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Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley
PowerPoint® Lectures for
University Physics, Twelfth Edition
– Hugh D. Young and Roger A. Freedman
Lectures by James Pazun
Chapter 24
Capacitance andDielectrics
Modified by P. Lam 7_15_2008
Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley
Topics for Chapter 24
• I. Capacitors and capacitance
• II. Energy stored in a charged capacitor
• III. Capacitors with dielectrics
Intermission
• IV. Dielectric breakdown
• V. Capacitors in series an parallel circuits
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I. Capacitor and capacitance
• If we pull a positive charge and
negative charge apart, it requires
energy; this energy is stored as
electrostatic potential energy in this
charge arrangement. If the charges
are allowed to move, they will move
toward each other and gain kinetic
energy (one can use this kinetic
energy to push something to do work
or to light up a light bulb)
• A capacitor is device which can
hold these separated charges.
Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley
How do we build a capacitor?• The basic components of a capacitor are two conductors
separated by a distance.
• The “parallel-plate capacitor is the easiest to analyze.
• Connect the plates to the two terminals of a battery (see figure
below). Electrons will move from the top plate to the battery
and then enter the bottom plate; leaving a net positive charge on
the top plate and net negative charge on the bottom plate. The
electrons will stop moving when the voltage difference between
the two plates (Vab) equal to the voltage of the battery ( ).
•Disconnect the battery and the capacitor holds the charge.
+
-
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Capacitance
+
-
How much charge (Q) a capacitor can hold for a given voltage
difference (Vab) between the plates depends on the size and shape of
the capacitor and it is called the capacitance (C).
Definition : C =Q
Vab
For parallel - plate capacitor with area (A) and separation (d),
C = o
Ad
(if d << A)
Derivation : d << A E =o
=Q
oA; Vab = Ed =
Qd
oAC =
Q
Vab
= o
A
d
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Unit of capacitance
Definition : C =Q
Vab
unit =Coulomb
voltFarad
Example : The separation of a parallel plate capacitor is 1 mm,
what is the area of the plate in order for C =1 Farad?
Answer : C = o
A
d; o = 8.85x10 12Farad /m
A =Cd
o
=(1Farad)(10 3m)
8.85x10 12Farad /m1x108m2 !!!
Common capacitances are pF =10-12Farad or μF =10-6Farad.
Note : Even if C =1μF, the area is still 100m2!!
(1)Large area can be accommodated by rolling the two plates into
a cylinder.
(2) Capacitance can also be increased by inserting an insulating
material(called dielectrics) between the plates (will be discussed later).
Electrolytic dielectrics capacitor can have a capacitance of 1 Farad and yet
of very small size.
Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley
Cylindrical capacitance - coaxial cable
Suppose we connect this capacitor to a battery;
when it is fully charged, the charge is Q.
If ra, rb << L, then we can use the infinite line cylinder
approximation.
Using Gauss's Law E =Q
2 rL o
for ra < r < rb
Vab =| Edra
b
|=Q
2 L o
ln rb
ra
CQ
Vab
= o
2 L
ln(rb /ra )
Note : Let d rb ra ,
ln(rb /ra ) = ln(1+d
ra)
d
ra if d << ra
C = o
2 L
ln(rb /ra ) o
2 raL
d= o
A
d (same formula as for parallel - plate)
A cylindrical capacitor is formed by two concentric
cylinders (eg. a coaxial cable); the inner conductor
is typically a solid cylinder, the outer conductor is a
thin cylindrical shell.
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Commercial capacitors
• Commercial capacitors forhome electronics have sizesrange from a grain of rice tothat of a large cigar.
• Capacitors like thosementioned above andpictured at right aremicrofarad capacitors.
• Note: Even though theexterior shape is a cylinder,it doesn’t mean that it iscylindrical capacitor; itcould be a rolled-upparallel-plate capacitor.
Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley
Note on capacitance
Altough the definition C involves Q and Vab : C =Q
Vab
The capacitance depends on the size and geometry of the
capacitor,
Example :
C = o
A
d for parallel - plate capacitor
C = o
2 L
ln(rb /ra ) for cylindrical capacitor
If we plot the charge (Q) vs the voltage across the capacitor (Vab),
then C is the slope of the linear graph.Q
Vab
slope=C
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Example
Example 24.2. The dimensions of a parallel-plate capacitor are
d=5mm and A=2 m2. A charge (Q) of 35.4μC is each plate
(a) What is the capacitance (C) ?
(b) What is the voltage (Vab) across the capacitor?
(c) What is the magnitude of the electric field between the plates?
(d) Now pull the plates apart to d=10mm, what are the values for
C, Q, Vab, and E? Do you have to apply energy to pull the plates
apart?
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II. Energy stored in a charged capacitor
Q
Vab
slope=CThe energy stored in a charged capacitor is
U =1
2QVab =
1
2
Q2
C=
1
2CVab
2
Refer to the example in the previous slide, find the energy stored in
the capacitor when d=5mm and the energy stored after you pull it
apart to d=10mm.
How much energy did you have to apply to pull the plates apart?
If the capacitor is connected to a battery (10 kV) while you are
pulling the plates apart, what is the stored energy after you have
pulled the plates to d=10mm?
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Capacitor with dielectrics
• The potential differencebetween the parallel plates ofa capacitor decreases when adielectric material is insertedbetween the plates becausethe dielectric weakens theelectric field (explanation innext slide).
Since CQ
V,
decreasing V while Q remains the same
increaseing C.
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Dielectric partially shields the electric field
E =Eo
K; K = dielectric constant of the material C =K o
A
d
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Table 24.1—Dielectric constants
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Intermission
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IV. Dielectric breakdown
• A very strong electrical field can exceed the strength of thedielectric to contain it. Table 24.2 at the bottom of the page listssome limits.
air ~1 3x106
•For a given capacitor separation(d), the dielectric strength sets alimit on the maximum voltage(max V=Emd) that can be sustainedby the capacitor.
Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley
V. Capacitors in series and parallel circuits
Series :
Q1 =Q2
V1 +V2 =V
Parallel :
Q1 +Q2 =Q
V1 =V2 =V
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Calculations regarding capacitance
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Energy stored in a charge capacitor re-visit.
The energy stored in a charged capacitor is
U =12Q2
C
Consider a parallel - plate capacitor, C = o
A
d
U =12Q2
C=
12Q2d
oA=
12
Q2
o2A2
d * A* o
E =o
=Q
oAU =
12 oE
2
*Volume
1
2 oE2
= energy density (energy/volume)
That is, one can think of energy being stored as separated
positive and negative charges OR being stored in the E - field.
The second interpretation is useful when we discuss
energy carried by electromagnetic waves.