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Introduction
Electric Potential Energy (PE)
• A conservative force potential energy (PE)
conservative force= - gradient of potential energy
conservativeF PE = −∇
• Gravity force near the Earth surface:gF mg=
• Gravity potential energy near the Earth surface :
gPE mgh=
h mg
A ball is exerted by the
gravity force and has
the potential energy ofmgh.
ground
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Introduction
Electric Potential Energy (Cont.)
• The Coulomb force is a conservative force.
• Work done by a conservative force is equal to thenegative of the change in potential energy.
work= - change of potential energy
ABW PE = −∆
• It is possible to define an electrical potential energyfunction with this force. The concept of electric potential energy is useful in the study of electricity
eF PE ⇔
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Work and Potential Energy (Cont.)
• The change of the electric potential energy is
independent of the path but the distance
ΔPE (path 1)= ΔPE (path 2)
Section 16.1
A B
Path 1
Path 2
• The Electric potential energy is a scalar quantity(magnitude), not a vector (magnitude and direction).
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Comparison between electric and gradational potential energy
0
0
AB e
e
e
W F d qEd
PE
PE
= =
= −∆ >
∆ <
0
0
AB g
g
g
W F d mgd
PE
PE
= =
= −∆ >
∆ <
Section 16.1
Electric potential energy differs significantly from gradationalpotential energy
—There are two kinds of electrical charges, positive and negative
— mass has no negative sign.
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Work and Potential Energy -Example
• There is a uniform field
between the two plates.
• As the charge moves from A to B, work is done by
electric force on it.
W AB = F x Δ x =q E x ( x f – x i)
=-ΔPE
Section 16.1
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Work and Potential Energy -Example (Cont.)
•
Note: E x and q can be positive ornegative
• The work-energy theorem―change in electric potential
energy (SI unit: Joule (J))
Section 16.1
0 0, 0,
0 0, 0,
0 0, 0,
0 0, 0,
x
x
AB x
x
x
if q E
if q E W qE x
if q E
if q E
> > >> <
ABPE W ∆ = −
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Potential Energy Differences in an Electric Field
Section 16.1
• A proton is released from rest. The
change in the electric potentialenergy associated with the proton:
ΔPE=-WAB=-F eΔy=-(eE Δy) < 0
Proton is released
E
A
B
∆y
+e
y
e
F eE
=
• An electron is fired in the samedirection. The change in the electricpotential energy associated with the
electron:ΔPE=-WAB=- (F eΔy)=-((-eE)Δy) > 0
An electron is fired in
the same direction
E
A
B
∆y
-e
y
eF eE = −
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• When an electron is released fromrest in a constant electric field, thechange in the electric potentialenergy associated with the electron
become more negative with time.
―True or false?
Section 16.1
ΔPE = - WAB = -((-e)(-E))∆
y < 0 (True)
An electron is released
E
A
B
∆y
-e
y
Potential Energy Differences in an Electric Field
eF eE = −
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• When an electron is fired in a uniformelectric field, find the initial speed v0 if its final speed has fallen by half.
• Apply conservation of energy,
Section 16.1
An electron is fired.
E
A
B
∆y
-e
y
Dynamics of Charged Particles
( ) ( )
2 2
2
2
2
1 10 0
2 2
1 1 10
2 2 2
8 ( ) 83 8
8 3 3 3
e f e i
e i e i
e i i
e e e
KE PE m v m v PE
m v m v PE
e E y eE yPE m v PE v
m m m
∆ + ∆ = → − + ∆ =
− + ∆ =
− − ∆ ∆∆
− = −∆ → = = =
eF eE = −
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Electric Potential and Potential Difference
• A field the field potential
―independent of the object in the field
• Gravitational field Gravitational potential
―independent of the object with mass m in the field
• Electric field Electric potential
―independent of the object with charge q in the field
Section 16.1
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Electric Potential Difference
• The electric potential difference ΔV between points A
and B is defined as the change in the potential energy(final value minus initial value) of a charge q moved from
A to B divided by the size of the charge.
ΔV = VB – V
A = ΔPE / q
SI unit: J/C or V (Volt)
Section 16.1
• Alternately, the electric potential difference is the work
per unit charge that would have to be done by some
force to move a charge from point A to point B in theelectric field.
• Potential difference is not the same as potential energy.
