Electromagnetism - deu.edu. 1 Electrics Electromagnetism Electromagnetism Magnetism is associated with charges in motion (currents): – microscopiccurrents in the atoms of magnetic

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    Electrics

    Electromagnetism

    ElectromagnetismMagnetismisassociatedwithchargesinmotion(currents):

    microscopic currentsintheatomsofmagneticmaterials. macroscopic currentsinthewindingsofanelectromagnet.

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    ElectromagnetismTheregionaroundamovingchargeisdisturbedbythechargesmotion.

    The Magnetic field Bdefines theeffect ofamagnet atapoint invector form.

    Themagneticfieldlinesshowthepath, anobjectwouldtakeifitwereattractedtoonepoleofmagnet(and,ofcourse,repelledbytheother).

    ElectromagnetismIfanisolatedchargeismoving,thespacecontainsbothanelectricfieldANDamagneticfield.Ifthechargeisstationary,onlyanelectricfieldispresent.

    Similartoelectricforceinstrengthanddirection,magneticb d h ` l ' ( h d h dobjectsaresaidtohave`poles'(northandsouth,instead

    ofpositiveandnegativecharge).However,magneticobjectsarealwaysfoundinpairs,thereexistsno isolatedmagnetic polesinNature.

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    Magnetic Force

    Electromagnetic ForceThemagneticfieldBisdefineddue to theLorentzForceLaw,andspecificallyfromthemagneticforceonamovingcharge:

    BEF

    BvqEqFLorentz

    Magnetic Force

    Electromagnetic Force1.TheforceisperpendiculartoboththevelocityvofthechargeqandthemagneticfieldB.

    BvqEqFLorentz

    2. ThemagnitudeoftheforceisF=qvB sin whereistheangle

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    Magnetic FieldsThemagnitudeofthemagneticfieldbymeasuringtheforceonamovingcharge:

    FB mag

    B

    Themagneticfield'sunitsareNs/Cm,whichareusuallyabbreviatedasTeslas,

    1 Tesla=1 Ns/Cm

    sinvqB g

    v

    1Tesla=1Ns/Cm.AsmallermagneticfieldunitistheGauss

    1Tesla=10,000Gauss

    Direction of Magnetic ForcesThedirectionofthemagneticforceonamovingchargeisperpendiculartotheplaneformedbyBandv.

    Todeterminethedirection,theh d l ( ) bF

    vTodepictavectororientedperpendiculartothepageweusecrossesanddots.

    Acrossindicatesavectorgoinginto

    RightHandRule(RHR) mustbeapplied

    thepage(thinkofthetailfeathersofanarrowdisappearingintothepage).

    Adotindicatesavectorcomingoutofthepage(thinkofthetipofanarrowcomingatyou,outofthepage).

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    Direction of Magnetic Forces (Right Hand Rule )

    Drawvectorsv andB withtheirtailsatthelocationofthechargeq.Point fingers of right handPointfingersofrighthandalongvelocityvectorv.CurlfingerstowardsMagneticfieldvectorB.ThumbpointsindirectionofmagneticforceF onq,perpendicular to both v andperpendiculartobothv andB.

    Motion of Charges in B FieldsIfachargedparticleismovingperpendiculartoauniformmagneticfield,itstrajectorywillbeacirclebecausetheforceF=qvB isalwaysperpendiculartothemotion,andtherefore centripetalthereforecentripetal.

    rmvmaF

    FF

    a

    maga

    2

    0

    Fromwhichwefindtheradiusofthecirculartrajectoryis:

    qvBFmag

    qBmvr

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    Magnetic Force on a Current-Carrying WireSinceelectriccurrentconsistsofacollectionofchargedparticlesin motion,whenplacedinamagneticfield,acurrentcarryingwirewillalsoexperienceamagneticforceforce.

    :

    )(

    l

    BlIFmag

    lengthvectorwithamagnitudelanddirectedalongthe directionoftheelectriccurrent.

    Torque on a Current LoopThemagneticforcesactingonsides1and3vanishbecausetheir

    lengthvectorsareparalleltoBandtheircrossproductsvanish.Ontheotherhand,themagneticforcesactingonsegments2and4arenonvanishing:

    Eventhoughthenetforceontheloopvanishes,theforcesF2and F4 will produce a torque which causes the loop to rotate

    :

    )(

    A

    BAImag

    andF4 willproduceatorquewhichcausesthelooptorotate

    Itisconvenienttointroducetheareavector isdefined with amagnitudeofarea ofthe loop and the directionnormaltotheplaneoftheloop.Thedirectionofthepositivesenseissetbytheconventionalrighthandrule.

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    Magnetic Field of CurrentThemagneticfieldlinesaroundalongwirewhichcarriesanelectriccurrentformconcentriccirclesaroundthewire.

    I

    The direction of the magnetic field is perpendicular to the

    r2IB 0

    Theconstant 0 isthepermeabilityoffreespace.

    Thedirectionofthemagneticfieldisperpendiculartothewire

    andisinthedirectionthefingersofyourrighthandwouldcurlifyouwrappedthemaroundthewirewithyourthumbinthedirectionofthecurrent.

    Magnetic Field of Current LoopElectriccurrentinacircularloopcreatesamagneticfieldwhichismoreconcentratedinthecenteroftheloopthanoutsidetheloop.

    Fi ld C f C LFieldatCenterofCurrentLoop

    RIB

    20

    FieldonAxisofCurrentLoop2

    0 2 IRB 3/2220

    )(2

    4 RzIRB

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    SolenoidAlongstraightcoilofwirecanbeusedtogenerateanearlyuniformmagneticfieldsimilartothatofabar magnet Such coils calledbarmagnet.Suchcoils,calledsolenoids,haveanenormousnumberofpracticalapplications.

