CHAPTER 29: Magnetic Fields and Sources

Transcription

1 CHAPTER 29: Magnetc Felds and Sources photo courtesy of Mr. Whte 1.)

2 Relatvty and Magnetsm --ewton s Classcal Mechancs demands that the speed of lght depend upon the relatve moton between the frame of reference of the lght source and the observer s frame (thnk of about a passng car s speed on the freeway how fast t passes you depends upon how fast you are gong). --Yet because alternatng magnetc felds (-flds) nduce electrc felds (E-flds), and alternatng E-flds nduce -flds, the only way an alternatng E-fld can coupled wth an alternatng -fld to produce an electromagnetc wave (.e., a wave n whch the alternatng E-fld feeds the -fld and the alternatng -fld feeds the E- fld), s f, accordng to Maxwell s equatons, the wave s velocty s 3x10 8 m/s, the speed of lght. --You can t have t both ways. The speed of lght s ether frame-of-reference dependent alla ewton, or t s a fxed value ndependent of frame-of-reference alla Maxwell. --Ths conundrum s what motvated Ensten to develop hs Theory of Specal Relatvty, a form of Mechancs that assume that the speed of lght s the same n all frames of reference. 2.)

3 --One of the stranger characterstcs of Specal Relatvty s that f you have an object that s approachng at relatvstc speeds, t wll length contract. And, n fact, ths length contracton phenomenon wll occur even at classcal speeds (though observng t at classcal speeds s dffcult). --Wth ths n mnd, consder a wre wth conventonal current flowng through t (.e., assume postve charge flow). If you fre a postve charge opposte the drecton of current flow, an nterestng thng s observed. The charge wll feel a force that motvates t to veer away from the wre. So what s gong on? path taken q + v q 3.)

4 As postve charge carrers move onto the wre (wth electrons assumed statonary), an equal number of postve charge carrers move off the wre. That means the number of postve and negatve charge stays even throughout tme, the wre stays electrcally neutral, and there s no good reason for the movng test charge to feel a force. ut t does. Ths led early theorsts to conclude that there must be a force, a magnetc force, affectng the movng charge... except that sn t what s really happenng here. To see ths, we have to look at the stuaton through the perspectve of the movng charge. q + q + v q e s statonary v + e - e - e - e - e - e - e - e - e - e - p + p + p + p + p + p + p + p + p + p + assume velocty of postve charge s v + 4.)

5 In the frame of reference attached to q +, q + s not movng. What s more, as far as q + s concerned, both the electrons and the wre are seen to be movng to the left wth velocty v q, and the protons are seen to be movng to the left wth velocty The sketch below show all of ths. v q + v +. electron and wre velocty: q + (statonary) v q e - v q + v + p + proton velocty: 5.)

6 What s mportant to notce here s that because the protons are movng faster than the electrons, they wll length contract more than wll the electrons. When they do so, from the movng charge s perspectve, there appears to be more protons on the wre than electrons. q + e - e - e - e - e - e - e - e - e - e - e - e - p + p + p + p + p + p + p + p + p + p + p + p + p + p + Protons length contract more than electrons. More protons means the wre looks electrcally postve and an electrc feld s set up pontng outward from the wre. 6.)

7 Wth more protons apparently on the wre, there s an electrc feld generated emanatng outward from the wre. It s that electrc feld that the responds to. q + E-feld due to preponderance of postve charge q + e - e - e - e - e - e - e - e - e - e - e - e - p + p + p + p + p + p + p + p + p + p + p + p + p + p + E-feld due to preponderance of postve charge 6.)

8 In short, what was descrbed by early researchers as a MAGETIC EFFECT was (and s) really a RELATIVISTIC EFFECT. Stll, the Classcal Theory of Magnetsm s a good theory n the everyday world (just as s the case wth ewtonan Mechancs), so that s what you wll be spendng the next several weeks learnng. 7.)

