PHY 208 - magnetism · 2018-04-28 · All magnetism is found to come from moving charges. Sometimes it is obvious how the charges are moving; other times, it’s more subtle. Magnetism

Magnets

1. Magnets-Introduction1. FieldLines&FieldVectors2. Movingchargescreatefields

2. Themagneticfield1. Calculatingthemagneticforce:2. CircularMotion3. ElectricandMagneticFields4. LorentzForceLaw5. TheHallEffect

3. CurrentsandMagneticFields4. Magneticfieldaroundawire5. TheBiotandSavartLaw6. Ampere'sLaw

1. Solenoid

Thelodestoneisanaturallyoccurringmineralcalledmagnetite.Itwasfoundtoattractcertainpiecesofmetal.Nooneknewwhy.SomeearlyGreekphilosophersthoughtthelodestonehadasoul.Today,we

haveabetterideaofwhat'sgoingon.

1. Magnets - Introduction

Allmagnetismisfoundtocomefrommovingcharges.Sometimesitisobvioushowthechargesaremoving;othertimes,it’smoresubtle.

MagnetismfromMagneticMaterials:Barmagnets,refrigeratormagnets,compasses

Magnetismformmovingcharges:currents,electromagnets,motors

Thefundamentalmechanismsneededtounderstandmagneticinteractionsinvolvesquantummechanics.Thus,forourpurposes,we'llhavetobecontentwithprovidingaquantitativedescriptionofthephenomenon,withoutmakingeffortstoexplainthedeeperwhyquestions.

PHY 208 - magnetism

updated on 2018-04-28 J. Hedberg | © 2018 Page 1

N

S

N

S

N

S

oppositepoles

N

S

N

S

similarpoles

Fig.1

N

S

N

S

N

S

NS

NSN

S

Fig.2

Fig.3

Some basics about magnets

Everymagnethasanorthpoleandsouthpole.

Theforcebetweentwoofthesamepolesisrepulsivewhiletheforcebetweentwooppositepolesisattractive.

Thesearenotthesamethingaspositiveandnegativecharges,althoughtheymaysoundsimilar.

(themagneticfieldisgiventheletterB)

Cuttingamagnetwillnevercreateaseparatenorthandsouthpole.You’llalwaysjustendupwithtwosmallermagnets.

Justlikeelectrostaticfields,wecanusefieldlinestoportraythemagneticfields.

Thesearethemagneticfieldlinesinthevicinityofabarmagnet.

TheypointfromNorthtoSouth.

PHY 208 - magnetism


Fig.4

Fig.5

magnetic field lines

Fig.6

Field Lines & Field Vectors

Thesearetwowaysofshowingthesameinformation.

Thefieldvectorscanbeusedtoshowthestrengthanddirectionataparticularpointinspace.

Theyaretangenttothefieldlines

Ifwesurroundabarmagnetwithmanylittlemagnets,theywilllineupalongthefieldlines.

Moving charges create fields

Currentinawirewillcreateamagneticfield.

Theshapeofthatmagneticfieldiscircularwiththewireatthecenterofthecircle.

Thestrengthofthefielddecreasesasthedistancefromthewireisincreased.

PHY 208 - magnetism


Fig.9

Right-hand rule

Fig.8

Let'sbendthewire

Nowwehavealoopwithacurrentinit.ThesameRightHandRulecanbeusedtofigureouttheBfieldnearthisloop.

Justletyourthumbpointinthedirectionoftheconventionalcurrentdirection.Thencurlyourfingersasshown.Themagneticfieldpointsinthedirectionofyourfingers.

PHY 208 - magnetism


Fig.10

Fig.12

Fig.13

Insidetheloop,theBfieldvectorsareallpointingthesamedirection.

Andwhatifwehavemanyloops,allparalleltoeachother?Thisislikeacoil,orsolenoid.Again,theRHRwillguideusinfiguringoutthedirectionofthemagneticfieldlines.

PHY 208 - magnetism


into the screen out of the screen

Fig.15

Fig.16

2. The magnetic field

Amagneticfieldisusuallygiventhesymbol .

TheSIunitsforthemagneticfieldstrengthareTeslas.

ATeslaisaratherlargemagneticfield,sosometimesweusetheunitofGauss,whichisgivenby:

Thedirectionsareoftenperpendiculartotheplaneofthepage(orscreen)

Fig.14

Firstlet'sagreeonacoordinatesystem

Thisonehas and onthehorizontalplane.pointupwards.

