1

Basic Concepts, Laws, and Principles

TOPICS DISCUSSED

Theneedtostudyelectricalandelectronicsengineering

Behaviourofmaterialsasconductors,semiconductors,andinsulators

Conceptofcurrent,resistance,potential,andpotentialdifference

Differencesbetweenelectricfieldandmagneticfield

Ohmslaw

Effectoftemperatureonresistance

Electromagnetismandelectromagneticinduction

Lawsofelectromagneticinduction

DynamicallyandstaticallyinducedEMF

Selfandmutualinductance

Electricalcircuitelements

1.1 INTRODUCTION

Weseeapplicationsofelectricityallaroundus.Weobservethepresenceofelectricityinnature.Itisindeedamazingaswellasinterestingtoknowhowmankindhasbeenabletoputelectricityforitsuse.Allelectronicandelectricalproductsoperateonelectricity.Beityourcomputersystem,cellphones,homeentertainmentsystem,lighting,heating,andairconditioningsystemsallareexamplesofapplicationsofelectricity.Applicationofelectricityislimitlessandoftenextendsbeyondourimagination.

Electricalenergyhasbeenacceptedastheformofenergywhichiscleanandeasytotransmitfromoneplacetotheother.Allotherformsofenergyavailableinnatureare,therefore,transformedintoelectricalenergyandthentransmittedtoplaceswhereelectricityistobeusedfordoingsomework.Electricalengineering,therefore,hasbecomeadiscipline,abranchofstudywhichdealswithgeneration,transmission,distribution,andutilizationofelectricity.

Electronicsengineeringisanoffshootofelectricalengineering,whichdealswiththetheoryanduseofelectronicdevicesinwhichelectronsaretransportedthroughvacuum,gas,orsemiconductors.Themotionofelectronsinelectronicdeviceslikediodes,transistors,thyristors,etc.arecontrolledbyelectricfields.Moderncomputersanddigitalcommunicationsystemsareadvancesofelectronics.Introductionofverylargescaleintegrated(VLSI)circuitshasledtotheminiaturizationofallelectronicsystems.

Electricalandelectronicengineeringare,therefore,veryexcitingfieldsofstudy.Apersonwhoisunawareofthecontributionofthesefieldsofengineeringandthebasicconceptsunderlyingtheadvancement,willonlyhavetoblamehimselforherselffornottakinganyinitiativeinknowingtheunknown.

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Inthischapter,wewillintroducesomebasicconcepts,laws,andprincipleswhichthestudentsmighthavestudiedinphysics.However,sincetheseformthebasisofunderstandingoftheotherchaptersinthisbook,itwillbegoodtostudythemagain.

1.2 ATOMIC STRUCTURE AND ELECTRIC CHARGE

Severaltheorieshavebeendevelopedtoexplainthenatureofelectricity.Themodernelectrontheoryofmatter,propoundedbyscientistsSirEarnestRutherfordandNielBohrconsiderseverymatteraselectricalinnature.Accordingtothisatomictheory,everyelementismadeupofatomswhichareneutralinnature.Theatomcontainsparticlesofelectricitycalledelectronsandprotons.Thenumberofelectronsinanatomisequaltothenumberofprotons.

Thenucleusofanatomcontainsprotonsandneutrons.Theneutronscarrynocharge.Theprotonscarrypositivecharge.Theelectronsrevolveroundthenucleusinellipticalorbitsliketheplanetsaroundthesun.Theelectronscarrynegativecharge.Sincethereareequalnumberofprotonsandelectronsinanatom,anatomisbasicallyneutralinnature.

Iffromabodyconsistingofneutralatoms,someelectronsareremoved,therewillbeadeficitofelectronsinthebody,andthebodywillattainpositivecharge.Ifneutralatomsofabodyaresuppliedsomeextraelectrons,thebodywillattainnegativecharge.Thus,wecansaythatthedeficitorexcessofelectronsinabodyiscalledcharge.

Chargeofanelectronisverysmall.Coulombistheunitofcharge.Thechargeofanelectronisonly1.60210 Coulomb(C).Thus,wecansaythatthenumberofelectronsperCoulombisthereciprocalof1.60210whichequalsapprox.6.2810 electrons.Therefore,chargeof6.28

10 electronsisequalto1C.Whenwesaythatabodyhasapositivechargeof1C,itisunderstoodthatthebodyhasadeficitof6.2810electrons.

Anychargeisanexampleofstaticelectricitybecausetheelectronsorprotonsarenotinmotion.Youmusthaveseentheeffectofchargedparticleswhenyoucombyourhairwithaplasticcomb,thecombattractssomeofyourhair.Theworkofcombingcausesfriction,producingchargeofextraelectronsandexcessprotonscausingattraction.

Chargeinmotioniscalledelectriccurrent.Anychargehasthepotentialofdoingwork,i.e.,ofmovinganotherchargeeitherbyattractionorbyrepulsion.Achargeistheresultofseparatingelectronsandprotons.Thechargeofelectronsorprotonshaspotentialbecauseitlikestoreturnbacktheworkthatwasdonetoproduceit.

1.3 CONDUCTORS, INSULATORS, AND SEMICONDUCTORS

Theelectronsinanatomrevolveindifferentorbitsorshells.TheshellsarenamedasK,L,M,N,etc.Thenumberofelectronsthatshouldbeinafilledinnershellisgivenby2n wherenisshellnumber1,2,3,4,etc.startingfromthenearestone,i.e.,firstshelltothenucleus.Ifn=1,thefirstshellwillcontaintwoelectrons.Ifn=2,thesecondshellwillcontaineightelectrons.Thisway,thenumberofelectronsintheshellsare2,8,18,32,etc.Thefilledoutermostshellshouldalwayscontainamaximumnumberofeightelectrons.Theoutermostshellofanatommayhavelessthaneightelectrons.Asforexample,copperhasanatomicnumberof29.Thismeans,copperatomhas29protonsand29electrons.TheprotonsareconcentratedinthenucleuswhiletheelectronsaredistributedintheK,L,M,andNshellsas2,8,18,and1electrons,respectively.Theoutermostshellofacopperatomhasoneelectrononlywhereasthisshellcouldhave8electrons.

Thepositionoccupiedbyanelectroninanorbitsignifiesitsenergy.Thereexistsaforceofattractionbetweentheorbitingelectronandthenucleusduetotheoppositechargetheofelectronandtheproton.Theelectronsintheinnerorbitsarecloselyboundtothenucleusthantheelectronsoftheouteroroutermostorbit.Iftheelectronisfarawayfromthenucleus,theforceofattractionisweak,andhencetheelectronsofoutermostorbitareoftencalledfreeelectrons.Forexample,acopperatomhasonlyoneatominthelastorbitwhichotherwisecouldhaveeightelectrons.

Inacopperwireconsistingoflargenumberofcopperatoms,theatomsareheldclosetogether.Theoutermostelectronsofatomsinthecopperwirearenotsureaboutwhichatomtheybelongto.Theycanmoveeasilyfromoneatomtotheotherinarandomfashion.Suchelectronswhichcanmoveeasilyfromoneatomtotheotherinarandomfashionarecalledfreeelectrons.Itisthemovementoffreeelectronsinamateriallikecopperthatconstitutesflowofcurrent.Here,ofcourse,thenetcurrentflowwillbezeroasthemovementofthefreeelectronsisinrandomdirections.Whenweapplyapotential,whichisnothingbutaforce,itwilldirecttheflowofelectronsinaparticulardirection,i.e.,fromapointofhigherpotentialtowardsapointoflowerpotential.Thus,currentflowisestablishedbetweentwopointswhenthereexistsapotentialdifferencebetweenthepoints.

Wheninamaterialtheelectronscanmovefreelyfromoneatomtoanotheratom,thematerialiscalledaconductor.Silver,copper,gold,andaluminiumaregoodconductorsofelectricity.Ingeneral,allmetalsare

19

19 18

18

18

2

goodconductorsofelectricity.Althoughsilveristhebestconductorofelectricity,thesecondbestconductor,i.e.,copper,ismostlyusedasconductorbecauseofthecostfactor.Inelectricalandelectronicengineeringfields,thepurposeofusingaconductorascarrierofelectricityistoallowelectriccurrenttoflowwiththeminimumofresistances,i.e.,theminimumofopposition.

Inamaterialwheretheoutermostorbitoftheatomsiscompletelyfilled,thematerialiscalledaninsulator.Insulatorslikeglass,rubber,mica,plastic,paper,air,etc.donotconductelectricityveryeasily.Intheatomsofthesematerials,theelectronstendtostayintheirownorbits.However,insulatorscanstoreelectricityandcanpreventflowofcurrentthroughthem.Insulatingmaterialsareusedasdielectricincapacitorstostoreelectriccharge,i.e.,electricity.

Carbon,silicon,andgermaniumhavingatomicnumbersof6,14,and32,respectively,arecalledsemiconductingmaterial.Thenumberofelectronsintheoutermostorbitoftheiratomsisfourinsteadofthemaximumofeight.Thus,intheoutermostorbitofasemiconductormaterial,therearefourvacantpositionsforelectrons.Thesevacantpositionsarecalledholes.Inamaterial,theatomsaresoclosetogetherthattheelectronsintheoutermostorbitorshellbehaveasiftheywereorbitingintheoutermostshellsoftwoadjacentatomsproducingabindingforcebetweentheatoms.Inasemiconductormaterialtheatomsformingabonding,calledcovalentbonding,sharetheirelectronsintheoutermostorbit,andtherebyattainastablestate.Theconditionislikeaninsulatorhavingalltheeightpositionsintheoutermostorbitfilledbyeightelectrons.However,insemiconductingmaterials,withincreaseintemperatureitispossibleforsomeoftheelectronstogainsufficientenergytobreakthecovalentbondsandbecomefreeelectrons,andcausetheflowofcurrent.

