Magnetic Fields. chapter

Transcription

1 chpter 29 Mgnetic Fields 29.1 Mgnetic Fields nd Forces 29.2 Motion of Chrged Prticle in Uniform Mgnetic Field 29.3 Applictions noling Chrged Prticles Moing in Mgnetic Field 29.4 Mgnetic Force Acting on Current-Crrying Conductor 29.5 Torque on Current Loop in Uniform Mgnetic Field 29.6 The Hll Effect Mny historins of science eliee tht the compss, which uses mgnetic needle, ws used in Chin s erly s the 13th century BC, its inention eing of Aric or ndin origin. The erly Greeks knew out mgnetism s erly s 800 BC. They discoered tht the stone mgnetite (Fe 3 O 4 ) ttrcts pieces of iron. Legend scries the nme mgnetite to the shepherd Mgnes, the nils of whose shoes nd the tip of whose stff stuck fst to chunks of mgnetite while he pstured his flocks. n 1269, Pierre de Mricourt of Frnce found tht the directions of needle ner sphericl nturl mgnet formed lines tht encircled the sphere nd pssed through two points dimetriclly opposite ech other, which he clled the poles of the mgnet. usequent experiments showed tht eery mgnet, regrdless of its shpe, hs two poles, clled north (N) nd south () poles, tht exert forces on other mgnetic poles similr to the wy electric chrges exert forces on one nother. Tht is, like poles (N N or ) repel ech other, nd opposite poles (N ) ttrct ech other. An engineer performs test on the electronics ssocited with one of the superconducting mgnets in the Lrge Hdron Collider t the Europen Lortory for Prticle Physics, operted y the Europen Orgniztion for Nucler Reserch (CERN). The mgnets re used to control the motion of chrged prticles in the ccelertor. We will study the effects of mgnetic fields on moing chrged prticles in this chpter. (CERN) 829

2 830 CHAPTER 29 Mgnetic Fields North Wind Picture Archies Hns Christin Oersted Dnish Physicist nd Chemist ( ) Oersted is est known for osering tht compss needle deflects when plced ner wire crrying current. This importnt discoery ws the first eidence of the connection etween electric nd mgnetic phenomen. Oersted ws lso the first to prepre pure luminum. The poles receied their nmes ecuse of the wy mgnet, such s tht in compss, ehes in the presence of the Erth s mgnetic field. f r mgnet is suspended from its midpoint nd cn swing freely in horizontl plne, it will rotte until its north pole points to the Erth s geogrphic North Pole nd its south pole points to the Erth s geogrphic outh Pole. 1 n 1600, Willim Gilert ( ) extended de Mricourt s experiments to riety of mterils. He knew tht compss needle orients in preferred directions, so he suggested tht the Erth itself is lrge, permnent mgnet. n 1750, experimenters used torsion lnce to show tht mgnetic poles exert ttrctie or repulsie forces on ech other nd tht these forces ry s the inerse squre of the distnce etween intercting poles. Although the force etween two mgnetic poles is otherwise similr to the force etween two electric chrges, electric chrges cn e isolted (witness the electron nd proton), wheres single mgnetic pole hs neer een isolted. Tht is, mgnetic poles re lwys found in pirs. All ttempts thus fr to detect n isolted mgnetic pole he een unsuccessful. No mtter how mny times permnent mgnet is cut in two, ech piece lwys hs north nd south pole. 2 The reltionship etween mgnetism nd electricity ws discoered in 1819 when, during lecture demonstrtion, Hns Christin Oersted found tht n electric current in wire deflected nery compss needle. 3 n the 1820s, further connections etween electricity nd mgnetism were demonstrted independently y Frdy nd Joseph Henry ( ). They showed tht n electric current cn e produced in circuit either y moing mgnet ner the circuit or y chnging the current in nery circuit. These osertions demonstrte tht chnging mgnetic field cretes n electric field. Yers lter, theoreticl work y Mxwell showed tht the reerse is lso true: chnging electric field cretes mgnetic field. This chpter exmines the forces tht ct on moing chrges nd on current-crrying wires in the presence of mgnetic field. The source of the mgnetic field is descried in Chpter Mgnetic Fields nd Forces n our study of electricity, we descried the interctions etween chrged ojects in terms of electric fields. Recll tht n electric field surrounds ny electric chrge. n ddition to contining n electric field, the region of spce surrounding ny 1 The Erth s geogrphic North Pole is mgneticlly south pole, wheres the Erth s geogrphic outh Pole is mgneticlly north pole. Becuse opposite mgnetic poles ttrct ech other, the pole on mgnet tht is ttrcted to the Erth s geogrphic North Pole is the mgnet s north pole nd the pole ttrcted to the Erth s geogrphic outh Pole is the mgnet s south pole. 2 There is some theoreticl sis for speculting tht mgnetic monopoles isolted north or south poles my exist in nture, nd ttempts to detect them re n ctie experimentl field of inestigtion. 3 The sme discoery ws reported in 1802 y n tlin jurist, Gin Domenico Romgnosi, ut ws oerlooked, proly ecuse it ws pulished in n oscure journl.

3 29.1 Mgnetic Fields nd Forces 831 moing electric chrge lso contins mgnetic field. A mgnetic field lso surrounds mgnetic sustnce mking up permnent mgnet. Historiclly, the symol B hs een used to represent mgnetic field, nd we use this nottion in this ook. The direction of the mgnetic field B t ny loction is the direction in which compss needle points t tht loction. As with the electric field, we cn represent the mgnetic field y mens of drwings with mgnetic field lines. Actie Figure 29.1 shows how the mgnetic field lines of r mgnet cn e trced with the id of compss. Notice tht the mgnetic field lines outside the mgnet point wy from the north pole nd towrd the south pole. One cn disply mgnetic field ptterns of r mgnet using smll iron filings s shown in Figure When we spek of compss mgnet hing north pole nd south pole, it is more proper to sy tht it hs north-seeking pole nd south-seeking pole. This wording mens tht the north-seeking pole points to the north geogrphic pole of the Erth, wheres the south-seeking pole points to the south geogrphic pole. Becuse the north pole of mgnet is ttrcted towrd the north geogrphic pole of the Erth, the Erth s south mgnetic pole is locted ner the north geogrphic pole nd the Erth s north mgnetic pole is locted ner the south geogrphic pole. n fct, the configurtion of the Erth s mgnetic field, pictured in Figure 29.3 (pge 832), is ery much like the one tht would e chieed y urying gigntic r mgnet deep in the Erth s interior. f compss needle is supported y erings tht llow it to rotte in the erticl plne s well s in the horizontl plne, the needle is horizontl with respect to the Erth s surfce only ner the equtor. As the compss is moed northwrd, the needle rottes so tht it points more nd more towrd the Erth s surfce. Finlly, t point ner Hudson By in Cnd, the north pole of the needle points directly downwrd. This site, first found in 1832, is considered to e the loction of the south mgnetic pole of the Erth. t is pproximtely mi from the Erth s geogrphic North Pole, nd its exct position ries slowly with time. imilrly, the north mgnetic pole of the Erth is out mi wy from the Erth s geogrphic outh Pole. Although the Erth s mgnetic field pttern is similr to the one tht would e set up y r mgnet deep within the Erth, it is esy to understnd why the source of this mgnetic field cnnot e lrge msses of permnently mgnetized mteril. The Erth does he lrge deposits of iron ore deep eneth its surfce, ut the high tempertures in the Erth s core preent the iron from retining ny permnent mgnetiztion. cientists consider it more likely tht the source of the Erth s mgnetic field is conection currents in the Erth s core. Chrged ions or electrons circulting in the liquid interior could produce mgnetic field just like N ACTVE FGURE 29.1 Compss needles cn e used to trce the mgnetic field lines in the region outside r mgnet. Mgnetic field pttern surrounding r mgnet Mgnetic field pttern etween opposite poles (N ) of two r mgnets Mgnetic field pttern etween like poles (N N) of two r mgnets () () (c) Henry Lep nd Jim Lehmn Figure 29.2 Mgnetic field ptterns cn e displyed with iron filings sprinkled on pper ner mgnets.