9 General Informaton Electrc Felds --electrc felds (abbrevated as E-flds), wth unts of newtons per coulomb or volt per meter, are modfed force felds (release a charge n an E-fld and t wll accelerate); --electrc felds are generated wth the presence of charge; --an electrc feld s drecton s defned as the drecton a postve charge wll accelerate f released n the feld; --electrc feld lnes: --go from postve to negatve charge; --dentfy the E-fld s drecton n a regon; --are closer together where E-flds are more ntense; Magnetc Felds --magnetc felds (abbrevated as -flds), wth unts of teslas n the MKS system, are OT modfed force felds (release a charge n a -fld and t wll just st there); --magnetc forces do exst when a charge moves through a -fld they are centrpetal and are governed by the relatonshp:! F = q! vx! ---felds are generated by charge n moton; --a -feld s drecton s defned as the drecton a compass ponts when placed n the feld; --magnetc feld lnes: --go from north to south pole, or crcle around current carryng wre; --dentfy the -fld s drecton n a regon; --are closer together where -flds are more ntense; 9.)

10 Termnology and the Compass Place a compass n the earth s magnetc feld. The compass end that orgnally ponted toward the north geographc hemsphere was called the north geographc seekng pole. earth compass north geographc pole south magnetc pole Wth tme, the word geographc was dropped leavng the north seekng pole, and wth even more tme, the seekng was dropped leavng us wth north pole. north magnetc pole south geographc pole north geographc seekng pole Problem s, north poles are attracted to south poles, whch means that gven the defnton, there must be a south magnetc pole n the north geographc hemsphere. ot very esthetcally pleasng, but that s lfe (and t ll swtch drectons n another 300,000 or so years due to slow oscllatory patterns n the earth s magma). 10.)

11 ar Magnets If magnetc felds are generated by charge n moton, where s the moton assocated wth a bar magnet? An atom s spn quantum number hghlghts the fact that electrons spn up or down, dependng. In most atoms, approxmately the same number of electrons spn n one drecton as the other, but n certan atoms (the ferromagnetc ones), they spn consderably more n one drecton than the other. electron electron spn about an axs producng a -fld The most promnent example of a ferromagnetc materal s ron, wth sx more electrons spnnng n one drecton than the other. As such, EACH IRO ATOM IS A MII-MAGET UTO ITSELF. 11.)

12 So how can a pece of ron be magnetc under some condtons and not under other condtons (there are, after all, ron nals that do not exhbt magnetc characterstcs at all). (The explanaton s called Ampere s Theory of Magnetsm.) Enter the magnetc doman. What happens s ths: Atoms wthn rregular, mcroscopc regons, called domans, algn themselves so that all of ther magnetc felds are n the same drecton. When the domans are themselves algned, the materal acts lke a magnet. When the domans are OT algned, the net magnetc effect s lost (n some cases, all that s needed to de-magnetze a magnetc s to have thermal agtaton shake the domans out of algnment). In any case, the frst sketch below s wthout algnment, the second wth algnment. domans unalgned (not realstc renderng) domans algned sketches courtesy of Mr. Whte 12.)

13 Magnetc Force When charge moves through a magnetc feld, t may or may not feel a force, dependng upon ts moton. If present, that force wll be:! F = q vx!!! The magntude s F = q v!!! sn θ, where q s the sze of the charge, v s the magntude of the velocty vector,! s the magntude of the magnetc feld and s the angle between the lne of the two vectors. θ The drecton s determned usng the rght-hand rule. sketch courtesy of Mr. Whte 13.)

14 Example 1: (courtesy of Mr. Whte) Identfy the mssng vector n these dagrams. Assume red sgnfes negatve charges, blue postve charges. The responses are n green v v F v v F v v F F F v F F F v 14.) v zp

15 Fne Prnt for! F = q vx!! The rght-hand-rule determne the drecton of force for a postve charge. Magnetc forces are centrpetal forces actng perpendcular to magnetc felds (whereas electrc forces act along electrc felds). That means magnetc forces DO O WORK on charges that feel ther effect. Magnetc forces are experence only by charges n moton. 15.)