    SolenoidEach loop ofthe coil isassumed to beseperate sources formagnetic fiels then the resultant magnetic field issimplysuperposition ofthe vector sum ofeach loop

    N InLNIB 00

    Nisthe #ofwindings and theListhe Length ofcoil.

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    SolenoidThedirectionofthemagneticfluxlinescanbefoundbyplacingthethumboftheright handinthedirectionofconventionalcurrent flowandnotingthedirectionofthefingersfingers

    Helmholtz CoilAHelmholtzcoilisadeviceforproducingaregionofnearlyuniformmagneticfield.

    AHelmholtzpairconsistsoftwoidenticalcircularmagneticcoils that are placed symmetrically one on each side of a

    RInB 0

    2/3

    54

    coils thatareplacedsymmetricallyoneoneachsideofacommonaxis,andseparatedbyadistanceequaltotheradiusofthecoil.

    Eachcoilcarriesanequalelectricalcurrentflowinginthesamedirection.

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    Maxwell Coil (Uniform)AMaxwellcoilisadeviceforproducingalargevolumeofalmostconstantorconstantgradientmagneticfield.

    AconstantfieldMaxwellcoilsetconsistsofthreecoilsoriented on the surface of a virtual sphereorientedonthesurfaceofavirtualsphere.

    Eachoftheoutercoilsshouldbeofradius 4/7R anddistancefromtheplaneof 3/7R thecentralcoilofradius R .Thenumberofampereturnsofeachofthesmallercoilsshouldequalexactly49/64 ofthemiddlecoil.

    Maxwell Coil (Gradient)AMaxwellcoilisadeviceforproducingalargevolumeofalmostconstantorconstantgradientmagneticfield.

    AconstantfieldMaxwellcoilsetconsistsofthreecoilsoriented on the surface of a virtual sphereorientedonthesurfaceofavirtualsphere.

    Eachoftheoutercoilsshouldbeofradius 4/7R anddistancefromtheplaneof 3/7R thecentralcoilofradius R .Thenumberofampereturnsofeachofthesmallercoilsshouldequalexactly49/64 ofthemiddlecoil.

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    Electromagnetic InductionImposinganelectricfieldonaconductorgivesrisetoacurrentwhichinturngeneratesamagneticfield.

    Onecouldtheninquirewhetherornotanelectricfieldld b d d b ti fi ldcouldbeproducedbyamagneticfield.

    In1831,MichaelFaradaydiscoveredthat,byvaryingmagneticfieldwithtime,anelectricfieldcouldbegenerated.

    Thephenomenonisknownaselectromagneticinduction.

    Faradays Law of InductionFaradayshowedthatnocurrentisregisteredinthegalvanometerwhenbarmagnetisstationarywithrespecttotheloop.

    H t i i d d i th l h l tiHowever,acurrentisinducedintheloopwhenarelativemotionexistsbetweenthebarmagnetandtheloop.

    Inparticular,theampermeterdeflectsinonedirectionasthemagnetapproachestheloop,andtheoppositedirectionasitmovesaway.

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    Magnetic FluxFaradaysexperimentdemonstratesthatanelectriccurrentisinducedintheloopbychangingthemagneticfield.Thecoilbehavesasifitwereconnectedtoanvoltage sourcevoltage source.

    Experimentallyitisfoundthattheinducedvoltagedependsontherateofchangeofmagneticfluxthroughthecoil.

    Themagneticfluxthrougha surfaceisgivenby:

    cosBAAB

    TheSIunitofmagneticfluxistheweber (Wb):1Wb =1Tm2

    Faradays Law of InductionTheinducedemf (voltage) inacoilisproportionaltothenegativeoftherateofchangeofmagneticflux:

    For acoil ofNwindings:dtd

    dtd

    N

    dt dt

    Ane.m.f. maybeinducedinthefollowingways:byvaryingthemagnitudeofBwithtime

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    Faradays Law of InductionTheinducedemf (voltage) inacoilisproportionaltothenegativeoftherateofchangeofmagneticflux:

    For acoil ofNwindings:dtd

    dtd

    N

    dt dt

    Ane.m.f. maybeinducedinthefollowingways:byvaryingthemagnitudeofA,i.e.,theareaenclosedbytheloopwithtime

    cos BAAB

    Faradays Law of InductionTheinducedemf (voltage) inacoilisproportionaltothenegativeoftherateofchangeofmagneticflux:

    For acoil ofNwindings:dtd

    dtd

    N

    dt dt

    Ane.m.f. maybeinducedinthefollowingways:varyingtheanglebetweenBandtheareavectorAwithtime

    cos BAAB

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    Lenzs Law Direction of Induction CurrentAninducedcurrentalwaysflowsinadirectionthatopposesthechangethatcausedit.

    In thisexamplethemagneticfieldintheupwarddirectionthroughtheloopisincreasing.

    Soacurrentisgeneratedintheloopwhichproducesandownwardmagneticfieldinsidethe loop to oppose the change.

    S

    N Iinducedthelooptoopposethechange.

    Motional E.M.F.Each charge within the

    conductor is movingand experiences amagnetic force

    qvBFmag

    Theseparatedchargeson theendsofthe conductorgiverisetoaninducedemf,calledamotionalemf.

    In equilibrium the electric force of repulsion from chargeInequilibrium,theelectricforceofrepulsion fromchargebuildupattheends,qE (where Eistheelectricfield),balancesFmag:

    BvE

    qEqvBFmag

    BLvBvLs

    VE

    MotionalEmf

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    The Electric GeneratorA generator converts mechanical energy to electricalenergy. Consider a current loop which rotates in aconstant magnetic field

    iBLW