Post on 31-Dec-2019
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CH 26
CurrentandResistance
I. ElectricCurrent:
A. Althoughanelectriccurrentisastreamofmovingcharges,notallmovingchargesconstituteanelectriccurrent.Ifthereistobeanelectriccurrentthroughagivensurface,theremustbeanetflowofchargethroughthatsurface.Twoexamplesaregiven.
1. Thefreeelectrons(conductionelectrons)inanisolatedlengthofcopperwireareinrandommotionatspeedsoftheorderof106m/s.Ifyoupassahypotheticalplanethroughsuchawire,conductionelectronspassthroughitinbothdirectionsattherateofmanybillionspersecond—butthereisnonettransportofchargeandthusnocurrentthroughthewire.However,ifyouconnecttheendsofthewiretoabattery,youslightlybiastheflowinonedirection,withtheresultthattherenowisanettransportofchargeandthusanelectriccurrentthroughthewire.
2. Theflowofwaterthroughagardenhoserepresentsthedirectedflowofpositivecharge(theprotonsinthewatermolecules)atarateofperhapsseveralmillioncoulombspersecond.Thereisnonettransportofcharge,becausethereisaparallelflowofnegativecharge(theelectronsinthewatermolecules)ofexactlythesameamountmovinginexactlythesamedirection.
B. Diagram
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C. Thefigurebelowshowsasectionofaconductor,partofaconductingloopinwhichcurrenthasbeenestablished.Ifchargedqpassesthroughahypotheticalplane(suchasaa’)intimedt,thenthecurrentithroughthatplaneisdefinedas:
1. Thechargethatpassesthroughtheplaneinatimeintervalextendingfrom0totis:
2. Understeady‐stateconditions,thecurrentisthesameforplanesaa’,bb’,andcc’andforallplanesthatpasscompletelythroughtheconductor,nomatterwhattheirlocationororientation.
3. TheSIunitforcurrentisthecoulombpersecond,ortheampere(A):
D. SampleProblem:
1. Anisolatedconductingspherehasa10cmradius.Onewirecarriesacurrentof1.0000020Aintoit.Anotherwirecarriesacurrentof1.0000000Aoutofit.Howlongwouldittakeforthespheretoincreaseinpotentialby1000V?
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II. ElectricCurrent,ConservationofCharge,andDirectionofCurrent
A. Acurrentarrowisdrawninthedirectioninwhichpositivechargecarrierswouldmove,eveniftheactualchargecarriersarenegativeandmoveintheoppositedirection.
1. Diagram
III. CurrentDensity
A. Themagnitudeofcurrentdensity,J,isequaltothecurrentperunitareathroughanyelementofcrosssection.Ithasthesamedirectionasthevelocityofthemovingchargesiftheyarepositiveandtheoppositedirectioniftheyarenegative.
B. IfthecurrentisuniformacrossthesurfaceandparalleltodA,thenJisalsouniformandparalleltodA.
Here, A is the total area of the surface.
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C. TheSIunitforcurrentdensityistheamperepersquaremeter
(A/m2).
D. Diagram
1. Figure26‐4showshowcurrentdensitycanberepresentedwithasimilarsetoflines,whichwecancallstreamlines.
2. Thecurrent,whichistowardtheright,makesatransitionfromthewiderconductoratthelefttothenarrowerconductorattheright.Sincechargeisconservedduringthetransition,theamountofchargeandthustheamountofcurrentcannotchange.
3. However,thecurrentdensitychanges—itisgreaterinthenarrowerconductor.
E. SampleProblem:
1. Afuseinanelectriccircuitisawirethatisdesignedtomelt,andtherebyopenthecircuit,ifthecurrentexceedsapredeterminedvalue.Supposethatthematerialtobeusedinafusemeltswhenthecurrentdensityrisesto440A/cm2.Whatdiameterofcylindricalwireshouldbeusedtomakeafusethatwilllimitthecurrentto0.50A?
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F. CurrentDensity,DriftSpeed
1. Whenaconductorhasacurrentpassingthroughit,theelectronsmoverandomly,buttheytendtodriftwithadriftspeedv
dinthedirectionopposite
thatoftheappliedelectricfieldthatcausesthecurrent.Thedriftspeedistinycomparedwiththespeedsintherandommotion.
2. Inthefigurebelow,theequivalentdriftofpositivechargecarriersisinthedirectionoftheappliedelectricfield,E.Ifweassumethatthesechargecarriersallmovewiththesamedriftspeedv
dandthatthecurrentdensityJis
uniformacrossthewire’scross‐sectionalareaA,thenthenumberofchargecarriersinalengthLofthewireisnAL.Herenisthenumberofcarriersperunitvolume.
3. ThetotalchargeofthecarriersinthelengthL,eachwithchargee,isthen
4. Thetotalchargemovesthroughanycrosssectionofthewireinthetimeinterval
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IV. ResistanceandResistivity:
A. WedeterminetheresistancebetweenanytwopointsofaconductorbyapplyingapotentialdifferenceVbetweenthosepointsandmeasuringthecurrentithatresults.TheresistanceRisthen
B. TheSIunitforresistancethatfollowsfromEq.26‐8isthevoltperampere.Thishasaspecialname,theohm(symbol):
C. Inacircuitdiagram,werepresentaresistorandaresistancewith
thesymbol
D. Picture
1. Remember,Isentthecolorcodesoutlastnight(email).
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E. Resistivity
1. Theresistivity,,ofaresistorisdefinedas:
a)
2. TheSIunitforis.m.
3. Theconductivityofamaterialisthereciprocalofitsresistivity:
a)
4. ExampleTable
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F. Note:Resistanceisapropertyofanobject.Resistivityisapropertyofamaterial.