16 Example 2: A proton movng upward wth speed 5x10 6 m/s n a magnetc feld feels a force of 8x10 14 to the west. When movng horzontally to the north, t feels no force. Fnd the magntude and drecton of the magnetc feld n ths regon. (courtesy of Mr. Whte) ecause no force s felt when the proton s movng northward, the magnetc feld must ether be toward the north or the south. Reverse engneerng the rght-hand-rule suggests f the feld s to the north*, the velocty vector would have to be out of the page to generate a force to the west, wth a magntude of: F = qv! sn90 o = F qv W ( 8x10 14 ) = ( 1.6x10 19 C) 5x10 6 m/s = 10 1 T ( ) *ote that f the feld was to the south, the velocty vector would be nto the page. 16.)

17 Example 3: A charge q of mass m s movng wth constant velocty v at rght angles to a magnetc feld. (dea courtesy of Mr. Whte) a.) What knd of moton wll t execute? ecause magnetc forces are centrpetal, the mass wll follow a crcular path. b.) What s the radus of the moton s path? F cent : q! vx! = m! a cent qvsn90 o = m v2 R R = mv q F v 17.)

18 c.) What s the perod of the moton? v F T = 1 ν 1 = ( ω ) = 1 ( v ) 2π R 2π = 2π ( v ) R 2π = v / mv q T = 2π m q 18.)

19 Lorentz Relatonshp When both magnetc and electrc forces act, the relatonshp looks lke:! F net = F E + F = q! E + q! vx! 19.)

20 Example 4 (the velocty trap): A velocty trap (or velocty selector) s an electrc feld and magnetc feld at rght angles to one another that selectvely allow charged partcles wth one velocty only to proceed down ts axs. It s made up of parallel plates bathed n a -fld (see sketch). -q v a.) Assumng the -fld s nto the page as shown, what path wll a negatve charge take f nothng addtonal s added to the system (.e., no E-fld s present). Accordng to the rght-hand-rule, a postve charge would feel a force upward, so a negatve charge wll feel a force downward. b.) Insert the E-fld requred to make the charge travel along a straght lne. The force exerted by the -fld s downward, so the force exerted by the E-fld must be upward. To create an upward force on a negatve charge, you need an E-fld that s downward. 20.)

21 c.) What velocty wll make t through the trap? Ths s a Lorentz relatonshp problem. The net force must be zero for ths to work, whch means: qe = qv -q v v = E 21.)

22 Example 5 (mass spectrometer): An unknown mass s volatalzes (made nto a gas), had ts molecules sngly onzed (had one electron strpped away), accelerated through a potental dfference to gve them velocty, and sent through a velocty trap made up of a 95,000 V/m E-fld and a.93 teslas -fld. The E molecules that make t through the trap move nto a regon n whch there s only the -fld. a.) What s the velocty of the molecules that make t through the trap? qe = qv v = E V = 9.5x104 m.93 T = 1.02x10 5 m/s v 22.)

23 b.) Draw n the path of the molecules n the far rght chamber. They wll crcle upward. c.) If the radus of the arc s observed to be.0667 meters, what was the mass of the partcle E (that s what these devces are desgned to do determne the mass of an unknown materal from whch the dentfcaton of the materal can be had). In the regon n whch there exsts only a -fld: qv = m v2 R m = qr v = 1.6x10 = 9.7x10 26 kg ( 19 C).93 T v R ( )( 6.67x10 2 m) ( 1.02x10 5 m/s) (Ths s the molecular weght of table salt.) 23.)

24 Pont of Order Magnetc forces exst on charges movng through magnetc felds OLY when the charges cut across magnetc feld lnes. Untl now, the only examples we ve vewed have been stuatons n whch the charges have cut across at rght-angles to the feld lnes. What happens when they cut across at an angle? --The velocty component perpendcular to the -fld wll generate a magnetc force that wll motvate the charge to crcle (magnetc forces are centrpetal n nature); --The parallel component wll smply provde momentum for the charge to contnue to move n that parallel drecton. --The net effect s that the charge wll helx along the -fld lnes. v v θ! v 24.)

25 -flds and Current Carryng Wres Consder a current movng nto the page that s postoned between the poles of a horseshoe magnet. What effect wll the wre feel due to beng n the -fld. S F --Accordng to the rght-hand-rule, conventonal current s the moton of postve charge. Postve charge nto the page n a magnetc feld to the left produces a magnetc force upward. --Mathematcally: otcng that a charge s velocty v s the dstance t travels L per unt tme, and current (q/t) s the charge q that passes by per unt tme, we can wrte:! F = q! vx! F! wre = q! L t =! Lx! x! 25.)