4 832 CHAPTER 29 Mgnetic Fields Figure 29.3 The Erth s mgnetic field lines. A south mgnetic pole is ner the Erth s north geogrphic pole. Geogrphic equtor Mgnetic xis outh mgnetic pole Axis of rottion North geogrphic 11 pole Mgnetic equtor N outh geogrphic pole North mgnetic pole A north mgnetic pole is ner the Erth s south geogrphic pole. current loop does, s we shll see in Chpter 30. There is lso strong eidence tht the mgnitude of plnet s mgnetic field is relted to the plnet s rte of rottion. For exmple, Jupiter rottes fster thn the Erth, nd spce proes indicte tht Jupiter s mgnetic field is stronger thn the Erth s. Venus, on the other hnd, rottes more slowly thn the Erth, nd its mgnetic field is found to e weker. nestigtion into the cuse of the Erth s mgnetism is ongoing. The direction of the Erth s mgnetic field hs reersed seerl times during the lst million yers. Eidence for this reersl is proided y slt, type of rock tht contins iron. Bslt forms from mteril spewed forth y olcnic ctiity on the ocen floor. As the l cools, it solidifies nd retins picture of the Erth s mgnetic field direction. The rocks re dted y other mens to proide time line for these periodic reersls of the mgnetic field. We cn define mgnetic field B t some point in spce in terms of the mgnetic force F B the field exerts on chrged prticle moing with elocity, which we cll the test oject. For the time eing, let s ssume no electric or grittionl fields re present t the loction of the test oject. Experiments on rious chrged prticles moing in mgnetic field gie the following results: Properties of the mgnetic force on chrged prticle moing in mgnetic field The mgnitude F B of the mgnetic force exerted on the prticle is proportionl to the chrge q nd to the speed of the prticle. When chrged prticle moes prllel to the mgnetic field ector, the mgnetic force cting on the prticle is zero. When the prticle s elocity ector mkes ny ngle u 2 0 with the mgnetic field, the mgnetic force cts in direction perpendiculr to oth nd B ; tht is, F B is perpendiculr to the plne formed y nd B (Fig. 29.4). The mgnetic force exerted on positie chrge is in the direction opposite the direction of the mgnetic force exerted on negtie chrge moing in the sme direction (Fig. 29.4). The mgnitude of the mgnetic force exerted on the moing prticle is proportionl to sin u, where u is the ngle the prticle s elocity ector mkes with the direction of B. We cn summrize these osertions y writing the mgnetic force in the form Vector expression for the mgnetic force on chrged prticle moing in mgnetic field FB 5 q 3 B (29.1) which y definition of the cross product (see ection 11.1) is perpendiculr to oth nd B. We cn regrd this eqution s n opertionl definition of the

5 29.1 Mgnetic Fields nd Forces 833 The mgnetic force is perpendiculr to oth nd B. F B u B F B F B B The mgnetic forces on oppositely chrged prticles moing t the sme elocity in mgnetic field re in opposite directions. Figure 29.4 () The direction of the mgnetic force F B cting on chrged prticle moing with elocity in the presence of mgnetic field B. () Mgnetic forces on positie nd negtie chrges. The dshed lines show the pths of the prticles, which re inestigted in ection mgnetic field t some point in spce. Tht is, the mgnetic field is defined in terms of the force cting on moing chrged prticle. Figure 29.5 reiews two right-hnd rules for determining the direction of the cross product 3 B nd determining the direction of FB. The rule in Figure 29.5 depends on our right-hnd rule for the cross product in Figure Point the four fingers of your right hnd long the direction of with the plm fcing B nd curl them towrd B. Your extended thum, which is t right ngle to your fingers, points in the direction of 3 B. Becuse FB 5 q 3 B, FB is in the direction of your thum if q is positie nd is opposite the direction of your thum if q is negtie. (f you need more help understnding the cross product, you should reiew ection 11.1, including Fig ) An lterntie rule is shown in Figure Here the thum points in the direction of nd the extended fingers in the direction of B. Now, the force FB on positie chrge extends outwrd from the plm. The dntge of this rule is tht the force on the chrge is in the direction you would push on something with your hnd: outwrd from your plm. The force on negtie chrge is in the opposite direction. You cn use either of these two right-hnd rules. The mgnitude of the mgnetic force on chrged prticle is F B 5 q B sin u (29.2) where u is the smller ngle etween nd B. From this expression, we see tht F B is zero when is prllel or ntiprllel to B (u 5 0 or 1808) nd mximum when is perpendiculr to B (u 5 908). Mgnitude of the mgnetic force on chrged prticle moing in mgnetic field (1) Point your fingers in the direction of nd then curl them towrd the direction of B. (2) Your upright thum shows the direction of the mgnetic force on positie prticle. B F B F B (1) Point your fingers in the direction of B, with coming out of your thum. B (2) The mgnetic force on positie prticle is in the direction you would push with your plm. Figure 29.5 Two right-hnd rules for determining the direction of the mgnetic force F B 5 q 3 B cting on prticle with chrge q moing with elocity in mgnetic field B. () n this rule, the mgnetic force is in the direction in which your thum points. () n this rule, the mgnetic force is in the direction of your plm, s if you re pushing the prticle with your hnd.

6 834 CHAPTER 29 Mgnetic Fields TABLE 29.1 ome Approximte Mgnetic Field Mgnitudes ource of Field Field Mgnitude (T) trong superconducting lortory mgnet 30 trong conentionl lortory mgnet 2 Medicl MR unit 1.5 Br mgnet urfce of the un urfce of the Erth nside humn rin (due to nere impulses) Electric nd mgnetic forces he seerl importnt differences: The electric force ector is long the direction of the electric field, wheres the mgnetic force ector is perpendiculr to the mgnetic field. The electric force cts on chrged prticle regrdless of whether the prticle is moing, wheres the mgnetic force cts on chrged prticle only when the prticle is in motion. The electric force does work in displcing chrged prticle, wheres the mgnetic force ssocited with stedy mgnetic field does no work when prticle is displced ecuse the force is perpendiculr to the displcement of its point of ppliction. The tesl From the lst sttement nd on the sis of the work kinetic energy theorem, we conclude tht the kinetic energy of chrged prticle moing through mgnetic field cnnot e ltered y the mgnetic field lone. The field cn lter the direction of the elocity ector, ut it cnnot chnge the speed or kinetic energy of the prticle. From Eqution 29.2, we see tht the unit of mgnetic field is the newton per coulom-meter per second, which is clled the tesl (T): N 1 T 5 1 C? m/s Becuse coulom per second is defined to e n mpere, N 1 T 5 1 A? m A non- mgnetic-field unit in common use, clled the guss (G), is relted to the tesl through the conersion 1 T G. Tle 29.1 shows some typicl lues of mgnetic fields. Quick Quiz 29.1 An electron moes in the plne of this pper towrd the top of the pge. A mgnetic field is lso in the plne of the pge nd directed towrd the right. Wht is the direction of the mgnetic force on the electron? () towrd the top of the pge () towrd the ottom of the pge (c) towrd the left edge of the pge (d) towrd the right edge of the pge (e) upwrd out of the pge (f) downwrd into the pge Exmple 29.1 An Electron Moing in Mgnetic Field An electron in n old-style teleision picture tue moes towrd the front of the tue with speed of m/s long the x xis (Fig. 29.6). urrounding the neck of the tue re coils of wire tht crete mgnetic field of mgnitude T, directed t n ngle of 608 to the x xis nd lying in the xy plne. Clculte the mgnetic force on the electron.

