Electric & Magnetic Particle Motion

7/27/2019 Electric & Magnetic Particle Motion

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Chapter2ParticleMotion inElectricandMagneticFieldsConsideringEandBtobegiven,westudythetrajectoryofparticlesundertheinfluenceofLorentzforce

F=q(E+vB) (2.1)2.1 ElectricFieldAlone

dvm =qE (2.2)

dtOrbitdependsonlyonratioq/m. UniformE uniformacceleration. Inone-dimensionz,Ez trivial. Inmultipledimensionsdirectlyanalogoustoparticlemovingunder influenceof

qgravity. AccelerationgravitygmE. Orbitsareparabolas. Energy isconservedtaking

z

x

__m

qE

Figure2.1: Inauniformelectricfield,orbitsareparabolic,analogoustogravity.intoaccountpotentialenergy

P.E.=q electricpotential (2.3)

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[Proofifneeded,regardlessofEspatialvariation,dv d

ddt

mdt.v

12

mv2

==

q.v=qd

dt(q)

dt (2.4)(2.5)

i.e. 1mv2 +q2 =const.]

Aparticlegainskineticenergyqwhenfallingthroughapotentialdrop-. Soconsidertheaccelerationandsubsequentanalysisofparticleselectrostatically: Howmuchdeflection

=0 a+ /2

E

L

Electrostatic AnalyserParticleSource

DetectingScreen

s

d

z

x

z0

a /2

Figure2.2: Schematicofelectrostaticaccelerationandanalysis.willtherebe? AfteraccelerationstageKE= 1mv2 =qs2 x

vx = 2qs . (2.6)m

SupposingEa,fieldofanalyser,tobepurelyz,thisvelocityissubsequentlyconstant. Withintheanalyser

dvz q q xm =qEa vz = Eat= Ea . (2.7)

dt m m vxSo

q t2 q 1x2z= vzdt= Ea = Ea . (2.8)

m 2 m 2v2xHenceheightatoutputofanalyseris

q 1L2 q 1L2 mzo = Ea = Ea

m 2v2 m 2 (2qs)x=

4

1Es

aL2 =+

4

1

a

sL

d2

(2.9)

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using Ea = a/d. Notice this is independent of q and m! We could see this directly byeliminatingthetimefromourfundamentalequationsnoting

dt

d=v

d

d(=v.) with v= 2q(

ms)

or v=

2m

qs +E

qs(2.10)

ifthereisinitialenergyEs. Soequationofmotionis mq

2q(s)m

dd 2q(s)m ddx=2

s d

d

sdxd =Ea = , (2.11)

whichisindependentofqandm. Trajectoryofparticleinpurelyelectrostaticfielddependsonlyonthefield(andinitialparticlekineticenergy/q). Ifinitialenergyiszero,cantdeduceanythingaboutq, m.

2.2 ElectrostaticAccelerationandFocussingAcceleratedchargedparticlebeamsarewidelyusedinscienceandineverydayapplications.Examples:X-raygenerationfrome-beams(Medical,Industrial)ElectronmicroscopesWelding. (e-beam)SurfaceionimplantationNuclearactivation(ion-beams)NeutrongenerationTelevisionand(CRT)MonitorsForapplicationsrequiring


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Clearly getting the required energy is simple. Ensure the potential difference is right andparticles are singly charged: Energy (eV) Potential V. More interesting question: Howtofocusthebeam? Whatdowemeanbyfocussing?

Object Image

Focal Point

(a) Optical Focussing

Lens

(b) Particle beam Focussing

Parallel Rays

Figure2.4: Analogybetweenopticalandparticle-beamfocussing.What is required of the Lens? To focus at a single spot we require the ray (particle

path) deviation from a thin lens to be systematic. Specifically, all initially parallel raysconvergetoapointifthelensdeviatestheirdirectionbysuchthat

r

Deviation Angle

Distance from axis

Parallel Rays

Figure2.5: Requirementforfocussingisthattheangulardeviationofthepathshouldbealinearfunctionofthedistancefromtheaxis.

r=ftan (2.12)andforsmallangles,,r=f . Thislineardependence(=r/f))ofthedeviation,ondistancefromtheaxis,r,isthekeyproperty. ElectrostaticLenswouldliketohave(e.g.)