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Potential Difference (Cont.)
• Another way to relate the potential energy differenceand the potential difference:
ΔPE = q ΔV
• Both electric potential energy and potential
difference are scalar quantities.
• A special case occurs when there is a uniform electric field .
∆V =ΔPE/q=-F e
Δx/q= -qEx
Δx/q= -E x ∆x
– Gives more information about units: N/C = V/m
since F e/q=E x=ΔV/Δx
Section 16.1
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Potential Energy Compared to Potential
• Electric potential is characteristic of the field only.
– Independent of any test charge that may be
placed in the field
• Electric potential energy is characteristic of thecharge-field system.
– Due to an interaction between the field and the
charge placed in the field
Section 16.1
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Electric Potential and Charge Movements
• When released from rest, positive charges acceleratespontaneously from regions of high potential to low potential.
ΔV =-E Δx=VB-VA0Δx
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Electric Potential of a Point Charge
• The point of zero electric potential is taken tobe at an infinite distance from the charge.
• The potential created by a point charge q at
any distance r from the charge is
Section 16.2
e
qV k
r =
0,
B A
B B
A
V V V
V when r
V V
∆ = −
→ → ∞
= −∆
VA VB=0
Ehigh low
r →∞
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Electric Field and Electric Potential Depend on
Distance
• The electric field is a vector
and is proportional to 1/r2
• The electric potential is a
scalar and is proportional
to 1/r
Section 16.2
2
0
ˆe
F q E k r
q r
≡ =
e
qV k
r =
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Electric Potential of Multiple Point Charges
• Superposition principle applies
• The total electric potential at some point P due to
several point charges is the algebraic sum of the
electric potentials due to the individual charges.
– The algebraic sum is used because potentials arescalar quantities.
Section 16.2
1 2 31 2 3
1 2 3
1 2 3
; ; ; ...
...
e e eP P P
P P P
P P P P
k q k q k qV V V
r r r
V V V V
= = =
= + + +
• P
q1
q2
q3
r1P r2Pr3P
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Dipole Example
• Two charges have equalmagnitudes and oppositecharges.
• Example of superposition
• Discussion:
Section 16.2
1 2
1 2
2 11 2
1 2
;e eP P
P P
P PP P P e
P P
k q k qV V
r r
r r V V V k q
r r
= = −
−= + =
q1=q
q2=-q
Pr1Pr2P
Potential of a dipole plotted on thevertical axis (in arbitrary units).
1
2
1 2
1 2
0;
0;
0 ;
0
P P
P P
P P P
P P P
V if r
V if r
V if r r
V if r and r
→ ∞ →
→ −∞ →
→ =
→ → ∞
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• The work required to bring q2
from infinity to P withoutacceleration (i.e., F ex=F e) is
Electrical Potential Energy of Two Charges
• V1 is the electric potential due to
q1 at some point P
Section 16.2
2 2 1 2 1( )exW PE q V q V V q V ∞∆ = ∆ = ∆ = − =
1 2
2 1 ; 0
e
q qPE q V k PE
r ∞= = =
1
1 e
qV k
r =
• This work is equal to the potentialenergy of the two particle system
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Electric Potential Energy of Two Charges (Cont.)
• If the charges have the same sign, PE is positive.
– Positive work must be done to force the twocharges near one another.
– The like charges would repel.
• If the charges have opposite signs, PE is negative.
– The electric force would be attractive. – Work must be done to hold back the unlike
charges from accelerating as they are broughtclose together.
Section 16.2
1 21 2
1 2
0 00 0
e for q qq qPE k for q qr
> >= <
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Electric Potential and Potential Energy (Ch 16.1)
ΔPE = q ΔV WAB=-ΔPE
For q>0: ΔPE 0: ΔPE >0 (PE A >PE B)
(gains energy)
For q
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Section 16.2
Brief Summary Of Potential and Potential Energy
(cont.)
O
● q>0
• If a positive charge is released in
the electric field, it will move
towards to lower electric potential.
q
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Problem Solving with Electric Potential
(Point Charges)
• Draw a diagram of all charges.
– Note the point of interest.
• Calculate the distance from each charge to the pointof interest.
• Use the basic equation V = k eq/r
– Include the sign
– The potential is positive if the charge is positive
and negative if the charge is negative.
