Fundamentals of Physics
Chapter 26 Capacitance
1.
2.
3.
4.
5.
6.
7.
8.
The Uses of Capacitors
Capacitance
Calculating the Capacitance
Capacitors in Parallel & Series
Energy Stored in an Electric Field
Capacitor with a Dielectric
Dielectrics: An Atomic View
Dielectrics & Gauss Law
Review & Summary
Chapter Questions
Exercises & Problems
2006
Physics 2112
Fundamentals of Physics
Chapter 25
1
Capacitor
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Physics 2112
Fundamentals of Physics
Chapter 25
2
Capacitors
A Capacitor accumulates, stores or supplies charge in response to a potential
difference being placed across it.
A parallel-plate capacitor:
Equal and opposite charges
on its plates.
The plates are conductors; hence
each plate is an equipotential
surface.
The Capacitor stores electric potential energy in the electric field between its plates.
2006
Physics 2112
Fundamentals of Physics
Chapter 25
3
Capacitance
A parallel-plate capacitor:
The charge on the plates is proportional to the
potential difference between the plates:
q = CV
C = capacitance
C only depends on the geometry of the plates,
such as their area and separation.
SI Unit: the farad
1F = 1C/V
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Physics 2112
Fundamentals of Physics
Chapter 25
4
Charging a Capacitor
Initially there is no charge on the
capacitor.
The power supply or battery does work
to move charges onto the plates of the
capacitor until the potential difference
across the capacitor reaches V.
Potential energy is being stored up in
the electric field between the plates.
q
CV
Battery
Switch
Capacitor
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Physics 2112
Fundamentals of Physics
Chapter 25
5
Calculating the Capacitance
Use Gauss Law to calculate the charge on the capacitor:
q
0
0
E dA
qenc
Calculate the potential difference across the capacitor:
V
Vf
Vi
E ds
Calculate the capacitance :
C
2006
Physics 2112
q
V
Fundamentals of Physics
Chapter 25
6
Calculating the Capacitance for Parallel Plates
qenc
0
q
0
C
C
2006
V
EA
V
q
V
E ds
Ed
EA
Ed
0
0
d
Physics 2112
A
0
= 8.85 x 10-12 F/m = 8.85 pF/m)
Fundamentals of Physics
Chapter 25
7
Calculating the Capacitance for Parallel Plates
Consider 2 Parallel Charged Planes:
E
2
2
0
0
0
+++++++++++++++++++++++++++++
E
2
2
0
A >> d
0
0
----------------------------E
2006
2
0
Physics 2112
2
0
0
Fundamentals of Physics
Chapter 25
8
Example: Variable Capacitor in an old radio
C
Area
Turning the knob changes the capacitance
Increasing or decreasing overlapping area of plates
Capacitance determines the resonant frequency of the radio
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Physics 2112
Fundamentals of Physics
Chapter 25
9
Example: Capacitance Switching in a Computer Keyboard
q
CV
Depressing the key changes the capacitance
Charge changes and electric current flows
Electronic Detection of Current
Signal to Computer
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Physics 2112
Fundamentals of Physics
Chapter 25
10
Cylindrical Capacitor
qenc
0
qenc
0
qenc
0
E
2006
V
E ds
E dA
V
E r dr
E 2 rL
V
E dA
q
2
q
2
0
Lr
V
0
q
2
0
L
Lr
ln
dr
q
2
b
0
L
a
dr
r
b
a
Physics 2112
Fundamentals of Physics
C
q
V
2
0
L
b
ln
a
Chapter 25
11
Capacitance of Concentric Spherical Shells
C
q
V
qenc
0
qenc
0
qenc
E
0
E dA
V
E ds
V
E r dr
E dA
E 4 r2
q
4
0
r
2
q
V
V
C
4
0
q
4
4
0
r
dr
2
1
a
1
b
q
b
4
0
q b a
4 0a b
ab
0
b
a
An Isolated Sphere; let b get very large: C
2006
Physics 2112
a
dr
r2
Fundamentals of Physics
4
0
a
Chapter 25
12
Capacitors in Parallel
Common potential difference on each capacitor
V
Va
Vb
Qi
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Physics 2112
Ci V
Fundamentals of Physics
Chapter 25
13
Capacitors in Parallel
Each capacitor in parallel has the same potential difference across it. The total
charge stored on the capacitors is the sum of the charges on all of the capacitors.
q
q1
q
C1 V
q
C1
q2
q3
C2 V
C2
C3 V
C3 V
Capacitors in parallel can be replaced with an equivalent capacitor that
has the same total charge q and the same potential difference V.
