Unit 6 Mgnetic foces 6.1 ntoduction. Mgnetic field 6. Mgnetic foces on moving electic chges 6. oce on conducto with cuent. 6.4 Action of unifom mgnetic field on flt cuent-cying loop. Mgnetic moment. Electic engine. 6.5 Hll Effect 6.6 Polems Ojectives Descie the effects of mgnetic field on moving electic chge. Clculte the mgnetic foce cting on conducto with cuent nd loop inside mgnetic field. ind the mgnetic moment of flt cuent-cying loop. Clculte the toque ppeing on flt cuent-cying loop inside unifom mgnetic field. Explin the Hll Effect. 6.1 ntoduction Although the knowledge of mgnetic popeties of some minels tces ck to the ncient Geece, until 1th centuy ny systemtic study of thei popeties is cied out. n this peiod, Piee of Micout expeiences with mgnets knowing two of thei popeties: The existence of two mgnetic poles ejecting ech othe if they e equl, nd ttcting ech othe if they e diffeent. The pesistence of oth poles fte eking the mgnet. The use of mgnets fo the eings (compss needle) gve nme to the poles of mgnet since oth poles oient ccoding the noth south teestil 1
poles. The pole tht oients to the noth teestil pole ws clled Noth Pole, nd simil sitution occus fo the South Pole. This ehvio llowed identify the Eth s mgnet nd, since the poles of the sme nme eject ech othe nd the ones of diffeent nme e ttcted; consequence of this fct is tht in the noth geogphic pole thee is mgnetic south pole, nd in the south geogphic pole thee is noth mgnetic pole. n the sme wy tht it hppen in electic fields, the e of the spce whee thee e mgnetic popeties is clled mgnetic field nd n osevle consequence of its existence is tht it cts on moving chge, s we ll see in the following section. 6. Mgnetic foce cting on moving chges. Mgnetic field When electosttic phenomen wee studied, it ws oseved tht n electic foce qe cts on n electic chge q when n electic field E exists. Howeve, the foce on electic chge q doesn t depend only of its position ut lso of its velocity v. Added to electic foce, nothe component exists, tht we ll cll mgnetic foce, hving the following fetues: t only cts if chge is moving t s pependicul to velocity of the chge Thee is fixed diection (fo ech point of spce) in such wy tht when chge is moving in this diection, no one foce cts on chge. The mgnitude of foce is diectly elted to the vlue of chge, to the vlue of speed nd to the vlue of fixed quntity fo ech point of the spce These fetues cy us to define the mgnetic field s the quntity gtheing the diection nd fixed module in ech point tht we hve quoted. Theefoe, we cn expess the mgnetic foce s: Mgnetic foce cting on n electic chge inside mgnetic field qv Eqution 6-1 t is necessy to ememe tht v is coss poduct, nd so the mgnetic foce will hve the following fetues: t s pependicul vecto to the two multiplied vectos, v nd. ts mgnitude is q v sinα, eing α the ngle etween nd v. om this fetue, when chge is moving pllel to mgnetic field on point, mgnetic foce is zeo. ts sense cn e otined fom the ight hnd ule (o scew ule). f the chge is negtive, the sense of the foce is the opposite.