Now,we'llconsiderauniformMagneticfield ,whichinthiscasepointsonlyinthedirection.

B

1Tesla =Newton

Coulomb meter/second

1 Tesla = Gauss104

x y z

B

+y

=B B0 j

PHY 208 - magnetism


Fig.17

Fig.18

Wecanaddachargedparticleinthe field.Sincethisparticleiselectricallycharged,andthereisnoelectricfield,therewillbenoforceontheparticle

Wecanalsoimpartavelocity totheparticle.Let'ssaythat isonlypointingthe direction.

Fig.19

Now,ifwehaveachargedparticlemovinginamagneticfield,thatparticlewillexperienceamagneticforce

givenby:

Thisisaforcedirected toboththemagneticfieldandthevelocityvectors: in

thiscase.

B

v

v +x

=v v0 i

FB

= q ×F B v B

90∘

k

PHY 208 - magnetism


Fig.21

1. Yourthumbpointsinthedirectionofthemovingparticle.

2. Indexfingerindirectionofmagneticfield.

3. Middlefingershowstheforce

v

v

The right hand rule

Fig.20

Quick Question 1

Whichwaywillthemagneticforcebedirectedontheparticleasitentersthefield?

1.Up2.Down3.Left4.Right5.Intothescreen6.Outofthescreen

Quick Question 2

Whichwaywillthemagneticforcebedirectedontheparticleasitentersthefield?

1.Up2.Down3.Left4.Right5.Intothescreen6.Outofthescreen

PHY 208 - magnetism


Fig.24

Let'snotforgetaboutthelittle intheequation:

Iftheqisnegative,thenthesign(andthereforethedirection)oftheforcevectorwillflip.

Calculating the magnetic force:

TheCrossproduct(orvectorproduct)isonewayofmultiplyingtwovectors:

Itproducesathirdvectorinadirectionperpendiculartoboththemultiplicands.

Magnitude of the cross product:

Definitionofthecrossproduct:

So,forthestrengthofthemagneticforceonachargedparticle:

Thistellsusthatiftheparticleismovinginthedirectionofthemagneticfield( ),therewillbenoforce.

q

= q ×F B v B

ijk

= j × k= k × i= i × j

(1)(2)(3)

k × j =i × k =j × i =

− i− j− k

(4)(5)(6)

A = ∥a × b∥ = ∥a∥ ∥b∥ sinθ.

F = qvBsinθ

θ = 0

PHY 208 - magnetism


Quick Question 3

Thesmallchargeinthecenterisfreetomove.Thetwolargerchargesarefixedinplace.Ifthesmallchargeisdirectlyhalfwaybetweenthetwolargecharges,whatdirectionwillitacceleratein.

1.+x2.-x3.+y4.-y5.+z6.-z7.Itwillremainatrestsincethenetforceonitiszero.

Quick Question 4

Ifyougentlynudgedthechargeinthemiddletowardsthepositivex(+x)direction,whichwaywoulditaccelerateafterthenudge?


Quick Question 5

Ifyougentlynudgedthechargeinthemiddletowardsthenegativex(–x)direction,whichwaywoulditaccelerateafterthenudge?


PHY 208 - magnetism


Quick Question 6

Ifyougentlynudgedthechargeinthemiddletowardsthepositivey(+y)direction,whichwaywoulditaccelerateafterthenudge?


Quick Question 7

Ifyougentlynudgedthechargeinthemiddletowardsthepositivey(+y)direction,whichwaywoulditaccelerateafterthenudge?


Quick Question 8

Ifyougentlynudgedthechargeinthemiddletowardsthepositivez(+z)direction,whichwaywoulditaccelerateafterthenudge?


PHY 208 - magnetism


Earnshaw's Theorum

"acollectionofpointchargescannotbemaintainedinastablestationaryequilibriumconfigurationsolelybytheelectrostaticinteractionofthecharges."

Quick Question 9

Anegatively-chargedparticletravelsparalleltomagneticfieldlineswithinaregionofspace.Whichoneofthefollowingstatementsconcerningtheparticleistrue?

1.Thereisaforcedirectedperpendiculartothemagneticfield.2.Thereisaforceperpendiculartothedirectioninwhichtheparticleismoving.3.Theforceslowstheparticle.4.Theforceacceleratestheparticle.5.Theforcehasamagnitudeofzeronewtons.

