Download Physics 1302.300 Spring 2017 Class 18—February 27, 2017

Physics1302.300Spring2017
Class18—February27,2017
MainTopics:
•Gauss’LawwithDielectrics
•Chapter26Summary
•Chapter31Circuits
Exam2Thursday/Friday
Gauss’LawforDielectrics
• ConsiderusingGauss’slaw
tocalculatetheelectricfield
insideadielectricmaterial
insertedintoacapacitor.
• Thefigureshowsthe
relevantgeometry.
• Thefluxthroughthe
Gaussiansurface(acylinder
inthiscase)iszeroexcept
throughtherightflat
surface:
 
∫ E ⋅ dA = EA
Physics1302.300Spring2017
 
qfree, enc − qbound, enc
∫ E ⋅ dA = EA =
∈0
2
Gauss’LawforDielectrics
• Usingtherelationshipbetweenthefreeandboundcharge
yields
  qfree, enc
∫ E ⋅ dA = ∈0κ
• Thisimportantconclusionrelatestheelectricfieldinsidethe
dielectrictothefreechargeonthecapacitorandthe
dielectricconstant.
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Chapter 26: Summary
Concepts:Capacitors
• A capacitor consists of a pair of conducting objects separated
by a nonconducting material or vacuum. The objects store
electric potential energy once charge has been transferred from
one to the other.
• A charge-separating device (such as a battery) has some
mechanism that moves charge carriers against an electric field.
The work done in this process increases the system’s electric
potential energy.
©2015PearsonEducation,Inc.
Chapter 26: Summary
Concepts:Capacitors
• A parallel-plate capacitor consists of two parallel conducting
plates of surface area A separated by a gap of width d. The
electric field is uniform between the plates.
• A coaxial capacitor consists of two coaxial conducting
cylinders of radii R1 and R2 > R1 and length ℓ >> R2.
• A spherical capacitor consists of two concentric conducting
spherical shells of radii R1 and R2 > R1.
©2015PearsonEducation,Inc.
Chapter 26: Summary
QuantitativeTools:Capacitors
• The conducting objects in a capacitor carry charges of equal
magnitude q but opposite sign. The charge separation produces
a potential difference of magnitude Vcap between the objects.
The capacitance C of this arrangement is
q
C≡
.
Vcap
• Capacitance is measured in farads F, where
1 F ≡ 1 C/V.
©2015PearsonEducation,Inc.
Chapter 26: Summary
QuantitativeTools:Capacitors
• The capacitance of a parallel-plate capacitor is
∈0 A
C=
.
d
• The capacitance of a coaxial capacitor is
2π∈0 
C=
.
ln( R2 /R1 )
• The capacitance of a spherical capacitor is
R1 R2
C = 4π∈0
.
R2 − R1
©2015PearsonEducation,Inc.
Chapter 26: Summary
Concepts:Electricfieldenergyandemf
• The energy density of an electric field is the energy per unit
volume stored in the field.
• The emf of any charge-separating device is the work per unit
charge done by nonelectrostatic interactions in separating
positive and negative charge carriers.
©2015PearsonEducation,Inc.
Chapter 26: Summary
QuantitativeTools:Electricfieldenergyandemf
• The electric potential energy UE stored in a capacitor is
2
q
2
U E = 12 = 12 CVcap
= 12 qVcap .
C
• In air or vacuum, the energy density uE of an electric field is
uE = 12 ∈0 E 2 .
• The emf
of a charge-separating device is
Wnonelectrostatic
≡
.
q
©2015PearsonEducation,Inc.
Chapter 26: Summary
Concepts:Dielectrics
• A dielectric is a polarizable nonconducting material. A polar
dielectric is made up of molecules that have a permanent
dipole moment, whereas a nonpolar dielectric consists of
molecules that do not have a dipole moment in the absence of
an electric field.
• A dielectric inserted between the plates of a capacitor becomes
polarized by the electric field of the capacitor. This
polarization gives the two faces of the dielectric thin layers of
charge of equal magnitude but opposite sign. This charge is
bound because the charge carriers are not free to move. The
charge on the capacitor plates is free because the charge
carriers are free to move.
©2015PearsonEducation,Inc.
Chapter 26: Summary
QuantitativeTools:Dielectrics
• The dielectric constant κ of a material between capacitor
plates is
V0
κ≡ ,
Vd
where V0 is the potential difference across the capacitor
without the dielectric and Vd is the potential difference with the
dielectric in place.
©2015PearsonEducation,Inc.
Chapter 26: Summary
QuantitativeTools:Dielectrics
• If a capacitor has capacitance C0 without a dielectric, its
capacitance with a dielectric is
Cd = κC0.
• If qfree is the free charge on a capacitor, then the bound charge
qbound is
κ –1
qbound =
qfree .
κ
• Gauss’s law in matter is
  qfree, enc
∫ κ E ⋅ dA = ∈0 ,
where qfree, enc is the free charge enclosed by the Gaussian surface.
©2015PearsonEducation,Inc.
Chapter31:ElectricCircuits
• Electriccircuitsconsistofcircuitelementsconnectedby
conductors.
• Inthischapter,wewillfocusontwotypesofcircuits:
– DCcircuitsinwhichelectricpotentialsaretime-invariant.Circuit
elementsinthesecircuitsareemf sources(batteries)andresistors.
– RCcircuitsaretime-dependent.Circuitelementsareemf sources,
resistorsandcapacitors.
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Chapter 31 Preview
Looking Ahead: The basic circuit
• Circuits that maintain a constant potential difference are
known as direct-current circuits or DC circuits.
• Circuits are designed to control the flow of charge carriers—
the current—in a device. The current then provides energy to
the device, allowing it to perform its function.
©2015PearsonEducation,Inc.
Chapter 31 Preview
Looking Ahead: The basic circuit
• An electric circuit is an
interconnection of electrical
components (circuit
elements).
• A loop is any closed
conducting path through a