Section 16.2
• P
q1
q2
q3
r1P r2P
r3P
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Problem Solving with Electric Potential (Cont.)
• Use the superposition principle when you have
multiple charges.
– Take the algebraic sum
• Remember that potential is a scalar quantity.
– So no components to worry about
Section 16.2
1 2 3 ...P P P PV V V V = + + + • P
q1
q2
q3
r1P r2Pr3P
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Conductors in Equilibrium
• All of the charge resides at the
surface.• E = 0 inside the conductor.
• The electric field just outside theconductor is perpendicular to
the surface.
Section 16.3
0 0
inside einside
inside
W F lPE V
q q q
qE l lq
∆ ⋅ ∆∆∆ = = − = −
⋅ ∆= − = − ⋅ ∆ =
• V inside=constant 0 E =
l∆
A
•
•
B
• The potential everywhere inside the conductor isconstant and equal to its value at the surface (why?).
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Potentials and Charged Conductors
• All points on the surface of a chargedconductor in electrostatic equilibriumare at the same potential.
Section 16.3
cos90 0
0
surface e surface surface
surface
W F l qE l E l
PE W V
q q
∆ = ⋅ ∆ = ⋅ ∆ = ∆ =
∆∆ = = − =
V surface=constant l∆
•The electric potential is constant everywhere on thesurface of a charged conductor in electrostaticequilibrium and equal to its value at the inside.
V inside=V surface
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More Information About the Unit for Electric
Potential Energy: The Electron Volt (eV)
• The electron volt (eV) is defined as the kineticenergy that an electron gains when accelerated through a potential difference of 1 V.
— PE = eΔV =1 eV = 1.6 x 10-19 J
—Electrons in normal atoms have energies of 10’sof eV.
—Excited electrons have energies of 1000’
s of eV.—High energy gamma rays have energies of millionsof eV.
Section 16.3
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Equipotential Surfaces
•
An equipotential surface is a surface on whichall points are at the same potential.
Section 16.4
0 B AV V V ∆ = − =
—No work is required to move acharge at a constant speed on anequipotential surface.
—The electric field at every pointon an equipotential surface isperpendicular to the surface.
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Equipotential surface
E
Equipotentials and Electric Fields Lines –
Positive Charge
• The equipotentials for a
point charge are a family of
spheres centered on the
point charge.
– In blue
• The field lines are
perpendicular to theelectric potential at all
points.
– In orangeSection 16.4
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Equipotentials and Electric Fields Lines – Dipole
• Equipotential lines are
shown in blue.
• Electric field lines are
shown in orange.• The field lines are
perpendicular to the
equipotential lines at all
points.
Section 16.4
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d
Equipotentials in the Two Parallel Plates
Section 16.4
y
•
A
•
B
WAB=qE Δxcos90˚=0
Δx
ΔV=VA-VB=0
• Equipotential lines are shown in blue.
• Electric field lines are shown in orange.
• The field lines are perpendicular to the
equipotential lines at all points.
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Application – Electrostatic Precipitator
• It is used to remove
particulate matterfrom combustion gases
• Reduces air pollution
•Can eliminateapproximately 90% by
mass of the ash and
dust from smoke
• Recovers metal oxides
from the stack
Section 16.5
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Application – Electrostatic Precipitator (Cont.)
• The wire is maintained at a
negative electric potential withrespect to the outer wall, so the
electric field is directed toward
the wire.
• The electric field near the wire
reaches a high enough value to
cause a discharge around the wire
and the formation of positiveions, electrons, and negative ions.
Section 16.5
E
+q/-q
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Application – Electrostatic Precipitator (Cont.)
•These negative charge particlesaccelerate and the dirt particles
become charged negatively by
collision and ion capture.
• These negatively charged dirt
particles are drawn to the outer
wall.
•
When the duct is shaken, theparticles fall loose and are
collected at the bottom
E
+q/-q
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Application – Electrostatic Air Cleaner
• Used in homes to reduce the discomfort of allergy
sufferers.
• It uses many of the same principles as the
electrostatic precipitator.
Section 16.5
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Application – Xerographic Copiers
• The process of xerography is used for making
photocopies.
• Uses photoconductive materials
– A photoconductive material is a poor conductorof electricity in the dark but becomes a good
electric conductor when exposed to light.
Section 16.5
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h h ( )
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The Xerographic Process (Cont.)