7 29.2 Motion of Chrged Prticle in Uniform Mgnetic Field cont. z OLUTON Conceptulize Recll tht the mgnetic force on chrged prticle is perpendiculr to the plne formed y the elocity nd mgnetic field ectors. Use one of the right-hnd rules in Figure 29.5 to conince yourself tht the direction of the force on the electron is downwrd in Figure Ctegorize We elute the mgnetic force using n eqution deeloped in this section, so we ctegorize this exmple s sustitution prolem. Figure 29.6 (Exmple 29.1) The mgnetic force F B cting on the electron is in the negtie z direction when nd B lie in the xy plne. x e F B 60 B y Use Eqution 29.2 to find the mgnitude of the mgnetic force: F B 5 q B sin u 5 ( C)( m/s)(0.025 T)(sin 608) N For prctice using the ector product, elute this force in ector nottion using Eqution Motion of Chrged Prticle in Uniform Mgnetic Field Before we continue our discussion, some explntion of the nottion used in this ook is in order. To indicte the direction of B in illustrtions, we sometimes present perspectie iews such s those in Figure f B lies in the plne of the pge or is present in perspectie drwing, we use green ectors or green field lines with rrowheds. n nonperspectie illustrtions, we depict mgnetic field perpendiculr to nd directed out of the pge with series of green dots, which represent the tips of rrows coming towrd you (see Fig. 29.7). n this cse, the field is leled B out. f B is directed perpendiculrly into the pge, we use green crosses, which represent the fethered tils of rrows fired wy from you, s in Figure n this cse, the field is leled B in, where the suscript in indictes into the pge. The sme nottion with crosses nd dots is lso used for other quntities tht might e perpendiculr to the pge such s forces nd current directions. n ection 29.1, we found tht the mgnetic force cting on chrged prticle moing in mgnetic field is perpendiculr to the prticle s elocity nd consequently the work done y the mgnetic force on the prticle is zero. Now consider the specil cse of positiely chrged prticle moing in uniform mgnetic field with the initil elocity ector of the prticle perpendiculr to the field. Let s ssume the direction of the mgnetic field is into the pge s in Actie Figure 29.8 (pge 836). As the prticle chnges the direction of its elocity in response to the mgnetic force, the mgnetic force remins perpendiculr to the elocity. As we found in ection 6.1, if the force is lwys perpendiculr to the elocity, the pth of the prticle is circle! Actie Figure 29.8 shows the prticle moing in circle in plne perpendiculr to the mgnetic field. Although mgnetism nd mgnetic forces my e new nd unfmilir to you now, we see mgnetic effect tht results in something with which we re fmilir: the prticle in uniform circulr motion! The prticle moes in circle ecuse the mgnetic force F B is perpendiculr to nd B nd hs constnt mgnitude qb. As Actie Figure 29.8 illustrtes, the rottion is counterclockwise for positie chrge in mgnetic field directed into the pge. f q were negtie, the rottion would e clockwise. We use the prticle under net force model to write Newton s second lw for the prticle: o F 5 F B 5 m Mgnetic field lines coming out of the pper re indicted y dots, representing the tips of rrows coming outwrd. Mgnetic field lines going into the pper re indicted y crosses, representing the fethers of rrows going inwrd. B out B in Figure 29.7 Representtions of mgnetic field lines perpendiculr to the pge.

8 836 CHAPTER 29 Mgnetic Fields When the elocity of chrged prticle is perpendiculr to uniform mgnetic field, the prticle moes in circulr pth in plne perpendiculr to B. z The mgnetic force F B cting on the chrge is lwys directed towrd the center of the circle. q ACTVE FGURE 29.8 B y F B q q r F B F B B in Helicl pth ACTVE FGURE 29.9 q A chrged prticle hing elocity ector tht hs component prllel to uniform mgnetic field moes in helicl pth. x Becuse the prticle moes in circle, we lso model it s prticle in uniform circulr motion nd we replce the ccelertion with centripetl ccelertion: F B 5 qb 5 m 2 r This expression leds to the following eqution for the rdius of the circulr pth: r 5 m (29.3) qb Tht is, the rdius of the pth is proportionl to the liner momentum m of the prticle nd inersely proportionl to the mgnitude of the chrge on the prticle nd to the mgnitude of the mgnetic field. The ngulr speed of the prticle (from Eq ) is 5 r 5 qb (29.4) m The period of the motion (the time interl the prticle requires to complete one reolution) is equl to the circumference of the circle diided y the speed of the prticle: T 5 2pr 5 2p 5 2pm qb (29.5) These results show tht the ngulr speed of the prticle nd the period of the circulr motion do not depend on the speed of the prticle or on the rdius of the orit. The ngulr speed is often referred to s the cyclotron frequency ecuse chrged prticles circulte t this ngulr frequency in the type of ccelertor clled cyclotron, which is discussed in ection f chrged prticle moes in uniform mgnetic field with its elocity t some ritrry ngle with respect to B, its pth is helix. For exmple, if the field is directed in the x direction s shown in Actie Figure 29.9, there is no component of force in the x direction. As result, x 5 0, nd the x component of elocity remins constnt. The mgnetic force q 3 B cuses the components y nd z to chnge in time, howeer, nd the resulting motion is helix whose xis is prllel to the mgnetic field. The projection of the pth onto the yz plne (iewed long the x xis) is circle. (The projections of the pth onto the xy nd xz plnes re sinusoids!) Equtions 29.3 to 29.5 still pply proided is replced y ' 5! y2 1 z2. Quick Quiz 29.2 A chrged prticle is moing perpendiculr to mgnetic field in circle with rdius r. (i) An identicl prticle enters the field, with perpendiculr to B, ut with higher speed thn the first prticle. Compred with the rdius of the circle for the first prticle, is the rdius of the circulr pth for the second prticle () smller, () lrger, or (c) equl in size? (ii) The mgnitude of the mgnetic field is incresed. From the sme choices, compre the rdius of the new circulr pth of the first prticle with the rdius of its initil pth. Exmple 29.2 A Proton Moing Perpendiculr to Uniform Mgnetic Field A proton is moing in circulr orit of rdius 14 cm in uniform 0.35-T mgnetic field perpendiculr to the elocity of the proton. Find the speed of the proton. OLUTON Conceptulize From our discussion in this section, we know the proton follows circulr pth when moing perpendiculr to uniform mgnetic field.

