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AP Physics Chapter 16
Electric Potential, Energy, andCapacitance
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Chapter 16: Electric Potential, Energy,
and Capacitance16.1 Electric Potential and Potential
Difference
16.2 Equipotential Surfaces and theElectric Field
16.3 Capacitance
16.4 Omitted16.5 Capacitors in Series and in Parallel
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Homework for Chapter 16
Read Chapter 16
HW 16.A: p.537-539: 4,8-13, 16,17, 20-23, 47,48, 51-53, 55-57.
HW 16.B: p.540-541: 64-70, 84-87, 89, 90.
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B L A C K H O L E S A R E WH ERE GO
D D I V I D E D B Y ZE R O
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If your car is traveling at the speed of light and
you turn your headlights on, what happens?
-StevenWright
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16.1: Electric Potential andPotential Difference
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Electric Potential Energy Difference
a) Movinga positive charge qo againsttheelectric fieldrequires positive workand
increasestheelectric potentialenergy.
Theforce requiredto movethe chargeisequaltotheelectric force:Fe = qo E
Thework done bythe force: Fe d= qo E d Theincreaseinthe chargeselectric potentialenergyisequaltothe workdone
onthe charge:Ue = UB UA = qo E d
The SI unitofelectric potentialenergyisthe joule (J).
b) Movinga mass m againstthegravitational fieldrequires positive workand
increasesthegravitational potentialenergy: Ug = UB UA = mgh
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Electric Potential Difference
** This is not the same as Electric Potential Energy Difference**
Theelectric potential difference (voltage) betweentwo pointsisthe work per
unit positive chargedone byanexternal forcein moving charge betweenthese
two points,
OR
the changeinelectric potentialenergy perunit positive charge.
V = W = Ue where qo isthe positivetest charge
qo qo
The SI Unitofelectric potentialdifferenceis: joule/coulomb (J/C)orvolt (V).
Electric potentialdifferenceis commonlyshorted fromV to just V.
Potentialdifferenceisdefined perunit charge,soitdoesnotdependonthe
amountof charge moved (potentialenergydifferencedoes).
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On Gold Sheet
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Potential Difference for a Uniform Field Between Two Parallel Plates
Assume we wouldliketo movea positivetest charge, qo,from thenegativeto
the positive plate. This move would beagainsttheelectric field, wouldrequirework,and wouldincreasethe charges potentialenergy. The potentialdifference
betweenthe platesis:
V = W = Ue where qo isthe positivetest chargeand
qo qo Ueisthe potentialenergygained
Ina uniform electric fieldE,the potentialdifferencein movingthe chargethrough
astraightlinedistanced is:
V = Ue = qoE d = E d Potential Difference
qo qo Parallel Plates
Whentalkingaboutelectric potential, wealways mustdefinethereference
value. Only changesinelectric potential (voltage)are meaningful.
ex: Thenegative plateis commonlyassignedavalueof zero,sovoltage
will be positive.
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Positive charges, when released, tend to move toward regions of low potential,
and negative charges tend to move toward regions of high potential.
Acceleratinga Charge
a) Movinga proton from thenegativetothe positive plateincreasesthe
protons potentialenergy.
b) Whenitisreleased from the positive plate,the protonaccelerates back
towardthenegative plate,gainingkinetic energyandlosingelectric potentialenergy.
c) The workdoneto movea proton betweenanytwo pointsintheelectric field
is INDEPENDENT OF PATH.
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Example 16.1: Anelectroninitiallyatrest,isacceleratedthrough anelectric
potentialdifferenceof 50.0 V.
a) Whatisthekinetic energyoftheelectron?
b) Whatisthespeedoftheelectron?
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Example 16.2:A 12-V battery maintainstheelectric potentialdifference between
tow parallel metal platesseparated by 0.10 m. Whatistheelectric field between
the plates?
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A positive point charge createsanelectric field.
Theelectric potential
increasesasyou move closer
tothe positive charge.
SincerB < rA, B isata higher
potentialthan A,andthe
potentialdifferenceis positive.
Electric potential increases (+V) as we get closer to positive charges or fartheraway from negative charges.
Electric potential decreases (-V) as we get farther away from positive charges
or nearer to negative charges.
