kg an length L. m, wth N. turns of wre wrappe aroun t longtunall, so that the plane of the wre col contans the long central as of the clner. The clner s release on a plane nclne L m at an angle u to the horzontal, wth the plane of the col parallel to the nclne plane. If there s a ertcal unform magnetc fel of magntue.5 T, what s the least current through the col that keeps the clner from rollng own the plane? θ Fgure 8-46 Problem 5. Fgure 8-44 Problem 46. G mamum alue of appromatel 4. cm, the magntue t of the torque on the loop changes. The mamum alue of t s N m.what s the current n the loop? 53 Proe that the relaton t NA sn u hols not onl for the rectangular loop of Fg. 8-9 but also for a close loop of an shape. (Hnt: Replace the loop of arbtrar shape wth an assembl of ajacent long, thn, appromatel rectangular loops that are nearl equalent to the loop of arbtrar shape as far as the strbuton of current s concerne.) Moule 8-8 The Magnetc Dpole Moment 54 A magnetc pole wth a pole moment of magntue. J/T s release from rest n a unform magnetc fel of magntue 5 mt. The rotaton of the pole ue to the magnetc force on t s unmpee. When the pole rotates through the orentaton where ts pole moment s algne wth the magnetc fel, ts knetc energ s.8 mj. (a) What s the ntal angle between the pole moment an the magnetc fel? (b) What s the angle when the pole s net (momentarl) at rest? 55 SSM Two concentrc, crcular wre loops, of ra r. cm an r 3. cm, are locate n an plane; each carres a clockwse current of 7. A (Fg. 8-48). (a) Fn the magntue of the net magnetc r pole moment of the sstem. (b) Repeat for reerse current n r the nner loop. 56 A crcular wre loop of raus 5. cm carres a current of.6 A. It s place so that the normal to ts plane makes an angle of 4. wth a Fgure 8-48 Problem 55. unform magnetc fel of magntue. T. (a) Calculate the magntue of the magnetc pole moment of the loop. (b) What s the magntue of the torque actng on the loop? 57 SSM A crcular col of 6 turns has a raus of.9 cm. (a) Calculate the current that results n a magnetc pole moment of magntue.3 A m. (b) Fn the mamum magntue of the torque that the col, carrng ths current, can eperence n a unform 35. mt magnetc fel. 58 The magnetc pole moment of Earth has magntue 8. J/T.Assume that ths s prouce b charges flowng n Earth s molten outer core. If the raus of ther crcular path s 35 km, calculate the current the prouce. 59 A current loop, carrng a current of 5. A, s n the shape of a rght trangle wth ses 3, 4, an 5 cm.the loop s n a unform magnetc fel of magntue 8 mt whose recton s parallel to the current n the 5 cm se of the loop. Fn the magntue of (a) the magnetc pole moment of the loop an (b) the torque on the loop. Fgure 8-47 Problem 5.