q α v -q α v igue 6-1. Mgnetic foce is pependicul to the plne contining the velocity nd the mgnetic field vecto The ight hnd ule o the scew ule hs een explined on Unit 0. t s useful to convet otting movement into line movement, detemining the coesponding senses. Refe to Unit 0 in ode to find moe detiled explntion out these ules. v The unit of mgnetic field in the ntentionl System is the tesl (T), honoing Cotin scientist Nikol Tesl (1856-194). om Eqution 6- it cn e stted tht thee is mgnetic field of one tesl on point of the spce when moving point chge of one coulom pependicully to mgnetic field with speed of one mete y second, on chge cts foce of one newton. o the mjoity of pplictions, 1T is too ig unit, s usul mgnetic fields e in nge of mt, eing usully used the guss (G): 1 G 10-4 T. As n exmple, the teestil mgnetic field mgnitude is oughly 0,6 G 60 µt, nd the mgnetic field used in spectoscopy of mgnetic nucle esonnce is the ode of 0.5 to 0T. As mgnetic field is vecto quntity, lines eing pllel to mgnetic field on ech point of the spce e the mgnetic field lines. These lines enle us visulize the diection of mgnetic field t ny point of spce. An exmple cn e seen in igue 6-. ) Mgnetic South pole ) Mgnetic Noth pole igue 6-. ) Lines of the mgnetic field in the vicinities of the Eth nd ) lines of the mgnetic field poduced y cicul cuent
Thee e two chcteistics of mgnetic field lines: The mgnetic field lines e pependicul to mgnetic foce on ech point. The mgnetic field lines e closed lines, since n only type of mgnetic pole (monopole) cn t e isolted. Until now, ny one hs een le to isolte noth o south mgnetic pole. f it would e possile, mgnetic field lines stting on monopole nd finishing on nothe monopole would exist, ut s monopole cn t e otined, mgnetic field lines e closed lines. f we tke in ccount not only the electosttic foces ut lso the mgnetic foces, the totl foce cting on chge q with velocity v inside n electic nd mgnetic field is qe qv, Eqution 6- clled Loentz s oce, expessed s function of E nd. Movement of point chges inside unifom mgnetic field Cse ) f point chge q hve pllel velocity v to the unifom mgnetic field, the mgnetic foce cting on chge will e zeo; so, the chge will move with Unifom Line Motion. Cse ) f point chge hs velocity v nd entes inside unifom mgnetic field pependicul to v (no electic field exists), mgnetic foce qv ppes on chge. As this foce is pependicul to movement (no wok poduces), the speed (modulus of velocity) q v will e constnt, nd cicul pth of dius will e descied y the chge. So, only centipetl cceletion cts on chge N igue 6-4. Chged pticle moving v in mgnetic field pependicul to plne of ppe 1 f we cll m the mss of chged pticle, mgnetic foce nd centipetl cceletion cn e elted though the thid Newton s lw, m, eing possile expess the ngul fequency of movement s function of mgnetic field, electic chge nd mss: mv mv v qv ω q q m. A Unifom Cicul Motion occus. 1 The pependicul vectos to plne of ppe e epesented with symol when they e going out fom ppe to ede nd with symol when they e going in fom ede to ppe. 4
Cse c) When velocity of pticle is neithe pependicul no pllel to mgnetic field, velocity cn e split in component in the sme diection thn mgnetic field nd nothe component pependicul. The fist one poduces unifom cicul motion (s it hs een explined) nd the second one, unifom line motion, esulting in helicoidlly pth; oth movements occu t the sme time. V T v N igue 6-4. Negtive chged pticle with helicoidlly movement inside mgnetic field Selecto of speed n selecto of speed, lot of chged pticles with diffeent speed ente inside mgnetic field. The ojective is to select n only speed, o now nge of speeds, septing those pticles not eing in the wished nge. o this pupose, the pticles with wide nge of speeds, fst, slow, etc., ente in n e with n electicl field nd pependicul (to the electic field) mgnetic field (igue 6-5). The electic field in the figue poduces foce pointing to down, whose mgnitude qe is not depending on the speed. The mgnetic field poduces pointing up foce depending on the speed, qv. n this wy, the foce pointing to up is diectly elted to the speed, nd pticles with high speed will e deflected to up. Pticles with low speed will e deflected to down due to the foce of electic field. Only those pticles veifying tht qe qv, will not e deflected, nd will pss hoizontlly, without deflection, long the selecto of speed. Then, this device clssifies the pticles ccoding thei speed, not poducing ny deflection on pticles with speed ve/. qv v E E v qe - - - - - - - igue 6-5. Chged pticles in selecto of speed. They en t deflected those pticles hving speed v E/ Vying the pplied electic field (it s esy to do with only chnge the diffeence of potentil etween pltes), chged pticles with diffeent speeds cn e selected, nd pticles fste o slowe cn e poduced. 5
