Download - Chapter S26
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Chapter 26Capacitance, dielectrics and
electric energy storage • Basic circuit devices
– Resistors
– Capacitors
– Inductors
– Power supply (Battery, Generator)
• Use our knowledge of electric fields, potentials, and energy to describe how capacitors work.
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Parallel Plate Capacitor
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Fig 26-4b, p.800
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Active Figure 26.4
(SLIDESHOW MODE ONLY)
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Potential difference and electric fields in a uniform electric field
+Q
-Qa
b
d
d d
bao o o0 0
ˆ ˆ ˆV j dxi dyj dy d
enclosedE
o
QE dA
o
AE A
o
E
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Potential difference and electric fields in a uniform electric field
+Q
-Qa
b
d
bao o
QdV d
A
baQ V
baQ CV
The constant of proportionality is called “capacitance.”For a parallel plate capacitor, the capacitance is:
0AC
d
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Factors affecting capacitance
• Size of the capacitor (A, d)• Geometric arrangement
– Plates– Cylinders
• Material between conductors– Air– Paper– Wax
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Units of capacitance
Coulomb CFarad F
Volt V
ba
QC
V
A Farad is a lot of capacitance. Typical capacitors are“micro, nano, pico-Farad
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Capacitance – Isolated Sphere
• Assume a spherical charged conductor
• Assume V = 0 at infinity
• Note, this is independent of the charge and the potential difference
4/ o
e e
Q Q RC πε R
V k Q R k
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Cylindrical capacitor
b a
enclosedE
o
QE dA
+Q on center conducting cylinder
-Q on outer conducting cylinder
o
QE 2 rL
o
QE
2 rL
L
b b
bao o oa a
Q Q dr Q bˆ ˆV r drr ln
2 rL 2 L r 2 L a
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Capacitance of a cylindrical capacitor
abQ CV
b a
L
o
ab
o
2 LQ QC
Q b bV ln ln2 L a a
L
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Example
• How strong is the electric field between the plates of a 0.80 F air gap capacitor if they are 2.0 mm apart and each has a charge of 72 C?
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Capacitors in Parallel
total 1 2 3Q Q Q Q
a b
abV V
1Q
3Q
2Qtotal eqQ C V
1C
2C
3C
1 1Q C V 2 2Q C V 3 3Q C V
eq 1 2 3C V C V C V C V
eq 1 2 3C C C C
1 2 3V V V V
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Capacitors in Series
1 2 3V V V V
a b
abV V
1Q 3Q2Q
eqeq
QQ C V V
C 1C 3C2C
11
QV
C
eq 1 2 3
Q Q Q Q
C C C C
eq 1 2 3
1 1 1 1
C C C C
22
QV
C 3
3
QV
C
total 1 2 3Q Q Q Q
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Capacitor circuit example
2 F
2 F3 F 4 F
What single capacitor can replace the four shown here?How much charge can the system hold?How much charge is on one of the 2 F capacitors?
3 V
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Energy Storage in Capacitors
bb
U dUV V
q dq
dU Vdq
(Like problem 23-50)
q Q
q 0
U Vdq
bao
QdV
A
bao
Q aV ln
2 L b
Parallel Plates
Concentric Cylinders
Qq Q 2 2
q 0 0
q 1 q QU dq
C C 2 2C
QV
CIn General
Change in potential energywhile charging capacitor
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Alternate Energy Expressions
2QU
2C Q
VC
2 22C V 1
U CV2C 2
21 Q 1U V QV
2 V 2
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Energy Density
Energy per unit volume: Uu
Volume
Consider a Parallel Plate Capacitor: 21U CV
2
0AC
d
V Ed
2 2 200
A1 1U E d Ad E
2 d 2
20
U U 1u E
Volume Ad 2
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Dielectrics•A dielectric is a nonconducting material that, when placed between the plates of a capacitor, increases the capacitance
•Materials with Dipoles that can align with an external electric Field. Dielectrics include rubber, plastic, and waxed paper
+ ++
--
-
+ ++
---
oE
indE
oDielectric o ind
EE E E
K
K is the Dielectric Constant
Measure of the degree of dipole alignment in the material
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Dielectrics
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Example values of dielectric constant
“Dielectric strength” is themaximum field in the dielectric before breakdown.(a spark or flow of charge)
max maxE V / d
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Effect of a dielectric on capacitance
oDielectric
EE
K o
Dielectric
EE d d
K
oDielectric
VV
K
Potential difference with a dielectric is less than the potential difference across free space
oo
Q QC K KC
V V Results in a higher capacitance.