•The negatively charged powder, called toner, is dusted
onto the photoconductor and adheres only to the
areas that contain the positively charged image (but
not neutralizes the positively charge image).
• It is then transferred to the surface of a sheet of
positively charged paper
• Finally, the toner is ‘fixed’ to the surface of the paper
by heat, resulting in a permanent copy of the original.
Section 16.5
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Application – Laser Printer
• The steps for producing a document on a laser printer
is similar to the steps in the xerographic process. – Steps a, c, and d are the same.
– The major difference is the way the image formson the selenium-coated drum.
• A rotating mirror inside the printer causes thebeam of the laser to sweep across the selenium-coated drum.
• The electrical signals form the desired letter in
positive charges on the selenium-coated drum.• Toner is applied and the process continues as in
the xerographic process.
Section 16.5
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Capacitance
Section 16.6
• Two parallel plates: ΔV=Ed=Qd/ ( A0ε0)
d
Q=σA0
• The capacitance, C, of a capacitor is defined as the
ratio of the magnitude of the charge on either
conductor (plate) to the magnitude of the potential
difference between the conductors (plates).
SI Units: Farad (F) – 1 F = 1 C / V
– A Farad is very large
– Often will see µF or pF
• Q/ ΔV=ε0 A0/d : characterize the
conductor and is independent of
Q and ΔV, called capacitance.
ΔV =Ed
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Parallel-Plate Capacitor
• The capacitor consists of twoparallel plates.
• Each has area A.
• They are separated by a distance d .
• The plates carry equal andopposite charges.
• When connected to the battery,
charge is pulled off from one plateand transferred to the other plate.
• The transfer stops when ∆Vcap =∆Vbattery
Section 16.7
• A capacitor is a device used in a variety of electric circuits.
-e
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Parallel-Plate Capacitor (Cont.)
• The capacitance of a device depends on the
geometric arrangement of the conductors.• For a parallel-plate capacitor whose plates are
separated by air:
Section 16.7
( ) 00/Q A A A
C V Ed d d
σ σ
ε σ ε = = = =∆
• The larger area can store more charge, therefore
enhances the capacitance.• The small plate separation can decrease the potential
difference and hence enhances the capacitance.
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Electric Field in a Realistic Parallel-Plate Capacitor
• The electric field between the plates is uniform.
– Near the center
– Nonuniform near the edges
•The field may be taken as constant throughout theregion between the plates.
Section 16.7
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Capacitors in Parallel
• When capacitors are first connected in the circuit,electrons are transferred from the left plates throughthe battery to the right plate, leaving the left platepositively charged and the right plate negativelycharged.
Section 16.8
-e -e
-e-e
+Q 1 -Q 1
+Q 2 -Q 2
• The capacitors reach their maximumcharge when the flow of chargeceases (completely charged).
• The flow of charges ceases whenthe voltage across the capacitorsequals that of the battery.
ΔV1
ΔV2
ΔV1=ΔV2 =ΔV
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Capacitors in Parallel (Cont.)
•
The potential differenceacross the capacitors is thesame:
– and each is equal to the
voltage of the batteryΔV1=ΔV2 =ΔV
Section 16.8
ΔV1=ΔV2
• The total charge, Q, isequal to the sum of thecharges on the capacitors.
Q = Q 1 + Q 2
ΔV1
ΔV2
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Capacitors in Parallel (Cont.)
•
The capacitors can bereplaced with one capacitor
with a capacitance of Ceq
Section 16.8
1 2 1 1 2 2Q Q Q C V C V = + = ∆ + ∆
1 2( ) eqQ C C V C V = + ∆ = ∆
1 2eqC C C = +
–
The equivalent capacitormust have exactly the
same external effect on
the circuit as the original
capacitors.
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Capacitors in Parallel (Cont.)
• Ceq = C1 + C2 + …
• The equivalent capacitance of a parallel
combination of capacitors is greater than any of
the individual capacitors.
Section 16.8
• Q = Q 1 + Q 2+…
• ΔV1=ΔV2 =…=ΔV
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Capacitors in Series
Section 16.8
• When a battery is connected to the circuit, electrons
are transferred from the left plate of C1 to the right plate of C2 through the battery.
-e-e
• The magnitude of the
charge must be the sameon the left plate of C1 and
the right plate of C2.