1.4 ELECTRIC FIELD AND MAGNETIC FIELD

Whenchargesareseparated,aspaceiscreatedwhereforcesareexertedonthecharges.Anelectricfieldissuchaspace.Dependinguponthepolarityofthecharges,theforceiseitherattractiveorrepulsive.Therefore,wecansaythatstaticchargesgenerateanelectricfield.Anelectricfieldinfluencesthespacesurroundingit.Electricfieldstrengthisdeterminedintermsoftheforceexertedoncharges.Acapacitorisareservoirofcharge.Thetwoparallelplatesofacapacitor,whenconnectedtoavoltagesource,establishesanelectricfieldbetweentheplates.Thepositiveterminal,orpoleofthevoltagesourcewilldrawelectronsfromplate1whereasthenegativepolewillpushextraelectronsontoplate2.Voltageacrossthecapacitorwillrise.Thecapacitorgetschargedequaltothevoltageofthesource.Thecapacitanceofacapacitorisameasureofitsabilitytostorecharge.Thecapacitanceofacapacitorisincreasedbythepresenceofadielectricmaterialbetweenthetwoplatesofthecapacitor.

Acurrentcarryingconductororacoilproducesmagneticfieldaroundit.Thestrengthofthemagneticfieldproduceddependsonthemagnitudeofthecurrentflowingthroughtheconductororthecoil.Thereispresenceofmagneticfieldaroundpermanentmagnetsaswell.

Amagnetisabodywhichattractsiron,nickel,andcobalt.Permanentmagnetsretaintheirmagneticproperties.Electromagnetsaremadefromcoilsthroughwhichcurrentisallowedtoflow.Theirmagneticpropertieswillbepresentaslongascurrentflowsthroughthecoil.

Thespacewithinwhichforcesareexertedbyamagnetiscalledamagneticfield.Itistheareaofinfluenceofthemagnet.

1.5 ELECTRIC CURRENT, RESISTANCE, POTENTIAL, AND POTENTIAL DIFFERENCE

1.5.1 Electric Current

Inanyconductingmaterial,theflowofelectronsformswhatiscalledcurrent.Electronshavenegativecharge.Chargeonanelectronisverysmall.ForthisreasonchargeisexpressedintermsofCoulomb.ChargeofoneCoulombisequaltoachargeof6.2810 electrons.Theexcessordeficitofelectronsinabodyiscalledcharge.Thus,electricalcurrentisexpressedasaflowofnegativecharge,i.e.,electrons.Anysubstancelikecopper,aluminum,silver,etc.whichhasalargenumberoffreeelectrons(i.e.,looslyboundelectronsintheoutermostorbitofitsatom)willpermittheflowofelectronswhenelectricalpressureintheformofEMF(electromotiveforce,i.e.,voltage)isapplied.

Sincethesematerialsconductelectricity,theyarecalledconductors.Theyeasilyallowelectriccurrenttoflowthroughthem.Thestrengthofcurrentwilldependupontheflowofchargeperunittime.Thisisexpressedas

wherechargeQismeasuredinCoulombandtime,tinseconds.Theunitofcurrent,therefore,isCoulombpersecond,when1Cofchargeflowsin1

18

sthemagnitudeofcurrentiscalledampere,namedafterAndrMarieAmpere.

Thus,1ampereofcurrentisequivalenttotheflowofchargeof1Coulombpersecond.

Inearlieryears,currentwasassumedtoflowfrompositivetonegativeterminals.Thisconventionisusedevennowalthoughitisknownthatcurrentisduetothemovementofelectronsfromthenegativetothepositiveterminal.

1.5.2 Resistance

Electricalresistanceisthehindranceoroppositiontotheflowofelectronsinagivenmaterial.Itismeasuredinunitcalledohm.Sincecurrentistheflowofelectrons,resistanceistheoppositionofferedbyamaterial,totheflowoffreeelectrons.Resistance,R,isdirectlyproportionaltothelengthofthematerial,andinverselyproportionaltotheareaofthecrosssectionofthematerial,throughwhichcurrentflows.Theresistanceofferedbyconductingmaterialslikecopperandaluminumislowwhereasresistanceofferedbysomeotherconductingmaterialslikenicrome,tungsten,etc.isveryhigh.Allthesematerialsarecalledconductingmaterials.However,thevaluesofresistivityofthesematerialsaredifferent.Theresistance,Rofamaterialisexpressedas

whereistheresistivity,isthelengthandAisthecrosssectionalareaoftheconductingmaterial.

Theresistivity,isalsocalledthespecificresistanceofthematerial.Themostconductingmaterial,silverhasthelowestvalueofresistivity,i.e.,0.01610 ohmm.Aftersilver,copperismostconducting.Theresistivityorspecificresistanceofcopperissomewhatmorethanthatofsilver,i.e.,0.01810 ohmm.Thatistosay,copperislessconductingthansilver.Wewillseealittlelaterwhyandhowthevalueofresistancechangeswithtemperature.

1.5.3 Potential and Potential Difference

EMFproducesaforceorpressurethatcausesthefreeelectronsinabodytomoveinaparticulardirection.TheunitofEMFisvolt.EMFisalsocalledelectricpotential.Whenabodyischarged(i.e.,eitherdefficiencyofelectronsorexcessofelectronsiscreated),anamountofworkisdone.Thisworkdoneisstoredinthebodyintheformofpotentialenergy.Suchachargedbodyiscapableofdoingworkbyattractingorrepellingothercharges.Theabilityofachargedbodytodoworkinattractingorrepellingchargesiscalleditspotentialorelectricalpotential.Workdonetochargeabodyto1Cisthemeasureofitspotentialexpressedinvolts:

Whenworkdoneis1jouleandchargemovedis1C,thepotentialiscalled1volt.Ifwesaythatapointhasapotentialof6volts,itmeansthat6Joulesofworkhasbeendoneinmoving1Cofchargetothatpoint.Inotherwords,wecansaythateveryCoulombofchargeatthatpointhasanenergyof6Joules.

Thepotentialdifferenceoftwopointsindicatesthedifferenceofchargedconditionofthesepoints.SupposepointAhasapotentialof6volts,andpointBhasapotentialof3volts.WhenthepointsAandBarejoinedtogetherbyaconductingwire,electronswillflowfrompointBtopointA.WesaythatcurrentflowsfrompointAtowardspointB.Thedirectionofcurrentflowistakenfromhigherpotentialtolowerpotentialwhiletheflowofelectronsareactuallyintheoppositedirection.Theflowofcurrentfromhigherpotentialtolowerpotentialissimilartotheflowofwaterfromahigherleveltoalowerlevel.

1.6 OHMS LAW

GeorgeSimonOhmfoundthatthevoltage,Vbetweentwoterminalsofacurrentcarryingconductorisdirectlyproportionaltothecurrent,Iflowingthroughit.Theproportionalityconstant,Ristheresistanceoftheconductor.Thus,accordingtoOhmslaw

6

6

Thisrelationwillholdgoodprovidedthetemperatureandotherphysicalconditionsdonotchange.

Figure1.1(a)ShowslinearrelationshipbetweenVandI(b)VIcharacteristicsfordifferentvaluesofR

OhmslawisnotapplicabletononlineardeviceslikeZenerdiode,voltageregulators,etc.OhmslawisexpressedgraphicallyonVandIaxiesasastraightlinepassingthroughtheoriginasshowninFig.1.1(a).

TherelationshipbetweenVandIhavebeenshownfordifferentvaluesofRinFig.1.1(b).HereinV=RI,Rindicatestheslopeoftheline.ThemorethevalueofRis,themorewillbetheslopeofthelineasshowninFig.1.1(b).

1.7 THE EFFECT OF TEMPERATURE ON RESISTANCE

Resistanceofpuremetalslikecopper,aluminum,etc.increaseswithincreaseintemperature.ThevariationofresistancewithchangeintemperaturehasbeenshownasalinearrelationshipinFig.1.2.

Thechangeinresistanceduetochangeintemperatureisfoundtobedirectlyproportionaltotheinitialresistance,i.e.,R R R .Resistance(R R )alsovariesdirectlyasthetemperatureriseandthischangealsodependsuponthenatureofthematerial.Thuswecanexpressthechangeinresistanceas,

R R R t

or,R R = R t,where iscalledthetemperaturecoefficientofresistanceat0C.

Figure1.2(a)Showsthevariationofresistancewithtemperature(b)resistancesattwodifferenttemperatures

Thisexpressioncanbeappliedforbothincreaseanddecreaseintemperature.FromthegraphofFig.1.2(a)itisseenthatresistanceofthematerialcontinuestodecreasewithdecreaseintemperaturebelow0C.Ifwegoondecreasingthetemperaturetoaverylowvalue,thematerialattainsastateofzeroresistance.Thematerialatthatstatebecomessuperconducting,i.e.,conductingwithnoresistanceatall.

t 0 0

t 0

t 0 0

t 0 0 0 0

Nowsupposeaconductorisheatedfromtemperaturet tot .Theresistanceoftheconductoratt isR andatt isR ashasbeenshowninFig.1.2(b).

Usingeq.(1.5),

Usingeq.(1.5),wecanwrite

Fromfig1.2(b)usingtherelationin(1.6),wecanwrite

Thus,ifresistanceatanytemperaturet isknown,theresistanceatttemperaturecanbecalculated.