Quick Question 10

Apositively-chargedparticleisstationaryinaconstantmagneticfieldwithinaregionofspace.Whichoneofthefollowingstatementsconcerningtheparticleistrue?

1.Theparticlewillnotmove.2.Theparticlewillaccelerateinthedirectionperpendiculartothefield.3.Theparticlewillaccelerateinthedirectionparalleltothefield.4.Theparticlewillaccelerateinthedirectionoppositetothefield.5.Theparticlewillmovewithconstantvelocityinthedirectionofthefield.

uniform B Field (into screen) charged particle

PHY 208 - magnetism


uniform B Field (into screen)

A

B

C

D


forcevector

Quick Question 11

Whichpath(s)couldtheparticletake?

1.nodeflection2.ParabolicTrajectory3.CircularTrajectory4.LinearTrajectory


Circular Motion

PHY 208 - magnetism


Circular Motion

Forthecaseofuniformcircularmotion,wehadfromNewton's2ndlaw:

Now,weknowtheforceduetothemagneticfield:

Thus,togetherweget:

Quick Question 12

Achargedparticleismovingthroughaconstantmagneticfield.Doesthemagneticfielddoworkonthechargedparticle?

1.yes,becausetheforceisactingastheparticleismovingthroughsomedistance2.no,becausethemagneticforceisalwaysperpendiculartothevelocityoftheparticle3.no,becausethemagneticfieldisavectorandworkisascalarquantity4.no,becausethemagneticfieldisconservative5.no,becausethemagneticforceisavelocity-dependentforce

Quick Question 13

Achargedparticleentersauniformmagneticfield.Whathappenstothekineticenergyoftheparticle?

1.itincreases2.itdecreases3.itstaysthesame4. itdependsonthedirectionofthevelocity5.itdependsonthedirectionofthemagneticfield

F = mv2

r

= q × ⇒ F = |q|vBF B v B

|q|vB =mv2

r

PHY 208 - magnetism


Usually,we'llbeconcernedwiththeradiusofmotion:

Let'snotforgetourfriendsperiodandfrequencytoo:

Anelectronisacceleratedfromrestthroughapotentialdifferenceof500V,theninjectedintoaUniformmagneticfield.Onceinthefield,iscompleteshalfarevolutionin2.0ns.Whatistheradiusoftheorbit.

WhatwillhappenifthereisacomponentoftheinitialvelocitynotperpendiculartotheB-field?

HelicalMotion!

r =mv

|q|B

T = =2πrv

2πm|q|B

f = =1T

|q|B

2πm

Example Problem #1:

PHY 208 - magnetism


Anelectronhasakineticenergyof22.5eV.Itmovesintoaregionwithauniformmagneticfieldgivenby455microTesla.Thevelocityoftheparticleissuchthatismakesangleof50degreeswiththemagneticfield.Whatisthepitchofthehelicalpath?

Electric and Magnetic Fields

Whathappensifthereisanelectricfield, ,andamagneticfield, ,inthesameregionofspace?

v

motion from electric field

motion from magnetic field

Answer:Thefieldswillbothaffecttheparticleasthewouldnormally.Twoforceswillexistandwecanjustaddthem.

Lorentz Force Law

Offically:TheLorentzForcelaw:

Thissumoftheforcesontheparticlewillaccountforitsdynamicalproperties.

The Hall Effect

Whathappenswhenthe isequaltothe ?

Example Problem #2:

E B

F = q (E + v × B)

FE FB

=FE FB

PHY 208 - magnetism


The Hall Effect - setup

Thechargeseparationwillleadtoapotentialdifferenceacrossthewidthofthedevice, .:.Thispotentialdifferencewillincrease,untiltheforceduetotheelectricfielditcreates

isequaltotheforcefromthemagneticfield:

Fortheforceduetothemagneticfield,wehave

Whatis ?It'sthedriftvelocityofthecharges:

( isthedensityofthecharges, isthecrosssectionalarea).

w

VHall

F = qE ⇒ E = V /w

= qvBFB

v

=vdI

nqA

n A

PHY 208 - magnetism


and so....

Let'ssaytheforcesareequal:

Theforcefromtheelectricfieldis:

Theforceduetothemagneticfieldis

Thus:

Whichthingsdoweknow?

:that'sameasurablequantity(useavoltmeter)

:currentwecanmeasure

:Themagneticfieldissomethingwecontrol(andknow)

:crosssectionalarea=

:chargeonanelectron,weknowthat

:densityofcharges:wedon'talwaysknowthat.

