PHY 208 - magnetism · 2018-04-28 · All magnetism is found to come from moving charges. Sometimes...
Transcript of PHY 208 - magnetism · 2018-04-28 · All magnetism is found to come from moving charges. Sometimes...
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.
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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.
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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.
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Fig.9
Right-hand rule
Fig.8
Let'sbendthewire
Nowwehavealoopwithacurrentinit.ThesameRightHandRulecanbeusedtofigureouttheBfieldnearthisloop.
Justletyourthumbpointinthedirectionoftheconventionalcurrentdirection.Thencurlyourfingersasshown.Themagneticfieldpointsinthedirectionofyourfingers.
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Fig.10
Fig.12
Fig.13
Insidetheloop,theBfieldvectorsareallpointingthesamedirection.
Andwhatifwehavemanyloops,allparalleltoeachother?Thisislikeacoil,orsolenoid.Again,theRHRwillguideusinfiguringoutthedirectionofthemagneticfieldlines.
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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
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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
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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
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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
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Quick Question 3
Thesmallchargeinthecenterisfreetomove.Thetwolargerchargesarefixedinplace.Ifthesmallchargeisdirectlyhalfwaybetweenthetwolargecharges,whatdirectionwillitacceleratein.
1.+x2.-x3.+y4.-y5.+z6.-z7.Itwillremainatrestsincethenetforceonitiszero.
Quick Question 4
Ifyougentlynudgedthechargeinthemiddletowardsthepositivex(+x)direction,whichwaywoulditaccelerateafterthenudge?
1.+x2.-x3.+y4.-y5.+z6.-z7.Itwillremainatrestsincethenetforceonitiszero.
Quick Question 5
Ifyougentlynudgedthechargeinthemiddletowardsthenegativex(–x)direction,whichwaywoulditaccelerateafterthenudge?
1.+x2.-x3.+y4.-y5.+z6.-z7.Itwillremainatrestsincethenetforceonitiszero.
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Quick Question 6
Ifyougentlynudgedthechargeinthemiddletowardsthepositivey(+y)direction,whichwaywoulditaccelerateafterthenudge?
1.+x2.-x3.+y4.-y5.+z6.-z7.Itwillremainatrestsincethenetforceonitiszero.
Quick Question 7
Ifyougentlynudgedthechargeinthemiddletowardsthepositivey(+y)direction,whichwaywoulditaccelerateafterthenudge?
1.+x2.-x3.+y4.-y5.+z6.-z7.Itwillremainatrestsincethenetforceonitiszero.
Quick Question 8
Ifyougentlynudgedthechargeinthemiddletowardsthepositivez(+z)direction,whichwaywoulditaccelerateafterthenudge?
1.+x2.-x3.+y4.-y5.+z6.-z7.Itwillremainatrestsincethenetforceonitiszero.
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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
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uniform B Field (into screen)
A
B
C
D
uniform B Field (into screen)
forcevector
Quick Question 11
Whichpath(s)couldtheparticletake?
1.nodeflection2.ParabolicTrajectory3.CircularTrajectory4.LinearTrajectory
uniform B Field (into screen)
Circular Motion
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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
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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:
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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
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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
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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
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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:
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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.
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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
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Theloopwillrotateuntilitreachesthisposition,atwhichpointtherewillnolongerbetorqueontheloop.
Whathappensifthelooprotatespasttheverticalposition?(wewouldexpecttosinceitshouldbeacceleratinguntilthatpointandwouldthenrotatepastvertical)
Hereisaloopofwire,withacurrentI,inauniformmagneticfield.
Wecangothroughanduseourforceonawireequationtofindthedirectionoftheforceateachpointalongthewire.
We’llseethatthetotalnetforceendsupbeingzero,butthatdoesn’tmeanthereisn’tatorque!
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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
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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
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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
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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
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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:
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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
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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
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+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
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Solenoid
Solenoid: Representations
Theidealizedsolenoidwillbeconsideredjustastackofcurrentloops.Insidethefieldiseffectivelyuniform.Outsidethesolenoid,thefieldismuchweaker,andnonuniform,sowe'llcallitnegligible.
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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
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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
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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
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