circuit.
• A power source is any device
that provides electric potential
energy to an electric circuit.
©2015PearsonEducation,Inc.
Chapter 31 Preview
Looking Ahead: The basic circuit
• The potential difference across the
terminals of the power source
drives charge carriers through the
circuit and thereby creates a current
in the circuit.
• The load in an electric circuit is all
the circuit elements connected to
the power source. In the load,
electric potential energy is
converted to other forms of energy.
©2015PearsonEducation,Inc.
Chapter 31 Preview
Looking Ahead: Current and resistance
• A circuit is in steady state when
the current has a constant value at
all points in the circuit. The
current continuity principle
states that in steady state, the
current is the same at all locations
along a single-loop circuit.
©2015PearsonEducation,Inc.
Chapter 31 Preview
Looking Ahead: Current and resistance
• The resistance of a circuit element is a measure of the
potential difference across that element for a given current in
it. The SI unit of resistance is the ohm Ω, where 1 Ω = 1 V/A.
The conductivity σ of a material is a measure of its ability to
conduct charge carriers through the material and is measured
in SI units of A/(V × m).
• When there is a current in a conductor, there must be an
electric field in the conductor to cause the current. In a
conductor of uniform cross section at steady state, this electric
field has the same magnitude everywhere and is parallel to the
walls of the conductor.
©2015PearsonEducation,Inc.
Chapter 31 Preview
Looking Ahead: Circuit loops
• A junction (node) is a location in a circuit where more than
two wires are connected.
• A branch in a circuit is a conducting path between two
junctions that is not intercepted by another junction.
• According to the branch rule, the current in a branch of a
multiloop circuit is the same throughout the branch.
©2015PearsonEducation,Inc.
Chapter 31 Preview
Looking Ahead: Circuit loops
• Two or more circuit elements are connected in series if there is only a
single current path through them and the charge carriers flow through one
element after the other.
• The potential difference across circuit elements connected in series is
equal to the sum of the individual potential differences across each circuit
element. Two or more circuit elements are connected in parallel if their
ends are connected to the same two junctions. The potential differences
across circuit elements connected in parallel are always equal.
©2015PearsonEducation,Inc.
ElectricCurrent(Chapter31)
• Movingchargescompriseanelectriccurrent.
• Themediumthroughwhichchargesmoveiscalleda
conductor.
• Conductorscanbesolids,liquids,gasesorevenvacuum.
Movingchargescanbeeitherpositiveornegative.
• Conductionbymetalsisprimarilyelectronmotionthrougha
fixedpositivelychargedlattice.
• Thematerialsmostcommonlyusedasconductorsaremetals,
particularlycopperand,insomecases,aluminum.During
WorldWarII,themagnetsusedtoseparate235Uforthe
HiroshimabombusedsomeofthesilverthebackedtheU.S.
dollarasconductors.
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Calutrons atOakRidge
ErnestLawrence
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FirstWorkingNuclearReactor
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FirstWorkingNuclearReactor
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ElectronConductioninMetals
• Electronconductioninmetalsisparameterizedbytwo
velocities:
– Theelectronshavearandomvelocity,whichisprimarilydetermined
bytheirtemperature.Highertemperaturesresultinlargerrandom
velocities.
– Theelectronsalsohaveadriftvelocity,whichissomeorganized
motionrelatedtoanelectricfield.
– [Metaphor:Imagineacrowdedpartywithdancingonthesideofa
hill.Therewillbealotoftoandfromotion,butthecrowdwillalso
graduallydriftdownthehillside.]
• Theunitsformeasuringcurrentareamperes.Oneampereis
oneCoulombpersecondpassingaparticularpoint.
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MoreonCurrent
• Thereareseveralquantitiesrelatedtocurrent.
– Current:chargeperunittimepassingapointonaconductor
(i, measuredinamperes)
– CurrentDensity:currentperunitofconductorcross-sectionalarea
(J, measuredinamperespersquaremeter)
– DriftVelocity:organizedmotionofchargecarriers(vd,measuredin
meterspersecond)
– ChargeperChargeCarrier:(q,measuredinCoulombs)
– NumberDensityofChargeCarriers:numberofchargecarriersperunit
volume(n,measuredinm-3)
– Cross-SectionalAreaofConductor(A,measuredinm2)
• Thekeyrelationshipsamongthesequantities:
J = nqvd
i = |J|A
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ContinuityofCurrent
• Thequantityofcurrentinaconductorisconstant.
• Iftheconductornarrowsatsomepoint,thecurrentdensity
mustincrease.Sincen andq areconstant,thedriftvelocity
mustalsoincrease.
• Ausefulmetaphorisawaterpipe.Thecross-sectionalareaof
awaterpipeiscommonlydecreasedatanozzleinorderto
increasethevelocityofthewaterflowand“shoot”thewater
further.
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AmericanWireGauge
• IntheUnitedStates,thediameterandcross-sectionalareaof
electricalwiresisusuallyspecifiedinaframeworkcalledAWG
orAmericanWireGauge.InAWG,thehigherthegauge,the
finerthewire.No.36gaugewireis0.005inchesindiameter,
No. 0000gaugewireis0.46inchesindiameter.
• Householdelectricalcircuitsareusuallyratedateither15or
20amperes.15amperecircuitsgenerallyuse14gaugewire
(cross-sectionalarea2.08 mm2).20amperecircuitsgenerally
use12gaugewire(area3.31 mm2).
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