7 PROLEMS Fgure 8-49 shows a current E loop ACDEFA carrng a current 5. A.The ses of the loop are parallel C D to the coornate aes shown, wth A. cm, C 3. cm, an FA z. cm. In unt-ector notaton, what s the magnetc pole moment of ths F loop? (Hnt: Imagne equal an opposte currents n the lne segment AD; A then treat the two rectangular loops ACDA an ADEFA.) Fgure 8-49 Problem 6. 6 SSM The col n Fg. 8-5 carres current. A n the recton ncate, s parallel to an z plane, has 3. turns an an area of 4. 3 m, an les n a unform magnetc fel : (.î 3.ĵ 4.kˆ ) mt. What are (a) the orentaton energ of the col n the magnetc fel an (b) the torque (n unt-ector notaton) on the col ue to the magnetc fel? (a) z Fgure 8-5 Problem 6. 6 In Fg. 8-5a, two concentrc cols, lng n the same plane, carr currents n opposte rectons. The current n the larger col s fe. Current n col can be are. Fgure 8-5b ges the net magnetc moment of the two-col sstem as a functon of. The ertcal as scale s set b m net,s. 5 A m, an the horzontal as scale s set b s. ma. If the current n col s then reerse, what s the magntue of the net magnetc moment of the two-col sstem when 7. ma? μ net ( 5 A m ) μ Fgure 8-5 Problem 6. net,s μ net,s (ma) (b) 63 A crcular loop of wre hang a raus of 8. cm carres a current of. A. A ector of unt length an parallel to the pole moment m : of the loop s gen b.6î.8ĵ. (Ths unt ector ges the orentaton of the magnetc pole moment ector.) If the loop s locate n a unform magnetc fel gen b : (.5 T)î (.3 T)kˆ, fn (a) the torque on the loop (n unt-ector notaton) an (b) the orentaton energ of the loop. 64 Fgure 8-5 ges the orentaton energ U of a magnetc pole n an eternal magnetc fel :, as a functon of angle f between the rectons of : an the pole moment. The ertcal as scale s set b U s. 4 J. The pole can be rotate about an ale wth neglgble frcton n orer to change f. Counterclockwse rotaton from f els poste alues of f, s an clockwse rotatons el negate alues. The pole s to be release at angle f wth a rotatonal knetc energ of J, so that t rotates counterclockwse. To what mamum alue of f wll t rotate? (In the language of Moule 8-3, what alue f s the turnng pont n the potental well of Fg. 8-5?) U ( 4 J) U s φ 8 U s 8 Fgure 8-5 Problem SSM ILW A wre of length 5. cm carrng a current of 4.5 ma s to be forme nto a crcular col an place n a unform magnetc fel : of magntue 5.7 mt. If the torque on the col from the fel s mamze, what are (a) the angle between : an the col s magnetc pole moment an (b) the number of turns n the col? (c) What s the magntue of that mamum torque? Atonal Problems 66 A proton of charge e an mass m enters a unform magnetc fel : î wth an ntal eloct : î ĵ. Fn an epresson n unt-ector notaton for ts eloct : at an later tme t. 67 A statonar crcular wall clock has a face wth a raus of 5 cm. S turns of wre are woun aroun ts permeter; the wre carres a current of. A n the clockwse recton. The clock s locate where there s a constant, unform eternal magnetc fel of magntue 7 mt (but the clock stll keeps perfect tme). At eactl : P.M., the hour han of the clock ponts n the recton of the eternal magnetc fel. (a) After how man mnutes wll the mnute han pont n the recton of the torque on the wnng ue to the magnetc fel? (b) Fn the torque magntue. 68 A wre lng along a as from to.5 m carres a current of. ma n the negate recton of the as. The wre full les n a nonunform magnetc fel that s gen b : (.3 T/m)î (.4 T/m)ĵ. In unt-ector notaton, what s the magnetc force on the wre? 69 Atom of mass 35 u an atom of mass 37 u are both sngl onze wth a charge of e. After beng ntrouce nto a mass spectrometer (Fg. 8-) an accelerate from rest through a potental fference V 7.3 kv, each on follows a crcular path n a unform magnetc fel of magntue.5 T. What s the stance between the ponts where the ons strke the etector? 7 An electron wth knetc energ.5 kev mong along the poste recton of an as enters a regon n whch a unform electrc fel of magntue kv/m s n the negate recton of the as. A unform magnetc fel : s to be set up to keep the electron mong along the as, an the recton of : s to be chosen to mnmze the requre magntue of :. In unt-ector notaton, what : shoul be set up? 7 Phscst S. A. Gousmt ese a metho for measurng the mass of hea ons b tmng ther pero of reoluton n a known magnetc fel. A sngl charge on of one makes 7. re n a 45. mt fel n.9 ms. Calculate ts mass n atomc mass unts.

