Current and Resistance - USNA 26... · Current and Resistance I. Electric Current: A. Although an...
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[SHIVOK SP212] February 8, 2016 Page 1 CH 26 Current and Resistance I. Electric Current: A. Although an electric current is a stream of moving charges, not all moving charges constitute an electric current. If there is to be an electric current through a given surface, there must be a net flow of charge through that surface. Two examples are given. 1. The free electrons (conduction electrons) in an isolated length of copper wire are in random motion at speeds of the order of 10 6 m/s. If you pass a hypothetical plane through such a wire, conduction electrons pass through it in both directions at the rate of many billions per second—but there is no net transport of charge and thus no current through the wire. However, if you connect the ends of the wire to a battery, you slightly bias the flow in one direction, with the result that there now is a net transport of charge and thus an electric current through the wire. 2. The flow of water through a garden hose represents the directed flow of positive charge (the protons in the water molecules) at a rate of perhaps several million coulombs per second. There is no net transport of charge, because there is a parallel flow of negative charge (the electrons in the water molecules) of exactly the same amount moving in exactly the same direction. B. Diagram
Transcript of Current and Resistance - USNA 26... · Current and Resistance I. Electric Current: A. Although an...
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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.