9 29.2 Motion of Chrged Prticle in Uniform Mgnetic Field cont. Ctegorize We elute the speed of the proton using n eqution deeloped in this section, so we ctegorize this exmple s sustitution prolem. ole Eqution 29.3 for the speed of the prticle: ustitute numericl lues: 5 qbr m p C T m kg m/s WHAT F? Wht if n electron, rther thn proton, moes in direction perpendiculr to the sme mgnetic field with this sme speed? Will the rdius of its orit e different? Answer An electron hs much smller mss thn proton, so the mgnetic force should e le to chnge its elocity much more esily thn tht for the proton. Therefore, we expect the rdius to e smller. Eqution 29.3 shows tht r is proportionl to m with q, B, nd the sme for the electron s for the proton. Consequently, the rdius will e smller y the sme fctor s the rtio of msses m e /m p. Exmple 29.3 Bending n Electron Bem n n experiment designed to mesure the mgnitude of uniform mgnetic field, electrons re ccelerted from rest through potentil difference of 350 V nd then enter uniform mgnetic field tht is perpendiculr to the elocity ector of the electrons. The electrons trel long cured pth ecuse of the mgnetic force exerted on them, nd the rdius of the pth is mesured to e 7.5 cm. (uch cured em of electrons is shown in Fig ) (A) Wht is the mgnitude of the mgnetic field? OLUTON Conceptulize This exmple inoles electrons ccelerting from rest due to n electric force nd then moing in circulr pth due to mgnetic force. With the help of Figures 29.8 nd 29.10, isulize the circulr motion of the electrons. Figure (Exmple 29.3) The ending of n electron em in mgnetic field. Ctegorize Eqution 29.3 shows tht we need the speed of the electron to find the mgnetic field mgnitude, nd is not gien. Consequently, we must find the speed of the electron sed on the potentil difference through which it is ccelerted. To do so, we ctegorize the first prt of the prolem y modeling n electron nd the electric field s n isolted system. Once the electron enters the mgnetic field, we ctegorize the second prt of the prolem s one similr to those we he studied in this section. Henry Lep nd Jim Lehmn Anlyze Write the pproprite reduction of the consertion of energy eqution, Eqution 8.2, for the electron electric field system: DK 1 DU 5 0 ustitute the pproprite initil nd finl energies: 1 1 2m e q DV2 5 0 ole for the speed of the electron: ustitute numericl lues: 5 Å 22q DV m e 5 Å C21350 V kg m/s continued

10 838 CHAPTER 29 Mgnetic Fields 29.3 cont. Now imgine the electron entering the mgnetic field with this speed. ole Eqution 29.3 for the mgnitude of the mgnetic field: ustitute numericl lues: (B) Wht is the ngulr speed of the electrons? B 5 m e er B kg m/s C m T OLUTON Use Eqution 10.10: 5 r m/s m rd/s Finlize The ngulr speed cn e represented s 5 ( rd/s)(1 re/2p rd) re/s. The electrons trel round the circle 24 million times per second! This nswer is consistent with the ery high speed found in prt (A). WHAT F? Wht if sudden oltge surge cuses the ccelerting oltge to increse to 400 V? How does tht ffect the ngulr speed of the electrons, ssuming the mgnetic field remins constnt? Answer The increse in ccelerting oltge DV cuses the electrons to enter the mgnetic field with higher speed. This higher speed cuses them to trel in circle with lrger rdius r. The ngulr speed is the rtio of to r. Both nd r increse y the sme fctor, so the effects cncel nd the ngulr speed remins the sme. Eqution 29.4 is n expression for the cyclotron frequency, which is the sme s the ngulr speed of the electrons. The cyclotron frequency depends only on the chrge q, the mgnetic field B, nd the mss m e, none of which he chnged. Therefore, the oltge surge hs no effect on the ngulr speed. (n relity, howeer, the oltge surge my lso increse the mgnetic field if the mgnetic field is powered y the sme source s the ccelerting oltge. n tht cse, the ngulr speed increses ccording to Eq ) The mgnetic force exerted on the prticle ner either end of the ottle hs component tht cuses the prticle to spirl ck towrd the center. Pth of prticle Figure A chrged prticle moing in nonuniform mgnetic field ( mgnetic ottle) spirls out the field nd oscilltes etween the endpoints. When chrged prticles moe in nonuniform mgnetic field, the motion is complex. For exmple, in mgnetic field tht is strong t the ends nd wek in the middle such s tht shown in Figure 29.11, the prticles cn oscillte etween two positions. A chrged prticle strting t one end spirls long the field lines until it reches the other end, where it reerses its pth nd spirls ck. This configurtion is known s mgnetic ottle ecuse chrged prticles cn e trpped within it. The mgnetic ottle hs een used to confine plsm, gs consisting of ions nd electrons. uch plsm-confinement scheme could fulfill crucil role in the control of nucler fusion, process tht could supply us in the future with n lmost endless source of energy. Unfortuntely, the mgnetic ottle hs its prolems. f lrge numer of prticles re trpped, collisions etween them cuse the prticles to eentully lek from the system. The Vn Allen rdition elts consist of chrged prticles (mostly electrons nd protons) surrounding the Erth in doughnut-shped regions (Fig ). The prticles, trpped y the Erth s nonuniform mgnetic field, spirl round the field lines from pole to pole, coering the distnce in only few seconds. These prticles originte minly from the un, ut some come from strs nd other heenly ojects. For this reson, the prticles re clled cosmic rys. Most cosmic rys re deflected y the Erth s mgnetic field nd neer rech the tmosphere. ome of the prticles ecome trpped, howeer, nd it is these prticles tht mke up the Vn Allen elts. When the prticles re locted oer the poles, they sometimes collide with toms in the tmosphere, cusing the toms to emit isile light. uch collisions re the origin of the eutiful uror orelis, or northern lights, in the northern hemisphere nd the uror ustrlis in the southern hemisphere. Aurors re usully confined to the polr regions ecuse the Vn Allen elts re nerest the Erth s surfce there. Occsionlly, though, solr ctiity cuses lrger numers of chrged prticles to enter the elts nd significntly distort the norml mgnetic

11 29.3 Applictions noling Chrged Prticles Moing in Mgnetic Field 839 field lines ssocited with the Erth. n these situtions, n uror cn sometimes e seen t lower ltitudes Applictions noling Chrged Prticles Moing in Mgnetic Field A chrge moing with elocity in the presence of oth n electric field E nd mgnetic field B experiences oth n electric force qe nd mgnetic force q 3 B. The totl force (clled the Lorentz force) cting on the chrge is F 5 qe 1 q 3 B (29.6) Velocity elector n mny experiments inoling moing chrged prticles, it is importnt tht ll prticles moe with essentilly the sme elocity, which cn e chieed y pplying comintion of n electric field nd mgnetic field oriented s shown in Actie Figure A uniform electric field is directed to the right (in the plne of the pge in Actie Fig ), nd uniform mgnetic field is pplied in the direction perpendiculr to the electric field (into the pge in Actie Fig ). f q is positie nd the elocity is upwrd, the mgnetic force q 3 B is to the left nd the electric force qe is to the right. When the mgnitudes of the two fields re chosen so tht qe 5 qb, the chrged prticle is modeled s prticle in equilirium nd moes in stright erticl line through the region of the fields. From the expression qe 5 qb, we find tht 5 E (29.7) B Only those prticles hing this speed pss undeflected through the mutully perpendiculr electric nd mgnetic fields. The mgnetic force exerted on prticles moing t speeds greter thn tht is stronger thn the electric force, nd the prticles re deflected to the left. Those moing t slower speeds re deflected to the right. The Mss pectrometer A mss spectrometer seprtes ions ccording to their mss-to-chrge rtio. n one ersion of this deice, known s the Binridge mss spectrometer, em of ions first psses through elocity selector nd then enters second uniform mgnetic field B0 tht hs the sme direction s the mgnetic field in the selector (Actie Fig on pge 840). Upon entering the second mgnetic field, the ions moe in semicircle of rdius r efore striking detector rry t P. f the ions re positiely chrged, the em deflects to the left s Actie Figure shows. f the ions re negtiely chrged, the em deflects to the right. From Eqution 29.3, we cn express the rtio m/q s m q 5 rb 0 Using Eqution 29.7 gies m q 5 rb 0B (29.8) E Therefore, we cn determine m/q y mesuring the rdius of curture nd knowing the field mgnitudes B, B 0, nd E. n prctice, one usully mesures the msses of rious isotopes of gien ion, with the ions ll crrying the sme chrge q. n this wy, the mss rtios cn e determined een if q is unknown. A rition of this technique ws used y J. J. Thomson ( ) in 1897 to mesure the rtio e/m e for electrons. Figure (pge 840) shows the sic Figure The Vn Allen elts re mde up of chrged prticles trpped y the Erth s nonuniform mgnetic field. The mgnetic field lines re in green, nd the prticle pths re dshed lck lines. B in F B ource E F e lit ACTVE FGURE A elocity selector. When positiely chrged prticle is moing with elocity in the presence of mgnetic field directed into the pge nd n electric field directed to the right, it experiences n electric force q E to the right nd mgnetic force q 3 B to the left.