EaEr = r (2.13)

abutthelenscanthavechargedsolidsinitsmiddlebecausethebeamsmustpassthroughso(initially)= 0 .E=0. ConsequentlypureEr isimpossible(0=.E= 1(rEr)/r=r

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2Ea/a Ea =0). Foran axisymmetric lens (/ =0)we musthave bothEr and Ez.Perhaps the simplest way to arrange appropriate Er is to have an aperture between two

E1

E2

Potential Contours

Figure2.6: Potentialvariationnearanaperturebetweentworegionsofdifferentelectricfieldgivesrisetofocussing.regions of unequal electric field. The potential contours bow out toward the lower fieldregion: givingEr.Calculating focal lengthofapertureRadialacceleration.

1 2

r

z

Figure2.7: Coordinatesnearanaperture.dvr q

= Er (2.14)dt m

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Sodvrdz =

1vz

dvrdt =

qm

Ervz (2.15)

But.E=0 1

r(rEr)

r +

Ezz

=0 (2.16)Neartheaxis,onlythelinearpartofEr isimportanti.e.

ErEr(r, z)r (2.17)

rr=0

r=0

So1 Er

rEr 2rr (2.18)r

andthusEr Ez

2 + =0 (2.19)r z

r=0andwemaywriteEr 21rEz/z. Then

dvr qr Ezdz =2mvz z , (2.20)

whichcanbeintegratedapproximatelyassumingthatvariationsinrandvz canbeneglectedinlenstoget

final qr 2vr = [vr]initial = [Ez]1 (2.21)2mvzTheangulardeviationistherefore

= vr = +qr [Ez2 Ez1] (2.22)vz 2mvz

2andthefocallengthisf =r/

f = 2mvz2 = 4E (2.23)q(Ez2 Ez1) q(Ez2 Ez1)

WhenE1 isanacceleratingregionandE2 iszeroorsmallthelensisdiverging. Thismeansthatjust depending on an extractor electrode to form an ion beam will give a divergingbeam. Needtodomorefocussingdownstream: moreelectrodes.2.2.1 ImmersionLensTwotubesatdifferentpotentialseparatedbygap InthiscasethegapregioncanbethoughtofasanaperturebutwiththeelectricfieldsE1, E2 thesame(zero)onbothsides. Previouseffectiszero. Howevertwoothereffects,neglectedpreviously,givefocussing:

1. vz isnotconstant.2. r isnotconstant.

Consideranacceleratinggap: q(2 1)


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ExtractionElectrode

DivergingSource

Figure2.8: Theextractionelectrodealonealwaysgivesadivergingbeam.

Region 1 Region 2

Tube

1 2

Figure2.9: AnImmersionLensconsistsofadjacentsectionsoftubeatdifferentpotentials.Effect (1) ions are converged in region 1, diverged in region 2. However because of z-acceleration, vz is higher in region 2. The diverging action lasts a shorter time. Henceoverallconverging.Effect (2) TheelectricfieldEr isweakeratsmallerr. Becauseofdeviation,r issmallerindivergingregion. Henceoverallconverging.