8 834 CHAPTER 8 MAGNETIC FIELDS 7 A beam of electrons whose knetc energ s K emerges from a thn-fol wnow at the en of an accelerator tube. A metal plate at stance from ths wnow s perpencular to the recton of the emergng beam (Fg. 8-53). (a) Show that we can preent the beam from httng the plate f we appl a unform magnetc fel such that n whch m an e are the electron mass an charge. (b) How shoul : be orente? 73 SSM At tme t, an electron wth knetc energ kev moes through n the poste recton of an as that s parallel to the horzontal component of Earth s magnetc fel :. The fel s ertcal component s ownwar an has magntue 55. mt. (a) What s the magntue of the electron s acceleraton ue to :? (b) What s the electron s stance from the as when the electron reaches coornate cm? 74 A partcle wth charge. C moes through a unform magnetc fel. At one nstant the eloct of the partcle s (.î 4.ĵ 6.kˆ ) m/s an the magnetc force on the partcle s (4.î ĵ kˆ ) N. The an components of the magnetc fel are equal. What s :? 75 A proton, a euteron (q e, m. u), an an alpha partcle (q e, m 4. u) all hang the same knetc energ enter a regon of unform magnetc fel :, mong perpencular to :. What s the rato of (a) the raus r of the euteron path to the raus r p of the proton path an (b) the raus r a of the alpha partcle path to r p? 76 anbrge s mass spectrometer, shown n Fg. 8-54, separates ons hang the same eloct. The ons, after enterng through slts, S an S, pass through a eloct selector compose of an electrc fel prouce b the charge plates P an P, an a magnetc fel : perpencular to the electrc fel an the on path. The ons that then pass uneate through the crosse E : an : fels enter nto a regon where a secon magnetc fel ests, where the are mae to follow crcular paths. A photographc plate (or a moern etector) regsters ther arral. Show that, for the ons, q/m E/r, where r s the raus of the crcular orbt. : A mk e, Fol wnow Electron beam Plate Tube Fgure 8-53 Problem 7. Plate 77 SSM In Fg. 8-55, an electron moes at spee m/s along an as through unform electrc an magnetc fels. The magnetc fel : s recte nto the page an has Fgure 8-55 Problem 77. magntue 5. T. In unt-ector notaton, what s the electrc fel? 78 (a) In Fg. 8-8, show that the rato of the Hall electrc fel magntue E to the magntue E C of the electrc fel responsble for mong charge (the current) along the length of r S S + P P' ' Fgure 8-54 Problem 76. the strp s E E C ner, where r s the resstt of the materal an n s the number enst of the charge carrers. (b) Compute ths rato numercall for Problem 3. (See Table 6-.) 79 SSM A proton, a euteron (q e, m. u), an an alpha partcle (q e, m 4. u) are accelerate through the same potental fference an then enter the same regon of unform magnetc fel :, mong perpencular to :. What s the rato of (a) the proton s knetc energ K p to the alpha partcle s knetc energ K a an (b) the euteron s knetc energ K to K a? If the raus of the proton s crcular path s cm, what s the raus of (c) the euteron s path an () the alpha partcle s path? 8 An electron s mong at 7. 6 m/s n a magnetc fel of strength 83. mt. What s the (a) mamum an (b) mnmum magntue of the force actng on the electron ue to the fel? (c) At one pont the electron has an acceleraton of magntue m/s. What s the angle between the electron s eloct an the magnetc fel? 8 A 5. mc partcle moes through a regon contanng the unform magnetc fel î mt an the unform electrc fel 3ĵ V/m. At a certan nstant the eloct of the partcle s (7î ĵ 7.kˆ ) km/s. At that nstant an n unt-ector notaton, what s the net electromagnetc force (the sum of the electrc an magnetc forces) on the partcle? 