12 840 CHAPTER 29 Mgnetic Fields P Velocity selector r Detector rry B in B 0, in E q ACTVE FGURE A mss spectrometer. Positiely chrged prticles re sent first through elocity selector nd then into region where the mgnetic field B 0 cuses the prticles to moe in semicirculr pth nd strike detector rry t P. Pitfll Preention 29.1 The Cyclotron s Not tte-of-the-art Technology The cyclotron is importnt historiclly ecuse it ws the first prticle ccelertor to produce prticles with ery high speeds. Cyclotrons re still in use in medicl pplictions, ut most ccelertors currently in reserch use re not cyclotrons. Reserch ccelertors work on different principle nd re generlly clled synchrotrons. pprtus he used. Electrons re ccelerted from the cthode nd pss through two slits. They then drift into region of perpendiculr electric nd mgnetic fields. The mgnitudes of the two fields re first djusted to produce n undeflected em. When the mgnetic field is turned off, the electric field produces mesurle em deflection tht is recorded on the fluorescent screen. From the size of the deflection nd the mesured lues of E nd B, the chrge-to-mss rtio cn e determined. The results of this crucil experiment represent the discoery of the electron s fundmentl prticle of nture. The Cyclotron A cyclotron is deice tht cn ccelerte chrged prticles to ery high speeds. The energetic prticles produced re used to omrd tomic nuclei nd therey produce nucler rections of interest to reserchers. A numer of hospitls use cyclotron fcilities to produce rdioctie sustnces for dignosis nd tretment. Both electric nd mgnetic forces ply key roles in the opertion of cyclotron, schemtic drwing of which is shown in Figure The chrges moe inside two semicirculr continers D 1 nd D 2, referred to s dees ecuse of their shpe like the letter D. A high-frequency lternting potentil difference is pplied to the dees, nd uniform mgnetic field is directed perpendiculr to them. A positie ion relesed t P ner the center of the mgnet in one dee moes in semicirculr pth (indicted y the dshed lck line in the drwing) nd rries ck t the gp in time interl T/2, where T is the time interl needed to mke one complete trip round the two dees, gien y Eqution The frequency of the pplied potentil difference is djusted so tht the polrity of the dees is reersed in the sme time interl during which the ion trels round one dee. f the pplied potentil difference is djusted such tht D 1 is t lower electric potentil thn D 2 y n mount DV, the ion ccelertes cross the gp to D 1 nd its kinetic energy increses y n mount q DV. t then moes round D 1 in semicirculr pth of greter rdius (ecuse its speed hs incresed). After time interl T/2, it gin rries t the gp etween the dees. By this time, the polrity cross the dees hs gin een reersed nd the ion is gien nother kick cross the gp. The motion continues so tht for ech hlf-circle trip round one dee, the ion gins dditionl kinetic energy equl to q DV. When the rdius of its pth is nerly tht of the dees, the energetic ion lees the system through the exit slit. The cyclotron s opertion Electrons re ccelerted from the cthode, pss through two slits, nd re deflected y oth n electric field (formed y the chrged deflection pltes) nd mgnetic field (directed perpendiculr to the electric field). The em of electrons then strikes fluorescent screen. Cthode lits Deflection pltes Mgnetic field coil Fluorescent coting Deflected electron em Undeflected electron em Lucent Technologies Bell Lortory, courtesy AP Emilio egre Visul Archies Figure () Thomson s pprtus for mesuring e/m e. () J. J. Thomson (left) in the Cendish Lortory, Uniersity of Cmridge. The mn on the right, Frnk Bldwin Jewett, is distnt reltie of John W. Jewett, Jr., couthor of this text.

13 29.4 Mgnetic Force Acting on Current-Crrying Conductor 841 The lck, dshed, cured lines represent the pth of the prticles. B P Alternting V D 1 D 2 After eing ccelerted, the prticles exit here. North pole of mgnet Lwrence Berkeley Ntionl L Figure () A cyclotron consists of n ion source t P, two dees D 1 nd D 2 cross which n lternting potentil difference is pplied, nd uniform mgnetic field. (The south pole of the mgnet is not shown.) () The first cyclotron, inented y E. O. Lwrence nd M.. Liingston in depends on T eing independent of the speed of the ion nd of the rdius of the circulr pth (Eq. 29.5). We cn otin n expression for the kinetic energy of the ion when it exits the cyclotron in terms of the rdius R of the dees. From Eqution 29.3, we know tht 5 qbr/m. Hence, the kinetic energy is K 5 1 2m 2 5 q 2 B 2 R 2 (29.9) 2m When the energy of the ions in cyclotron exceeds out 20 MeV, reltiistic effects come into ply. (uch effects re discussed in Chpter 39.) Osertions show tht T increses nd the moing ions do not remin in phse with the pplied potentil difference. ome ccelertors oercome this prolem y modifying the period of the pplied potentil difference so tht it remins in phse with the moing ions Mgnetic Force Acting on Current-Crrying Conductor f mgnetic force is exerted on single chrged prticle when the prticle moes through mgnetic field, it should not surprise you tht current-crrying wire lso experiences force when plced in mgnetic field. The current is collection of mny chrged prticles in motion; hence, the resultnt force exerted y the field on the wire is the ector sum of the indiidul forces exerted on ll the chrged prticles mking up the current. The force exerted on the prticles is trnsmitted to the wire when the prticles collide with the toms mking up the wire. One cn demonstrte the mgnetic force cting on current-crrying conductor y hnging wire etween the poles of mgnet s shown in Figure (pge 842). For ese in isuliztion, prt of the horseshoe mgnet in prt () is remoed to show the end fce of the south pole in prts () through (d) of Figure The mgnetic field is directed into the pge nd coers the region within the shded squres. When the current in the wire is zero, the wire remins erticl s in Figure When the wire crries current directed upwrd s in Figure 29.17c, howeer, the wire deflects to the left. f the current is reersed s in Figure 29.17d, the wire deflects to the right. Let s quntify this discussion y considering stright segment of wire of length L nd cross-sectionl re A crrying current in uniform mgnetic field B s in