Foradeceleratinggapyoucaneasilyconvinceyourselfthatbotheffectsarestillconverging.[Timereversalsymmetryrequiresthis.] Onecanestimatethefocallengthas

1 3f

16

2q2 E

2z

dz (forweakfocussing) (2.24)r=0

but numerical calculations give the values in figure 2.10 where 1 = E/q. Here E is theenergy inregion1. Effect(2)above,thatthe focussingordefocussingdeviation isweakerat points closer to the axis, means that it is a general principle that alternating lenses ofequal converging and diverging power give a net converging effect. This principle can beconsideredtobethebasisfor

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Figure 2.10: Focal length of Electrostatic Immersion Lenses. Dependence on energy perunitcharge()inthetworegions,fromS.Humphries19862.2.2 AlternatingGradientFocussingIdeaistoabandonthecylindricallysymmetricgeometrysoastoobtainstrongerfocussing.ConsideranelectrostaticconfigurationwithEz =0and

dEx dExEx = x with =const. (2.25)dx dxSince.E=0,wemusthave

dEx dEy dEy dExdx + dx = 0 dy =const Ey = dx y (2.26)

Thissituationarisesfromapotential 1dEx

= x2 2 (2.27)y2 dx

so equipotentials arehyperbolas x2 y2 =const. If qdEx/dx is negative, thenthisfield isconverginginthex-direction,butdEy/dy=dEx/dx,soitis,atthesametime,diverginginthey-direction. Byusingalternatingsectionsof+veand-vedEx/dxanetconvergingfocuscanbeobtained inboththexandy directions. Thisalternatinggradientapproach isveryimportant for high energy particle accelerators, but generally magnetic, not electrostatic,fieldsareused. Sowellgointoitmorelater.

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Image removed due to copyright restrictions.


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2.3 UniformMagneticfielddv

m =q(vB) (2.28)dt

TakeB in z-direction. Neveranyforcein z-dir. vz =constant. Perpendiculardynamicsareseparate.

v B

zy

B

v

x

Figure2.11: Orbitofaparticleinauniformmagneticfield.

2.3.1 Brute forcesolution:vx = q vyB vy = qvxB (2.29)

m m 2 2qB qBvx =

m vx vy =

m vy (2.30)

SolutionqB qB

vx =vsin t vy =vcos tmm qB m mqB , (2.31)x=v cos t+x0 y=v sin t+y0

qB m qB mthe equation of a circle. Center (x0, y0) and radius (vm/qB) are determined by initialconditions.2.3.2 PhysicsSolution

1. Magneticfield forcedoesnoworkonparticlebecauseFv. Consequentlytotal |v|isconstant.

2. Forceisthusconstant,tov. Givesrisetoacircularorbit.v2 Force vB mv3. Centripetalaccelerationgivesr = mass =qm i.e. r= qB. Thisradius iscalledthe

Larmor(orgyro)Radius.4. Frequency of rotation v = qB is called the Cyclotron frequency (angular fre

r mquency,s1,notcycles/sec,Hz).

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Whenweaddtheconstantvz wegetahelicalorbit. Cyclotronfrequency=qB/mdependsonlyonparticlecharacterq, mandB-strengthnotv(nonrelativistically,seeaside). LarmorRadiusr=mv/qBdependsonparticlemomentummv. All(non-relativistic)particleswithsame q/m have same . Different energy particles have different r. This variation can beusedtomakemomentumspectrometers.2.3.3 RelativisticAsideRelativisticdynamicscanbewritten

ddtp=q([E+]vB) (2.32)

whererelativisticmomentumisp=mv= m0v

1v2

c2. (2.33)

Massmisincreasedbyfactor 12v2

= 1c2 (2.34)

relative to restmass m0. Since forE=0the velocity |v| = const, isalso constant, andso ism. Thereforedynamicsofaparticle inapurelymagneticfieldcanbecalculatedasifitwere non-relativistic: m dv/dt=q(vB), exceptthattheparticle hasmass greater byfactor thanitsrestmass.2.3.4 MomentumSpectrometersParticlespassingverticallythroughslittakedifferentpathsdependingonmv/q. Bymea-

x

v

Different

Momenta

Unform

B

Slit Detection Plane

Figure2.12: Differentmomentumparticlesstrikethedetectionplaneatdifferentpositions.suringwhereaparticlehitsthedetectionplanewemeasureitsmomentum/q :

mv mv Bx2

qB =x : q = 2 . (2.35)43


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Whymakethedetectionplaneadiameter? Becausedetectionpositionis leastsensitivetovelocitydirection. Thisisaformofmagneticfocussing. Ofcoursewedontneedtomakethefull360,soanalysercanbereducedinsize.