26 Example 6: (courtesy of Mr. Whte): A 12 cm length of wre carryng a 30-Amp current s placed n a magnet feld at an angle of 60 relatve to the feld s drecton (as shown). If the feld s 0.90 T, what force does the wre feel, and n whch drecton? a.) On the sketch, show the drecton of the force on the wre. Accordng to the r.h.r., t s out of the page. I F b.) Determne the magntude of the force. F = L = 30 A = 2.81 ( )(.12 m).9 T ( )sn60 o 26.)

27 Example 7: (courtesy of Mr. Whte): A 10 cm 10 cm square loop of wre hangs vertcally, as shown here. When the current n the wre s A counterclockwse, a scale supportng the wre measures a downward force of 3.48x10 2. Fnd the magntude of the magnetc feld. I Accordng to the r-h-r, the force wll be downward. Its magntude wll be: F = L 3.48x10 2 =.245 A = 1.42 T ( ).10 m ( )sn90 o ote: Ths could easly have been turned nto a torque problem... just sayn )

28 Example 8: Two metal ramps (n red) at an angle θ are d unts apart and are bathed n a downward -fld. A battery wre s connected to each ramp makng a crcut. A metal rod would slde frctonless down the ramp f t were not for the magnetc force provded by the -fld. In fact, n ths case the rod s motonless. Assumng the net resstance of the system s R: a.) What must the polarty of the battery be f the rod s to stay statonary on the ramp (agan, assume the rod s frctonless). For equlbrum, you can see from observaton that you need a component of the magnetc force to the left to counter the component of gravty to the rght. Reverse engneerng F! = Lx!! F mg yelds the need for a current nto the page to generate a magnetc force n that drecton... So the polarty of the battery must be hgh-sde on left. d θ θ 3-d vew: V sde vew: 28.)

29 b.) How bg must the battery voltage be to effect ths stuaton? Dong a f.d.b. and breakng the forces nto components, we can determne : dsn90 o θ F y : mg + cosθ = 0 = mg cos θ F x : d + snθ = 0 = sn θ d mg cos θ = d mg tan θ = d ( ) sn θ V = R = mg From Ohm s Law: mg tan θ d θ R 29.)

30 Another Pont of Order The relatonshp for the force on a wre n a -fld looks lke:! F wre =! Lx! but ths only works when L s a straght wre, s constant and the angle between the two vectors remans the same throughout. If any of those parameters vary, you need to use an ntegral. Specfcally: --defne a dfferental dsplacement ds along the wre; --cross ds nto the magnetc feld evaluated at ds; --execute the ntegral:! F wre = a b d! sx! a ds b 30.)

31 Galvanometer and Torque on a Current-Carryng Loop Consder a current-carryng loop of wdth a and heght c, pnned at the top and bottom, bathed n a magnetc feld. How wll the current respond to the -fld? Look down from the top and gnore the bllowng of magnetc feld generated by the bar magnets. In that vew, the current moves across the upper secton of wre as shown, moves down nto the page n the secton on the rght, and moves out of the page n F the secton on the lower left. The r.h.r. suggests forces on those sectons as shown. pn c pn pn top vew n a S S 3-d vew out F 31.)

32 A torque s beng generated about the pn due to the two forces (ah, revew!). That torque s:! τ = 2! rx! F = 2! r! F snθ = 2 a 2 ( c)sn90 0 = ( ac)sn90 0 Ths s for one wre. If there were wres n the loop, and f we note that ac s the area A of the loop, the net torque could be wrtten at:! τ = Asn90 0 Although ths s not somethng I thnk the AP folks are lkely to ever test on, f we defne a vector drected perpendcular to the face of the col (thumb of rght-hand crclng the loop n the drecton of the current dentfes orentaton) wth a! magntude equal to µ = A, (ths s called the magnetc moment), then the torque due to the loop becomes:! τ =! µx! c F pn pn a F 3-d vew S! µ S 32.)

