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SourceofMagneticFieldsandMagnetisminMaterials

I. MagneticFieldduetoaCurrent

A. Wevenowtalkedalotabouthowmovingchargesaswellascurrentswillfeelaforceinthe

presenceofamagneticfield.Buthowaremagneticfieldscreatedanyway?

1. Wementionedatthebeginningoftheprevioussectionofnotesthattheonlythingthat

cancreateamagneticfield(atleastuntillaterinthesemester)isamovingcharge.

2. Socurrentsactuallyalsocreatemagneticfieldssincetheyarejustmanymovingcharges.

3. Thepropertiesofmagneticfieldscreatedbycurrentsareveryinteresting,butwellstart

withasimplecaseofasinglewirecarryingaconstantcurrent.

4.

Inthatcase,firsttodeterminethedirectionofthemagneticfieldcreated,weuseyet

anotherright

hand

rule

(RHR).

This

is

the

RHR

for

the

magnetic

field

due

to

currents:

5. Pointyourthumbinthedirectionofthecurrent(mustuseyourrighthand!),thencurl

yourfingersaroundyourthumbandyourfingersrepresentthemagneticfielddirection.

Seethefigurebelowtogetabettersenseofit.

6. Thisisaninterestingresult:eventhoughthecurrentisstraight,itproducesamagnetic

fieldthatwrapsaroundthecurrentinacircularmanner.

7. Asfarasmagnitudeisconcerned,wecantalkaboutthestrengthofmagneticfieldsfora

coupleparticularcases.Thefirstisforalong,straightwirecarryingaconstantcurrent:

,whereisthecurrentandisthedistancefromthewire.isanewconstant

calledthepermeabilityoffreespace,essentiallyameasureofhowfreespace(vacuum)

respondstomagneticeffects. 4 10/

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8.

Fromthisequationitisclearthatasthecurrentmagnitudeincreases,sodoesthe

magneticfield(seemsreasonable),andalsothattheeffectofdistanceisonlyaninverse

relationship(1/r,not1/r2or1/r

3).

9. Thesecondcasewecanlookatisforaspecialgeometrycalledasolenoid.Asolenoidis

justmanywindings(usuallycircular,butnotalways)ofasinglewiresothatmanycurrent

loopsare

stacked

right

next

to

each

other

and

sometimes

also

on

top

of

each

other.

See

thefigurebelow:

10.Eachloophasthesamecurrentflowinginit(theyareallfromthesamepieceofwire),so

theyallproducethesamemagneticfieldinthesamedirection.Thekeyisthattheywillall

addtogethertomakeaverystrongmagneticfieldontheinsideofthesolenoid(asshown

inthefigure).

11.

Foranidealsolenoid,meaningonethatisverylong,themagneticfieldontheoutsideis

essentially=0,

or

is

at

least

very

small

compared

to

that

on

the

inside.

12.Youcanonlyhavetwopossibledirectionsforthemagneticfieldinsideasolenoid:itwill

alwaysbealongthelengthofthesolenoid,justeithertotherightortotheleft.To

determinethedirection,useyetanotherrighthandrule(!):Curlthefingersofyourright

handinthedirectionofthecurrentflowingintheloops,thenyourthumbpointsinthe

directionofthemagneticfield(asseeninthefigureabove:B=totheleft(xdirection).

13.Thepurposeofcreatingasolenoidgeometryistogetaverystrongmagneticfieldstrength

foragivencurrentflowinginasinglepieceofwire.

14.

Theformula

for

the

magnitude

of

the

magnetic

field

of

a

solenoidis

as

follows:

.isthenumberofturns,i.e.thenumberofwindingsthesolenoidhas

(thenumberofcurrentloops).

15.Thisexpressionworksevenforrealsolenoidsofagivenlength,butitisstillan

approximationifthesolenoidisntverylongorinfinitelylong.Sooftenyoullseethe

secondversioninsteadwhichuses /,ortheturnsperunitlength.

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16.

Anothergeometrywecanhaveforacurrentproducingamagneticfieldisashort,straight

wire.Inthiscasewecannotignoretheeffectsoftheendsofthewirebecausethewireis

notlong.Soitbecomesimportanttobeabletoincludehowthepointofinterest

comparestothetwoendsofthewire:itmaynotbethesamedistancefromeachendof

thewire:

TheendresultcanonlybecalculatedusingtheBiotSavartLaw,andisasfollows:

4sin sin

Oneanglewillbepositiveandonewillbenegativesincetheyarebothreferenced

fromthepoint.Foralongwirethetwoanglesapproach90inmagnitudeandthis

formulabecomesthesameasforalongwire.

17.Ifweinsteadbendthewireintoacircularshape,themagneticfieldatthecenterofthe

circle(orring,orcurrentloop)willbe:

; /

II.