UnformB

v

Different

Momenta

Unform

B

Slit Detection Plane

(b)

v

Slit Focus

Different Angles

(a)

Entrance

Figure2.13: (a)Focussingisobtainedfordifferentinputanglesbyusing180degreesoforbit.(b)Theotherhalfoftheorbitisredundant.Even

so,

it

may

be

inconvenient

to

produce

uniform

B

of

sufficient

intensity

over

sufficiently

largeareaifparticlemomentumislarge.2.3.5 HistoricalDayDream(J.J.Thomson1897)Cathoderays: howtotelltheirchargeandmass?ElectrostaticDeflectionTells only their energy/q = E/q and we have no independent way to measure E since thesamequantityE/qjustequalsacceleratingpotential,whichisthethingwemeasure.MagneticDeflectionTheradiusofcurvatureis

mvr= (2.36)

qBSocombinationofelectrostaticandelectromagneticgivesus

1 2mv mv2 =M1 and =M2 (2.37)

q qHence

2M1 qM22 = m. (2.38)Wecanmeasurethecharge/massratio. Inordertocompletethejobanindependentmeasureofq(orm)wasneeded. Millikan(1911-13). [ActuallyTownsendinJ.J.Thomsonslabhadanexperimenttomeasureqwhichwaswithinfactor2correct.]

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2.3.6 PracticalSpectrometerIn fusion research fast ion spectrum is often obtained by simultaneous electrostatic andelectromagnetic analysis E

1 parallel to B. This allows determination of E/q and q/m velocity of particle [E = mv2]. Thus e.g. deuterons and protons can be distinguished.2

y

x

B

E

Particle y

x

Figure2.14: EparalleltoBanalyserproducesparabolicoutputlocusasafunctionofinputvelocity. Thelociaredifferentfordifferentq/m.

qHowever,He4 andD2 havethesamem soonecantdistinguishtheirspectraonthebasis

ofionorbits.2.4 DynamicAcceleratorsInadditiontotheelectrostaticaccelerators,thereareseveraldifferenttypesofacceleratorsbasedontime-varyingfields. WiththeexceptionoftheBetatron,theseareallbasedonthegeneralprincipleofarrangingforaresonancebetweentheparticleandtheoscillatingfieldssuchthatenergyiscontinuallygiventotheparticle. Simpleexample

1+ 2+ 2- 3+1- 3-

1 2 3

Figure2.15: Sequenceofdynamicallyvaryingelectrodepotentialsproducescontinuousacceleration. Valuesat3timesareindicated.Particle isacceleratedthroughsequenceofelectrodes3attimes(1)(2)(3). Thepotentialofelectrodeisraisedfromnegativeto+vewhileparticleisinsideelectrode. SoateachgapitseesanacceleratingEz. Canbethoughtofasasuccessivemovingpotentialhill:Waveofpotentialpropagatesatsamespeedasparticleso it iscontinuouslyaccelerated.Historicallyearliestwidespreadacceleratorbasedonthisprinciplewasthecyclotron.

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z

Figure2.16: Oscillatingpotentialsgiverisetoapropagatingwave.+ -

Poles

Electric

MagneticPoles

Orbit

Plan

Elevation

B

Figure2.17: SchematicofaCyclotronaccelerator.2.4.1 Cyclotron

qBTake advantage of the orbit frequency in a uniform B-field =m . Apply oscillating

potentialtoelectricpoles,atthisfrequency. Eachtimeparticlecrossesthegap(twice/turn)itseesanacceleratingelectricfield. Resonantfrequency

f = = qB = 1.52107B Hz (2.39)2 m2

15.2MHz/Tforprotons. Ifmagnetradius isRparticle leavesacceleratorwhen itsLarmorradiusisequaltoR

mv=R 1mv2 =1q2B2R2 (2.40)