Calculation of at different temperatures

Wehaveseen,

If and arethetemperaturecoefficientsofresistanceatt andtdegrees,respectively,then

Thus,wecanwrite,

Therefore,

1 2

1 1 2 2

1 2

1 2 1 2

Temperaturecoefficientofresistance,at20CandspecificresistanceofcertainmaterialhavebeenshowninTable1.1.

Table1.1TemperatureCoefficientandSpecificResistanceofDifferentMaterials

Material Temp.coeff.ofresistance Specificresistanceinmicro

ohm

Silver 0.004 0.016

Copper 0.0039 0.018

Aluminium 0.0036 0.028

Iron 0.005 0.100

Brass 0.0015 0.070

Lead 0.0042 0.208

Tin 0.0046 0.110

Carbon 0.00045 66.67

Itistobenotedthatcarbonhasanegativetemperaturecoefficientofresistance.Thismeans,theresistanceofcarbondecreaseswithincreaseintemperature.

Bythistimeyoumustbewonderingastowhyresistanceinmostmaterialsincreaseswithincreaseintemperaturewhileresistanceinsomedecreaseswithincreaseintemperature.

Thechargedparticlesinsideamaterialisinthestateofvibration.Temperatureriseinmostmaterialsincreasesthisvibrationinsidethematerialobstructingtheflowofelectrons.Obstructiontotheflowofelectronsiscalledresistance.Atlowertemperaturesthevibrationgetsreduced,andhencetheresistance.

1.8 WORK, POWER, AND ENERGY

1.8.1 Work

Whenaforceisappliedtoabodycausingittomove,andifadisplacement,discausedinthedirectionoftheforce,then

IfforceisinNewtonsanddisinmeters,thenworkdoneisexpressedinNewtonmeterwhichiscalledJoules.

1.8.2 Power

Poweristherateatwhichworkisdone,i.e.,rateofdoingwork.Thus,

TheunitofpowerisJoules/secondwhichisalsocalledWatt.Whentheamountofpowerismore,itisexpressedinKilowatt,i.e.,kW.

1kW=110 W

Wehaveearlierseenineq.(1.3),thatelectricalpotential,Visexpressedas

20

3

ThusinacircuitifIisthecurrentflowing,andVistheappliedvoltageacrosstheterminals,power,Pisexpressedas

Thuselectricalpowercanbeexpressedas

1.8.3 Energy

Energyisdefinedasthecapacityfordoingwork.Thetotalworkdoneinanelectricalcircuitiscalledelectricalenergy.Whenavoltage,Visapplied,thecharge,Qwillflowsothat

IfpowerisinkWandtimeisinhour,theunitofenergywillbeinKilowatthourorkWh.

1.8.4 Units of Work, Power, and Energy

InSIunit,workdoneisthesameasthatofenergy.

Mechanical work or energy

WhenaForce,FNewtonactingonabodymovesitinthedirectionoftheforcebyadistancedmeters:

Workdone=FDNmorJoules

WhenaforceFNewtonisappliedtangentiallyonarotatingbodymakingaradiusrmeters,then

IfNisexpressedinrevolutionsperminute(rpm)

Whenabodyofmassmkgisliftedtoaheighthmetersagainstthegravitationalforcegm/sec ,workdoneisconvertedintopotentialenergyofthebody.

Electrical energy

Asmentionedearlier,workdoneinanelectricalcircuitisitsenergy.

IfelectricalpowerisexpressediskWandtimeinhour,then

WewillnowconvertkWhintoCalories

Since1Calorie=4.2Joules

Thermal energy

InSIunit*thermalenergyisexpressedincalories.Onecalorieindicatestheamountofheatrequiredtoraisethetemperatureof1gmofwaterby1C.Thisheatisalsocalledthespecificheat.Ifmisthemassoftheliquid,Sisthespecificheat,andtisthetemperatureriserequired,thentheamountofheatrequired,Hisexpressedas

Example1.1Acopperwirehasresistanceof0.85ohmsat20C.Whatwillbeitsresistanceat40C?Temperaturecoefficientofresistanceofcopperat0Cis0.004C.

Solution:

2

Example1.2Theheatingelementofanelectricheatermadeofnicromewirehasvalueofresistivityof110 Ohmm.Thediameterofthewireis0.2mm.Whatlengthofthisnicromewirewillmakearesistanceof100Ohms?

Solution:

Substitutingthevalues,lengthofwire,is

Example1.3Itisrequiredtoraisethetemperatureof12kgofwaterinacontainerfrom15Cto40Cin30minthroughanimmersionrodconnectedtoa230Vsupplymains.Assuminganefficiencyofoperationas80percent,calculatethecurrentdrawnbytheheatingelement(immersionrod)fromthesupply.Alsodeterminetheratingoftheimmersionrod.Specificheatofwateris4.2kiloJoules/kg/C.

Solution:

OutputorEnergyspentinheatingthewater,His

H=ms(t t )

Wheremisthemassofwaterandsisthespecificheatofwater.

Here,H=124.210 (4015)Joules

=12610 Joules

Weknow,efficiency

So,Energyinputtoimmersionrod

Thetimeofoperationoftheheaterrod

2 1

6

3

4

Currentdrawnfrom230Vsupply

P=VI=870Watts.

Example1.4Amotordrivenwaterpumplifts64m ofwaterperminutetoanoverheadtankplacedataheightof20metres.Calculatethepowerofthepumpmotor.Assumeoverallefficiencyofthepumpas80percent.

Solution:

Workdone/min=mghJoules

m=6410 kg(1m ofwaterweights1000kg)

g=9.81m/sec

h=20m

Substitutingvalues

Inputpowerofthepumpmotor

=261.3KW

Example1.5Aresidentialflathasthefollowingaverageelectricalconsumptionsperday:

1. 4tubelightsof40wattsworkingfor5hoursperday2. 2filamentlampsof60wattsworkingfor8hoursperday3. 1waterheaterrated2kWworkingfor1hourperday4. 1waterpumpof0.5kWratingworkingfor3hoursperday.

Calculatethecostofenergypermonthif1kWhofenergy(i.e.,1unitof

energy)costs

Solution:

Totalkilowatthourconsumptionofeachloadfor30daysarecalculatedas:

3

3 3

2

TotalkWhconsumedpermonth

=24kWh+28.8kWh+60kWh+45kWh

=157.5kWh

OnekWhofenergycosts

Thetotalcostofenergypermonth=157.53.50

Example1.6Anelectrickettlehastoraisethetemperatureof2kgofwaterfrom30Cto100Cin7minutes.Thekettleishavinganefficiencyof80percentandissuppliedfroma230Vsupply.Whatshouldbetheresistanceofitsheatingelement?

Solution:

Outputenergyofthekettle=mst

Supplyvoltage,V=230Volts.

Power,P=1.74kW=1740Watts.

V=230V

Example1.7Calculatethecurrentflowingthrougha60Wlampona230Vsupplywhenjustswitchedonatanambienttemperatureof25C.Theoperatingtemperatureofthefilamentmaterialis2000Canditstemperaturecoefficientofresistanceis0.005perdegreeCat0C.

Solution:

Thisresistanceofthefilamentisat2000C.LetuscallitR =881.6Ohms.

Attheinstantofswitching,resistanceisatroomtemperature,i.e.,at25C.LetuscallitasR .

WeknowR wehavetocalculateR given =0.005ohm/C.

Weknow,

Weknowtherelation,

Thecurrentflowingthroughthe60Wlampattheinstantofswitchingwillbecorrespondingtoitsresistanceat25C.

Example1.8Acoilhasaresistanceof18at20Cand20at50C.Atwhattemperaturewillitsresistancebe21Ohms?

Solution:

weknow,

Wecanwrite,

substituting,

2000

25

2000 25 0

Example1.9Theresistanceofawireincreasesfrom40at20Cto50at70C.Calculatethetemp.coefficientofresistanceat0C.

Solution:

given

or,

or,

Example1.10Aresistanceelementofcrosssectionalareaof10mmandlength10mdrawsacurrentof4Aat220Vsupplyat20C.Calculatetheresistivityofthematerial.Whatcurrentwillbedrawnwhenthetemperaturerisesto60C?Assume =0.0003/C.

Solution:

a=10mm

=1010 m

V=IR

or,

Thisresistance,wecallasR

SincewehavetocalculateR ,wehaveto

Now,

20

20

60 60

2

2

6 2

Current,

1.9 ELECTROMAGNETISM AND ELECTROMAGNETIC INDUCTION

1.9.1 Introduction

Electromagnetismisthestudyofinteractionbetweenelectriccurrentandmagneticfield,andforcesproducedthereof.Thissectionwillincludedescriptionsofmagneticfieldaroundcurrentcarryingconductors,magneticfieldproducedbyacurrentcarryingcoil,forceproducedonacurrentcarryingconductororacoilwhenplacedinamagneticfield.

ADanishscientist,Oerstedintheearlynineteenthcenturydiscoveredthattherewasamagneticfieldaroundacurrentcarryingconductor.Linesofforceintheformofconcentriccirclesexistedonaperpendicularplanearoundacurrentcarryingconductor.Thismeant,magnetismcouldbecreatedbyelectriccurrent.Itwasalsoobservedthatthedirectionoflinesofforcegotchangedwhenthedirectionofcurrentflowingthroughtheconductorwaschanged.AfewyearsafterthediscoveryofOersted,Faraday,anotherscientistfromEnglanddiscoveredthatamagneticfieldcancreateanelectriccurrentinaconductor.Whenthereisachangeinfluxlinkageinaconductororacoil,EMFisinducedinit.ThisphenomenoniscreditedtoFaradaywhoestablishedfamouslawsofelectromagneticinduction.Youwillobservethatmostoftheelectricalmachinesanddeviceshavebeendevelopedutilizingtheobservationsanddiscoveriesmadeasmentionedabove.

