Download - Current and Resistance - USNA 26... · Current and Resistance I. Electric Current: A. Although an electric current is a stream of moving charges, not all moving charges constitute

[SHIVOK SP212] February 8, 2016

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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.