qB 2 2mIf iron is used for magnetic pole pieces then B


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2.4.2 LimitationsofCyclotronAcceleration:RelativityMass increase (1v2/c2)21 breaks resonance, restricting maximum energy to 25MeV(protons). Improvement: sweep oscillator frequency (downward). Synchrocyclotron allowed energy up to 500MeV but reduced flux. Alternatively: Increase B with radius.Leads to orbit divergence parallel to B. Compensate with azimuthally varying field forfocussingAVF-cyclotron. Advantagecontinuousbeam.2.4.3 SynchrotronVaryboth frequencyandfield intimetokeepbeam inresonanceatconstantradius. Highenergyphysics(to800GeV).2.4.4 LinearAcceleratorsAvoid limitations of electron synchrotron radiation. Come in 2 main types. (1) Induction(2) RF (linacs) with different pros and cons. (RF for highest energy electrons). Electronacceleration: v=cdifferentproblemsfromion.2.5 MagneticQuadrupoleFocussing(AlternatingGra-

dient)Magneticfocussingispreferredathighparticleenergy. Why? Itsforceisstronger.MagneticforceonarelativisticparticleqcB.ElectricforceonarelativisticparticleqE.E.g. B = 2T cB = 6108 same force as an electric field of magnitude6108V /m =0.6MV/mm! Howevermagnetic force isperpendiculartoBsoanaxisymmetric lenswould

liketohavepurelyazimuthalB field B=B. Howeverthiswouldrequireacurrentright

vB

v B^

Figure2.18:

Impossible

ideal

for

magnetic

focussing:

purely

azimuthal

magnetic

field.

wherethebeamis:

B.d=oI. (2.41)Axisymmetricmagneticlensisimpossible. Howeverwecanfocusinonecartesiandirection(x, y)atatime. Thenusethefactthatsuccessivecombinedfocus-defocushasanetfocus.

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2.5.1 PreliminaryMathematicsConsider

z =0 purely transverse field (approx)Bx, By. This can berepresented byB= AwithA=zAso A= (zA) =z A(sincez=0).Inthevacuumregionj=0(nocurrent)so

0 = B= (z A) =z2A+ ( (2.42)z.)A=0

i.e. 2A = 0. A satisfies Laplaces equation. Notice then that solutions of electrostaticproblems, 2=0 are also solutions of (2-d) vacuum magnetostatic problems. The samesolutiontechniqueswork.2.5.2 MultipoleExpansionPotentialcanbeexpandedaboutsomepointinspaceinakindofTaylorexpansion. Chooseoriginatpointofexpansionandusecoordinates(r, ),x=rcos,y=rsin.

1 A 1 2A2A=

rrrr +r 2 =0 (2.43)2LookforsolutionsintheformA=u(r).w().Theserequire

d2wd2 =const.w (2.44)

andd du

r r =const.u. (2.45)dr dr

Hencewsolutionsaresinesandcosinesw=cosn or sinn (2.46)

where n2 is the constant in the previous equation and n integral to satisfy periodicity.Correspondingly

u=rn or lnr , rn (2.47)Thesesolutionsarecalledcylindricalharmonicsor(cylindrical)multipoles:

1 lnrrncosn rncosn (2.48)r

nsin

n

rn

sinn

Ifourpointofexpansionhasnosourceatit(nocurrent)thentheright-handcolumnisruledoutbecausenosingularityatr=0ispermitted. Theremainingmultipolesare

1 constantirrelevanttoapotentialrcos(=x) uniformfield,Axr2cos2=r2(cos2sin2) =x2 y2 non-uniformfieldHigherorders neglected.

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Thesecondordersolution,x2y2 iscalledaquadrupolefield(althoughthisissomethingof a misnomer). [Similarly r3cos3 hexapole, r4cos octupole.] We already dealtwiththispotentialintheelectriccase.

