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February 8, 2016 CH 26 CurrentandResistance
I.
ElectricCurrent:
A. Althoughanelectriccurrentisastreamofmovingcharges,notall
movingchargesconstituteanelectriccurrent.Ifthereistobean
electriccurrentthroughagivensurface,theremustbeanetflowof
chargethroughthatsurface.Twoexamplesaregiven.
1.
Thefreeelectrons(conductionelectrons)inanisolatedlengthof
copperwireareinrandommotionatspeedsoftheorderof106m/s.Ifyou
passahypotheticalplanethroughsuchawire,conductionelectronspass
throughitinbothdirectionsattherateofmanybillionspersecond—but
thereisnonettransportofchargeandthusnocurrentthroughthewire.
However,ifyouconnecttheendsofthewiretoabattery,youslightlybiasthe
flowinonedirection,withtheresultthattherenowisanettransportof
chargeandthusanelectriccurrentthroughthewire.
2.
Theflowofwaterthroughagardenhoserepresentsthedirectedflow
ofpositivecharge(theprotonsinthewatermolecules)atarateofperhaps
severalmillioncoulombspersecond.Thereisnonettransportofcharge,
becausethereisaparallelflowofnegativecharge(theelectronsinthewater
molecules)ofexactlythesameamountmovinginexactlythesamedirection.
B. Diagram
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February 8, 2016 C. Thefigurebelowshowsasectionofaconductor,partofa
conductingloopinwhichcurrenthasbeenestablished.Ifchargedq
passesthroughahypotheticalplane(suchasaa’)intimedt,thenthe
currentithroughthatplaneisdefinedas:
1.
Thechargethatpassesthroughtheplaneinatimeintervalextending
from0totis:
2.
Understeady‐stateconditions,thecurrentisthesameforplanesaa’,
bb’,andcc’andforallplanesthatpasscompletelythroughtheconductor,no
matterwhattheirlocationororientation.
3.
TheSIunitforcurrentisthecoulombpersecond,ortheampere(A):
D. SampleProblem:
1.
Anisolatedconductingspherehasa10cmradius.Onewirecarriesa
currentof1.0000020Aintoit.Anotherwirecarriesacurrentof
1.0000000Aoutofit.Howlongwouldittakeforthespheretoincreasein
potentialby1000V?
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February 8, 2016 II.
ElectricCurrent,ConservationofCharge,andDirectionofCurrent
A. Acurrentarrowisdrawninthedirectioninwhichpositivecharge
carrierswouldmove,eveniftheactualchargecarriersarenegative
andmoveintheoppositedirection.
1.
Diagram
III.
CurrentDensity
A. Themagnitudeofcurrentdensity,J,isequaltothecurrentperunit
areathroughanyelementofcrosssection.Ithasthesamedirectionas
thevelocityofthemovingchargesiftheyarepositiveandtheopposite
directioniftheyarenegative.
B. IfthecurrentisuniformacrossthesurfaceandparalleltodA,then
JisalsouniformandparalleltodA.
Here, A is the total area of the surface.
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February 8, 2016 C. TheSIunitforcurrentdensityistheamperepersquaremeter
2
(A/m ).
D. Diagram
1.
Figure26‐4showshowcurrentdensitycanberepresentedwitha
similarsetoflines,whichwecancallstreamlines.
2.
Thecurrent,whichistowardtheright,makesatransitionfromthe
widerconductoratthelefttothenarrowerconductorattheright.Since
chargeisconservedduringthetransition,theamountofchargeandthusthe
amountofcurrentcannotchange.
3.
However,thecurrentdensitychanges—itisgreaterinthenarrower
conductor.
E. SampleProblem:
1.
Afuseinanelectriccircuitisawirethatisdesignedtomelt,and
therebyopenthecircuit,ifthecurrentexceedsapredeterminedvalue.
Supposethatthematerialtobeusedinafusemeltswhenthecurrentdensity
risesto440A/cm2.Whatdiameterofcylindricalwireshouldbeusedtomake
afusethatwilllimitthecurrentto0.50A?
