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February 8, 2016 CH 27 Circuits
I.
“Pumping”Charges:
A. Inordertoproduceasteadyflowofchargethrougharesistor,one
needsa“chargepump,”adevicethat—bydoingworkonthecharge
carriers—maintainsapotentialdifferencebetweenapairof
terminals.
B. Suchadeviceiscalledanemf,or.
C. Acommonemfdeviceisthebattery,usedtopowerawidevariety
ofmachinesfromwristwatchestosubmarines.Theemfdevicethat
mostinfluencesourdailylivesistheelectricgenerator,which,by
meansofelectricalconnections(wires)fromageneratingplant,
createsapotentialdifferenceinourhomesandworkplaces.
D. Someotheremfdevicesknownaresolarcells,fuelcells.Anemf
devicedoesnothavetobeaninstrument—livingsystems,ranging
fromelectriceelsandhumanbeingstoplants,havephysiologicalemf
devices.
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February 8, 2016 II.
Work,Energy,andEmf:
A. SimpleCircuit:
1.
Inanytimeintervaldt,achargedqpassesthroughanycrosssectionof
thecircuitshown,suchasaa’.Thissameamountofchargemustentertheemf
deviceatitslow‐potentialendandleaveatitshigh‐potentialend.
2.
TheemfdevicemustdoanamountofworkdWonthechargedqto
forceittomoveinthisway.
3.
Wedefinetheemfoftheemfdeviceintermsofthiswork:
B. Anidealemfdeviceisonethathasnointernalresistancetothe
internalmovementofchargefromterminaltoterminal.Thepotential
differencebetweentheterminalsofanidealemfdeviceisexactly
equaltotheemfofthedevice.
C. Arealemfdevice,suchasanyrealbattery,hasinternalresistance
totheinternalmovementofcharge.Whenarealemfdeviceisnot
connectedtoacircuit,andthusdoesnothavecurrentthroughit,the
potentialdifferencebetweenitsterminalsisequaltoitsemf.However,
whenthatdevicehascurrentthroughit,thepotentialdifference
betweenitsterminalsdiffersfromitsemf.
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February 8, 2016 III.
CalculatingtheCurrentinaSingle‐LoopCircuit:
A. Circuit:
2
B. TheequationP=i Rtellsusthatinatimeintervaldtanamountof
2
energygivenbyi Rdtwillappearintheresistor,asshowninthe
figure,asthermalenergy.
C. Duringthesameinterval,achargedq=idtwillhavemoved
throughbatteryB,andtheworkthatthebatterywillhavedoneon
thischarge,is
D. Fromtheprincipleofconservationofenergy,theworkdonebythe
(ideal)batterymustequalthethermalenergythatappearsinthe
resistor:
E. Therefore,theenergyperunitchargetransferredtothemoving
chargesisequaltotheenergyperunitchargetransferredfromthem.
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February 8, 2016 IV.
CalculatingtheCurrentinaSingle‐LoopCircuit,PotentialMethod:
A. LoopRule(Kirchoff’svoltagelaw):Thealgebraicsumofthe
changesinpotentialencounteredinacompletetraversalofanyloop
ofacircuitmustbeZERO.
B. Circuit:
1.
Inthefigure,letusstartatpointa,whosepotentialisVa,andmentally
goclockwisearoundthecircuituntilwearebackata,keepingtrackof
potentialchangesaswemove.
2.
Ourstartingpointisatthelow‐potentialterminalofthebattery.Since
thebatteryisideal,thepotentialdifferencebetweenitsterminalsisequalto
E.
3.
Aswegoalongthetopwiretothetopendoftheresistor,thereisno
potentialchangebecausethewirehasnegligibleresistance.
4.
Whenwepassthroughtheresistor,however,thepotentialdecreases
byiR.
5.
Wereturntopointabymovingalongthebottomwire.Atpointa,the
potentialisagainVa.Theinitialpotential,asmodifiedforpotentialchanges
alongtheway,mustbeequaltoourfinalpotential;thatis
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February 8, 2016 C. Forcircuitsthataremorecomplexthanthatofthepreviousfigure,
twobasicrulesareusuallyfollowedforfindingpotentialdifferences
aswemovearoundaloop:
1.
ResistanceRule:Foramovethrougharesistanceinthedirectionofthe
current,thechangeinpotentialis–iR;intheoppositedirectionitis+iR.
2.
EMFRule:Foramovethroughanidealemfdeviceinthedirectionof
theemfarrow,thechangeinpotentialis+ E; in the opposite direction it is – E. V.