Potential Difference Due to Point Charges
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On Gold Sheet
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Electric Potential
V =kq electric potentialduetoa point charge
r ( V = 0 atr=g)
Unlikeelectric field,electric potentialisascalarquantity. Whenadding
potentialsdueto point charges, justaddthem algebraically (includingthe + or
signs).
V =k i qi
ri
Note:onthegoldsheet, Coulombslaw constantis writtenas 1/4TIo.
Anotherdifference betweenelectric fieldandelectric potentialisnotable:
electric fieldis proportionalto 1/r2
electric potentialis proportionalto 1/r
On Gold Sheet
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On Gold Sheet
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Electric Potential Energy of Various Charge Configurations
a) A positive point charge q1 is fixedinspaceandasecond positive charge q2 is
pushedtowardit from averylargedistance (r=g
)toadistancer12.
Thereisanincreasein potentialenergy because positive work must bedone
to bringthe mutuallyrepelling charges closertogether.
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Electric Potential Energy of Various Charge Configurations
Formorethantwo charges,thesystemselectric potentialenergyisthesum ofthe
energiesofeach pair.
Electric Potential Energy followsthe Law
of Conservationof Energy. Forexample,
considertwo protonsthatare heldneareach otheratrest. Work wasdoneto
bringthem close,andstoredinthe form
ofelectric potentialenergy. If werelease
them,they will flyapart. Theelectric
potentialenergy will be convertedto
kinetic energy. Thisisverysimilartotwomasses connected bya compressed
spring.
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Example 16.3: A chargeof 5.0 nC isat (0,0)andasecond chargeof -2.0 nC isat
(3.0m, 0m). Ifthe potentialistakento be zeroatinfinity,
a) whatistheelectric potentialat point P (0,4.0) m?b) whatisthe potentialenergyofa 1.0 nC chargeat point P?
c) whatisthe workrequiredto bringa chargeof 1.0 nC charge from infinityto
point P?
d) whatisthetotal potentialenergyofthethree chargesystem?
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Summary:
Theelectric potentialdifference betweentwo pointsisthe work perunit positive
chargedone byanexternal forcein moving charge betweenthosetwo points.
Electric potentialdifferenceisthe changeinelectric potentialenergy perunit
positive charge.
Voltageissynonymous with electric potentialdifference.
V = W = Ue electric potentialdifference (voltage)definitionqo qo
V = Ed electric potentialdifference between parallel plates
V =kq electric potentialduetoa point charge (V = 0 atr=g)r
U = U12 + U23 + U13 + electric potentialenergyofa configurationof point
charges
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2. Whatisthedifference betweenelectrostatic potentialenergyandelectrostatic
potential?
Answer: Potentialisthe potentialenergy perunit charge: V = Ue / qo
3. Whatisthedifference betweenelectric potentialdifferenceandvoltage?
Answer:nodifference.
Check forUnderstanding:
1.The SI unitofelectric potentialdifferenceisthe
a) jouleb)newton
c)newton-meter
d) joule percoulomb
Answer:d
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Check forUnderstanding:
4. Theelectrostatic potentialenergyoftwo point charges
a)isinversely proportionaltotheirseparationdistance
b)isavectorquantity
c)isalways positive
d) has unitsofnewton percoulomb
Answer: a,since potentialenergyisinversely proportionaltothedistance betweenthe charges.
5. Anelectronisreleasedinaregion wherethereisavaryingelectric
potential. Theelectron will
a) movetowardthelowerpotentialregion
b) movetowardthe higherpotentialregion
c)remainatrest
Answer: b, becausetheelectron hasanegative charge
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I lived in a house that ran on static electricity...
If you wanted to run the blender, you had to
rub balloons on your head. If you wanted to
cook, you had to pull off a sweater real quick.
-StevenWright
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16.2: Equipotential Surfaces andthe Electric Field
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Construction of Equipotential Surfaces Between Parallel Plates
Considera positive charge moving from A to A
perpendiculartoanelectric field.
Sincetheelectric fieldis perpendiculartothe
displacement,no workisdone bythe field.
Ifno work wasdone,thentheelectric potential
energyofthe charge mustnot have changed.
We can concludeall pointson path I havethe
sameelectric potentialenergy,andthereforethe
same potential.
We canextendthistoall pointsonthe planeparalleltothe plates containing path I.
Such a plane,is calledanequipotential surface,
orsimply anequipotential.
Since path II startsandstopsonthesameequipotential,no work wasdone.