33 Asde from gvng physcs teachers an excuse for havng students do torque calculatons n an E&M secton, the real usefulness of all of ths qute cool. If you put a pnned col n a magnetc feld, attach a sprng to t to provde a restorng torque, then attach a ponter hung over a scale, you have the makngs of a meter. 5 0 S 5 And, n fact, that s exactly how a GALVAOMETER s made t s a col suspended n a magnetc feld wth a sprng attached to t to provde a restorng torque, so that when you put 5x10 4 amps through t, the torque provded by the movng current n the -fld coupled wth the restorng torque sees the ponter swng full deflecton over the scale... and that s very cool )

34 Example 8: (courtesy of Mr. Whte): A rectangular col of dmensons 5.40 cm x 8.50 cm conssts of 25 turns of wre and carres a current of 15.0 ma. a.) Calculate the magnetc moment of the col, puttng the drecton onto the sketch.! µ = A ( ).054 m = ( 25) 15x10 3 A = 1.72x10 3 A m 2 ( )(.080 m)! µ Crclng the rght hand n the drecton of the current yelds a thumb-pont leftward (the drecton should be perpendcular to the face of the col) 34.)

35 b.) If a magnetc feld of T s appled parallel to the plane of the loop, what s the magntude of the torque actng on the loop?! τ =! µx! =! µ! snθ ( ) 0.35 T = 1.72x10 3 T m 2 ( )sn90 o = 6.02x10 4 m (drecton downward) c.) Calculate the magntude of the torque on the col when the T -fld makes an angle of 60! o and 0 o wth µ.! τ =! µ! snθ ( ) 0.35 T = 1.72x10 3 T m 2 = 5.21x10 4 m ( )sn60 o! µ!! µ!! µ!! τ = 0 as angle! between and µ! s zero. 35.)

36 Electrc Motors (courtesy of Mr. Whte) Electrc motors convert electrcal energy to knetc energy, and are created by placng a current-carryng loop n an external magnetc feld. There are a number of dfferent ways of dong ths, but here s one common type: S I S ote from Fletch: So current n motvates rotaton (hence a motor). 36.)

37 Electrc Generators (courtesy of Mr. Whte) Electrc generators convert mechancal energy (work provded by an external source) to electrcal energy. Motors and generators, n most cases, have the same physcal structure. (ote from Fletch: In other words, as far as energy converson goes, a motor s just a generator run n reverse. ) S S ote from Fletch: So rotaton motvates a current (hence a generator). 37.)

38 Other Devces A lttle more sophstcated verson of a motor requred one bt of nformaton that would normally not be covered untl next chapter. It s charge n moton that generates magnetc felds. Wth current carryng cols, the generated magnetc felds are down the axs and through the face of the col. A handy trck to determne the drecton of a current carryng wre s -fld s to lay your rght hand on the col wth your fngers followng the drecton of current n the col. The drecton n whch your extended rght-thumb ponts dentfes the drecton of the col s -fld. drecton of -fld ths end of col acts lke a north pole ote that wth the -fld extendng along the axs as t does, the col s ends look lke north and south poles. Wth that, consder the followng: 38.)

39 o S Follow the current from the battery, through the brushes to the col, then determne the -fld due to the col s current (see below). o S Fngers of rght hand curl along lne of current; thumb dentfes drecton of - feld (wth - feld lnes extng ths end, so t s a orth Pole) V o 39.)

40 o S otce the attractons between the poles... these cause rotaton... S o V o 40.)

41 o More attractons, more rotaton... S S o V o 41.)

42 rushes not n contact current n cols about to change drecton momentum carres moton... S V o 42.)

43 o otce the repulson between the poles... stll gettng rotaton... S S o V o 43.)

44 S o ack to attracton... S o V o Ths s a DC motor )

45 What s t? What you are lookng at here s the crcut for an old-fashoned door bell. See f you can follow through to see how the mechansm works (note the drecton of the magnetc feld n the green horseshoe electromagnet, and the drecton of the nduced magnetc feld n the blue bar opposte the poles of the horseshoe magnetc, when the current flows). ron bell 45.)

