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Today’sConcepts:
A)Oscilla*onFrequency
B)Energy
C)Damping
Electricity & MagnetismLecture 19
Electricity&Magne/smLecture19,Slide1
“Whydocapacitorsstartoffwithachargeofzerowhentheswitchisopened?shouldn'ttheystartoffwithcharge?“
Italldependsonhowthecircuitwasstarted.Youhavetodeterminetheini*alcondi*onsfromtheproblemstatement
WhatistheuseofLCandRLCcircuits?
“There are a lot of formulas in this section”“Please explain the meaning of cos(ωt + x)”
Differen/alEqua/onsDoDetermineMuchBehaviourinPhysics.Wewillshowcorrespondingequa/onsin
mechanicstoday
“Allofthisstuff,IwishwecouldgoslowerbutIknowwecant.”
TOOTRUE:Hanginthere,we’lldoourbesttoworkontheissues….
Your Comments
Electricity&Magne/smLecture19,Slide2
Weshouldinvitetheprelecturenarratorovertoourclassforapizzaparty.Imean,he'sprobablynotbusy,justavoiceactorlivinginhisMom'sbasement.
Prof. Gary Glading
Prac*calTestisonFriday,July8! Wewillhaveaprac*sesessionsonMondayandWednesday! CoversDCCircuitsandtheOscilloscope(LCorRCorLRCCircuit)! 50min! smartPhysicswillcon*nuecoveringthesetopicstheore*cally! Prac*seexamishere:! Youmayalsoprac*seoncircuitsofyourown(eg.LCRcircuitsandresonance)
Some Practical Test Stuff
Electricity&Magne/smLecture19,Slide3
C
I
L Q
LC Circuit
Electricity&Magne/smLecture19,Slide4
+
− +
−
CircuitEqua/on:
where
“tank”
CL
k
x
m
F = -kx a
Samethingifweno/cethatand
Electricity&Magne/smLecture19,Slide5
CLI
+ +
−−
Time Dependence
Electricity&Magne/smLecture19,Slide6
Wikipedia Animation“LCCircuit”
Whatisthepoten/aldifferenceacrosstheinductoratt = 0?A)VL = 0 B)VL = Qmax/C C)VL = Qmax/2C
Pendulum.
At/met = 0 thecapacitorisfullychargedwithQmaxandthecurrentthroughthecircuitis0. L C
ThevoltageacrossthecapacitorisQmax/CKirchhoff'sVoltageRuleimpliesthatmustalsobeequaltothevoltageacrosstheinductor
sinceVL = VC
CheckPoint 2
Electricity&Magne/smLecture19,Slide7
dI/dtiszerowhencurrentismaximum
CheckPoint 4
At/met = 0 thecapacitorisfullychargedwithQmaxandthecurrentthroughthecircuitis0. L C
Whatisthepoten/aldifferenceacrosstheinductorwhenthecurrentismaximum?A)VL = 0 B)VL = Qmax/C C)VL = Qmax/2C
Electricity&Magne/smLecture19,Slide8
Howmuchenergyisstoredinthecapacitorwhenthecurrentisamaximum?A)U = Qmax
2/(2C) B)U = Qmax
2/(4C) C)U = 0 TotalEnergyisconstant!ULmax = ½ LImax
2
UCmax = Qmax2/2C
I = max when Q = 0
CheckPoint 6
At/met = 0 thecapacitorisfullychargedwithQmaxandthecurrentthroughthecircuitis0. L C
Electricity&Magne/smLecture19,Slide9
Thecapacitorischargedsuchthatthetopplatehasacharge+Q0 andthebo]omplate−Q0. At/met = 0, theswitchisclosedandthecircuitoscillateswithfrequencyω = 500 radians/s.
WhatisthevalueofthecapacitorC?
A)C = 1 x 10-3 F B) C = 2 x 10-3 F C) C = 4 x 10-3 F
L C
L = 4 x 10-3 Hω = 500 rad/s+ +
−−
CheckPoint 8
Electricity&Magne/smLecture19,Slide10
Prelecture
Whatisthecurrentthroughthecircuit,attheinstantthechargeonthecapacitorisQmax/2?
A)I<Imax/2B)I=Imax/2C)I>Imax/2
Att=0thecapacitorisfullycharged.ExpressionsforthechargeandcurrentasafuncKonofKmeareshowntotheright.
Whichplotbestrepresentstheenergyintheinductorasafunc*onof*mestar*ngjusta_ertheswitchisclosed?
Currentischanging⇒ULisnotconstant
Ini/alcurrentiszero
L C
closedatt = 0+Q0
−Q0
CheckPoint 10
Energypropor/onaltoI2⇒Ccannotbenega/ve
Electricity&Magne/smLecture19,Slide11
Whentheenergystoredinthecapacitorreachesitsmaximumagainforthefirst/meaaert = 0,howmuchchargeisstoredonthetopplateofthecapacitor?
A) +Q0B)+Q0 /2C) 0D)−Q0/2E)−Q0
CheckPoint 12
L C
closedatt = 0+Q0
−Q0
CurrentgoestozerotwiceduringonecycleElectricity&Magne/smLecture19,Slide12
Qismaximumwhencurrentgoestozero
…butanswerlookskindofcomplicatedConceptmakessense…
JustlikeLCcircuitbuttheoscilla/onsgetsmallerbecauseofR
Add R: Damping
Electricity&Magne/smLecture19,Slide13
Physics Truth #1:
Eventhoughtheanswersome*meslookscomplicated…
thephysicsunderthehoodiss*llverysimple!