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Equipotential Surfaces Between Parallel Plates
Onceyou movetoa higherpotential (A to B),you can
stayonthatnew equipotential by movingperpendicularlytotheelectric field. (B to B).
The changein potentialisindependentof path,since
thesame changeoccursvia path I asvia path II.
Since no work is required to move a charge along an equipotential surface,then it must be generally true that equipotential surfaces are always at rightangles to the electric field.
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Gravitational Potential Energy as an Analogy
Raisinganobjectaway from theearth resultsinanincreaseintheobjects
potentialenergy.
Atagiven height,its potentialenergyis constantaslongasitremainsonthat
equipotentialsurface.
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Topographic Maps: A Gravitational Analogy for Equipotential Surfaces
Considerasymmetrical hill with slicesatdifferentelevations. Each sliceisa planeof constantgravitational potential.
Hereisanoverheadview,ortopographic
map,ofthe hill.
The contours, wherethe planesintersectthe
surface,representgravitationalequipotentials.
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Topographic Maps: A Gravitational Analogy for Equipotential Surfaces
The potential V arounda
point charge formsasymmetrical hill.
V is constantat fixed
distances from q.
Electricalequipotentialsarounda point chargearespherical,orintwodimensions, circular
slices.
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Equipotentials of an Electric Dipole
Equipotentialsare perpendicularto
electric fieldlines.
Noticethat V1 > V2,sinceequipotential
surface 1 is closertothe positive charge
thanissurface 2.
Activity: Learn by Drawing, p. 522.
Animation:
http://regentsprep.org/Regents/physics/phy
s03/aequilines/default.htm
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Electric Field From Potential
Ina uniform electric field,such asone
betweentwo parallel plates,the potential
difference betweenanytwoequipotentialplanes,separated byadistance xis
V = E x
Foragiventraveldistance x, movement
perpendiculartotheequipotentialsvieldsmaximum potentialgain.
By findingthedirectionof maximum
potential change, weare findingthedirection
oppositethatofthe E field.
E = - V electric field
x max from potentialOn Gold Sheet
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Electric Field from Potential
E = - V electric fieldx max from potential
The minussignindicatesthat E isinthedirectionoppositethatin which V
increases mostrapidly,orinthedirection V decreases mostrapidly.
The unitsofelectric fieldarevolts permeter(V/m). Thisisdimensionallyequivalentto N/C, which welearnedaboutin Chapter15. (Proveit!)
Answer: V = J/C = J = Nm = N
m m Cm Cm C
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Example 16.4: Two parallel plates,separated by 0.10 m,are connectedtoa 6.0 V
battery. Anelectronisreleased from restatthenegative plate.
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The Electron Volt
electronvolt (eV) - thekinetic energyacquired byanelectron
acceleratedthrough a potentialdifferenceofexactly 1 V.
1 eV = 1.60 x 10-19 J
Theelectronvoltisa convenient waytoexpresstypicalenergiesontheatomic
scale.
Theelectronvoltisa unitofenergy,notvoltage: e V = q V = Ue
You mayalsoencounter keV: kiloelectron volts, meaning1000 eV
MeV: megaelectron volts, meaning106 eV
GeV: gigaelectron volts, meaning109
eV
** Warning electronvoltisnotan SI unit. You must convert backto joules before
you can usethenumberina formula**
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Summary
Equipotential surfaces (equipotentials)aresurfaceson which a charge hasa
constantelectric potential (V),and constantelectric potentialenergy (Ue).
Equipotentials are perpendiculartotheelectric fieldatall points.
Ittakesno workto movea chargealonganequipotential.
E isinthedirectionoppositethatin which V increases mostrapidly,orinthe
directionin which V decreases mostrapidly.
Anelectronvolt (eV)isthekinetic energygained byanelectronaccelerated
from restthrough a potentialdifferenceof 1 V.
1 eV = 1.60 x 10-19 J
E = - V relationship between potentialandelectric field
x max
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Check forUnderstanding
1) Equipotential surfacesarethosesurfaceson which
a) the potentialis constant
b) theelectric fieldis zero
c) the potentialis zero
Answer:a
2) Equipotential Surfacesare
a) paralleltotheelectric field
b) perpendiculartotheelectric field
c)atanyangle with respecttotheelectric field
Answer: b
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Check forUnderstanding
3) Between charged parallel plates, which equipotentials havea higherpotential:
a)theonesnearthe positive plate
b)theonesnearthenegative plate
c)theonesnearthe middle?