46 What s t? Ths s the desgn for a loud speaker (actually, the col and magnet are usually swtched n real lne). Ths s how t works: a.) Let s assume we want to project a 256 Hz (mddle C) sound wave nto the room. The sgnal would look lke the sne wave shown below. S col cone (constraned at the edges) 46.)

47 b.) Durng the frst half of the cycle, let s assume the drecton of the tme-varyng current through the col s drected as shown on the sketch. There are two thngs to note:.) eng snusodal, the current wll ncrease to a maxmum, then wll drop back down to zero whereupon the drecton wll change and the current wll agan ncrease to a maxmum n the opposte drecton, then proceed back to zero. Ths pattern wll contnue through tme. S col ( t) cone (constraned at the edges) 47.)

48 .) Wth the current movng n the drecton noted, the drecton of the nduced magnetc feld n the col (alternate rght-hand rule) wll leave the sde of the col closest to the permanent magnet a orth Pole (see sketch). c.) The north pole of the permanent magnet wll nteract wth the nduced north pole of the current carryng col, and the net effect wll be a repulson experenced by both the col AD the cone. As the cone s fxed at t s outsde edge, ths wll flex the cone outward wth the amount of flex beng dependent upon the sze of the current at the gven nstant. S col ( t) cone (flexes out) cone (constraned at the edges) d.) As the cone flexes outward, t wll compress ar nto a hgh pressure regon. That pressure rdge wll travel away from the cone at approxmately 330 m/s, or the speed of sound n ar. 48.)

49 e.) As the current proceeds down toward zero, the cone wll relax, pullng back. f.) When the current drecton changes, the drecton of the nduced magnetc feld n the col wll change and the pullng back wll proceed through equlbrum and nto a flexng nward. The degree of flexng wll, agan, depend upon how much current to movng through the col at a gven nstant. g.) As the cone flexes nward, t wll create a rarfed regon of ar generatng a low pressure regon. That regon wll travel away from the cone at approxmately 330 m/s, or the speed of sound n ar. S col ( t) S cone (flexes n) cone (constraned at the edges) h.) Ths flexng outward, then nward, then outward wll occur at the current frequency, or 256 Hz n our example, and the pressure varatons wll pass by your ear at a frequency of 256 Hz. That, n turn, wll wggle the lttle hars n your ears creatng electrcal mpulses that your bran wll nterpret as sound. Clever of nature, eh? 49.)

50 HALL EFFECT An experment to prove that negatve charges moves through electrcal crcuts when current flows. How normal current flows through the plate. ROAD, THI, METAL PLATE R V o 50.)

51 What happens when the plate s permeated by -fld. x x x x x plate permeated by -fld x x x x x x x x x x x x x x x R V o 51.)

52 ... f POSITIVE charge s assumed to flow through crcut? postve charge flow suggests upper sde of plate wll be hgh voltage sde x x x x x postve charge flow x x x x x x x x x x x x x x x R postve charge flow V o 52.)

53 ... f EGATIVE charge s assumed to flow through crcut? negatve charge flow suggests upper sde of plate wll be low voltage sde x x x x x negatve charge flow x x x x x x x x x x x x x x x R negatve charge flow V o 53.)

54 So back to the postve charge-flow assumpton: _ If there exsts a predomnance of postve charge on the upper sde of the plate makng that sde of the x x x x x plate hgher voltage than the x x x x x bottom sde, then placng a voltmeter wth ts + and termnals x x x x x as postoned would produce a meter readng that was sensble x x x x x (that s, the needle would swng n R the approprate drecton). V In fact, f ths experment was done, the needle would swng n the wrong drecton. In other words, the the meter s hooked up wrong. What are accumulatng on the upper sde of the plate are not postve charges, they are EGATIVE charges. In short, t s negatve charge that flows through crcuts. V + 54.)

55 3 Types of Magnetsm Ferromagnetsm - materal has a permanent magnetc moment, due to mcroscopc domans n whch moments are algned. Paramagnetsm - materals has a small magnetc susceptblty, that only becomes evdent when placed n an external magnetc feld. Damagnetsm - materal does not have permanent magnetc moments. In the presence of an external magnetc feld, a weak magnetc moment s nduced n a drecton opposte to the external feld. 55.)