1.9.2 Magnetic Field Around a Current-carrying Conductor

InFig.1.3isshownaconductorcarryingacurrent,I.Linesofforceareestablishedaroundtheconductoronaperpendicularplane.InFig.1.3(a)magneticfieldaroundalongconductorhasbeenshown.Thelinesofforceareestablishedonaperpendicularplane.InFig.1.3(b)and(c),thecrosssectionalviewsofacurrentcarryingconductorhavebeenshown.Thecrossatthecentreoftheconductorindicatesthatcurrentisenteringtheconductorwhichisplacedperpendiculartotheplaneofthepaper.Thelinesofforceintheformofconcentriccirclesareontheplaneofthepaper.ThedirectionofcurrentthroughtheconductorisreversedinFig.1.3(c).Thedotatthecentreoftheconductorindicatesthatthecurrentiscomingtowardstheobserver.Thedirectionofthelinesforcearoundtheconductoralsogetreversed.

Thedirectionoffluxlinesaroundacurrentcarryingconductorisdeterminedbyapplyingthecorkscrewrulewhichisstatedbelow.

Figure1.3(a)Alongcurrentcarryingconductor(b)crosssectionalviewofaconductorwithfluxaroundit(c)crosssectionalviewoftheconductorwiththedirectionofcurrentreversed(d)resultantmagneticfieldproducedbytwocurrentcarryingconductors

CorkScrewRule:Considerarighthandscrewheldononeendofacurrentcarryingconductorandisrotatedintheclockwisedirection.Iftheadvancementofthescrewindicatesthedirectionofcurrent,thedirectioninwhichthescrewisrotatedwillindicatethedirectionofthelinesofforcearoundtheconductor.

InFig.1.3(d)hasbeenshownthattwocurrentcarryingconductorsplacedsidebysideproducearesultantmagneticfield.

1.9.3 Magnetic Field Around a Coil

Acoilisformedbywindingawireofcertaincrosssectionaroundaformer(ahollowcylindermadeofsomenonmagneticmateriallikebakelite,plastics,etc).Suchacoilisoftencalledasolenoid.Whencurrentisallowedtoflowthroughsuchacoil,amagneticfieldisproducedbythecoil.Thedirectionoffluxproducedbyacurrentcarryingcoilisdeterminedbyapplyingtherighthandgriprule.InFig.1.4(a)isshownacurrentcarryingcoil.Ifweholdthecoilbyourrighthandinsuchaway

thatthefourfingersbendtowardsthedirectionofthecurrentflowthroughthecoilturns,thethumbwillindicatethedirectionoftheresultantfluxproduced.

Figure1.4(a)Righthandgripruleappliedtodeterminedirectionoffluxproducedbyacurrentcarryingcoil(b)magneticfieldproducedbyacurrentcarryingcoil

Thefourfingersbendinthedirectionofcurrentthroughthecoil.Thedirectioninwhichthethumbpointsisthedirectionoffluxproduced.InFig.1.4(b),wehaveshownthecrosssectionalviewofthesamecoil.Forthedirectionofcurrentflowthroughthecoil,crosssectionshavebeenshownbyputtingcrossanddotconvention.Theuppersideofthecoilturns1,2,3,4,5willindicatethatcurrentisenteringwhiletheywillcomeoutfromtheothersideasshowninthebottomconductorcrosssections.ByapplyingthecorkscrewrulealsowecandeterminethedirectionoftheresultantmagneticfieldandshowthepositionsofNorthandSouthpolesformed.Ifthedirectionofcurrentflowthroughthecoilisreversed,thedirectionofthemagneticlinesofforcewillbeopposite,andhencethepositionsofNorthandSouthpoleswillchange.

IfweapplysomealternatingvoltageacrossthecoilasshowninFig.1.5,thepolarityofpowersupplywillchangeineveryhalfcycleoftheappliedvoltage.Ifasinusoidalacsupplyisprovided,boththemagnitudeaswellasthedirectionofcurrentflowwillchange.Asaresult,themagnitudeofthemagneticfieldproducedwillchangestartingfromzerovaluereachingitsmaximumvalue,thengettingreducedagaintozero,andthenbecomingnegative.Thedirectionoffluxproducedwillchangeineveryhalfcycleofcurrentflow.Suchamagneticfieldwhosemagnitudeasalsoitsdirectionchangesiscalledapulsatingalternatingmagneticfield.Incaseofdcsupply,themagneticfieldproducedwillbeofconstantmagnitudeandfixedpolarity.

Figure1.5ACsupplytoacoilproducesanalternatingmagneticfieldofvaryingmagnitude

1.9.4 A Current-carrying Conductor Placed in a Magnetic Field

Whenaconductorcarryingcurrentisplacedinamagneticfielditexperiencesaforce.Theforceactsinadirectionperpendiculartoboththemagneticfieldandthecurrent.

InFig.1.6aconductorisshownplacedperpendiculartothedirectionofmagneticfield.Suchaconductorincrosssectionalviewhasbeenshownbyasmallcircle.Thedotinsidethesmallcircleindicatesthatcurrentisflowingtowardstheobserver.Theconductorwillexperienceaforceintheupwarddirectionashasbeenshown.Ifthedirectionofcurrentthroughtheconductorisreversed,theforceontheconductorwillbeinthedownwarddirection.

Theforceontheconductorwilldependupontheflux, orfluxdensity,

B whereAistheareaofthemagneticpoles.Theforcewillalsodependupontheeffectivelengthoftheconductorinthemagneticfield,i.e.,onthemagnitudeofcurrentflowing,i.e.,I.Theforcedevelopedisexpressedas

Herethecurrentcarryingconductorandthemagneticfieldsareatrightanglestoeachother.If,however,theconductorisinclinedwiththemagneticfieldbyanangle,thenthelengthoftheconductorperpendiculartothemagneticfieldistobeconsideredasshowninFig.1.7.ThelengthoftheconductorperpendiculartothemagneticfieldisSin.Thus,thegeneralexpressionforforceFis

ThedirectionoftheforceisdeterminedbyapplyingFlemingslefthandrulewhichisstatedas:

Figure1.6Forceexperiencedbyaconductorcarryingcurrentinamagneticfield

Figure1.7Forceonacurrentcarryingconductor

Flemings left-hand rule

ThethreefingersofthelefthandarestretchedasshowninFig.1.6.Iftheforefingerpointstowardsthedirectionofthelinesofforce,andthemiddlefingerpointstowardthecurrentflowingthroughtheconductor,thenthethumbwillpointtowardsthedirectionofforceexperiencedbytheconductor.

1.9.5 A Current-carrying Coil Placed in a Magnetic Field

Nowwewillconsideracoilplacedinamagneticfield.Acoilhastwocoilsideswhichlieinthemagneticfield.Thesecoilsidesarecalledconductors.Thus,acoilhastwoconductors.Ifacoilhastwoturns,thenumberofconductorswillbefour.SeeFig.1.8(aandb).InFig.1.8(c)hasbeenshownasingleturncoilplacedinamagneticfield.Thedirectionofcurrentthroughthecoilhasalsobeenshown.ThedirectionofthemagneticfieldisfromNorthpoletoSouthpole.Thedirectionofcurrentincoilsideaisupward,i.e.,towardstheobserver.IfweapplyFlemingslefthandrule,wefindthatcoilsideawillexperienceanupwardforce.

Similarly,byapplyingthesamerule,weobservethatcoilsidewillexperienceadownwardforce.ThetwoforcesactingsimultaneouslyonthecoilwilldevelopatorquewhichwilltrytorotatethecoilalonganaxisxxintheclockwisedirectionashasbeenshowninFig.1.8(c).Thecoilwillrotatebyanangleof90.TheNorthpoleofthemagneticfieldproducedbythecurrentcarryingcoilwillfacethestationarySouthpoleasshowninFig.1.9.

ThetwomagneticfieldsgetalignedasshowninFig.1.9(b).IfitispossibletochangethedirectionofcurrentinthecoilwhenitchangesitspositionfromDDaxistoXXaxis,thecoilwillcontinuetodeveloptorqueintheclockwisedirection.Wewillgetcontinuousrotationofthecoil.Thisisthebasicprincipleofdirectcurrentelectricmotorwhichwillbediscussedindetailinaseparatechapter.

Figure1.8(a)Acoilhavingoneturn(b)acoilhavingtwoturns(c)asingleturncoilcarryingcurrentisplacedinamagneticfield(d)thecoilsidesofthecurrentcarryingcoilinthemagneticfieldexperienceforce

Figure1.9(a)Acurrentcarryingcoilinamagneticfieldexperiencesatorque(b)magneticfieldproducedbythecurrentcarryingcoilandthestationarymagneticfieldgetaligned

1.10 LAWS OF ELECTROMAGNETIC INDUCTION

Faraday,onthebasisoflaboratoryexperiments,establishedthatwheneverthereaischangeinthemagneticfluxlinkagebyacoil,EMFisinducedinthecoil.ThemagnitudeoftheEMFinducedisproportionaltotherateofchangeoffluxlinkages.Faradayslawsofelectromagneticinductionarestatedas:

Firstlaw:EMFisinducedinacoilwhenevermagneticfieldlinkingthatcoilischanged.