A= x2 y2 =2xx2yy . (2.49)So

z A=2xy2yx (2.50)Forceonlongitudinallymovingcharge:

F = qvB=qv( A) (2.51)= qv(z A) =q(v.z)A qvzA (2.52)

Magneticquadrupoleforceisidenticaltoelectricquadrupoleforcereplacing Avz (2.53)

Consequentlyfocussinginx-direction defocussinginy-directionbutalternatinggradientsgivenetfocussing. Thisisbasisofallstrongfocussing.2.6 ForceondistributedcurrentdensityWehaveregardedtheLorentzforcelaw

F =q(E+vB) (2.54)as

fundamental.

However

forces

are

generally

measured

in

engineering

systems

via

the

interaction of wires or conducting bars with B-fields. Historically, of course, electricity and

magnetismwerebasedonthesemeasurements. Acurrent(I)isaflowofcharge: Coulombs/sAmp. Acurrentdensityj isaflowofchargeperunitareaA/m2. Thecharge iscarriedbyparticles:

j= niviqi (2.55)species i

HencetotalforceoncurrentcarriersperunitvolumeisF= niqi(vi B) =jB (2.56)

iAlso,forafinewirecarryingcurrentI, if itsarea is,thecurrentdensityaveragedacrossthesectionis

Ij= (2.57)

Volumeperunitlengthis. Andtheforce/unitlength=jB. =IBperpendiculartothewire.

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2.6.1 ForcesondipolesWe saw that the field of a localized current distribution, far from the currents, could beapproximatedasadipole. Similarlytheforcesonalocalizedcurrentbyanexternalmagneticfieldthatvariesslowlyintheregionofcurrentcanbeexpressedintermsofmagneticdipole.[Sameistrueinelectrostaticswithanelectricdipole].Total force

F= jBd3x (2.58)whereBisanexternalfieldthatisslowlyvaryingandsocanbeapproximatedas

B(x) =B0 + (x.)B (2.59)wherethetensorB(Bj/xi)issimplyaconstant(matrix). Hence

F

=j

Bo +j(x.B)d3x

= jd3x Bo + j(x.B)d3x (2.60)Thefirsttermintegraliszeroandthesecondistransformedbyourpreviousidentity,whichcanbewrittenas

x (x j)d3x = 2x. jxd3x =2x. xjd3x (2.61)foranyx. UsethequantityBforx(i.e. xi Bj)givingxi

12 (x

j)d3x B=m B= j(x.)Bd3x (2.62)

This tensor identity is then contracted by an internal cross-product [ijk Tjk ] to give thevectoridentity

(m )B= j[(x.)B]d3x (2.63)Thus

F= (m )B=(m.B)m(.B) (2.64)(remembertheoperatesonlyonBnotm). Thisistheforceonadipole:

F=(m.B) . (2.65)TotalTorque(Momentof force)is

M = x (jB)d3x (2.66)= j(x.B)B(x.j)d3x (2.67)

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Bhereis(tolowestorder)independentofx:B0 sosecondtermiszerosince 1 x.jd3x =

2 . |x|2j |x|2.j d3x = 0. (2.68)Thefirsttermisofthestandardformofouridentity.

1M=B. xjd3x =

2B (x j)d3x (2.69)M=mB Momentonadipole. (2.70)

y

x

zdx

dy I

Figure2.19: Elementarycircuitforcalculatingmagneticforce.

2.6.2 ForceonanElementaryMagneticMomentCircuitConsideraplanerectangularcircuitcarryingcurrentI havingelementaryareadxdy=dA.Regardthisasavectorpointing inthez directiondA. Theforceonthiscurrent inafieldB(r)isFsuchthat

BzFx = Idy[Bz(x+dx)Bz(x)]=Idydx (2.71)

xBz

Fy = Idx[Bz(y+dy)Bz(y)]=Idydx (2.72)y

Fz = Idx[By(y+dy)By(y)]Idy[Bx(x+dx)Bx(x)]Bx By Bz

= Idxdyx + y =Idydx z (2.73)

(using .B = 0). Hence, summarizing: F = IdydxBz. Now definem = IdA = Idydxzandtakeitconstant. Thenclearlytheforcecanbewritten

F=(B.m) (2.74)orstrictly(B).m.