Amperes

Law

A. CalculatingmagneticfieldswiththeBiotSavartLawisnoteasyinmanycases.Somuch

likewithGausssLawforcalculatingelectricfields,wecanuseAmperesLawtocalculate

magneticfieldsincertainhighsymmetrysituations.

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1. AmperesLawsimplysaysthattheonlythingthatcancreatemagneticfieldsiscurrent,

sotheremustbearelationshipbetweenthetwo:

2.

Touse

Amperes

Law

do

the

following:

a. Chooseasurfacetoenclosethecurrentthatisproducingthemagneticfield.

b. Thendrawthesurfacesuchthatthepatharoundtheedgeofthesurfacehasthe

samedirectionasthemagneticfieldthismakestheleftsideofAmperesLaw

becomeverysimpleandthedotproductgoesaway.

c. Asanexample,foralong,straightwirethemagneticfieldcirclesaroundthewire,

soanappropriatesurfaceisacircledrawnaroundthewirewitharadiusatthe

distancewherethemagneticfieldistobecalculated

III. MagnetisminMaterials:Paramagnetism,Diamagnetism,andFerromagnetism

A. Believeitornot,everysinglekindofmaterialthereishasatleastsomelevelofmagnetic

propertiesmeaningitcanproduceamagneticfieldorinteractwithmagneticfields(or

both).

1. Wetypicallyonlythinkofmaterialswithmagneticpropertiesasthosethatobviouslyreact

tomagnets,orareactuallymagnetsontheirown.Forsure,thosekindsofmaterials

definitelyhavemagneticpropertiesandtheyfallintothethirdclass:ferromagnetic

materials.

2. ButALLmaterialshavesomemagneticbehavior.Theothertwokindsareparamagnetism

anddiamagnetism.Theybothonlyshowup,orbecomemeasurableandusefulwhen

themagneticfieldsmaterialsinteractwithareverystrong.

3. SmallermagneticfieldslikethatduetotheEarthandthoseassociatedwithsmall

permanentmagnetsarenotstrongenoughtoshowparamagnetismordiamagnetisminan

obviousway.

4. Paramagnetismcomesfromthefactthatelectrons,whicharecharges,tendtospin

aroundan

axis

as

they

orbit

the

nucleus

of

an

atom

(just

like

the

Earth

spins

on

its

axis

as

it

orbitstheSun).Thisspinningmotionproducesacurrent(currentisjustamoving

charge),whichadmittedlyisaverysmallcurrent,butanycurrentproducesamagnetic

field.Sospinningelectronsproduceasmallmagneticfield.

5. Wecanalsomodelthisspinningelectronasamagneticdipole,meaningaNorthandSouth

pole,andwhenyouconsiderthatanyrealmaterialismadeofmany,manyelectrons

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(atoms),allthesesmalldipolescaninteractwithamagneticfield(anexternalone)to

produceanoverallmagneticbehaviorthisisparamagnetism.

6. Onethingweneedtoqualifythough:whenconsideringanindividualelement,sayfor

exampleAluminum,howmanyelectronsdoesAlhave?Ithas13,anoddnumber.That

meansitwillhave12electronsthatarepairedupwithrespecttotheirspin,butone

remainingthatisnotpairedup.

7. Thepairedupspinsoffseteachother:oneelectronspinsinonedirection,theother

electronspinsintheoppositedirection,sotheiroverallmagneticdipolecontributionadds

tobecome=0.

8. Buttheunpairedelectronsmagneticdipolecontributionremains,soAlcanhave

paramagneticbehavior.Asageneralrule,elementswithanoddnumberofelectronswill

beparamagnetic.Elementswithanevennumberofelectronswillnotbeparamagnetic.

9.

Diamagnetismcomesfromadifferenteffectoftheelectronsmotioninsideanatom.

Electronsalsoorbitthenucleus,sothisisalsoamovingchargeandwillalsoproducea

smallcurrent.Theorbitingelectroncreatesitsowneffectofamagneticdipole.

10.Inthiscase,thetinymagneticfieldscreatedbytheorbitingelectronswillactuallyinteract

withexternalmagneticfieldsinarepulsiveway,notanattractiveway.Bythatwemean,

themagneticdipolescreatedbytheorbitingelectronswillalignoppositetoanyexternal

magneticfield.

11.Sinceallatomshaveorbitingelectrons,essentiallyallelementsofferthepossibilityfor

diamagnetism.However,

it

is

very,

very

weak,

so

ifan

element

has

an

odd

number

of

electronsandthereforewillbeparamagnetic,thiswilldominateanddiamagnetismwill

notshowup.

12.Somematerialsarestronglydiamagneticthoughandcanbemadetolevitateinan

externalmagneticfieldduetotherepulsiveeffectofdiamagnetism.Also,livingthingscan

bemadetolevitateinVERYstrongmagneticfields(likearound15Torso)duetothe

diamagneticeffectfinallyshowinglargescalepropertiesforsuchlargemagneticfields(see

thelevitatingfroginyourtextbook!).