14 842 CHAPTER 29 Mgnetic Fields Figure () A wire suspended erticlly etween the poles of mgnet. () through (d) The setup shown in () s seen looking t the south pole of the mgnet so tht the mgnetic field (green crosses) is directed into the pge. When there is no current in the wire, the wire remins erticl. When the current is upwrd, the wire deflects to the left. When the current is downwrd, the wire deflects to the right. The erge mgnetic force exerted on chrge moing in the wire is q d B. N B in B in B in F B 0 q B in d The mgnetic force on the wire segment of length L is L B. L Figure A segment of current-crrying wire in mgnetic field B. Force on segment of current-crrying wire in uniform mgnetic field A The mgnetic force on ny segment d s is d s B nd is directed out of the pge. B d s Figure A wire segment of ritrry shpe crrying current in mgnetic field B experiences mgnetic force. c d Figure The mgnetic force exerted on chrge q moing with drift elocity d is q d 3 B. To find the totl force cting on the wire, we multiply the force q d 3 B exerted on one chrge y the numer of chrges in the segment. Becuse the olume of the segment is AL, the numer of chrges in the segment is nal, where n is the numer of chrges per unit olume. Hence, the totl mgnetic force on the segment of wire of length L is FB 5 1q d 3 B 2nAL We cn write this expression in more conenient form y noting tht, from Eqution 27.4, the current in the wire is 5 nq d A. Therefore, FB 5 L 3 B (29.10) where L is ector tht points in the direction of the current nd hs mgnitude equl to the length L of the segment. This expression pplies only to stright segment of wire in uniform mgnetic field. Now consider n ritrrily shped wire segment of uniform cross section in mgnetic field s shown in Figure t follows from Eqution tht the mgnetic force exerted on smll segment of ector length d s in the presence of field B is df B 5 d s 3 B (29.11) where df B is directed out of the pge for the directions of B nd d s in Figure Eqution cn e considered s n lterntie definition of B. Tht is, we cn define the mgnetic field B in terms of mesurle force exerted on current element, where the force is mximum when B is perpendiculr to the element nd zero when B is prllel to the element. To clculte the totl force FB cting on the wire shown in Figure 29.19, we integrte Eqution oer the length of the wire: FB 5 3 d s 3 B (29.12) where nd represent the endpoints of the wire. When this integrtion is crried out, the mgnitude of the mgnetic field nd the direction the field mkes with the ector d s my differ t different points.

15 29.5 Torque on Current Loop in Uniform Mgnetic Field 843 Quick Quiz 29.3 A wire crries current in the plne of this pper towrd the top of the pge. The wire experiences mgnetic force towrd the right edge of the pge. s the direction of the mgnetic field cusing this force () in the plne of the pge nd towrd the left edge, () in the plne of the pge nd towrd the ottom edge, (c) upwrd out of the pge, or (d) downwrd into the pge? Exmple 29.4 Force on emicirculr Conductor A wire ent into semicircle of rdius R forms closed circuit nd crries current. The wire lies in the xy plne, nd uniform mgnetic field is directed long the positie y xis s in Figure Find the mgnitude nd direction of the mgnetic force cting on the stright portion of the wire nd on the cured portion. OLUTON Conceptulize Using the right-hnd rule for cross products, we see tht the force F 1 on the stright portion of the wire is out of the pge nd the force F 2 on the cured portion is into the pge. s F 2 lrger in mgnitude thn F 1 ecuse the length of the cured portion is longer thn tht of the stright portion? Ctegorize Becuse we re deling with current-crrying wire in mgnetic field rther thn single chrged prticle, we must use Eqution to find the totl force on ech portion of the wire. y R d u u u d s Figure (Exmple 29.4) The mgnetic force on the stright portion of the loop is directed out of the pge, nd the mgnetic force on the cured portion is directed into the pge. B x Anlyze Notice tht d s is perpendiculr to B eerywhere on the stright portion of the wire. Use Eqution to find the force on this portion: To find the mgnetic force on the cured prt, first write n expression for the mgnetic force df 2 on the element d s in Figure 29.20: From the geometry in Figure 29.20, write n expression for ds: ustitute Eqution (2) into Eqution (1) nd integrte oer the ngle u from 0 to p: F d s 3 R B 5 3 B dx k^ 5 2RB k^ (1) df 2 5 d s 3 B 52B sin u ds k^ (2) ds 5 R du F p p RB sin u du k^ 52RB 3 sin u du k^ 52RB 32cos u 4 p 0 k^ 0 2R 5 RB 1cos p2cos 02k^ 5 RB k^ 5 22RB k^ 0 Finlize Two ery importnt generl sttements follow from this exmple. First, the force on the cured portion is the sme in mgnitude s the force on stright wire etween the sme two points. n generl, the mgnetic force on cured current-crrying wire in uniform mgnetic field is equl to tht on stright wire connecting the endpoints nd crrying the sme current. Furthermore, F 1 1 F is lso generl result: the net mgnetic force cting on ny closed current loop in uniform mgnetic field is zero Torque on Current Loop in Uniform Mgnetic Field n ection 29.4, we showed how mgnetic force is exerted on current-crrying conductor plced in mgnetic field. With tht s strting point, we now show tht torque is exerted on current loop plced in mgnetic field.

16 844 CHAPTER 29 Mgnetic Fields B No mgnetic forces ct on sides nd ecuse these sides re prllel to B. B O Torque on current loop in mgnetic field 2 ides nd re perpendiculr to the mgnetic field nd experience forces. The mgnetic forces F 2 nd F 4 exerted on sides nd crete torque tht tends to rotte the loop clockwise. F 2 F 4 Figure () Oerhed iew of rectngulr current loop in uniform mgnetic field. () Edge iew of the loop sighting down sides nd. The purple dot in the left circle represents current in wire coming towrd you; the purple cross in the right circle represents current in wire moing wy from you. Consider rectngulr loop crrying current in the presence of uniform mgnetic field directed prllel to the plne of the loop s shown in Figure No mgnetic forces ct on sides nd ecuse these wires re prllel to the field; hence, L 3 B 5 0 for these sides. Mgnetic forces do, howeer, ct on sides nd ecuse these sides re oriented perpendiculr to the field. The mgnitude of these forces is, from Eqution 29.10, F 2 5 F 4 5 B The direction of F2, the mgnetic force exerted on wire, is out of the pge in the iew shown in Figure nd tht of F4, the mgnetic force exerted on wire, is into the pge in the sme iew. f we iew the loop from side nd sight long sides nd, we see the iew shown in Figure 29.21, nd the two mgnetic forces F2 nd F4 re directed s shown. Notice tht the two forces point in opposite directions ut re not directed long the sme line of ction. f the loop is pioted so tht it cn rotte out point O, these two forces produce out O torque tht rottes the loop clockwise. The mgnitude of this torque t mx is t mx 5 F F B B2 2 5 B where the moment rm out O is /2 for ech force. Becuse the re enclosed y the loop is A 5, we cn express the mximum torque s t mx 5 AB (29.13) This mximum-torque result is lid only when the mgnetic field is prllel to the plne of the loop. The sense of the rottion is clockwise when iewed from side s indicted in Figure f the current direction were reersed, the force directions would lso reerse nd the rottionl tendency would e counterclockwise. Now suppose the uniform mgnetic field mkes n ngle u, 908 with line perpendiculr to the plne of the loop s in Actie Figure For conenience, let s ssume B is perpendiculr to sides nd. n this cse, the mgnetic forces F1 nd F3 exerted on sides nd cncel ech other nd produce no torque ecuse they pss through common origin. The mgnetic forces F2 nd F4 cting on sides nd, howeer, produce torque out ny point. Referring to the edge iew shown in Actie Figure 29.22, we see tht the moment rm of F2 out the point O is equl to (/2) sin u. Likewise, the moment rm of F4 out O is lso equl to (/2) sin u. Becuse F 2 5 F 4 5 B, the mgnitude of the net torque out O is t5f 2 2 sin u 1F 4 2 sin u 5 B 2 sin u 1 B sin u 5 B sin u 2 5 AB sin u where A 5 is the re of the loop. This result shows tht the torque hs its mximum lue AB when the field is perpendiculr to the norml to the plne of the loop (u 5 908) s discussed with regrd to Figure nd is zero when the field is prllel to the norml to the plne of the loop (u 5 0). A conenient ector expression for the torque exerted on loop plced in uniform mgnetic field B is t 5 A 3 B (29.14) where A, the ector shown in Actie Figure 29.22, is perpendiculr to the plne of the loop nd hs mgnitude equl to the re of the loop. To determine the direction of A, use the right-hnd rule descried in Figure When you curl the fingers of your right hnd in the direction of the current in the loop, your