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February 8, 2016 F. CurrentDensity,DriftSpeed
1.
Whenaconductorhasacurrentpassingthroughit,theelectronsmove
randomly,buttheytendtodriftwithadriftspeedvdinthedirectionopposite
thatoftheappliedelectricfieldthatcausesthecurrent.Thedriftspeedistiny
comparedwiththespeedsintherandommotion.
2.
Inthefigurebelow,theequivalentdriftofpositivechargecarriersisin
thedirectionoftheappliedelectricfield,E.Ifweassumethatthesecharge
carriersallmovewiththesamedriftspeedvdandthatthecurrentdensityJis
uniformacrossthewire’scross‐sectionalareaA,thenthenumberofcharge
carriersinalengthLofthewireisnAL.Herenisthenumberofcarriersper
unitvolume.
3.
ThetotalchargeofthecarriersinthelengthL,eachwithchargee,is
then
4.
Thetotalchargemovesthroughanycrosssectionofthewireinthe
timeinterval
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February 8, 2016 IV.
ResistanceandResistivity:
A. Wedeterminetheresistancebetweenanytwopointsofa
conductorbyapplyingapotentialdifferenceVbetweenthosepoints
andmeasuringthecurrentithatresults.TheresistanceRisthen
B. TheSIunitforresistancethatfollowsfromEq.26‐8isthevoltper
ampere.Thishasaspecialname,theohm(symbol):
C. Inacircuitdiagram,werepresentaresistorandaresistancewith
thesymbol
D. Picture
1.
Remember,Isentthecolorcodesoutlastnight(email).
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February 8, 2016 E. Resistivity
1.
Theresistivity,,ofaresistorisdefinedas:
a)
2.
TheSIunitforis.m.
3.
Theconductivityofamaterialisthereciprocalofitsresistivity:
a)
4.
ExampleTable
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February 8, 2016 F. Note:Resistanceisapropertyofanobject.Resistivityisa
propertyofamaterial.
G. EquationforResistance
1.
Proof:
a)
Ifthestreamlinesrepresentingthecurrentdensityareuniform
throughoutthewire,theelectricfield,E,andthecurrentdensity,J,will
beconstantforallpointswithinthewire.
b)
c)
d)
Thus
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February 8, 2016 H. ResistanceandResistivity,VariationwithTemperature:
1.
Therelationbetweentemperatureandresistivityforcopper—andfor
metalsingeneral—isfairlylinearoveraratherbroadtemperaturerange.For
suchlinearrelationswecanwriteanempiricalapproximationthatisgood
enoughformostengineeringpurposes:
2.
Graph
I.
Sampleproblem:
1.
Acommonflashlightbulbisratedat0.30Aand2.9V(thevaluesofthe
currentandvoltageunderoperatingconditions).Iftheresistanceofthe
tungstenbulbfilamentatroomtemperature(20°C)is1.1Ω,whatisthe
temperatureofthefilamentwhenthebulbison?
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February 8, 2016 V.
Ohm’sLaw:
A. Ohm’sLawisanassertionthatthecurrentthroughadeviceis
ALWAYSdirectlyproportionaltothepotentialdifferenceappliedto
thedevice.
B. AconductingdeviceobeysOhm’sLawwhentheresistanceofthe
deviceisindependentofthemagnitudeandpolarityoftheapplied
potentialdifference.
C. AconductingmaterialobeysOhm’sLawwhentheresistivityofthe
materialisindependentofthemagnitudeanddirectionoftheapplied
electricfield.
D. Diagrams/graphs
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February 8, 2016 E. AMacroscopicViewofOhm’sLaw:
1.
Itisoftenassumedthattheconductionelectronsinametalmovewith
asingleeffectivespeedv ,andthisspeedisessentiallyindependentofthe
eff
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temperature.Forcopper,v =1.6x10 m/s.
eff
2.