3. Currents and Magnetic Fields

Sincecurrentsarereallynothingmorethanmovingcharges,thereshouldbeaforceonacurrentcarryingwireifamagneticfieldispresent.

Indeed,itshouldfollowthesameformatasbefore.Inthiscase,wehaveawireoflength withacurrenttravelinginthedirectionof .Since ,

becomestheforceonacurrentcarryingwire.

=FE FB

= qFEV

w

= qFBIB

nqA

=V

w

BI

nqA

=V

w

BI

nqA

V

I

B

A w ∗ h

q

n

L

L q = It = I L

vd

= q ×FB vd→

B

= I ×FB L B

PHY 208 - magnetism


3 Ω15 V

+x

+y

10 cm

Theforce:

Themagnitudeofthisforceisjust:

isthelengthofthewireand istheconventionalcurrent.

istheanglebetweenthewireandthefield.(Inthisfigureitwillbe ).

And,justtobesure,theforceontheelectroncurrentisalsopointinginthesamedirection.(Otherwise,wewouldhaveproblems)

Thiscircuitispartiallyexposedtoa50mTmagneticfield,asshown.Whatisthenetforceonthecircuit.

= I ×FB L B

= IlBsinθFB

l I

θ

π/2

Example Problem #3:

PHY 208 - magnetism


NS NSV +x+z

+y

1.Thewiremovestowardthenorthpoleofthemagnet(-x).2.Thewiremovestowardthesouthpoleofthemagnet(+x).3.Thewiremovesoutofthescreen(towardus,+z).4.Thewiremovesintothescreen(awayfromus,-z).5.Thewiredoesn’tmove.

Quick Question 14

Aportionofaloopofwirepassesbetweenthepolesofamagnetasshown.Whentheswitchisclosedandacurrentpassesthroughthecircuit,whatisthemovement,ifany,ofthewirebetweenthepolesofthemagnet?

Quick Question 15

WhatwillhappentothismetalwireintheuniformBField?

1.Itwillnotmove2.Itwillacceleratetotheright3.Itwillacceleratetotheleft4.Itwillaccelerateup5.Itwillacceleratedown

Rotations in a B Field

Here'sasimple(oversimplified)currentloopinauniformmagneticfield.

PHY 208 - magnetism


Perspectiveview Topview:

Let'slabeleachofthesidesoftheloop

Ifwegothroughandusetherulesaboutmagneticforceonacurrentcarryingwire,we'llseethattheforceswillbegivenby:

1. noforce( )2. pointedup( )3. noforce( )4. pointeddown( )

Andthus,we'llexpecttoseesomerotationoftheloop(i.e.atorquemustbepresent).

θ = 0θ = 90

θ = 180θ = 270

PHY 208 - magnetism


Theloopwillrotateuntilitreachesthisposition,atwhichpointtherewillnolongerbetorqueontheloop.

Whathappensifthelooprotatespasttheverticalposition?(wewouldexpecttosinceitshouldbeacceleratinguntilthatpointandwouldthenrotatepastvertical)

Hereisaloopofwire,withacurrentI,inauniformmagneticfield.

Wecangothroughanduseourforceonawireequationtofindthedirectionoftheforceateachpointalongthewire.

We’llseethatthetotalnetforceendsupbeingzero,butthatdoesn’tmeanthereisn’tatorque!

PHY 208 - magnetism


L/2

loop

µ

loop

uniform B field

Thethetotaltorquewillbethesumofthetwotorquesfromthetopandthebottomoftheloop.(theonesontheside=0,since )

Addingthetorquesfromthetopandbottomsides:

But,weknowthetorquesinceweknowtheforceanddistanceawayfromtheaxis.

Whichreducesto:

Ifthereweremultipleloops,say ,inacoil,wecouldjustaddthetorquefromeachone()

Magnetic Dipole

Wecanmakethatexpressionalittlemorecompactbydefiningthemagneticdipolemoment:

where isthecurrentthroughtheloop, istheareaoftheloop,and isthenumberofturns.

Thedipolemomentisavector.Itpointparalleltothesurfacenormaloftheloop.

Thedirection(upordown)isgivenbytherotationofthecurrent.