G. EquationforResistance
1. Proof:
a) Ifthestreamlinesrepresentingthecurrentdensityareuniformthroughoutthewire,theelectricfield,E,andthecurrentdensity,J,willbeconstantforallpointswithinthewire.
b)
c)
d) Thus
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H. ResistanceandResistivity,VariationwithTemperature:
1. Therelationbetweentemperatureandresistivityforcopper—andformetalsingeneral—isfairlylinearoveraratherbroadtemperaturerange.Forsuchlinearrelationswecanwriteanempiricalapproximationthatisgoodenoughformostengineeringpurposes:
2. Graph
I. Sampleproblem:
1. Acommonflashlightbulbisratedat0.30Aand2.9V(thevaluesofthecurrentandvoltageunderoperatingconditions).Iftheresistanceofthetungstenbulbfilamentatroomtemperature(20°C)is1.1Ω,whatisthetemperatureofthefilamentwhenthebulbison?
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V. Ohm’sLaw:
A. Ohm’sLawisanassertionthatthecurrentthroughadeviceisALWAYSdirectlyproportionaltothepotentialdifferenceappliedtothedevice.
B. AconductingdeviceobeysOhm’sLawwhentheresistanceofthedeviceisindependentofthemagnitudeandpolarityoftheappliedpotentialdifference.
C. AconductingmaterialobeysOhm’sLawwhentheresistivityofthematerialisindependentofthemagnitudeanddirectionoftheappliedelectricfield.
D. Diagrams/graphs
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E. AMacroscopicViewofOhm’sLaw:
1. Itisoftenassumedthattheconductionelectronsinametalmovewithasingleeffectivespeedv
eff,andthisspeedisessentiallyindependentofthe
temperature.Forcopper,veff=1.6x10
6m/s.
2. Whenweapplyanelectricfieldtoametalsample,theelectronsmodifytheirrandommotionsslightlyanddriftveryslowly—inadirectionoppositethatofthefield—withanaveragedriftspeedv
d.Thedriftspeedinatypical
metallicconductorisabout5x10‐7m/s,lessthantheeffectivespeed(1.6x10
6
m/s)bymanyordersofmagnitude.
3. ThemotionofconductionelectronsinanelectricfieldisacombinationofthemotionduetorandomcollisionsandthatduetoE.
4. IfanelectronofmassmisplacedinanelectricfieldofmagnitudeE,the
electronwillexperienceanacceleration:
5. Intheaveragetimebetweencollisions,theaverageelectronwill
acquireadriftspeedofvd=a.
6. Thus
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VI. PowerinElectricCircuits:
A. Inthefigurebelow,thereisanexternalconductingpathbetweenthetwoterminalsofthebattery.Asteadycurrentiisproducedinthecircuit,directedfromterminalatoterminalb.Theamountofchargedqthatmovesbetweenthoseterminalsintimeintervaldtisequaltoidt.
B. ThischargedqmovesthroughadecreaseinpotentialofmagnitudeV,andthusitselectricpotentialenergydecreasesinmagnitudebytheamount
C. ThepowerPassociatedwiththattransferistherateoftransferdU/dt,givenby
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D. Theunitofpoweristhevolt‐ampere(VA).
E. Sampleproblem:
1. InFig.below(a),a20Ωresistorisconnectedtoabattery.Figurebelow
(b)showstheincreaseofthermalenergyEthintheresistorasafunctionoftimet.TheverticalscaleissetbyEth,s=2.50mJ,andthehorizontalscaleissetbyts=4.0s.Whatistheelectricpotentialacrossthebattery?
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VII. Semiconductors:
A. Puresiliconhasahighresistivityanditiseffectivelyaninsulator.However,itsresistivitycanbegreatlyreducedinacontrolledwaybyaddingminuteamountsofspecific“impurity”atomsinaprocesscalleddoping.
B. Asemiconductorislikeaninsulatorexceptthattheenergyrequiredtofreesomeelectronsisnotquitesogreat.Theprocessofdopingcansupplyelectronsorpositivechargecarriersthatareverylooselyheldwithinthematerialandthusareeasytogetmoving.Also,bycontrollingthedopingofasemiconductor,onecancontrolthedensityofchargecarriersthatareresponsibleforacurrent.
C. Theresistivityinaconductorisgivenby:
D. Inasemiconductor,nissmallbutincreasesveryrapidlywithtemperatureastheincreasedthermalagitationmakesmorechargecarriersavailable.Thiscausesadecreaseofresistivitywithincreasingtemperature.Thesameincreaseincollisionratethatisnotedformetalsalsooccursforsemiconductors,butitseffectisswampedbytherapidincreaseinthenumberofchargecarriers.
E. Table
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VIII. Superconductors:
A. In1911,DutchphysicistKamerlinghOnnesdiscoveredthattheresistivityofmercuryabsolutelydisappearsattemperaturesbelowabout4K.Thisphenomenoniscalledsuperconductivity,anditmeansthatchargecanflowthroughasuperconductingconductorwithoutlosingitsenergytothermalenergy.
B. Oneexplanationforsuperconductivityisthattheelectronsthatmakeupthecurrentmoveincoordinatedpairs.Oneoftheelectronsinapairmayelectricallydistortthemolecularstructureofthesuperconductingmaterialasitmovesthrough,creatingnearbyashort‐livedconcentrationofpositivecharge.Theotherelectroninthepairmaythenbeattractedtowardthispositivecharge.Suchcoordinationbetweenelectronswouldpreventthemfromcollidingwiththemoleculesofthematerialandthuswouldeliminateelectricalresistance.Newtheoriesappeartobeneededforthenewer,highertemperaturesuperconductors.