Secondlaw:ThemagnitudeoftheinducedEMFisproportionaltotherateofchangeoffluxlinkage.

Therateofchangeoffluxlinkageisexpressedas whereNisthenumberofturnsofthecoillinkingtheflux.Thus,theinducedEMF,eisexpressedas

TheminussignisintroducedinaccordancewithLenzslawwhichisstatedbelow.

Lenzslaw:ThislawstatesthattheinducedEMFduetochangeoffluxlinkagebyacoilwillproduceacurrentinthecoilinsuchadirectionthatitwillproduceamagneticfieldwhichwillopposethecause,thatisthechangeinfluxlinkage.

Thestudentsmayconductanexperimentinthelaboratory,similartothatdonebyFaraday,whichisexplainedbelow.

IfthemagnetshowninFig.1.10(a)isquicklybroughtnearthecoil,therewillbedeflectioninthegalvanometerindicatingEMFinducedinthecoilandcurrentflowinthecircuit.Ifthemagnetisheldstationarynearthecoil,althoughthereisfluxlinkingthecoil,therewillbenoinducedEMFsincethereisnochangeinthefluxlinkage.TheinducedEMFwillbethereonlyifthereisincreaseordecreaseinfluxlinkagebythecoil.Itwillbeobservedthatwhenthemagnetistakenawayquicklytherewillbedeflectioninthegalvanometer.ItmayalsobenotedthatEMFwillalsobeinducedinthecoilwhenthecoilismovedkeepingthemagnetstationary.

Figure1.10Faraday'sexperimentonelectromagneticinduction.(a)Amagnetissuddenlybroughtnearacoil(b)determinationofthedirectionofcurrentproducedinthecoil

Thedirectionofcurrentflowingthroughthecoilcanbedeterminedbyapplyingtherighthandgriprule.Theruleisexplainedasfollows.

Right-hand-grip rule

Holdthecoilwithyourrighthandwiththethumbopposingthedirectionofmovementofthemagnet.Theotherfourfingerswillindicatethedirectionofcurrentflowthroughthecoil.Thismeansthatthecurrentinducedinthecoilwillproducefluxinthedirectionofthethumb,thusopposingthefluxproducingtheinducedEMFinthecoil.SeeFig.1.10(b).

1.11 INDUCED EMF IN A COIL ROTATING IN A MAGNETIC FIELD

NowwewillconsideracoilrotatedinastationarymagneticfieldasshowninFig.1.11.

Hereacoil,havingtwosides(conductors)isrotatedinauniformmagneticfieldasshowninFig.1.11.Becauseoftherotationofthecoilinthemagneticfield,fluxlinkagebythecoilchanges,i.e.,thenumberoflinesofforcepassingthroughthecoilchanges.Becauseofchangeoffluxlinkage,EMFisinducedinthecoil.ThedirectionoftheinducedEMFintheconductorscanbedeterminedbyapplyingFlemingsrighthandrule(FRHR).

Figure1.11(a)EMFisinducedinacoilwhenrotatedinamagneticfield(b)determinationofdirectionofinducedEMF

FRHRstatesthatwhenwestretchthethreefingersoftherighthandperpendiculartoeachother,iftheforefingerpointstowardsthefluxlinesfromNorthpoletoSouthpole,andthethumbshowsthedirectionofmovementoftheconductor,thenthemiddlefingerwillrepresentthedirectionoftheinducedEMForcurrentintheconductor.InFig.1.11(b)isshownthedirectionoftheinducedEMFincoilsideaboftherotatingcoilabcd.Thiscoilsideisshowngoingupwards.ThemagneticfielddirectionisfromNorthpoletoSouthpole.Hence,thedirectionoftheinducedEMFwillbefrombtoaasdeterminedbyapplyingFRHR.Thestrongerthemagneticfieldis,themorewillbethemagnitudeofEMFinduced.Themorethespeedofrotationofthecoilis,themorewillbethe

magnitudeoftheEMFinduced.Thisisbecause willincreaseifboth

aswellastherateofchangeoflinkageof arechanged.ThemagnitudeoftheEMFinducedwillalsobedirectlyproportionaltothenumberofturnsoftherotatingcoil,orthenumberofcoilsconnectedinseries.TheEMFinducedcanalsobeconsideredintermsoffluxcutbyaconductor(coilside)persecond.

HereinFig.1.11,thenumberofpolesistwo.Wecanalsohavefourpoles,sixpoles,etc.Whenaconductorrotatesinsuchmagneticfield,itcutsthelinesofforce.Thenumberoflinesofforcecutbyaconductorinone

revolution,whentherearetwopoles,is2 Webers,where isthefluxperpole.IftherearesayPnumberofpoles,fluxcutbyaconductorinone

revolutionwillbeP Webers.Ifthecoilmakesnrevolutionspersecond,thetimetakenbyaconductortomakeonerevolutionwillbe1/nseconds.Thus,fluxcutpersecondwillbetheEMFinduced,ewhichis

1.12 EMF INDUCED IN A CONDUCTOR

Intermsoflengthofconductor,andvelocityoftheconductor,vinamagneticfieldoffluxdensity,B,theEMFinducedinaconductor,eiscalculatedas

Toestablishtheaboverelation,letusconsiderasingleconductorrepresentedbyasmallcircle(crosssectionalview)ismovedinamagneticfieldofstrengthBWb/m asshowninFig.1.12.

2

Figure1.12EMFinducedinaconductormovinginamagneticfield

Lettheconductorcutthefluxatrightanglesbymovingadistancedxmeter.Theareasweptbythemovingconductorisdxm .ThefluxdensityisBWb/m .

Thetimetakentomoveadistancedxmisdtseconds.

Inducedemf,e=Fluxcutpersecond

Since thelinearvelocityvoftheconductor,

IftheconductormovesinadirectionmakingananglewiththedirectionofmagneticfieldasshowninFig.1.12,theinducedEMFwillbeasstatedearlierineq.1.26.

1.13 DYNAMICALLY INDUCED EMF AND STATICALLY INDUCED EMF

Whenemfisinducedinacoilorconductorbyvirtueofmovementofeithertheconductororthemagneticfield,theemfiscalleddynamicallyinducedEMFashasbeenexplainedinsection1.11.

WhenEMFisinducedinastationarycoilbychangingitsfluxlinkageduetochangeincurrentflowthroughthecoil,suchemfiscalledstaticallyinducedEMF.

Ifacoilcarriesacurrent,fluxisestablishedaroundthecoil.Ifthecurrentischangedquickly,thefluxlinkagebythecoilwillchangeasshowninFig.1.13(a).

2

2

Figure1.13(a)ChangeinfluxlinkageinacoilduetoswitchingONandswitchingOFFofdccurrent(b)changeinfluxlinkageduetoalternatingcurrentsupply(c)inducedemfincoils1and2duetochangingfluxproducedbyalternatingcurrentflowingincoil1

InFig.1.13(a),acoilofcertainnumberofturnsiswoundonaformer,i.e.,itscore.CurrentissuppliedfromabatterybyclosingaswitchS.Iftheswitchiscontinuouslyturnedonandoff,fluxlinkagebythecoilwillchange.TherateofchangeofthefluxlinkagewillinduceEMFinthecoil.

AsimilareffectwillbethereifanacsupplyisappliedacrossthecoilasshowninFig.1.13(b).Thedirectionofcurrentinthecoilisshownforthepositivehalfcycleofthealternatingcurrent.Thedirectionofcurrentwillchangeineveryhalfcycle,andhencethedirectionoffluxproducedwillchangeineveryhalfcycle.Themagnitudeofcurrentchangescontinuouslysinceasinusoidalcurrentisflowing.Thischangingcurrentwillcreateachangingfluxlinkage,therebyinducingEMFinthecoilinboththecasesasshowninFig.1.13(a)and(b).NotethatinFig.1.13(a),iftheswitchSiskeptclosed,asteadydirectcurrent,i.e.,aconstantcurrentwillflowthroughthecoil.Thisconstantcurrentwillproduceaconstantflux.Therewillbenochangeinfluxlinkagebythecoilwithrespecttotime,andhencenoEMFwillbeinducedinthecoil.Thus,thenecessaryconditionfortheproductionofinducedEMFisthatthereshouldbeachangeinfluxlinkageandnotmerelyfluxlinkagebyacoil.

1.14 SELF-INDUCED EMF AND MUTUALLY INDUCED EMF

TheEMFinducedinacoilduetochangeinfluxlinkagewhenachangingcurrentflowsthroughthecoiliscalledselfinducedEMF.

AsshowninFig.1.13(c),whenasecondcoilisbroughtnearacoilproducingchangingflux,EMFwillbeinducedinthesecondcoilduetochangeincurrentinthefirstcoil.ThisiscalledmutuallyinducedEMF.Infact,EMFwillbeinducedinboththecoilsasboththecoilsarelinkingachangingflux.However,inthesecondcoilEMFisinducedduetochangingfluxcreatedbycoil1.ThemagnitudeoftheinducedEMFwilldependupontherateofchangeoffluxlinkageandthenumberofturnsoftheindividualcoils.TheinducedEMFinthetwocoils,e ande willbe

whereN andN arethenumberofturnsofcoil1andcoil2,respectively.

YouwillstudyinaseparatechapterhowtransformersarebuiltutilizingthebasicprincipleofmutuallyinducedEMF.