17 29.5 Torque on Current Loop in Uniform Mgnetic Field 845 F 2 2 A u O sin u 2 u F 4 B (1) Curl your fingers in the direction of the current round the loop. m A (2) Your thum points in the direction of A nd m. When the norml to the loop mkes n ngle u with the mgnetic field, the moment rm for the torque is (/2) sin u. ACTVE FGURE An edge iew of the loop in Figure with the norml to the loop t n ngle u with respect to the mgnetic field. Figure Right-hnd rule for determining the direction of the ector A. The direction of the mgnetic moment m is the sme s the direction of A. thum points in the direction of A. Actie Figure shows tht the loop tends to rotte in the direction of decresing lues of u (tht is, such tht the re ector A rottes towrd the direction of the mgnetic field). The product A is defined to e the mgnetic dipole moment m (often simply clled the mgnetic moment ) of the loop: m ; A (29.15) The unit of mgnetic dipole moment is the mpere-meter 2 (A? m 2 ). f coil of wire contins N loops of the sme re, the mgnetic moment of the coil is m coil 5 N A (29.16) Using Eqution 29.15, we cn express the torque exerted on current-crrying loop in mgnetic field B s t 5 m 3 B (29.17) This result is nlogous to Eqution 26.18, t 5 p 3 E, for the torque exerted on n electric dipole in the presence of n electric field E, where p is the electric dipole moment. Although we otined the torque for prticulr orienttion of B with respect to the loop, the eqution t 5 m 3 B is lid for ny orienttion. Furthermore, lthough we deried the torque expression for rectngulr loop, the result is lid for loop of ny shpe. The torque on n N-turn coil is gien y Eqution y using Eqution for the mgnetic moment. n ection 26.6, we found tht the potentil energy of system of n electric dipole in n electric field is gien y U 52p? E. This energy depends on the orienttion of the dipole in the electric field. Likewise, the potentil energy of system of mgnetic dipole in mgnetic field depends on the orienttion of the dipole in the mgnetic field nd is gien y Mgnetic dipole moment of current loop Torque on mgnetic moment in mgnetic field U 52m? B (29.18) This expression shows tht the system hs its lowest energy U min 5 2mB when m points in the sme direction s B. The system hs its highest energy U mx 5 1mB when m points in the direction opposite B. Potentil energy of system of mgnetic moment in mgnetic field

18 846 CHAPTER 29 Mgnetic Fields The torque on current loop cuses the loop to rotte; this effect is exploited prcticlly in motor. Energy enters the motor y electricl trnsmission, nd the rotting coil cn do work on some deice externl to the motor. For exmple, the motor in n cr s electricl window system does work on the windows, pplying force on them nd moing them up or down through some displcement. We will discuss motors in more detil in ection Quick Quiz 29.4 (i) Rnk the mgnitudes of the torques cting on the rectngulr loops (), (), nd (c) shown edge-on in Figure from highest to lowest. All loops re identicl nd crry the sme current. (ii) Rnk the mgnitudes of the net forces cting on the rectngulr loops shown in Figure from highest to lowest. Figure (Quick Quiz 29.4) Which current loop (seen edge-on) experiences the gretest torque, (), (), or (c)? Which experiences the gretest net force? c Exmple 29.5 The Mgnetic Dipole Moment of Coil A rectngulr coil of dimensions 5.40 cm cm consists of 25 turns of wire nd crries current of 15.0 ma. A T mgnetic field is pplied prllel to the plne of the coil. (A) Clculte the mgnitude of the mgnetic dipole moment of the coil. OLUTON Conceptulize The mgnetic moment of the coil is independent of ny mgnetic field in which the loop resides, so it depends only on the geometry of the loop nd the current it crries. Ctegorize We elute quntities sed on equtions deeloped in this section, so we ctegorize this exmple s sustitution prolem. Use Eqution to clculte the mgnetic moment: m coil 5 NA 5 (25)( A)( m)( m) (B) Wht is the mgnitude of the torque cting on the loop? OLUTON A? m 2 Use Eqution 29.17, noting tht B is perpendiculr to m coil: t 5 m coil B 5 ( A? m 2 )(0.350 T) N? m Exmple 29.6 Rotting Coil Consider the loop of wire in Figure mgine it is pioted long side, which is prllel to the z xis nd fstened so tht side remins fixed nd the rest of the loop hngs erticlly in the grittionl field of the Erth ut cn rotte round side (Fig ). The mss of the loop is 50.0 g, nd the sides re of lengths m nd m. The loop crries current of 3.50 A nd is immersed in erticl uniform mgnetic field of mgnitude T in the positie y direction (Fig c). Wht ngle does the plne of the loop mke with the erticl?

19 29.6 The Hll Effect cont. OLUTON Conceptulize n the edge iew of Figure 29.25, notice tht the mgnetic moment of the loop is to the left. Therefore, when the loop is in the mgnetic field, the mgnetic torque on the loop cuses it to rotte in clockwise direction round side, which we choose s the rottion xis. mgine the loop mking this clockwise rottion so tht the plne of the loop is t some ngle u to the erticl s in Figure 29.25c. The grittionl force on the loop exerts torque tht would cuse rottion in the counterclockwise direction if the mgnetic field were turned off. The loop hngs erticlly nd is pioted so tht it cn rotte round side. Ctegorize At some ngle of the loop, the two torques descried in the Conceptulize step re equl in mgnitude nd the loop is t rest. We therefore model the loop s rigid oject in equilirium. g m c y x The mgnetic torque cuses the loop to rotte in clockwise direction round side, wheres the grittionl torque is in the opposite direction. m g u 2 cos u y 2 sin u Figure (Exmple 29.6) () The dimensions of rectngulr current loop. () Edge iew of the loop sighting down sides nd. (c) An edge iew of the loop in () rotted through n ngle with respect to the horizontl when it is plced in mgnetic field. B x Anlyze Elute the mgnetic torque on the loop out side from Eqution 29.17: Elute the grittionl torque on the loop, noting tht the grittionl force cn e modeled to ct t the center of the loop: B t 5 m 3 B 5 2mB sin 190 2u2k^ 52AB cos u k^ 52B cos u k^ g t 5 r 3 mg 5 mg sin u k^ 2 From the rigid ody in equilirium model, dd the torques nd set the net torque equl to zero: t 52B cos u k^ 1 mg sin u k^ ole for u: ustitute numericl lues: B cos u5mg 2 u5tn 21 2B mg u5tn 21 c sin u tn u 52B mg A m T kg m/s 2 d Finlize The ngle is reltiely smll, so the loop still hngs lmost erticlly. f the current or the mgnetic field B is incresed, howeer, the ngle increses s the mgnetic torque ecomes stronger The Hll Effect When current-crrying conductor is plced in mgnetic field, potentil difference is generted in direction perpendiculr to oth the current nd the mgnetic field. This phenomenon, first osered y Edwin Hll ( ) in 1879, is known s the Hll effect. The rrngement for osering the Hll effect consists of flt conductor crrying current in the x direction s shown in Figure (pge 848). A uniform mgnetic field B is pplied in the y direction. f the chrge crriers re electrons moing in the negtie x direction with drift elocity d, they experience n upwrd mgnetic force F B 5 q d 3 B, re deflected upwrd, nd ccumulte t the upper edge of the flt conductor, leing n excess of positie chrge t