13.

Ferromagnetismis

the

final

class

of

magnetic

behavior.

This

is

by

far

the

most

familiar

formandtheonethatweseeanddealwithinoureverydaylives.

14.Ferromagnetismisalsorelatedtoelectronspin,butthistimeweareinterestedin

elementsthathaveseveralunpairedelectronsallwiththesamespin.ExamplesincludeFe,

Co,NiandrareearthelementslikeNd(neodymium).Alloftheseelementshaveseveral

unpairedelectronsthatallhavethesamespin.

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15.

Theseunpairedspinstendtoaligntogether,evenwithoutanyexternalmagneticfieldto

changetheirorientation.Sincetherearesomanyunpairedelectronswiththesamespin

andtheytendtoaligntogether,theycanadduponthelargescaletoproducestrong

magneticpropertiesandinteractions.

16.Thekeyingredientthatexplainswhysuchferromagneticmaterialsreacttoexternal

magneticfieldsandcanbemadetoproducetheirownverystrongmagneticfieldsisthe

conceptofferromagneticdomains.

17.Ifwecouldlookatapieceofferromagneticmaterialonthevery,verysmallscale,we

wouldfindthatalloftheirtinymagneticdipoleswerealignedinthesamedirectiondue

toeachatomhavingseveralunpairedelectronswithrespecttospin.BUTthereisafurther

qualification

18.Ifwezoomedoutalittlebitfromthisvery,verysmallscale,wewouldactuallyfindthat

thereweresectionsofthematerialthatweredifferentandwererepresentedbydistinct

borders.We

call

these

separate

regions

domains.

19.Thedomainsexistbecauseinanyrealmaterial,particularlyferromagneticmaterialswhich

areusuallycrystallinesolidsmadeofatomsthatcometogetherinaveryrepeatableand

crystallinestructure,therearemanyindividualcrystalsthatstacktogetherandmeetat

boundariestomakethefullsizematerial.Itisdifficulttomakearealsizedmaterialthatis

asinglecrystalmeaningittrulyisasinglerepetitivecrystalthroughoutwithoutany

boundarieswheretwocrystalscometogether.

20.Thesesmallcrystalscomingtogetheratboundariesrepresentthedomains.Eachindividual

domainhasallofitsmagneticdipolesalignedtogether.BUTifwelookatmanydomainsall

atonce,wewillfindthatoveralltheytendtobealignedinaveryrandomway(seepicture

above).Soonthelargescale,undermostconditions,theoverallmagneticeffectof

ferromagneticmaterialsisnotstrongbecausethedomainsarequiterandomintheir

direction.

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21.Thekeyideacomeswhensuchferromagneticmaterialsinteractwithanexternalmagnetic

field.Ofcourse,themagneticdipoles(andthenbyextensionthemagneticdomains)will

interactwiththismagneticfield,andtheywillinfactveryeasilyalignalltogetherinthe

samedirection.

22.SonowweDOhaveaverystrongmagneticeffectonthelargescalebecauseallofthetiny

magneticdipolesareinfactalignedinthesamedirection.Theferromagneticmaterialhas

becomelikeamagnet.

23.Theferromagneticmaterialwillstaythiswaysolongasitisintheexternalmagneticfield.

Butifitisremovedfromthemagneticfield,theindividualdomainswillgoback(forthe

mostpart)totheiroriginalorientationandtheoverallmagneticpropertiesarelost.A

magneticfieldcaneasilymovethedomains,butthedomainswillthengobackoncethe

magneticfieldisremoved.

24.

Thisistrueforanyferromagneticmaterial,butweDOhavetwodistinctclassesof

ferromagneticmaterials:

a. Thefirstclassarecalledsoftferromagneticmaterials.Thoseinthisgroupreact

exactlyasdescribedabovein2123.Theycanbeaffectedbyanexternalmagnetic

field,butifitisremovedthedomainsmostlygobackandits(permanent)magnetic

propertiesarelost.Fe(iron)isthebestexampleofasoftferromagneticmaterial.

b. Thesecondclassarecalledhardferromagneticmaterials.Theserespondmuchthe

sameasin21and22above,butoncetheexternalmagneticfieldisremoved,the

domainsactually

stay

pretty

much

like

they

were.

They

do

NOT

tend

to

go

back

to

a

randomorientation.

c. SopermanentmagnetslikeCo,Ni,andNdareallexamplesofhardferromagnetic

materials.

d. Theydontactuallyexistaspermanentmagnetsontheirown(naturally).Tomake

thembecomepermanentmagnets,theymustfirstbeplacedinaverystrongmagnetic

fieldtoalignalltheirdomains,andthenthosedomainswillmostlystayandyouhavea

permanentmagneticpropertyofthematerial.