1.15 SELF-INDUCTANCE OF A COIL

ConsideracoilofNturnswoundonacoreofmagneticmaterial.LetanalternatingcurrentipassthroughthecoilasshowninFig.1.14.

1 2

1 2

Figure1.14Inductanceofacoil

Theemfinduced,ewillbe

whereisthepermeabilityofthecorematerialisthelengthoffluxpathAistheareaofthecoil.substituting,

Liscalledthecoefficientofselfinductanceorsimplyselfinductanceofthecoil.

Inductanceofacoilis,therefore,dependentuponthepermeabilityofthecorematerial.Ifweputironasthecorematerialinsteadofanynonmagneticmaterial,orairasthecore,theinductancewillincreasemanytimes.The(permcability),isexpressedas

=

where istherelativepermeabilityand isthepermeabilityoffreespace.Therelativepermeabilityofironmaybeashighas2000timesthanthatofair.Hence,anironcorecoilmayhaveaninductancevalue2000timesmorethanthatofanaircoreone,otherdimensionsremainingthesame.Again,inductance,Lisinverselyproportionaltothelengthofthefluxpathanddirectlyproportionaltotheareaofthecorematerialorthecoil.Inductanceisproportionaltothesquareofthenumberofturns.Tohaveaninductanceofalargevalue,thenumberofturnsshouldbehigh.

Theinductance,Lcanbeexpressedintermsoftherateofchangeofthefluxwithrespecttocurrentflowinginthecoilas

Forasmallincrementofdi,lettheincreaseoffluxbed .Therefore,

o r

r o

If andihavealinearrelationship,

Rememberthatreluctanceistheinverseofthepermeability.Lowreluctancewillgiverisetoahighvalueofinductance.Thatiswhyinordertoproducehighvalueinductance,thenumberofturnsshouldbehighandthereluctancetothefluxpathshouldbelow.Thecoreshouldbemadeofhighpermeabilitymateriallikeiron.

Consideringasmallincreaseofiproducingasmallincreasein asd

Inductanceisthepropertyofacoilcapableofinducingemfinitselfduetochangingcurrentthroughit.

The formulae so far derived are

1. Forceonacurrentcarryingconductorinamagneticfield

F=BINewtons.

Iftheconductorisinclinedatananglewiththemagneticfield,

F=BISinNewtons.

2. InducedEMFinacoilwherethereischangeoffluxlinkageorchangeincurrent,

3. InducedEMFinaconductorrotatinginamagneticfield,

e=P nV

wherePisthenumberofpoles, isthefluxperpoleandnistherevolutionspersecond.

4. InducedEMFinaconductormovinginamagneticfieldinaperpendiculardirection,

e=BvV

whereBisthefluxdensityinWb/m ,isthelengthoftheconductorinmandvisthevelocityinm/sec.

Iftheconductorismovingatanangleofwiththemagneticfield,theinducedEMFis

e=BvSinV.

5. InducedEMFinacoil,

Thus,wecansaythatacoilhasaninductanceof1Henryifacurrentof1Ampereflowingthroughthecoilproducesafluxlinkageof1Wbturn.

1.16 MUTUAL INDUCTANCE

ConsidertwocoilshavingN andN numberofturnsplacedneareachotherasshowninFig.1.15.Letachangingcurrent,i ,flowthroughcoil1.

Thefluxproducedbyi inN is .Sincecoil2isplacednearcoil1,a

partofthefluxproducedbycoil1willbelinkedbycoil2.Letflux

linkedbycoil2is =K whereK 1.

Ifmagneticcouplingbetweenthetwocoilsisverytight,i.e.,verygood,thewholefluxproducedbycoil1willlinkthecoil2,inwhichcasethecoefficientofthecouplingK willbe1.TheinducedEMFincoil2ise .

Figure1.15Mutualinductanceoftwocoils

From(i)and(ii),

1 2

1

1 1 1

2

2 1 1 1

1 2

2

Similarly,ifwecalculatetheinducedEMFincoil1,duetochangeincurrentincoil2,wecanfindtheinducedEMFe incoil1as

Now,multiplyingtheexpressionforMasin(iii)and(iv)above,

Againfrom(iii),

From(iii)and(vi),

1

Fromeq.(1.38)wecandefinethemutualinductanceMbetweentwocoilsasthefluxlinkageinonecircuitduetochangeperunitofcurrentintheothercircuit.

Similarly,consideringcurrentchangeinthesecondcoil

1.17 INDUCTANCE OF COILS CONNECTED IN SERIES HAVING A COMMON CORE

WehavetwocoilshavingselfinductanceL andL connectedinseries.InFig.1.16(a),theyproducefluxinthesamedirection,andinFig.1.16(b),theconnectionissuchthattheyproducefluxintheoppositedirections.

Sincethetwocoilsareconnectedinseries,thesamecurrentflowsthroughthem.

Ifthereisachangeincurrentdiamperesintimedtseconds,theEMFinducedincoil1duetoitsselfinductanceL is

Similarly,theEMFinducedincoil2duetoitsselfinductance,L is

Duetomutualinductance,theEMFinducedincoil1duetochangeincurrentincoil2andviceversaareexpressedasEMFinducedincoil1duetochangeincurrentincoil2is

Figure1.16Coilsconnectedinseriesin(a)commulativelyin(b)differentially

EMFinducedincoil2duetochangeincurrentincoil1is

1 2

1

2

Nowletthetotalequivalentinductanceofthesinglecircuitcomprisingcoil1andcoil2astheyareconnectedasinFig.1.16(a)beL

TheEMFinducedinthewholecircuitwill,therefore,be

Thus,equatingtheexpressionforein(iv)withthetotalEMFsasin(i),(ii),(iii),and(iv):

WhenthecoilsaredifferentiallyconnectedasinFig.1.16(b),theEMF

inducedincoil1duetodiintimedtincoil2,i.e., inoppositiontotheEMFinducedincoil1duetoitsselfinductance.SimilaristhecaseoftheEMFinducedincoil2duetomutualinductance.Thus,forthedifferentiallyconnectedcoil

Thus,thetotalinductanceofaninductivelycoupledseriesconnectedcoilcircuitcanbeexpressedas

Dotconventionisusedtodeterminethesignofinducedvoltage.

Ifweusedotconvention,itwillnotberequiredtoknowthewaythecoilshavebeenactuallywound.

Example1.11Thetotalinductanceoftwocoilsconnectedinseriescumulatativelyis1.6Handconnecteddifferentiallyis0.0.4H.Theselfinductanceofonecoilis0.6H.Calculate(a)themutualinductanceand(b)thecouplingcoefficient.

Solution:

Substitutingthegivenvalues

From(i)and(ii)

e

1.18 ENERGY STORED IN A MAGNETIC FIELD

Letusconsideracoilsuppliedwithanalternatingvoltagevduetowhichanalternatingcurrentflowsthroughthecoil.Whencurrentincreasesfromitszerovalue,themagneticfieldstartsincreasingandreachesitsmaximumvaluewhencurrentreachesitsmaximumvalue.Whencurrentstartsdecreasing,thefieldgoesondecreasingandgraduallybecomeszero.Then,inthenegativecycleifthecurrentflows,thefieldgetsestablishedintheoppositedirection,whichcollapseswhencurrentagainreacheszero.Thiswaythefieldisestablishedandthencollapsesineveryconsecutivehalfcycleofcurrentflow.Whenthefieldisestablished,energyintheformofamagneticfieldisstoredandwhenthefieldcollapses,thesameenergyisreturnedtothesupplysource.Assuch,noenergyisconsumedbythepurelyinductivecoil.Therefore,energystoredisequaltotheenergysupplied.

ThisinducedEMFopposestheappliedvoltagefromwhichitisproduced.ThisisduetoLenzslaw,sothat

e=vor,v=e

Thus,

foracurrentchangefrom0toI

EnergystoredinacoilofinductanceLis

Figure1.17Magneticfieldenergy

Example1.12Aconductoroflength0.5misplacedinamagneticfieldofstrength0.5Wb/m .Calculatetheforceexperiencedbytheconductorwhenacurrentof50Aflowsthroughit.Iftheforcemovestheconductoratavelocityof20m/sec,calculatetheEMFinducedinit.

Solution:

Force,Fonacurrentcarryingconductorplacedinamagneticfieldisgivenas

F=BINewton

Substitutingthevalues,

F=0.5Wb/m 500.5m=12.5N

InducedEMF,einaconductormovinginamagneticfieldisgivenas

e=BlvV

Substitutingthegivenvalues,

e=0.5Wb/m 0.5m20m/sec=5Wb/sec=5V

Example1.13Anironcoredtoroidalcoilhas100turns.Themeanlengthofthefluxpathis0.5mandthecrosssectionalareaofthecoreis10cm .Calculatetheinductanceofthecoil.Assumerelativepermeabilityofironas2000.Alsocalculatetheinducedemfinthecoilwhencurrentof5Aisreversedin10ms.

Solution:

Theexpressionforinductanceintermsofitsparametersis

Currentinthecoilischangedfrom+5Ato5Ain1010 secs.Totalchangeofcurrentis10A.

Puttingthegivenvaluesweget,

2

2

2

2

3

Example1.14Thereismutualmagneticcouplingbetweentwocoilsofnumberofturns500and2000,respectively.Only50%ofthefluxproducedbythecoilof500turnsislinkedwiththecoilof1000turns.Calculatethemutualinductanceofthetwocoils.AlsocalculatetheEMFinducedinthecoilof1000turnswhencurrentchangesattherateof10A/secondintheothercoil.Theselfinductanceofthecoilof500turnsin200mH.