Exmple 6-1 A poton moves in the diection of the xis x nd positive sense in egion whee the electic nd mgnetic field e pependicul etween them. f the electic field is j kv/m sized, nd the mgnetic field 50k mt sized, Which is the speed of the no deflected potons? f the potons e moving slowe, whee will e they deflected to? Solution The no deflected potons e those eing n electic foce qe equl to the mgnetic foce qv. So: v E 10 50 10 60 km/s o lowe speeds, mgnetic foce qv will e lowe, nd theefoe they will e deflected to the y positive xis. z y qe E qv v x 6. oce cting on n element of cuent n pevious section we hve seen tht foce cts on n electic chge moving inside mgnetic field. So, on conducto flowed y n electic cuent inside mgnetic field, will lso ct mgnetic foce. This foce will e the esulting foce cting on ech chge is moving on conducto. Let s conside conducto flowed y cuent, plced inside d mgnetic field, s on igue 6-. n n intevl of time dt, the electic chges move long the conducto α length d dl d lv l equl to the poduct of dift dt speed y the time dl v dt igue 6-6. oce on n element of cuent The electic chge contined inside the tken piece of conducto (length d l ) could e computed s the totl chge hs cossed coss section of conducto in time dt, nd so dqdt. Then, the elementy foce d cting on the element d l will e: d dq( v ) dt( v ) ( dl ) The poduct d l is clled n element of cuent. t s necessy to note tht this eqution only povides the elementy foce cting on n element of cuent. To clculte the foce cting on piece of conducto flowed y n electic cuent, it will e necessy to integte this expession. 6
oce cting on conducto cying electic chge inside mgnetic field ( dl ) l l A Eqution 6- n the cse of cuents in unifom mgnetic field, the mgnetic field cn go out of the integl, so this expession ecomes l d dl l la eing l the vecto joining the stting point of conducto nd the finishing point, s it s shown on igue 6-7. oce cting on conducto cying electic chge inside unifom mgnetic field l Eqution 6-4 A A l A consequence of this esult is tht the foce cting on closed conducto (with ny shpe) inside unifom mgnetic field is zeo, ecuse l vecto is lso zeo. igue 6-7 nside unifom mgnetic field doesn t mtte the shpe of conducto to clculte the net foce on conducto. Exmple 6- Clculte the foce cting on conductos on figue, plced inside mgnetic field k T. Solution y (m) A () A () (c) ) The cuent goes fom (0,10) to (8,7), A l ( 8,7) (0,10) (8, ) x (m) nd using the Eqution 6-4 7
i j k l 8 0 18i 48 j N 0 0 ) n the sme wy, fo ny shpe of conducto, l ( 8,11) (9,) ( 1,9) i j k l 1 9 0 6i 4 j N 0 0 c) Since it s closed conducto, l (0,0) nd 0. nd 6.4 Action of unifom mgnetic field on flt loop. Mgnetic moment A pticul cse of foce on conducto flowed y cuent is tht of flt loop with cuent inside mgnetic field. Let s conside ectngul loop like tht of igue 6-8, with sides of length nd. An intensity of cuent is flowing long the loop, nd it s plced inside unifom mgnetic field. The foce cting on ech side of loop comes fom Eqution 6-4, ut the net foce cting on loop is zeo, since loop is closed conducto. Since the vecto length fo two pllel sides is the sme ut with opposite sign, foces on sides of loop e cncelled y couples: 1 ( ) 1 ( 0 4 S 4 ) 1 4 S θ igue 6-8. Rectngul loop of sufce S, inside unifom mgnetic field n these equtions, nd vectos hve een chosen in such wy tht the sense of thei coss poduct is elted to the sense of the intensity of cuent ccoding with the ight hnd o scew ule. esides, this e vecto is pependicul to the loop nd its mgnitude equls the e of the loop S. 8
9 Although the net foce is zeo, the loop is sujected to toque ; if we compute the toque due to ech foce with espect the cente of loop: 1 1 τ τ τ 4 4 τ The net toque, due to the set of fou foces, tking in ccount tht these foces e cncelled y couples nd the eqution fo the doule coss poduct: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ) ( 4 1 4 1 τ τ τ τ τ t s necessy to ememe tht s well s the esult of foce on system is n cceletion, the esult of toque on system is n ngul cceletion, nd theefoe, the loop will tun ound its xis due to the computed net toque τ. Since S : S τ The poduct of intensity of cuent y the e vecto is clled mgnetic moment of the loop: S µ t s mesued in Am Eqution 6-5 To finish, the toque cting on loop emins finlly: µ τ t s mesued in Nm Eqution 6-6 Theefoe, when the mgnetic moment vecto o the e of the loop vecto e pllel to the pplied mgnetic field, the toque will e zeo nd the loop will e in dynmic equiliium (stle lnce if they e pllel nd unstle lnce if they e ntipllel). f the loop is plced with its mgnetic moment in ny othe diection, toque will ppe tuning the loop until mgnetic moment vecto emins pllel to pplied mgnetic field. Equtions fo squed loop cn e extpolted to ny othe loop with ny shpe, with the sme esult. The only equiement is tht the loop must e flt loop (it should e possile to plce the loop in plne). These equtions fil if the loop wsn t flt. Anywy, these equtions e lso pplicle to coil of N tuns, only multiplying y N: mgnetic moment of coil S N µ, nd toque NS µ τ On unit 0, definition of toque cn e found.