Answer:a
4) Atagiven pointonanequipotentialsurface,theelectric field pointsdirectly
tothe
a)next highestequipotential
b)thenextlowestequipotential
c) paralleltotheequipotentialsurface
Answer: b
HW 16.A: p.537-539:4,8-13, 16,17, 20-23,47,48, 51-53, 55-57.
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I heard that in relativity theory space and time
are the same thing.
Einstein discovered this when he kept showing
up three miles late for his meetings.
-StevenWright
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16.3: Capacitance
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Capacitance
A capacitorconsistsoftwo conductors.
Capacitorsstore charge,andthereforeelectric energy,inthe form ofanelectric field.
capacitance (C) - a quantitative measureof how effectivea capacitor
isinstoring charge.
C = Q definitionof capacitance
V
where Q isthe magnitudeofthe chargeoneitherplate (the plates haveequal butopposite charge),and V isthe potentialdifferenceacrossthe plates
Note, from this pointon we will use V forV; it means potentialdifference.
assorted capacitors
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Metal plates
+Q
Capacitor and Circuit Diagram
Two metal platesare charged bya batterytoa charge Q = CV, where C isthe
capacitance.
Workisdonein chargingthe capacitor,andenergyisstoredintheelectric field.
Noticethesymbols used fora battery (V)anda capacitor(C).
parallel linesare equal inlength
the longer line of the batterysymbol is the positive terminal
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Capacitance
The battery worksasa pump toremoveelectrons from the positive plateand
transferthem through the wiretothenegative plate.
The battery chargesthe capacitoruntilthe potentialdifference betweenthe
platesisequaltothevoltageofthe battery.
Whenthe batteryisdisconnected from the capacitor,theelectric potential
energyisstoredintheelectric field. Thisstoredenergy canthen be usedtodo
work.
The SI unitof capacitanceis coulomb pervolt (C/V),or farad (F).
Itis more commontoseethe microfarad ( 1 QF = 10-6 F)
orthe picofarad (1 pF = 10-12 F)
The farad wasnamed forthe English scientistMichael Faraday (1791-1867),anearlyinvestigatorofelectrical phenomena who firstintroducedthe conceptofthe
electric field.
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Capacitance
Capacitancedependsonlyonthesize,shapeandspacingofthe plate
arrangement,as wellatthe material betweenthe plates (dielectric).
A common capacitoristhe parallel plate capacitor. It consistsoftwo metal plates
ofarea A andseparated byadistanced. The formulais:
C =IoA capacitanceofa parallel-plate
d capacitor(inair) Ioisthepermittivityof freespace.Itisa fundamental constant which describes
theelectrical propertiesofavacuum. Itsvalueinairisessentiallythesame.
Io =8.85 x 10-12 C2/(Nm2)
IoisrelatedtoCoulombs constantby:
k= 1 = 9.00 x 109 Nm2/C2
4TIo
On Blue Sheet
On Blue Sheet
On Gold Sheet
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Capacitance
A plotof chargevs.voltage fora charging capacitorisastraightline with slope C.
Q = CV( y= mx + b)
The workdone bythe batteryisstoredinthe capacitoras potentialenergy, Uc.
This workisthearea underthe curve. Therefore, Uc = Q V.
Theequivalent formsofthisequationare:
Uc = QV = Q2 = CV2 energystorage2C ina capacitor
The form CV2 is usuallythe most practical,sincethe capacitanceandthe
appliedvoltageareoftenknownorcan be measured mosteasily.
Voltage
Q
(charge)
slope=
CapacitanceOn Gold Sheet
On Gold Sheet
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Example 16.5:A parallel-plate capacitorconsistsof platesofarea 1.5 x 10-4 m2
andseparated by 2.0 mm. The capacitoris connectedtoa 12 volt battery.
a) Whatisthe capacitance?b) Whatisthe chargeonthe plates?
c) How much energyisstoredinthe capacitor?
d) Whatistheelectric field betweenthe plates?
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If youre not part of the solution, youre part
of the precipitate.
-StevenWright
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16.5: Capacitors in Series and inParallel
Capacitors can be connectedintwo basic ways:inseriesorin parallel.
Inseries capacitorsare connected headtotail.