OtherSingle‐LoopCircuits,InternalResistance:
A. Figure
1.
Thefigureaboveshowsarealbattery,withinternalresistancer,wired
toanexternalresistorofresistanceR.Accordingtothepotentialrule,
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February 8, 2016 VI.
OtherSingle‐LoopCircuits,ResistancesinSeries:
A. Kirchoff’sVoltagelaw:Thealgebraicsumofthechangesin
potentialencounteredinacompletetraversalofanyloopofacircuit
mustbeZERO.
1.
Inotherwords:WhenapotentialdifferenceVisappliedacross
resistancesconnectedinseries,theresistanceshaveidenticalcurrentsi.The
sumofthepotentialdifferencesacrosstheresistancesisequaltotheapplied
potentialdifferenceV.
2.
3.
Thus,
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February 8, 2016 B. Resistancesconnectedinseriescanbereplacedwithan
equivalentresistanceReqthathasthesamecurrentiandthesame
totalpotentialdifferenceVastheactualresistances.
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February 8, 2016 VII.
Potentialbetweentwopoints:
A. Tofindthepotentialbetweenanytwopointsinacircuit,startat
onepointandtraversethecircuittotheotherpoint,followanypath,
andaddalgebraicallythechangesinpotentialyouencounter.
B. Thecircuit:
1.
Goingclockwisefroma:
2.
Goingcounterclockwisefroma:
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February 8, 2016 VIII. Potentialacrossarealbattery:
A. Circuit:
1.
Iftheinternalresistancerofthebatteryinthepreviouscasewere
zero,VwouldbeequaltotheemfEofthebattery—namely,12V.
2.
However,sincer=2.0,VislessthanE.
3.
Theresultdependsonthevalueofthecurrentthroughthebattery.If
thesamebatterywereinadifferentcircuitandhadadifferentcurrent
throughit,Vwouldhavesomeothervalue.
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February 8, 2016 IX.
GroundingaCircuit:
A. Circuit:
1.
Thisisthesameexampleasinthepreviousslide,exceptthatbattery
terminalaisgroundedinFig.27‐7a.Groundingacircuitusuallymeans
connectingthecircuittoaconductingpathtoEarth’ssurface,andsucha
connectionmeansthatthepotentialisdefinedtobezeroatthegrounding
pointinthecircuit.
2.
InFig.27‐7a,thepotentialataisdefinedtobeV =0.Therefore,the
a
potentialatbisV =8.0V.
b
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February 8, 2016 X.
Power,Potential,andEmf:
A. ThenetratePofenergytransferfromtheemfdevicetothecharge
carriersisgivenby:
whereVisthepotentialacrosstheterminalsoftheemfdevice.
B. But,therefore
1.
ButP istherateofenergytransfertothermalenergywithintheemf
r
device:
C. ThereforethetermiEmustbetheratePemfatwhichtheemfdevice
transfersenergybothtothechargecarriersandtointernalthermal
energy.
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February 8, 2016 XI.
Example,Singleloopcircuitwithtworealbatteries:
A.
1.
Solution:
B. Graph:
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February 8, 2016 C.
1.
Solution:
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February 8, 2016 XII.
Multi‐loopCircuits:
A. Junction/NodeRule[Kirchhoff’scurrentlaw]:Thealgebraicsum
ofthecurrentsenteringanyjunction,or“Node”,mustbeequaltothe
algebraicsumofthecurrentsleavingthatjunction/Node
B. Circuit:
1.
Considerjunctiondinthecircuit.Incomingcurrentsi andi ,andit
1
3
leavesviaoutgoingcurrenti .Sincethereisnovariationinthechargeatthe
2
junction,thetotalincomingcurrentmustequalthetotaloutgoingcurrent:
2.
ThisruleisoftencalledKirchhoff’sjunctionrule(orKirchhoff’s
currentlaw).
3.
NoticetherearemultipleKirchoff’sVoltageLawloopspossible:
a)
Fortheleft‐handloop,
b)
Fortheright‐handloop,
c)
Andfortheentireloop(orakaouterloop),
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February 8, 2016 XIII. Multi‐loopCircuits,ResistorsinParallel:
A. WhenapotentialdifferenceVisappliedacrossresistance
connectedinparallel,theresistancesallhavethatsamepotential
differenceV.
B. Resistancesconnectedinparallelcanbereplacedwithan
equivalentReqthathasthesamepotentialdifferenceVandthesame
totalcurrentiastheactualresistances.