Exmple 6- Along the loop of the figue of sides, nd c, flows n intensity in the shown diection. The loop is plced inside mgnetic field j. ind: ) Mgnetic foces on sides of the loop. ) Mgnetic moment of the loop. c) Toque cting on loop. z Solution c y x ) The foce on conducto of length l with cuent inside mgnet- l. Theefoe on ech side of loop we ll hve: ic field is ( ) ( ) ( k cj ) j ) i ( k j ) i c ( j j ) 0 ) Mgnetic moment of loop is defined s S 1 1 µ. As S c ( i ) ci 1 1 µ S c( i) ci c) And the toque 1 1 τ µ ci j ck Theefoe, the toque will ty to tun the loop ound the OZ xis (in clockwise sense when it s seen fom upside). Exmple 6-4 A coil is mde up y 100 ectngul loops hving 6x4 cm of e. t s tuned s it ppes on pictue, foming 7º ngle with the y xis. A,5 A cuent flows long coil. ) Which is the mgnetic moment of the coil? ) Which is the toque cting on coil when T mgnetic field is pplied in the diection of the y xis in positive sense? 6 cm z 4 cm 100 vuelts y α x 10
Solution ) The e vecto is pependicul to the plne of coil, eing its sense ccoding to the scew ule fo the intensity of cuent. S S(cos7i sen7 j) 4(0.8i 0.6 j)cm x S And mgnetic moment µ NS i j k ) τ µ 0,48 0,6 0 0,96kNm 4 50 4 10 (0.8i 0.6 j) (0.48i 0.6 j 0 0 )Am 7º Theefoe, the toque will tun the coil in counteclockwise sense. y Electic engine An electic engine is mde up y coil (N loops) le to tun ound fixed xis. When cuent flows long this coil nd mgnetic field is pplied to coil, toque cts N S S τ NSsenα Mgnetic field cn e pplied in some diffeent wys, ut fo little engines pemnent mgnet cn e used; so, mgnetic field cn e supposed s unifom. This toque will poduce tuning to lign the e vecto pllel to mgnetic field (α0 nd τ0); equiliium position would e eched fte few oscilltions due to the ineti of coil. ut when equiliium position is eched, the sense of cuent on coil is chnged though the ushes (two hlf ings t the end of coil, whee teminls of ttey e gliding). f sense of cuent is inveted, e vecto is inveted, nd mgnetic moment - igue 6-5. Electic engine. The mgnetic field doesn t chnge its diection, pefoming the e vecto whole tun Clockwise tun m m α α Equiliium, inetil tuning α0 m Clockwise tuning fte the ushes hve chnged the sense of cuent 11
emins opposite to mgnetic field (ntipllel) (α180º nd τ0). n this sitution, the coil tuns hlf of tun in ode to get mgnetic moment pllel to mgnetic field. When this position is eched, the ushes chnge the sense of cuent, nd the pocess is epeted, in such wy tht equiliium is neve eched, nd engine is lwys tuning. The ineti of coil enles tht the engine don t stop t the exct position whee toque is zeo. The vege toque long hlf of tun cn e clculted s τ 1 And the tnsfomed powe y the engine is τ om this eqution, ssuming thee en t mechnicl losses nd using the given definition fo contelectomotive foce of ecepto Whee k is constnt only depending on the geomety nd nume of tuns of engine,. Then, contelectomotive foce is linely elted to nd (ngul speed). 6.5 Hll Effect Hll Effect is poduced when conducto cies n electic cuent in mgnetic field. As we ledy know, this sitution poduces foce cting on conducto (section 6.), ut lso poduces the Hll Effect. Hll Effect is vey difficult to detect, since it poduces diffeences of potentil in the ode of µv (10-6 V) nd usully it s neglected. ut in some cses, Hll Effect is used to uild mesuement instuments, s hppen with instuments to mesue mgnetic fields (teslmetes). esides, Hll Effect demonsttes tht it isn t tue tht positive chges moving in sense ws completely equivlent to negtive chges moving in opposite sense. Even this equivlence hs een useful when studying diect cuent, Hll Effect leds to diffeent esults when positives o negtives chges e supposed. Hll Effect is the esulting equiliium stte etween mgnetic nd n electic field, s we e going to study. 1