• As this negative charge accumulates on the right plate of C2, an equivalent amount of negative
charge is removed from the left plate of C1, leavingit with an excess positive charge.
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Capacitors in Series (Cont.)
Section 16.8
• All of the right plates gain charges of –Q and allthe left plates have charges of +Q.
• left plate of C2 are notconnected to the battery but
connected together and areelectrically neutral. Theircharge are induced by the left plate of C1 and the right plate
of C2.
-e
• How about the charge on the right plate of C1 and
the left plate of C2?
??
induced
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Capacitors in Series (Cont.)
• An equivalent capacitor can befound that performs the samefunction as the seriescombination.
Section 16.8
QΔV
1 2
1 2
1 21 2
1 2
;
1 1 1
eq
eq
Q QV V
C C
Q Q Q
V V V C C C
C C C
∆ = ∆ =
∆ = ∆ + ∆ = + =
= +
•
The potential differences add up to the battery voltage.
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Capacitors in Series (Cont.)
•
• The equivalent capacitance of a series combination is
always less than any individual capacitor in the
combination.
Section 16.8
• Q=Q1=Q2=…
P bl S l i St t
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Problem-Solving Strategy
• Combine capacitors following
the formulas. – When two or more capacitors
are connected in parallel , thepotential differences across
them are the same V1=V2. – The charge on each capacitor
is proportional to itscapacitance: Qi =C i ΔV .
– The capacitors add directly togive the equivalentcapacitance.
Section 16.8
Ceq = C1 + C2 + …
Q = Q 1 + Q 2
V1=V2
Parallel
Problem-Solving Strategy (Cont.)
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g gy ( )
• Be careful with the choice of units.
• Combine capacitors following theformulas.
– When two or more unequalcapacitors are connected in
series, they carry the sameQ 1=Q 2 charge, but the potentialdifferences across them are notthe same.
–
The capacitances add asreciprocals and the equivalentcapacitance is always less thanthe smallest individual capacitor.
Section 16.8
Q 1=Q 2
Series
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Problem-Solving Strategy (Cont.)
• A complicated circuit can often be reduced to oneequivalent capacitor.
—Replace capacitors in series or parallel withtheir equivalent.
—Redraw the circuit and continue.
Section 16.8
•To find the charge (Q) on, or the potential difference(ΔV ) across, one of the capacitors, start with yourfinal equivalent capacitor C eq and work back throughthe circuit reductions.
• Repeat the process until there is only one singleequivalent capacitor.
P bl S l i S E i S
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Problem-Solving Strategy, Equation Summary
• Use the following equations when working through
the circuit diagrams:
– Capacitance equation: C = Q / ∆V; Q=CΔV; ΔV=Q/C
– Capacitors in parallel: Ceq = C1 + C2 + …
–
Capacitors in parallel all have the same voltagedifferences as does the equivalent capacitance:
– Capacitors in series: 1/Ceq = 1/C1 + 1/C2 + …
– Capacitors in series all have the same charge, Q, asdoes their equivalent capacitance:
Section 16.8
V1=V2=V3=…
Q 1=Q 2=Q 3=…
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Energy Stored in a Capacitor
• A capacitor can store energy
Section 16.9
— How to calculate the stored energy?
― The battery transfers charges
from one plate to another when
they are connected to a battery.
-e
-e
-e
— The stored energy is the same as the work required to movecharge onto the plates done by
the battery.battery storeW E ∆ =
E St d i C it (C t )
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Energy Stored in a Capacitor (Cont.)
Section 16.9
;i i i iW PE V Q∆ = ∆ = ∆ ∆
• The work ∆W i required to transfer
charge ∆Qi from one plate to theother must be done against thepotential ∆Vi of the capacitor:
i
ii
QV
C
∆∆ =
ΔVi1
2iiW W Q V ∆ = ∆ = ∆∑
• The total work done by the battery :
∆V
∆Vi - ∆Qi ∆Qi
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Energy Stored in a Capacitor (Cont.)
• Energy stored = ΔW=½ QΔV
Section 16.9
• The stored energy is the same as the work required
to move charge onto the plates.
Q
ΔV QC
V
=
∆ Q=CΔV
ΔV=Q/C
• From the definition of capacitance, this can be
rewritten in different forms.