Another right hand rule

Themagneticdipolemomentwillpointinthedirectionofthethumb!

sin(0) = 0

τ = +τtop τbottom

τ = ILB L sinθ + ILB L sinθ12

12

τ = I Bsinθ = IABsinθL2

N

× N

μ

= NIAμ

I A

N

PHY 208 - magnetism


µ

loop

uniform B field

Thus,wecansimplifytheexpressionalittlebitmore:

whichshouldbesomewhatreminiscentofthetorqueonanelectricdipoleinanelectricfield.

Quick Question 16

Considertherelationshipsbetweenthedirectionsofthetorqueactingonamagneticdipoleinamagneticfield,themagneticfield,andthemagneticdipolemoment.Whichoneofthefollowingstatementsregardingthesedirectionsistrue?

1.Thetorqueisparalleltoboththemagneticfieldandthedipolemoment.2.Thetorqueisperpendiculartothemagneticfield,butparalleltothedipolemoment.3.Thetorqueisparalleltothemagneticfield,butperpendiculartothedipolemoment.4.Thetorqueisperpendiculartoboththemagneticfieldandthedipolemoment.

Quick Question 17

Whatwillthiscurrentloopdo?

1.RotateClockwise2.RotateCounterclockwise3.Nothing4.MoveUp5.MoveDown6.MoveLeft7.MoveRight

τ = ×μ B

PHY 208 - magnetism


magnetic field lines

4. Magnetic field around a wire

Wesawthattherewasamagneticforcecreatedbyamovingchargeinamagneticfield.

Sincephysicsisusuallysymmetric,wewouldexpecttheretobesomemagneticeffectsfromamovingcharge.

Indeed,ifwesetchargesinmotionthroughawire,thenamagneticfieldiscreatedaroundthewire.

Another representation

Theelectricfieldstrengthfromapointchargewasgivenby:

Let'sjustnaivelywriteasimilarlawformagneticfields:

E =1

4πϵ0

q

r2r

B = avectorfordirection1

someconstantssomethingtodowithmovingcharges

somethingtodowiththesquareofthedistance

PHY 208 - magnetism


Let'saskabouttheBfieldatapointnearawire.

Wecanquantifythemovingchargesby

Forthedirectionofthings,we'llusetheradiallydirectedvector .

Usingthecrossproduct,wecanestablishthedirection.

5. The Biot and Savart Law

Inamannersimilartoelectrostatics,wecanfigureoutthemagneticfieldatadistance awayfromamovingcharge(i.e.current)bythefollowing:

This isthepermeabilityconstantandisgivenby: =

qv

r

×v r

r

dB =μ0

4πi ×ds→

r

r2

μ0

= 4π ×μ0 10−7 T ⋅ m/A 1.26 × 10−6 T ⋅ m/A

PHY 208 - magnetism


0

P

+x

+y

Showthatthefieldadistance awayfromaninfinitelylongwirewithcurrent isgivenby:

Findthemagneticfieldatthecenterofthecirculararc.

Theforcebetweentwowires.

attractive force between the wires

wire 1 wire 2

repulsive force between the wires

wire 1 wire 2

Example Problem #4:

d I

B =Iμ0

2πd

Example Problem #5:

PHY 208 - magnetism


wireAmperian Loop

6. Ampere's Law

ThemagneticanaloguetoGauss'Law.

Theloopintegraloftheproduct aroundanyclosedloopisequaltothecurrentsenclosed

What is a line integral?

Field around a wire

Let'stakeasimplecaseofthecurrentcarryingwire.

WeconstructandAmperiansurfacearoundthewireasshown.

ThelastexpressionisjustBtimesthecircumferenceoftheAmperianloop.

So,accordingtoAmpere'slaw,thisshouldbeequaltothecurrentenclosedbytheloop(timesaconstant)

∮ B ⋅ ds = μ0 Ienc

B ⋅ ds

∮ ⋅ dB s

∮ B ⋅ ds = ∮ B cosθ ds = B∮ ds

B× 2πr

B =Iμ0

2πr

PHY 208 - magnetism


wire

Amperian Loop

+x-x

+y

-y

d d

1.

2.

3.

4.5.

Field Inside a wire

Wecanusethistoalsofindthemagneticfieldinsideacurrentcarryingwire.

First,we'llassumethatthecurrentisuniformwithinthewire.