20 848 CHAPTER 29 Mgnetic Fields When is in the x direction nd B in the y direction, oth positie nd negtie chrge crriers re deflected upwrd in the mgnetic field. z d t d B F B c F B d B y x Figure To osere the Hll effect, mgnetic field is pplied to current-crrying conductor. The Hll oltge is mesured etween points nd c. the lower edge (Fig ). This ccumultion of chrge t the edges estlishes n electric field in the conductor nd increses until the electric force on crriers remining in the ulk of the conductor lnces the mgnetic force cting on the crriers. When this equilirium condition is reched, the electrons re no longer deflected upwrd. A sensitie oltmeter connected cross the smple s shown in Figure cn mesure the potentil difference, known s the Hll oltge DV H, generted cross the conductor. f the chrge crriers re positie nd hence moe in the positie x direction (for rightwrd current) s shown in Figures nd 29.27, they lso experience n upwrd mgnetic force q d 3 B, which produces uildup of positie chrge on the upper edge nd lees n excess of negtie chrge on the lower edge. Hence, the sign of the Hll oltge generted in the smple is opposite the sign of the Hll oltge resulting from the deflection of electrons. The sign of the chrge crriers cn therefore e determined from mesuring the polrity of the Hll oltge. n deriing n expression for the Hll oltge, first note tht the mgnetic force exerted on the crriers hs mgnitude q d B. n equilirium, this force is lnced y the electric force qe H, where E H is the mgnitude of the electric field due to the chrge seprtion (sometimes referred to s the Hll field). Therefore, q d B 5 qe H E H 5 d B f d is the width of the conductor, the Hll oltge is DV H 5 E H d 5 d Bd (29.19) Therefore, the mesured Hll oltge gies lue for the drift speed of the chrge crriers if d nd B re known. We cn otin the chrge-crrier density n y mesuring the current in the smple. From Eqution 27.4, we cn express the drift speed s d 5 (29.20) nqa where A is the cross-sectionl re of the conductor. ustituting Eqution into Eqution gies DV H 5 Bd (29.21) nqa When the chrge crriers re negtie, the upper edge of the conductor ecomes negtiely chrged nd c is t lower electric potentil thn. The chrge crriers re no longer deflected when the edges ecome sufficiently chrged tht there is lnce etween the electric force nd the mgnetic force. When the chrge crriers re positie, the upper edge of the conductor ecomes positiely chrged nd c is t higher potentil thn. B c q d B d q E H V H =+=+1.50 V B c q d B q E H d V H =+=+2.50 V Figure The sign of the Hll oltge depends on the sign of the chrge crriers.

21 29.6 The Hll Effect 849 Becuse A 5 td, where t is the thickness of the conductor, we cn lso express Eqution s DV H 5 B nqt 5 R HB (29.22) t where R H 5 1/nq is clled the Hll coefficient. This reltionship shows tht properly clirted conductor cn e used to mesure the mgnitude of n unknown mgnetic field. Becuse ll quntities in Eqution other thn nq cn e mesured, lue for the Hll coefficient is redily otinle. The sign nd mgnitude of R H gie the sign of the chrge crriers nd their numer density. n most metls, the chrge crriers re electrons nd the chrge-crrier density determined from Hlleffect mesurements is in good greement with clculted lues for such metls s lithium (Li), sodium (N), copper (Cu), nd siler (Ag), whose toms ech gie up one electron to ct s current crrier. n this cse, n is pproximtely equl to the numer of conducting electrons per unit olume. This clssicl model, howeer, is not lid for metls such s iron (Fe), ismuth (Bi), nd cdmium (Cd) or for semiconductors. These discrepncies cn e explined only y using model sed on the quntum nture of solids. The Hll oltge Exmple 29.7 The Hll Effect for Copper A rectngulr copper strip 1.5 cm wide nd 0.10 cm thick crries current of 5.0 A. Find the Hll oltge for 1.2-T mgnetic field pplied in direction perpendiculr to the strip. OLUTON Conceptulize tudy Figures nd crefully nd mke sure you understnd tht Hll oltge is deeloped etween the top nd ottom edges of the strip. Ctegorize We elute the Hll oltge using n eqution deeloped in this section, so we ctegorize this exmple s sustitution prolem. Assuming one electron per tom is ille for conduction, find the chrge-crrier density in terms of the molr mss M nd density r of copper: ustitute this result into Eqution 29.22: n 5 N A V 5 N Ar M DV H 5 B nqt 5 MB N A rqt kg/mol215.0 A211.2 T2 ustitute numericl lues: DV H mol kg/m C m mv uch n extremely smll Hll oltge is expected in good conductors. (Notice tht the width of the conductor is not needed in this clcultion.) WHAT F? or lrger? Wht if the strip hs the sme dimensions ut is mde of semiconductor? Will the Hll oltge e smller Answer n semiconductors, n is much smller thn it is in metls tht contriute one electron per tom to the current; hence, the Hll oltge is usully lrger ecuse it ries s the inerse of n. Currents on the order of 0.1 ma re generlly used for such mterils. Consider piece of silicon tht hs the sme dimensions s the copper strip in this exmple nd whose lue for n is electrons/m 3. Tking B T nd ma, we find tht DV H mv. A potentil difference of this mgnitude is redily mesured.

22 850 CHAPTER 29 Mgnetic Fields Definitions ummry The mgnetic dipole moment m of loop crrying current is m ; A (29.15) where the re ector A is perpendiculr to the plne of the loop nd 0 A 0 is equl to the re of the loop. The unit of m is A? m 2. Concepts nd Principles The mgnetic force tht cts on chrge q moing with elocity in mgnetic field B is FB 5 q 3 B (29.1) The direction of this mgnetic force is perpendiculr oth to the elocity of the prticle nd to the mgnetic field. The mgnitude of this force is F B 5 0 q 0 B sin u (29.2) where u is the smller ngle etween nd B. The unit of B is the tesl (T), where 1 T 5 1 N/A? m. f chrged prticle moes in uniform mgnetic field so tht its initil elocity is perpendiculr to the field, the prticle moes in circle, the plne of which is perpendiculr to the mgnetic field. The rdius of the circulr pth is r 5 m (29.3) qb where m is the mss of the prticle nd q is its chrge. The ngulr speed of the chrged prticle is 5 qb m (29.4) f stright conductor of length L crries current, the force exerted on tht conductor when it is plced in uniform mgnetic field B is FB 5 L 3 B (29.10) where the direction of L is in the direction of the current nd 0 L 0 5 L. f n ritrrily shped wire crrying current is plced in mgnetic field, the mgnetic force exerted on ery smll segment d s is df B 5 d s 3 B (29.11) To determine the totl mgnetic force on the wire, one must integrte Eqution oer the wire, keeping in mind tht oth B nd d s my ry t ech point. The torque t on current loop plced in uniform mgnetic field B is t 5 m 3 B (29.17) The potentil energy of the system of mgnetic dipole in mgnetic field is U 52m? B (29.18) Ojectie Questions Ojectie Questions 2 through 4 in Chpter 11 cn e ssigned with this chpter s reiew for the ector product. 1. A sptilly uniform mgnetic field cnnot exert mgnetic force on prticle in which of the following circumstnces? denotes nswer ille in tudent olutions Mnul/tudy Guide There my e more thn one correct sttement. () The prticle is chrged. () The prticle moes perpendiculr to the mgnetic field. (c) The prticle moes prllel to the mgnetic field. (d) The mgnitude of the mgnetic field chnges with time. (e) The prticle is t rest.