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Applying Physics (Maximum Energy Design)
•
How should three capacitors and two batteries beconnected so that the capacitors will store the
maximum possible energy?
Section 16.9
• To have maximum capacitance, the three capacitors
must be parallel. Why?
( )221 1
2 2
eq eq Max Max Max E C V E C V = ∆ → = ∆
1 2 3 Max iC C C C C = = + +∑• To have maximum the potential difference, the two
batteries must be series. Why?
1 2 Max iV V V V ∆ = ∆ = ∆ + ∆∑
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Energy Stored in a Capacitor (Cont.)
Section 16.9
•
Maximum energy stored/maximum operatingvoltage—before discharging the capacitor
• In general, capacitors act as energy reservoirs that
can be slowly charged and then discharged quickly toprovide large amounts of energy in a short pulse.
+Q
-Q
( )2
max max
1
2 E C V = ∆
max
max
QQC
V V = =∆ ∆
Discharge
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Capacitors with Dielectrics
Section 16.10
0 0
AC
d ε =
0 0
AC C
d κε κ = =
κ ≥ 1
2
0
1
2
E C V = ∆
2
0 0 012
E C V = ∆0
Capacitors with Dielectrics (Cont )
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Capacitors with Dielectrics (Cont.)
• A dielectric is an insulating material that, when placed
between the plates of a capacitor, increases thecapacitance.
Section 16.10
– Dielectrics include rubber, plastic, orwaxed paper.
• C = κ C o = κ εo( A/d ) ≥C 0 where κ ≥ 1
– The capacitance is multiplied by the
factor κ when the dielectric completelyfills the region between the plates.
– κ is called the dielectric constant .
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Dielectric Strength
• For any given plate separation, there is a
maximum electric field E max that can be
produced in the dielectric before it breaks
down and begins to conduct.
• This maximum electric field is called the
dielectric strength.
Section 16.10
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Section 16.10
Capacitors with Dielectrics (Examples)
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Capacitors with Dielectrics (Examples)
Section 16.10
C1
C2
A/2
A/2
Two capacitors in series:
1 2
1 1 1
eqC C C = +
1 1
1 1 0 0 0
/ 2;
2 2
A AC C
d d
κ κ
κ ε ε = = =
2 22 2 0 0 0
/ 2
2 2
A AC C
d d
κ κ
κ ε ε = = =
1 2eqC C C = +
Two capacitors in parallel:
1 1 0 1 0 1 02 2 ;
/ 2
A AC C
d d κ ε κ ε κ = = =
2 2 0 2 0 2 02 2 ;
/ 2
A AC C
d d
κ ε κ ε κ = = =
d/2 d/2
C1 C2
0 0
AC
d ε =
An Atomic Description of Dielectrics
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An Atomic Description of Dielectrics
• Polarization occurs when there is a separation
between the average positions of its negative chargeand its positive charge.
Section 16.10
• In a capacitor, the dielectric becomes polarized because it is in an electric field that exists between
the plates.
More Atomic Description
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Section 16.10
More Atomic Description
• The field generated by the two plates produces an induced polarization in the dielectric material.
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More Atomic Description (Cont.)
Section 16.10
•
The electric field from the polarizeddielectric will partially cancel the
electric field from the charge on the
capacitor plates.
0net ind E E E = +
0 0
0
;net
E E σ σ
ε ε
ε ε
= < = >
This decreases the net field inside the
capacitor, and decreases the potential
difference across the capacitor.
0net net V E d V ∆ = < ∆
More Atomic Description (cont )
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More Atomic Description (cont.)
Section 16.10
• To return the capacitor to its original
potential difference, more charge isneeded.
Q (with dielectric)>Q (vacuum)
• The net effect of the dielectric is to
increase the amount of charge stored on
a capacitor, and then increase the
capacitance.0
0
QQ
C C V V = > =∆ ∆
Q
0net V V ∆ < ∆
0
0
C C C
C κ κ = ⇒ =
net V V ∆ = ∆
Application – Computers
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Application – Computers
• Computers use capacitors in
many ways.
– Some keyboards usecapacitors at the bases of thekeys.
– When the key is pressed, thecapacitor spacing decreases and the capacitanceincreases.
– The key is recognized by thechange in capacitance.
Section 16.10
Commercial Capacitor Designs
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Commercial Capacitor Designs