Solution:

Figure1.18

Mutualinductance,

InducedEMFinthesecondcoil,e is

Example1.15Acurrentof5Aflowingthroughacoilof500turnsproducesafluxof1mWb.Anothercoilisplacednearthiscoilandcurrentinthiscoilissuddenlyreversedin10ms.Asaresult,theEMFinducedinthesecondcoilismeasuredas50V.Calculateselfandmutualinductanceofthecoilsassumingacoefficientofcouplingas60percent.

Solution:

Selfinductanceofcoil1is

Usingtheformula,M=K

Example1.16TwocoilsofnumberofturnsN =1000andN =400,respectively,areplacedneareachother.Theyaremagneticallycoupledinsuchawaythat75percentofthefluxproducedbytheoneof1000turnslinkstheother.Acurrentof6Aproducesafluxof0.8mWbinN andthesameamountofcurrentproducesafluxof0.5mWbinthecoilofNturns.DetermineL ,L ,M,andKforthecoils.

Solution:

2

1 2

1

2

1 2

Usingtherelation,M=K

substitutingvalues,

So,

Selfinductanceofcoil1=0.133H

Selfinductanceofcoil2=0.033H

Mutualinductanceofthecoils=0.04H

Coefficientofcoupling=0.606

1.19 ELECTRICAL CIRCUIT ELEMENTS

Resistors,inductors,andcapacitorsarethethreebasiccircuitparametersorcircuitcomponentsofanyelectricalnetwork.Resistorscanbewirewoundtypeorcarbonmouldedtype.Whencurrentflowsinaresistance,heatisproduced,whichisdissipated.Theheatisproducedbecausefrictionbetweenmovingfreeelectronsandatomsobstructthefreeflowofelectronsproducingelectriccurrent.Aresistorisanelementthatdissipatesenergyasheatwhencurrentflowsthroughit.

Inductorsaremadeofacoilhavinganumberofturns.Thecoreofthecoilmaybeairoramagneticmaterial,whichisplacedinsidethecoil.Whenthecoiliswoundonanironcore,theinductorformediscalledanironcoreinductorcoil.Inductanceofaninductorisdirectlyproportionaltothesquareofthenumberofturnsofthecoilused.Inductorstoresenergybecauseofcurrentflowingthroughit.

Acapacitorconsistsoftwoconductorsorconductingplatesbetweenwhichadielectricisplaced.Thecapacitanceofacapacitorisitsabilitytostoreelectriccharge.Differenttypesofcapacitorsareavailable.Theyarenamedaccordingtothedielectricplacedbetweentheconductors.Commontypesofcapacitorsareair,mica,paper,ceramic,etc.

1.19.1 Resistors

Wirewoundresistorsaremadeofwiresofconstantan,manganinornichromewoundonaceramictube.Theseresistancesareavailableinrangesvaryingfromafractionofanohmtothousandsofohms.

ThepowerratingalsovariesfromafractionofaWatttofewkiloWatts.Whilespecifyingaresistance,bothresistancevalueandpowerdissipatingvaluemustbementioned.Electroniccircuitsrequireresistorsofaccuratevalues.Thevalueofresistorsusedinelectroniccircuitsisquitehigh,oftheorderofkiloohms.Sincecarbonhashighresistivity,carbonresistorsaremadewithcopperleads.TheirpowerratingvariesfromafractionofaWatttoseveralWatts.Colorcodeisusedtoindicatethevalueofsuchresistors.

1.19.2 Inductors

TheabilityofacoiltoinduceEMFinitselfwhenthecurrentthroughitchangesiscalleditsinductance.TheunitofinductanceisHenry.1Henryofinductancecauses1Volttobeinducedwhencurrentchangesattherateof1Amperepersecond:

whereLisinHenry,eisinVolt,and isinAmperepersecond.

Whensteadydirectcurrentflowsthroughaninductor,itwillnotaffectthecircuitasthereisnochangeincurrent.Inductorsareoftwotypesvizaircoretypeandironcoretype.Inductorsarealsocalledchokes.Inductorsareavailableinallcurrentranges.Aircoreinductorsarewoundonbakeliteorcardboardrodsandareextensivelyusedinelectroniccircuitsinmillihenryandmicrohenryranges.Highvalueinductorsaremadeofironcore.Theyaremainlyusedinacpowersupplyoffrequencyof50Hz.

Thedetailsofselfandmutualinductancehavebeendiscussedearlier.

1.19.3 Capacitors

Acapacitor,initssimplestform,consistsoftwothinparallelplatesofconductingmaterialseparatedbyadielectricmaterial.Acapacitoriscapableofstoringchargewhenavoltageisappliedacrossthecapacitorplates.Ifavoltagesource,sayabattery,isconnectedacrossthetwoplatesofaparallelplatecapacitorasshowninFig.1.19,electronsfromthenegativeterminalofthevoltagesourceaccumulateonplateAofthecapacitor.TheotherplateBloseselectronsasitisconnectedtothepositiveterminalofthesourceofvoltage.Thisway,theexcesselectronsproducenegativechargeononesideofthecapacitorwhiletheoppositesidewillhavepositivecharge.Thedielectricmaterialplacedinbetweentheplatesholdthechargebecausethefreeelectronscannotflowthroughaninsulator(i.e.,thedielectricmateriallikeair,paper,ormica).Storageofchargebyacapacitormeansthatthechargeremainsinplaceevenafterthevoltagesourceisdisconnected.Capacitanceofacapacitoristheabilitytostorecharge.Charginganddischargingarethetwomaineffectsofcapacitors.Whenavoltageisapplied,thereisaccumulationofchargeinthecapacitorandasaresultvoltageisbuiltupacrosstheterminalsofthecapacitor.Thisiscalledchargingofthecapacitor.Thecapacitorvoltagebecomesequaltotheappliedvoltagewhenthecapacitorisfullycharged.Thevoltageacrossthecapacitorremainsevenafterthevoltagesourceisdisconnected.Thecapacitordischargeswhenaconductingpathisprovidedacrosstheplateswithoutanyappliedvoltageconnected.

Figure1.19Acapacitorstoreschargeinthedielectricmaterialplacedbetweentheconductingplates

Themorethechargingvoltageis,themoreistheaccumulationofchargeinthecapacitor.Theamountofcharge,Qstoredinacapacitoris,therefore,proportionaltothechargingvoltage,V.Acapacitorwithalargeareaoftheparallelplatescanstoremorecharge.Capacitanceofacapacitoralsodependsonthedistancebetweentheplatesandthetypeofdielectricusedbetweentheplates.Alargecapacitor,obviously,willstoremorecharge.Thus,wecanwrite

Q=CVCoulombs

whereQisthechargestoredinCoulombs,Visthevoltageappliedacrosstheplates,andCisthecapacitanceofthecapacitorinFarads.Thecapacitanceofaparallelplatecapacitorisexpressedas

whereistheabsolutepermittivityconstant,Cisthecapacitance,Aistheareaoftheplateanddisthedistancebetweentheplates.

Thetermabsolutepermittivityisexpressedas

=

where isthepermittivityconstantofvacuumand istherelativepermittivityofthedielectricmaterialplacedbetweenthetwoplates.

Thevalueof hasbeencalculatedexperimentallyas8.8510 Faradpermeter.

Therefore,thecapacitanceofaparallelplatecapacitorcanbeexpressedas

1.20 ENERGY STORED IN A CAPACITOR

Wehaveknownthatwhenacapacitorisswitchedontoadcsupply,thechargeqcanbeexpressedasq=Cv,whereatanyinstantqisthechange,visthepotentialdifferenceacrossthecapacitorplates,andCisthecapacitanceofthecapacitor.

PotentialdifferenceofvvoltsacrossthecapacitormeansvJoulesofworkhastobedoneintransferring1Coulombofchangefromoneplatetotheother.Ifasmallchargedqistransferredthentheworkdonedwcanbeexpressedas

dw=vdq=Cvdv

ThetotalworkdoneinraisingthepotentialofthecapacitortothesupplyvoltageofVvoltcanbeexpressedas

Thisworkdoneisstoredintheelectrostaticfieldsetupbetweentheplatesofthecapacitorintheformofenergy.Thus,theenergystored,Eisexpressedas

Example1.17Thecurrentthrougha100mHinductorchargesfrom0to200mAin4s.WhatisthevalueoftheinducedEMFintheinductororthechoke?

Solution:

Itisobservedthatahighvoltageisinducedinthechokebecauseofveryfastchangeofcurrentflowthroughit.Inatubelightcircuit,ahighvoltageisinducedinthechokebythesamemethodandisusedtoionizethegasinsidethetubelight,andthusstartthetubelight.

Example1.18SelfinductancesoftwocoilsareL =2HandL =8H.ThecoilL producesamagneticfluxof80mWbofwhichonly60WbarelinkedwithcoilL .Calculatethemutualinductanceofthetwocoils.

Solution:

Thecoefficientofcoupling,Kisgivenas

o r

o r

o

1 2

1

2

12

MutualinductanceMiscalculatedas

Example1.19Calculatethecapacitanceofacapacitormadeoftwoparallelplatesof3m havingadistancebetweentheplatesof1cm.Thedielectricisairbetweentheplates.

Solution:

=265510 F

Notethatalthoughtheareaoftheplatesislarge,thevalueofcapacitanceisverysmall.Insteadofairasthedielectric,ifweplacemicaorpaperbetweentheplates,capacitancewillincrease.Ifwealsoreducethedistancebetweentheplates,thecapacitancewillincrease.