8 In a Hall-effect eperment, a current of 3. A sent lengthwse through a conuctor. cm we, 4. cm long, an mm thck prouces a transerse (across the wth) Hall potental fference of mv when a magnetc fel of.5 T s passe perpencularl through the thckness of the conuctor. From these ata, fn (a) the rft eloct of the charge carrers an (b) the number enst of charge carrers. (c) Show on a agram the polart of the Hall potental fference wth assume current an magnetc fel rectons, assumng also that the charge carrers are electrons. 83 SSM A partcle of mass 6. g moes at 4. km/s n an plane, n a regon wth a unform magnetc fel gen b 5.î mt. At one nstant, when the partcle s eloct s recte 37 counterclockwse from the poste recton of the as, the magnetc force on the partcle s.48kˆ N. What s the partcle s charge? 84 A wre lng along an as from to. m carres a current of 3. A n the poste recton. The wre s mmerse n a nonunform magnetc fel that s gen b : (4. T/m ) î (.6 T/m ) ĵ. In unt-ector notaton, what s the magnetc force on the wre? 85 At one nstant, : (.î 4.ĵ 6.kˆ ) m/s s the eloct of a proton n a unform magnetc fel : (.î 4.ĵ 8.kˆ ) mt. At that nstant, what are (a) the magnetc force F : actng on the proton, n unt-ector notaton, (b) the angle between : an F :, an (c) the angle between : an :? 86 An electron has eloct : (3î 4ĵ) km/s as t enters a unform magnetc fel : 6î mt. What are (a) the raus of the helcal path taken b the electron an (b) the ptch of that path? (c) To an obserer lookng nto the magnetc fel regon from the entrance pont of the electron, oes the electron spral clockwse or counterclockwse as t moes?

9 PROLEMS Fgure 8-56 shows a homopolar generator, whch has a sol conuctng sk as rotor an whch s rotate b a motor (not shown). Conuctng brushes connect ths emf ece to a crcut through whch the ece res current. The ece can prouce a greater emf than wre loop rotors because the can spn at a much hgher angular spee wthout rupturng. The sk has raus R.5 m an rotaton frequenc f 4 Hz, an the ece s n a unform magnetc fel of magntue 6. mt that s perpencular to the sk. As the sk s rotate, conucton electrons along the conuctng path (ashe lne) are force to moe through the magnetc fel. (a) For the ncate rotaton, s the magnetc force on those electrons up or own n the fgure? (b) Is the magntue of that force greater at the rm or near the center of the sk? (c) What s the work per unt charge one b that force n mong charge along the raal lne, between the rm an the center? () What, then, s the emf of the ece? (e) If the current s 5. A, what s the power at whch electrcal energ s beng prouce? rush Conuctng path Fgure 8-56 Problem 87. rush 88 In Fg. 8-57, the two ens of a U-shape wre of mass m. g an length L. cm are mmerse n mercur (whch s a conuctor).the wre s n a unform fel of magntue. T. A swtch (unshown) s rapl close an then reopene, senng a pulse of current through the wre, whch causes the wre to jump upwar. If jump heght h 3. m, how much charge was n the pulse? Assume that the uraton of the pulse s much less than the tme of flght. Conser the efnton of mpulse (Eq. 9-3) an ts relatonshp wth momentum (Eq. 9-3). Also conser the relatonshp between charge an current (Eq. 6-). Hg 89 In Fg. 8-58, an electron of mass m, charge e, an low (neglgble) spee enters the regon between V two plates of potental fference V an plate separaton, ntall heae rectl towar the top plate. A unform magnetc fel of Fgure 8-58 Problem 89. magntue s normal to the plane of the fgure. Fn the mnmum alue of such that the electron wll not strke the top plate. 9 A partcle of charge q moes n a crcle of raus r wth spee. Treatng the crcular path as a current loop wth an aerage current, fn the mamum torque eerte on the loop b a unform fel of magntue. 9 In a Hall-effect eperment, epress the number enst of charge carrers n terms of the Hall-effect electrc fel magntue E, the current enst magntue J, an the magnetc fel magntue. 9 An electron that s mong through a unform magnetc fel has eloct : (4 km/s)î (35 km/s)ĵ when t eperences a force F : (4. fn)î (4.8 fn)ĵ ue to the magnetc fel. If, calculate the magnetc fel :. L Fgure 8-57 Problem 88. m