In parallel capacitors havealltheirheads hookedtogether,andall
theirtails hookedtogether.
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Capacitors in Series
All capacitorsinseries havethesame charge.
Thesum ofthevoltagedropsisequaltothevoltageofthe battery.
Thetotal capacitanceisequivalentto Cs,ortheequivalentseries capacitance.
Cs isalwayslessthanthatofthesmallest capacitorinthe combination.
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Capacitors in Series
When capacitorsare wiredinseriesthe chargeisthesameonallthe plates.
Q = Q1 = Q2 = Q3 =
Thevoltagedrop acrossallthe capacitors must beequaltothevoltageacrossthebattery.
Therefore,thesum oftheindividualvoltagedropsacrossthe capacitorsisequalto
thevoltageofthe battery. V =V1 + V2 + V3 +
Theindividualvoltagesarerelatedtotheindividual charges byV1 =Q, V2 =Q, V3 =Q, andV =Q
C1 C2 C3 Cs
Substituting Q/C forV: Q = Q + Q + Q +
Cs C1 C2 C3
Dividing both sidesoftheequation by Q: 1 = 1 + 1 + 1 +...
equivalent series capacitance Cs C1 C2 C3
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Capacitors in Parallel
Whenthe capacitorsarein parallel,thevoltagesacrossthe capacitorsare
thesame.
Thetotal chargeisequaltothesum ofthe chargesontheindividual
capacitors.
Thetotal capacitanceisequivalentto Cp.
Cp isalwayslargerthanthatofthelargest capacitorinthe combination.
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Capacitors in Parallel
When capacitorsare wiredin parallelthevoltagesacrossthe capacitorsarethe
same,each equaltothevoltageofthe battery
V = V1 = V2 = V3 =
Thetotalstored chargeisequaltothesum ofthe chargesoftheindividual
capacitors.
Qtotal = Q1 + Q2 + Q3 +
Theindividual chargesaregiven by Q1 = C1V, Q2 = C2V, and Qtotal = CpV
Substituting CV forQ: CpV = C1V + C2V + C3V +
Divide both sidesoftheequation by V:
equivalent parallel capacitance Cp = C1 + C2 + C3 +
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On Gold Sheet
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Example 16.7: Capacitors C1 and C2 arein parallel. This combinationisinseries
with C3. The positiveterminalofa 12.0 V batteryis connectedto C3.
C1 = 6.00 QF, C2 =8.00 QF, C3 = 14.0 QF
a) Whatistheequivalent capacitance?b) Whatisthe chargeofeach capacitor?
c) Whatisthevoltageacrosseach capacitor?
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Summary
A capacitorstores charge,andthereforeelectric energy,inthe form ofanelectric
field.
Capacitanceisa quantitative measureof how effectivea capacitorisinstoring
charge.
Theequivalentseries capacitanceisalwayslessthanthatofthesmallest
capacitoroftheseries combination.
Theequivalent parallel capacitanceisalwayslargerthanthatofthelargest
capacitorinthe parallel combination.
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Summaryof Equations
C = Q definitionof capacitance
V
C =IoA capacitanceofa parallel-plate
d capacitor(inair)
Uc = QV = Q2 = CV2 energyina2C charged capacitor
1 = 1 + 1 + 1 +... equivalent series capacitance
Cs C1 C2 C3
Cp = C1 + C2 + C3 + equivalent parallel capacitance
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Check forUnderstanding
1. Capacitance has unitsof
a. farads
b. joules
c. coulombs pervolt
d. both a.and c.
Answer:d.
2. Toincreasethe capacitanceandtheenergy-storage capabilityofa parallel-plate
capacitor, we can
a.increasethe plateseparationdistance
b.increasethe platearea
c.evacuatethespace betweenthe plates
d.noneoftheabove
Answer: b,asthe capacitanceisdirectly proportionaltothe platearea,
C =IoA
d
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Check forUnderstanding
3. Capacitorsinseries havethesame
a. voltage
b. charge
c. energystorage
Answer: b
4. Capacitorsin parallel havethesamea.voltage
b. charge
c.energystorage
Answer:a
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Check forUnderstanding
5. Underwhat conditions wouldtwo capacitorsinseries havethesamevoltage?
Answer: Whentheyareequalin capacitance.
HW 16.B: p.540-541: 64-70,84-87,89,90.