Let s conside conducto flowed y cuent in mgnetic field pependicul to cuent, s cn e seen on igue 6-7. n ccodnce with Eqution 6-, foce pependicul to mgnetic field nd velocity of chges ppes ( M qv ); if we suppose tht electons (negtive chge) e moving, this foce is pointing to down, nd electons will e deflected to down when moving. So, in lowe side of conducto we ll find lot of electons (negtive chge), nd lck of them in uppe side (positive chge). So, n electic field (E) ppes on conducto, pointing to down (fom positive chge to negtive chge), nd the electons moving on conducto will suffe n electic foce to up ( E qe). This electic field poduces diffeence of potentil etween the uppe nd the lowe side of conducto (in the pependicul diection oth to cuent s to mgnetic field), clled C Hll voltge (V H ). f we suppose electic field s unifom, V E d l E. This voltge is vey little (of the ode of µv), nd cn e detected with only vey ccute instuments (its polity is tht shown on pictue). n the equiliium stte, mgnetic foce nd electic foce e in equiliium nd the electons E VH move without deflection. Then, it ll e: qv qe v v As dift speed (v ) is elted to intensity of cuent in ccodnce to eqution. (nqsv ), eing S the coss section of conducto pependicul to intensity of cuent, then V v z y x VHnqS H Eqution 6-7 f we tke piece of conducto, n, q, S nd e constnts. The intensity of cuent cn e mesued with n mmete nd so, mgnetic field is diectly elted to V H. Mesuing this Hll voltge with n ccute voltmete, mgnetic field cn e computed, nd teslmete cn e uilt. ut if we conside tht chges moving inside conducto e positive chges moving in opposite sense (to ight), then the mgnetic foce lso cts to down, ppeing positive chge on lowe side of conducto nd negtive chge on uppe side. So, polity of Hll voltge is opposite to tht found when negtive chges moving on conducto wee supposed. This diffeence supposing the chge cies e positive o negtive chges moving in opposite sense will e useful when we study semiconducto mteils. v E M C A E igue 6-6. Hll Effect on conducto H A V H - 1
Exmple 6-5 A piece of silve with 5 cm of width () nd 0,5 mm of thickness (e) is plced in T mgnetic field, such s it ppes on the figue. Which is the Hll voltge if 00 A e flowing long conducto? You cn suppose tht silve hs 0,65 fee electons y tom (vege vlue). Dt: Density of silve 10,5 g/cm ; mol mss of silve 107,9 g/mole; Avogdo s nume 6,0 10 tom/mole. z y x Solution e As we e deling with conducto, negtives chges will move inside, nd the mgnetic nd electic foces cting on electons e those dwn on pictue: M 0 q(v ) qe x z y v E M E E H - V H n ode to pply Eqution 6-7, the density of electons y unit of volume (n) on silve must e computed. t cn e done fom dt of density (ρ), mol mss (m m ) of silve nd Avogdo s nume (Av): g ρ( ) e e tom 0,65 6,0 10 10,5 ( ) 0,65( ) ( ) cm n Av,8 10 cm tom mol g m ( ) 107,9 m mol And fom Eqution 6-7: 8 e m V 5 10 00 H 11 µ V 8 19 nqs,8 10 1,6 10 5 10 0,5 10 14