Example1.20A25microfaradcapacitorisswitchedontoatimevaryingvoltagesource.Thevoltagewaveissuchthatvoltageincreasesattherateof10Vpersecond.Calculatethechargeaccumulatedinthecapacitoratanelapseof1secondandtheamountofenergystoredinthecapacitor.

Solution:

1.21 CAPACITOR IN PARALLEL AND IN SERIES

Whenweconnecttwocapacitorsinparallel,theplateareasareadded.Thetotalcapacitance,therefore,getsaddedup.WhencapacitancesC ,C ,C ,etc.areconnectedinparallel,thetotalcapacitanceC becomesequalto

C =C +C +C +

Figure1.20(a)Equivalentofcapacitorsconnectedinparallel(b)equivalentofcapacitorsconnectedinseries

ThisisshowninFig.1.20(a).

Seriesconnectionofcapacitors,asshowninFig.1.20(b),isequivalenttoincreasingtheeffectivedistancebetweentheplatesorthethicknessofthedielectricused.Thecombinedcapacitanceislessthantheindividualvalue.

1 2

3 r

r 1 2 3

2

12

Thevalueofacapacitorisalwaysspecifiedineithermicrofaradorpicofarad.Thereareavarietyofwaysinwhichmanufacturersindicatethevalueofacapacitor.

1.22 Review Questions

1. Giveanoverviewofthescopeofelectricalandelectronicsengineering.

2. Chargeinmotioniscalledcurrent.Explainwiththehelpofatomictheory.

3. Distinguishbetweenconductors,semiconductors,andinsulators.4. DistinguishbetweenWork,Power,andEnergy.5. Differentiatebetweentemperaturecoefficientofresistanceand

specificresistance.6. Distinguishbetweenanelectricfieldandamagneticfield.7. Definethefollowingterms:Volt,Ampere,Ohm.8. Explainwhytwoparallelcurrentcarryingconductorsattracteach

otherwhencurrentinthemflowinthesamedirection.9. StateFlemingsRightHandRule.

10. ExplainthattheEMFinducedinacoildependsuponthefluxandthespeedofrotationofthecoil.

11. DistinguishbetweenstaticallyinducedEMFanddynamicallyinducedEMF.

12. Explainwhyanironcorecoilwillhavemoreinductancethananaircorecoilofthesamenumberofturns.

13. Whatisthemeaningofcoefficientofcouplingbetweentwocoils?Whenisthisvalueequaltounityandequaltozero?

14. WhatareFaradayslawsofelectromagneticinduction?15. WhatistheLenzslaw?Giveanexample.16. Whatisthemagnitudeofforceexperiencedbyacurrentcarrying

conductorplacedinamagneticfield?17. Howdoyoudeterminethedirectionofforcedevelopedina

currentcarryingconductorplacedinamagneticfield?18. Whatarethefactorsonwhichinductanceofacoildepends?19. Whydoestheinductanceofacoilincreaseifthecorehasa

magneticmaterialinsteadofair?20. Derivethefollowingexpressionforselfinductanceofacoil

21. Youhavetomakeaninductanceofhighvalue.Howwillyouproceed?

22. WhatisFlemingsRightHandRule?Whereisitused?23. Whatruledoyouapplytodeterminethedirectionofforceona

currentcarryingconductorplacedinmagneticfield?24. Whatisthemagnitudeofforceonacurrentcarryingconductor

placedinamagneticfield?25. Showthattheenergystoredinamagneticfieldproducedbyan

inductoris .26. Distinguishbetweenselfinductanceandmutualinductance.27. Explainwhyinductanceofacoilincreasesifanironpieceforms

itscoreinsteadofairoranynonmagneticmaterial.

28. Establishtherelation, fortwoadjacentcoilslinkingflux.

29. Onwhatfactorsdoesthereluctanceofamagneticmaterialdepend?

30. Whatisthecorkscrewrule?Wheredoyouuseit?31. Twoadjacentconductorsarecarryingcurrentintheopposite

directions.Showthattherewillbeforceofrepulsionbetweentheconductors.

32. Whencapacitorsareconnectedinparallel,theirequivalentcapacitanceisincreased.Explainwhy?

33. Explainwhycapacitorsarecalledenergystoragedevices.34. Whatisthemeaningofrelativepermittivityordielectric

constant?Whatisitsunit?35. Writethreeformulaeofelectricalpower.36. Provethat1kWhisequalto3.610 Joules.37. Themostimportantpropertyofacapacitorisitsabilitytoblock

steadydcvoltagewhilepassingacsignals,explain.38. DefinetheFaradunitofcapacitance.39. Howisenergystoredinacapacitor?Onwhatfactorsdoesit

depend?40. Whatarethephysicalfactorsthataffectthecapacitanceofa

capacitor?41. TwocoilsofN =50andN =500turns,respectively,arewound

sidebysideonanironringofcrosssectionalareaof50cm andmeanlengthof120cm.Calculatethemutualinductancebetweenthecoils,selfinductanceofthecoils,andthecoefficientofcouplingassumingpermeabilityofironas1000.

1 2

6

2

[Ans0.13H,0.013H,1.3H,1.0]

42. TwocoilsofN =1500andN =200turnsarewoundonacommonmagneticcircuitofreluctance2510 AT/Wb.Calculatethemutualinductancebetweenthecoils.

[Ans1.2H]

43. Twocoilshaveamutualinductanceof400mH.CalculatetheEMFinducedinonecoilwhencurrentinthesecondcoilvariesatarateof6000Amperespersecond.

[Ans2.4V]

44. Twosimilarcoilshaveacouplingcoefficientof0.4.Whenthecoilsareconnectedinseriescumulatively,thetotalinductancebecomesequalto140mH.Calculatetheselfinductanceofeachcoil.

[Ans50mH]

45. Twocoilswhenconnectedinseriescumulatativelyshowtohaveatotalinductanceof2.4Handwhenconnectedinseriesbutdifferentiallyshowatotalinductanceof0.4H.Theinductanceofonecoilwhenisolatediscalculatedasequalto0.8H.Calculate(a)themutualinductanceand(b)thecoefficientofcouplingbetweenthecoils.

[AnsM=0.5H,0.75]

46. Calculatetheinductanceofacoilhaving100turnswoundonamagneticcoreofpermeabilityequalto1000,meanlengthof0.25m,andcrosssectionalareaof10cm .

[AnsL=50.24mH]

47. Aconductoroflength25cmisplacedinauniformmagneticfieldofstrength0.5Wb/m .CalculatetheEMFinducedintheconductorwhenitismovedattherateof10m/sec(a)paralleltothemagneticfield,(b)perpendiculartothemagneticfield.

[Ans(a)0V(b)1.25V]

MultipleChoiceQuestions

1. ThenumberofelectronsperCoulombisequalto1. 1.602102. 6.28103. 1.602104. 6.2810 .

2. Ininsulatorstheoutermostorbitoftheiratomsisfilledwith1. 4electrons2. 8electrons3. 1electron4. 18electrons.

3. Intheatomsofsemiconductingmaterialslikesiliconandgermaniumtheoutermostorbithas

1. 1electron2. 2electrons3. 8electrons4. 4electrons.

4. Whichofthefollowingexpressionsisincorrect?

1. Current,2. Charge=currenttime

3.4. Volt=joulesperCoulomb.

5. Whichisthefollowingexpressionsdoesnotrepresentpower?1. I R

2.3. VI

4. .6. Whichofthefollowingisnottheunitofpower?

1 24

2

2

19

18

18

19

2

1. Joules/second2. Watthour3. KW4. VoltAmpere

7. AconductoroflengthanddiameterdhasresistanceofRohms.Ifthediameterisreducedtoonethirdandlengthincreasedbythreetimes,theresistanceoftheconductorwillbe

1. 3R2. 6R3. 9R4. 27R.

8. Whichofthefollowingexpressionsisincorrect?

1.

2.

3.

4. .9. Whichofthefollowingexpressionsisincorrect?

1.

2.

3.

4. .10. Inductanceofanaircorecoilwillincreaseifthecoreismadeof

1. copper2. aluminium3. iron4. porcelain.

11. Whichofthefollowingstatementsisnottrue?1. Inductanceofacoilwillincreasebyfourtimesifthe

numberoftermseqisdoubled2. inductanceofacoilwillincreaseiftheareaofcross

sectionofthecoil,i.e.,thefluxpathisincreased3. inductanceofacoilwillincreasedifthelengthofflux

pathisincreased4. inductanceofacoilwillincreaseifthecoreismadeupof

materialhavinghigherpermeability.12. ThedirectionoftheinducedEMFinthecoilsidesofacoil

rotatinginamagneticfieldcanbedeterminedbyapplying1. Flemingslefthandrule2. Righthandgriprule3. Flemingslefthandrule4. Corkscrewrule.

13. Whichofthefollowingisnottheunitofenergy?1. kWh2. Joules/second3. Watthour4. Joules.

14. Selfinductanceoftwomagneticallycoupledcoilsare8Hand2H,respectively.Whatcoefficientofcouplingwillmaketheirmutualinductanceequalto4H?

1. K=0.52. K=0.253. 0.14. 1.0.

15. Whichofthefollowingeq.isincorrectwithrespectofincreaseinresistancewithincreaseintemperatureofaconductingmaterial?

1.

2.

3.

4. .

AnswerstoMultipleChoiceQuestions

1. (b)2. (b)3. (d)4. (c)5. (d)6. (b)7. (d)8. (b)9. (a)

10. (c)11. (c)

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12. (c)13. (b)14. (d)15. (d)

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