Allows more charge to be stored before breakdown voltage.
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Effect of the dielectric constant
Parallel Plate Capacitoro o
o o
A K AC C KC
d d
oK
AC
d
Material permittivity measures degree to which the material permits induced dipolesto align with an external field
2 2 2o 0 0
1 1 1u E u K E E
2 2 2
Example modificationsusing permittivity
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d
3
What is new capacitance?
Example – Parallel Plate Capacitor
+Q
-Q
d
A
oAC
d
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DipolesThe combination of two equal charges of opposite sign, +Q and –Q, separated by a distance l
p Q2a p
2a
-Q +Q
1p
2p
1 2p p p
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Dipoles in a Uniform Electric Field
p
2a
-Q
+Q F QE
F QE
E
F a sin F a sin QEa sin QEa sin Q2aEsin pEsin
x y z
ˆ ˆ ˆi j k
r F x y z
F F F
r F sin
p E
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Work Rotating a Dipole in an Uniform Electric Field
p
2a
-Q
+Q F QE
F QE
E
dW d pEsin d dU
oLet: U( =90 ) 0 U pE cos
U p E
0U pE sin d pE cos U
0U 0
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Example P26.9
When a potential difference of 150 V is applied to the plates of a parallel-plate capacitor, the plates carry a surface charge density of 30.0 nC/cm2. What is the spacing between the plates?
0 AQ Vd
12 2 2
09 2 4 2 2
8.85 10 C N m 150 V4.42 m
30.0 10 C cm 1.00 10 cm m
Vd
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Example P26.21
Four capacitors are connected as shown in Figure P26.21.
(a) Find the equivalent capacitance between points a and b.
(b) Calculate the charge on each capacitor if ΔVab = 15.0 V.
1
2.50 F
2.50 6.00 8.50 F
1 15.96 F
8.50 F 20.0 F
s
p
eq
C
C
C
1 1 115.0 3.00sC
5.96 F 15.0 V 89.5 CQ C V
89.5 C4.47 V
20.0 F15.0 4.47 10.53 V
6.00 F 10.53 V 63.2 C on 6.00 F
QV
C
Q C V
89.5 63.2 26.3 C
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Example P26.27
Find the equivalent capacitance between points a and b for the group of capacitors connected as shown in Figure P26.27. Take C1 = 5.00 μF, C2 = 10.0 μF, and C3 = 2.00 μF.
1
1
2
1
1 13.33 F
5.00 10.0
2 3.33 2.00 8.66 F
2 10.0 20.0 F
1 16.04 F
8.66 20.0
s
p
p
eq
C
C
C
C
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Example P26.35
A parallel-plate capacitor is charged and then disconnected from a battery. By what fraction does the stored energy change (increase or decrease) when the plate separation is doubled?
2 12d d 2 112
C C stored energy doubles,
. Therefore, the
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Example P26.43Determine (a) the capacitance and (b) the maximum potential difference that can be applied to a Teflon-filled parallel-plate capacitor having a plate area of 1.75 cm2 and plate separation of 0.040 0 mm.
12 4 2110
5
2.10 8.85 10 F m 1.75 10 m8.13 10 F 81.3 pF
4.00 10 m
AC
d
6 5max max 60.0 10 V m 4.00 10 m 2.40 kVV E d
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Example P26.59A parallel-plate capacitor is constructed using a dielectric material whose dielectric constant is 3.00 and whose dielectric strength is 2.00 × 108 V/m. The desired capacitance is 0.250 μF, and the capacitor must withstand a maximum potential difference of 4 000 V. Find the minimum area of the capacitor plates.
8 maxmax 2.00 10 V m
VE
d
60 0.250 10 FA
Cd
62max
12 80 0 max
0.250 10 40000.188 m
3.00 8.85 10 2.00 10
C VCdA
E
3.00