6.6 Polems 1. ind the mgnetic foce cting on poton moving t 4 10 6 m/s in the positive sense of the X xis, inside T mgnetic field in the positive sense of Z xis. (Dt: q(p) 1,6 10-19 C). Sol: -1,8 10-1 j N. Point the initil diection of the deflection of chged pticles on pictue. ) ) c) ent to up to ight d) e) 45º exiting f) to ight Negtive chge exiting Positive chge enteing Sol: ) upwds; ) pependicully to ppe, going out; c) doesn t deflect; d) pependicully to ppe, enteing; e) doesn t deflect; f) downwds.. A undle of electons move etween the pltes of cpcito with diffeence of potentil V. etween pltes thee is unifom mgnetic field pependicul to the electic field. f pltes of cpcito e septed distnce d, clculte the speed of the electons not deflecting when they move etween pltes. Sol: vv/d E Y X 4. A long conducto, pllel to the X xis, cies cuent of 10 A in the positive sense of X xis. Thee is unifom mgnetic field of T in the diection nd sense of the Y xis. ind the foce y unit of length cting on conducto. Sol: 0k N/m 5. y the segment of conducto in the figue flows cuent A fom P to Q. t exists mgnetic field 1k T. ind the totl foce cting on conducto nd pove tht it is the sme tht if the entie conducto ws stight segment fom P to Q. 8i -6 j 10 Sol: ( ) N Z Y P Q C X PC cm CQ 4 cm 15
6. Along conducto AC of figue flows cuent of 10 A (it s pt of n electic cicuit), eing le to glide long two veticl ods. Compute the necessy unifom mgnetic field, pependicul to the plne of the figue, in ode tht the mgnetic foce on A C conducto could equilite the gvittionl foce. Which should e the sense of mgnetic field? The length of conducto is 10 cm nd its mss, 0 g. Sol: 0,196 T exiting fom ppe to ede. 7. Along flt conducto of ity shpe flows cuent, eing the conducto inside unifom mgnetic field pependicul to the plne of conducto. Pove tht the totl foce cting on the piece of conducto going fom point to nothe is Lx,whee L is the vecto going fom to. 8. The figue shows one of the ectngul loops of 10 cm y 5 cm of coil with 0 tuns. The coil hs hinges on side long Z xis, nd 0,1 A cuent is flowing. Which is the toque cting on coil (modulus, diection nd sense) if it is foming 0º egding the diection of unifom mgnetic field 0,5 j T? Sol: - 4, 10 - k Nm X Z 0º Y 9. n ode to mesue mgnetic field, coil with 00 tuns of 14 cm of e nd foming n ngle of 0º with the field is used. When flowing n intensity of 0,7 A, toque of 980 10-6 Nm is mesued. Compute. Sol: 5,7 10 - T. Z S 0º Y 10. Let s conside the ectngul loop on pictue, with sides nd, nd flowed y n intensity in the shown sense. The loop is in no unifom mgnetic field 0 k. Clculte the foces cting on sides 1 nd. x Sol: i ln( ) 1 0 0 j Y 1 X 16
GLOSSARY Mgnetic foce on moving chge. t s the foce cting on chge q when moving with speed v in mgnetic field. qv Tesl: On point of the spce thee is mgnetic field of 1 tesl if point chge of 1 coulom moving pependicully to the mgnetic field with speed of one mete y second, suffe foce of one newton pependicully to mgnetic field nd velocity. oce on n element of cuent: oce cting on conducto of length dl flowed y cuent in mgnetic field d ( dl ) Mgnetic moment of loop: it s the poduct of the intensity flowing long the loop y the e vecto. ts sense is tht of the e vecto, tht is, the esulting to pply the ule of the scew o the ight hnd to the sense of intensity of cuent on loop. m S Toque on loop: t s the coss poduct etween the mgnetic moment of the loop nd the mgnetic field τ µ Hll Effect. Voltge ppeing when n electic cuent flows long conducto in mgnetic field. This voltge is vey little nd it s diectly elted to intensity of cuent nd mgnetic field V H v eing v the dift speed of chge cies, the mgnetic field nd the width of conducto. 17