Whenweapplyanelectricfieldtoametalsample,theelectronsmodify
theirrandommotionsslightlyanddriftveryslowly—inadirectionopposite
thatofthefield—withanaveragedriftspeedv .Thedriftspeedinatypical
d
‐7
6
metallicconductorisabout5x10 m/s,lessthantheeffectivespeed(1.6x10 m/s)bymanyordersofmagnitude.
3.
Themotionofconductionelectronsinanelectricfieldisacombination
ofthemotionduetorandomcollisionsandthatduetoE.
4.
IfanelectronofmassmisplacedinanelectricfieldofmagnitudeE,the
electronwillexperienceanacceleration:
5.
Intheaveragetimebetweencollisions,theaverageelectronwill
acquireadriftspeedofv =a.
d
6.
Thus
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February 8, 2016 VI.
PowerinElectricCircuits:
A. Inthefigurebelow,thereisanexternalconductingpathbetween
thetwoterminalsofthebattery.Asteadycurrentiisproducedinthe
circuit,directedfromterminalatoterminalb.Theamountofcharge
dqthatmovesbetweenthoseterminalsintimeintervaldtisequaltoi
dt.
B. Thischargedqmovesthroughadecreaseinpotentialof
magnitudeV,andthusitselectricpotentialenergydecreasesin
magnitudebytheamount
C. ThepowerPassociatedwiththattransferistherateoftransfer
dU/dt,givenby
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February 8, 2016 D. Theunitofpoweristhevolt‐ampere(VA).
E. Sampleproblem:
1.
InFig.below(a),a20Ωresistorisconnectedtoabattery.Figurebelow
(b)showstheincreaseofthermalenergyEthintheresistorasafunctionof
timet.TheverticalscaleissetbyEth,s=2.50mJ,andthehorizontalscaleisset
byts=4.0s.Whatistheelectricpotentialacrossthebattery?
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February 8, 2016 VII.
Semiconductors:
A. Puresiliconhasahighresistivityanditiseffectivelyaninsulator.
However,itsresistivitycanbegreatlyreducedinacontrolledwayby
addingminuteamountsofspecific“impurity”atomsinaprocess
calleddoping.
B. Asemiconductorislikeaninsulatorexceptthattheenergy
requiredtofreesomeelectronsisnotquitesogreat.Theprocessof
dopingcansupplyelectronsorpositivechargecarriersthatarevery
looselyheldwithinthematerialandthusareeasytogetmoving.Also,
bycontrollingthedopingofasemiconductor,onecancontrolthe
densityofchargecarriersthatareresponsibleforacurrent.
C. Theresistivityinaconductorisgivenby:
D. Inasemiconductor,nissmallbutincreasesveryrapidlywith
temperatureastheincreasedthermalagitationmakesmorecharge
carriersavailable.Thiscausesadecreaseofresistivitywithincreasing
temperature.Thesameincreaseincollisionratethatisnotedfor
metalsalsooccursforsemiconductors,butitseffectisswampedby
therapidincreaseinthenumberofchargecarriers.
E. Table
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February 8, 2016 VIII. Superconductors:
A. In1911,DutchphysicistKamerlinghOnnesdiscoveredthatthe
resistivityofmercuryabsolutelydisappearsattemperaturesbelow
about4K.Thisphenomenoniscalledsuperconductivity,anditmeans
thatchargecanflowthroughasuperconductingconductorwithout
losingitsenergytothermalenergy.
B. Oneexplanationforsuperconductivityisthattheelectronsthat
makeupthecurrentmoveincoordinatedpairs.Oneoftheelectrons
inapairmayelectricallydistortthemolecularstructureofthe
superconductingmaterialasitmovesthrough,creatingnearbya
short‐livedconcentrationofpositivecharge.Theotherelectroninthe
pairmaythenbeattractedtowardthispositivecharge.Such
coordinationbetweenelectronswouldpreventthemfromcolliding
withthemoleculesofthematerialandthuswouldeliminateelectrical
resistance.Newtheoriesappeartobeneededforthenewer,higher
temperaturesuperconductors.
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