Sincethecurrentisuniformwithinthewire,thecurrentenclosedwilljustbegivenbytheratioofthelooptothetotalwire:

Whichyieldsfor :

Quick Question 18

Thedrawingshowstwolong,straightwiresthatareparalleltoeachotherandcarryacurrentofmagnitude towardyou.Thewiresareseparatedbyadistance2dandareequidistantfromtheorigin.Whichoneofthefollowingexpressionscorrectlygivesthetotalmagneticfieldattheoriginofthex,ycoordinatesystem?

∮ B ⋅ ds = B(2πr)

= IIencπr2

πR2

B

B = ( ) rIμ0

2πR2

I

= +B Iμ0

2dj

= −B Iμ0

2πdj

= +B Iμ0

πdj

= −B Iμ0

πdj

= 0B

PHY 208 - magnetism


+x-x

+y

-y

d d

1.

2.

3.

4.5.

Quick Question 19

Thedrawingshowstwolong,straightwiresthatareparalleltoeachotherandcarryacurrentofmagnitude inthedirectionsshown.Thewiresareseparatedbyadistance2dandareequidistantfromtheorigin.Whichoneofthefollowingexpressionscorrectlygivesthetotalmagneticfieldattheoriginofthex,ycoordinatesystem?

Rings of current

Solenoid

Thesolenoidisessentiallyastackofcurrentcarryingrings.

Solenoid

Uponapplyinganelectricalcurrentthroughthesolenoid,magneticfieldswillbesetupinsideandout.

I

= +B Iμ0

2dj

= −B Iμ0

2πdj

= +B Iμ0

πdj

= −B Iμ0

πdj

= 0B

PHY 208 - magnetism


Solenoid

Solenoid: Representations

Theidealizedsolenoidwillbeconsideredjustastackofcurrentloops.Insidethefieldiseffectivelyuniform.Outsidethesolenoid,thefieldismuchweaker,andnonuniform,sowe'llcallitnegligible.

PHY 208 - magnetism


Amperian loop

h

a b

cd

Amperian loop

h

a b

cd

1.Thereisnoeffectonthewire.2.Thewireispusheddownward.3.Thewireispushedupward.4.Thewireispushedtowardtheleft.5.Thewireispushedtowardtheright.

Ampere's Law on the solenoid

Let'sintegratearoundtheloop:

Thecurrentenclosedwillbegivenbythenumberofwires,N,timesthecurrentineachone:

(where isthenumberofturnsperunitlength.

Ampere'sLawthengivenus:

Quick Question 20

Awire,connectedtoabatteryandswitch,passesthroughthecenterofalongcurrent-carryingsolenoidasshowninthedrawing.Whentheswitchisclosedandthereisacurrentinthewire,whathappenstotheportionofthewirethatrunsinsideofthesolenoid?

∮ B ⋅ ds = B ⋅ ds + B ⋅ ds + B ⋅ ds +∫ b

a

∫ c

b

∫ d

c

∫

∮ B ⋅ ds = Bh

= IN = InhIenc

n

Bh = Inh ⇒μ0 = nIBsol μo

PHY 208 - magnetism


Amperian Loop

A1meterlongMRIsolenoidgeneratesa1.2Tmagneticfield.Tocreatethisfield,itsends100Athroughhowmanyturns?

Toroids

Ampere's Law and the Toroid

We'llmakeanAmperianloopinside

Toevaluate aswegoaroundtheloop(clockwise),wecanrelyonthesymmetryofthetoroidtotellusthattheanglebetween andwillbe .

Thus,

where isthenumberofwindingsaroundthecircle.

Example Problem #6:

B ⋅ ds

B ds

0∘

B(2πr) = μIN

N

=BtorINμ0

2πr

PHY 208 - magnetism


I

N

S

Loops as dipole

Biot-SavardLawcanpredictavalueforthemagneticfieldstrengthalongtheaxisoftheloop:

Which,throughalittlerearrangementcanbewrittenas

Aparticlewithcharge+1Cmoveswithavelocityequalto .Ifamagneticfieldgivenby: isapplied,whatwilltheforcebeontheparticle?(Expresstheanswerinunitvectornotation)

B(z) =Iμ0 R2

2( +R2 z2)(3/2)

B(z) = =NIAμ0

2πz3

μ0

2πμ

z3

Example Problem #7:

v = 1 + 2 + 1i j k

B = 1 + 2 + 1i j k

PHY 208 - magnetism


Documents

PHY 208 - magnetism · 2018-04-28 · All magnetism is found to come from moving charges. Sometimes it is obvious how the charges are moving; other times, it’s more subtle. Magnetism