Electricity&Magne/smLecture19,Slide14
Theelementsofacircuitareverysimple:
Thisisallweneedtoknowtosolveforanything!
Electricity&Magne/smLecture19,Slide15
Startwithsomeini*alV, I, Q, VL Nowtakea*ny*mestepdt (1 ms)
Repeat…
A Different Approach
Electricity&Magne/smLecture19,Slide16
Theswitchinthecircuitshownhasbeenclosedforalong/me.Att = 0,theswitchisopened.
WhatisQMAX,themaximumchargeonthecapacitor?
ConceptualAnalysisOnceswitchisopened,wehaveanLCcircuitCurrentwilloscillatewithnaturalfrequencyω0
StrategicAnalysisDetermineini*alcurrentDetermineoscilla*onfrequencyω0
Findmaximumchargeoncapacitor
Calculation
C
R
LV
Electricity&Magne/smLecture19,Slide17
WhatisIL,thecurrentintheinductor,immediatelyaaertheswitchisopened?Takeposi/vedirec/onasshown.
A)IL < 0 B) IL = 0 C) IL > 0
beforeswitchisopened:allcurrentgoesthroughinductorindirec*onshown
Theswitchinthecircuitshownhasbeenclosedforalong/me.Att = 0,theswitchisopened.
Calculation
C
R
LV
IL
Currentthroughinductorimmediatelya_erswitchisopenedisthesameas
thecurrentthroughinductorimmediatelybeforeswitchisopened
Electricity&Magne/smLecture19,Slide18
Theenergystoredinthecapacitorimmediatelyaaertheswitchisopenediszero.
A)TRUEB)FALSE
VC = 0
BUT:VL = VC sincetheyareinparallel
dIL/dt ∼ 0 ⇒ VL = 0beforeswitchisopened:
VCcannotchangeabruptly
a_erswitchisopened:
IMPORTANT:NOTEDIFFERENTCONSTRAINTSAFTERSWITCHOPENEDCURRENTthroughINDUCTORcannotchangeabruptlyVOLTAGEacrossCAPACITORcannotchangeabruptly
Calculation
C
R
LV
ILTheswitchinthecircuitshownhasbeenclosedforalong/me.Att = 0,theswitchisopened.
IL (t = 0 +) > 0
VC = 0
VC = 0
UC = ½ CVC2 = 0 !
Electricity&Magne/smLecture19,Slide19
C
R
LV
IL
Whatisthedirec/onofthecurrentimmediatelyaaertheswitchisopened?
A)clockwiseB)counterclockwise
Beforeswitchisopened:CurrentmovesdownthroughL
A_erswitchisopened:Currentcon*nuestomovedownthroughL
IL (t = 0 +) > 0 VC (t = 0 +) = 0
Theswitchinthecircuitshownhasbeenclosedforalong/me.Att = 0,theswitchisopened.
Calculation
Currentthroughinductorimmediatelya_erswitchisopenedisthesameas
thecurrentthroughinductorimmediatelybeforeswitchisopened
Electricity&Magne/smLecture19,Slide20
A)B)C)D)
C
R
LV
Calculation
VL = 0Beforeswitchisopened: VL = 0
IL
IL IL
Theswitchinthecircuitshownhasbeenclosedforalong/me.Att = 0,theswitchisopened. C
R
LV
Whatisthemagnitudeofthecurrentrightaaertheswitchisopened?
Currentthroughinductorimmediatelya_erswitchisopenedisthesameas
thecurrentthroughinductorimmediatelybeforeswitchisopened
IL (t = 0 +) > 0 VC (t = 0 +) = 0
V = ILRElectricity&Magne/smLecture19,Slide21
A)B)C)D)
WhatisQmax,themaximumchargeonthecapacitorduringtheoscilla*ons?
IL (t = 0 +) = V/R VC (t = 0 +) = 0Hint:Energyisconserved
CalculationTheswitchinthecircuitshownhasbeenclosedforalong/me.Att = 0,theswitchisopened.
C
R
LV
IL
Electricity&Magne/smLecture19,Slide22
C
WhenQismax(andIis0)
LQmax
CL
Imax
WhenIismax(andQis0)
Follow-Up
IsitpossibleforthemaximumvoltageonthecapacitortobegreaterthanV ?
A)YESB)NO
Wecanrewritethiscondi*onintermsoftheresonantfrequency:
WewillseetheseformsagainwhenwestudyACcircuits!
C
R
LV
IL
Theswitchinthecircuitshownhasbeenclosedforalong/me.Att = 0,theswitchisopened.
Imax = V/R
Electricity&Magne/smLecture19,Slide23
VmaxcanbegreaterthanVIF:
OR
FlipItPhysics Problem
AtwhatKme,doesthecurrentthroughtheinductor I3reachavaluethatis63%ofitsmaximumvalue?
ApplyKirchhoff'srulesto(1)outerloop,(2)innerloopand(3)thejuncKon.
FlipItPhysics Problem
EliminateI1andI2and.FindtheequaKonforI3