C. Circuit:
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February 8, 2016 XIV. Multi‐loopCircuits:
A. SummaryTable
XV.
Example,ResistorsinParallelandinSeries:
A. Circuit
1.
Solutioninsteps:
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February 8, 2016 2.
Solution:
3.
Solution:
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February 8, 2016 XVI. Example,Realbatteriesinseriesandparallel:
A. ElectricFishforexample
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February 8, 2016 1.
(a)IfthewatersurroundingtheeelhasresistanceRw=800,how
muchcurrentcantheeelproduceinthewater?
2.
a)
Solution:
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February 8, 2016 XVII. Multi‐loopcircuitandsimultaneousloopequations:
A. Example
1.
Solution:
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2.
Solutioncontinued:
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February 8, 2016 [SHIVOK SP212]
February 8, 2016 XVIII. AmmeterandVoltmeter:
A. Aninstrumentusedtomeasurecurrentsiscalledanammeter.Itis
essentialthattheresistanceRAoftheammeterbeverymuchsmaller
thanotherresistancesinthecircuit.
B. Ameterusedtomeasurepotentialdifferencesiscalleda
voltmeter.ItisessentialthattheresistanceRVofavoltmeterbevery
muchlargerthantheresistanceofanycircuitelementacrosswhich
thevoltmeterisconnected.
C. Diagramofhowtoconnectammetersandvoltagemeters.
1.
AmmetersareconnectedinSERIESsothattheymeasurethecurrent
youareinterestedin.
2.
Voltagemetersareconnectedinparallelsothattheymeasurethe
voltageyouareinterestedin.
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February 8, 2016 XIX. RCCircuits,ChargingaCapacitor:
A. Circuit:
1.
Itturnsoutthat:
2.
Weknowthat:
3.
4.
5.
Graph
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February 8, 2016 6.
Acapacitorthatisbeingchargedinitiallyactslikeanordinary
connectingwirerelativetothechargingcurrent.Alongtimelater,itactslikea
brokenwire.
B. RCCircuits,TimeConstant:
1.
TheproductRCiscalledthecapacitivetimeconstantofthecircuitand
isrepresentedwiththesymbol:
2.
Attimet==(RC),thechargeontheinitiallyunchargedcapacitor
increasesfromzeroto:
3.
Duringthefirsttimeconstantthechargehasincreasedfromzeroto
63%ofitsfinalvalueCE.
4.
i=
XX.
RCCircuits,DischargingaCapacitor:
A. Assumethatthecapacitorofthefigureisfullychargedtoa
potentialV0equaltotheemfofthebatteryE.
B. Atanewtimet=0,switchSisthrownfromatobsothatthe
capacitorcandischargethroughresistanceR.
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February 8, 2016 1.
2.
3.
Graph
Figure:Thisshowsthedeclineofthechargingcurrentinthecircuit.The
curvesareplottedforR=2000,C=1F,andE=10V;thesmalltriangles
representsuccessiveintervalsofonetimeconstant.
4.
i=
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February 8, 2016 C. SampleProblem:
1.
Figurebelowshowsthecircuitofaflashinglamp,likethoseattachedto
barrelsathighwayconstructionsites.ThefluorescentlampL(ofnegligible
capacitance)isconnectedinparallelacrossthecapacitorCofanRCcircuit.
Thereisacurrentthroughthelamponlywhenthepotentialdifferenceacross
itreachesthebreakdownvoltageVL;thenthecapacitordischargescompletely
throughthelampandthelampflashesbriefly.Foralampwithbreakdown
voltageVL=72.0V,wiredtoa95.0Videalbatteryanda0.150μFcapacitor,
whatresistanceRisneededfortwoflashespersecond?
2.
Solution:
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February 8, 2016 D. SampleProblem#2&3:bothproblemsdeal with the RC circuit shown below.
The capacitor is initially uncharged.
1.
Attimet =0,theswitchisplacedinpositiona.Thetimeatwhichthe
capacitorreaches70%ofitsmaximumchargeisclosestto
A. 0.023 s.
B. 1.02 s.
C. 0.051 s.
D. 1.07 s.
E. 0.072 s.
2.
Oncethecapacitorisfullycharged,theswitchisplacedinpositionb.At
t =0.020saftertheswitchisplacedinpositionb,themagnitudeofthevoltage
acrosstheRESISTORisclosestto
A. 0.
B. 8.6 V.
C. 6.2 V.
D. 12.0 V.
E. 3.4 V.
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