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S

N

m

M

B

Figure2.20: Momentonabarmagnetinauniformfield.

S N

N Sm

m

Net Force:

B

|B|

Figure2.21: Amagneticmomentintheformofabarmagnetisattractedorrepelledtowardthestrongerfieldregion,dependingonitsorientation.2.6.3 ExampleSmallbarmagnet: archetypeofdipole. InuniformBfeelsjustatorquealigningitwithB.Inauniformfield,nonetforce.

Non-uniformfield: Ifmagnettakesitsnaturalrestingdirection,mparalleltoB,forceisF=m|B| (2.75)

Abarmagnetisattractedtohighfield. AlternativelyifmparalleltominusBthemagnetpointsotherway

F=m|B| repelledfromhigh|B|. (2.76)Samewouldbetrueforanelementarycircuitdipole. It isattracted/repelledaccordingtowhether itactsto increaseordecreaseB locally. Achargedparticlemoving in itsLarmororbitisalwaysdiamagnetic: repelledfromhigh|B|.

2.6.4 IntuitionThereissomethingslightlynon-intuitiveaboutthenaturalbehaviorofanelementarywirecircuitandaparticleorbitconsideredassimilartothiselementarycircuit. Theircurrentsflow in opposite directions when the wire is in its stable orientation. The reason is thatthestrengthofthewiresustainsitagainsttheoutwardmagneticexpansionforce,whiletheparticleneedsaninwardforcetocausethecentripetalacceleration.

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Paramagnetic Attracted

Diamagnetic Repelled

B

|B|

Net Force:

Figure 2.22: Elementary circuit acting as a dipole experiences a force in a non-uniformmagneticfield.

I

Orbiting ParticleWire Loop

j B j B^ outwards. ^ inwards.

To balance accelerationStable Orientation

qv

Figure2.23: Differencebetweenawireloopandaparticleorbitintheirnaturalorientation.2.6.5 AngularMomentumIfthe localcurrent ismadeupofparticleshavingaconstantratioofchargetomass: q/Msay (Notational accidentm is magnetic moment). Then the angular momentum isL =

iMixi vandmagneticmomentism= 1 qixi vi. So2qm= L. Classical (2.77)

2MThiswouldalsobetrueforacontinuousbodywithconstant(chargedensity)/(massdensity)(/m). Elementaryparticles,e.g. electronsetc.,have spinwithmomentsm,L.Howevertheydonotobeytheaboveequation. Instead

qm=g L (2.78)

2MwiththeLandeg-factor(2 forelectrons). This isattributedtoquantumandrelativisticeffects. However the classical value might not occur if /m were not constant. So weshouldnotbesurprisedthatg isnotexactly1forparticlesspin.2.6.6 PrecessionofaMagneticDipole(formedfromchargedpar-

ticle)The result of a torquemB is a change in angular momentum. Sincem=gLq/2M wehave

dL q=mB=g (LB)) (2.79)

dt 2M53


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m B^

m B^

mL

Figure2.24: PrecessionofanangularmomentumLandalignedmagneticmomentmaboutthemagneticfield.ThisistheequationofacirclearoundB. [Comparewithorbitequation dv = qvB]. The

dt mdirectionofLprecesseslikeatiltedtoparounddirectionofB withafrequency

qB=g (2.80)

2MForan electron (g =2) this is equaltothe cyclotron frequency. Forprotonsg = 22.79[Writtenlikethisbecausespinis 1]. Forneutronsg= 2(1.93).

2Precessionfrequencyisthus

electronf = = (28GHz)(B/Tesla) (2.81)

2proton

= (43MHz)(B/Tesla) (2.82)2

This isthe(classical)basisofNuclearMagneticResonancebutofcoursethatreallyneedsQM.

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Electric & Magnetic Particle Motion