Ceq
q
V
C1
C2
C3
n
Ceq
Ci
i 1
2006
Physics 2112
Fundamentals of Physics
Chapter 25
14
Capacitors in Series
Common amount of charge on each capacitor:
isolated from the circuit
Each plate has the same charge
2006
Physics 2112
Fundamentals of Physics
Chapter 25
15
Capacitors in Series
For capacitors in series, each capacitor holds the same
charge. The sum of the potential differences across all the
capacitors is equal to the applied potential difference V.
q
Vi
Ci
V
V
q
V1
V2
V3
1
C1
1
C2
1
C3
Capacitors in series can be replaced with an equivalent
capacitor that has the same total charge q and the same
potential difference V.
1
Ceq
1
Ceq
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Physics 2112
1
C1
n
i 1
1
C2
1
C3
1
Ci
Fundamentals of Physics
Chapter 25
16
Example
Series Capacitors
V
18V
Q
Ci Vi
12 Vi
V4
1
Ceq
1
C2
1
C4
1
Ceq
1
2 F
1
4 F
Q
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V4
V2
Ceq
2006
V2
6 Vi
3
4 F
1.33 F
Ceq V
Fundamentals of Physics
24 C
Chapter 25
17
Example
Series and Parallel Capacitors
n
in parallel
in series
Ceq
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Physics 2112
Ceq
Cn
2
4
6
i 1
1
Ceq
n
i 1
1
Ci
1
6
1
3
1
2
2 F
Fundamentals of Physics
Chapter 25
18
Example
C1 = 3.55 mF
V0 = 6.30 V
Battery removed and C2
attached
C2 = 8.95 mf
Switch is closed, find V
2006
Physics 2112
Fundamentals of Physics
Chapter 25
19
Energy Stored in Electric Field
The work required to move an element of charge through a potential difference V is:
dW
W
V dq
q
dq
C
V dq
1
2
q2
C
This work is stored as electrostatic potential energy in the capacitor:
U
2006
W
1
2
Physics 2112
q2
C
1
2
CV 2
Fundamentals of Physics
Chapter 25
20
Energy Stored in an Electric Field
This work done in charging a capacitor is stored as
electrostatic potential energy in the capacitor:
U
W
C
0
1
2
q2
C
1
2
CV 2
A
d
This energy is stored in the electric field between the plates
of the capacitor. The energy density, u, is the potential
energy per unit volume of electric field:
u = ½
0
E2
Where ever an electric field exists, u is the electric potential energy per unit volume at
that point.
2006
Physics 2112
Fundamentals of Physics
Chapter 25
21
Example 4
Isolated Sphere has R = 6.85
q = 1.25 nC
a) What is potential energy stored in sphere
b) What is the energy density at the surface of the
sphere?
2006
Physics 2112
Fundamentals of Physics
Chapter 25
22
Capacitor with a Dielectric
If an insulator (aka a dielectric) is used in a capacitor, the capacitance is
increased by a factor k, the dielectric constant.
V
Cair
V
C =
Cair
In a region filled with a material with dielectric constant ,
all electrostatic equations containing 0 are modified by
replacing 0 by 0.
2006
Physics 2112
Fundamentals of Physics
Chapter 25
23
Example 5
A parallel-plate capacitor whose capacitance is 13.5 pF is charged by a
battery to a potential difference V=12.5 V between its plates. The
charging battery is now disconnected and a porcelain slab ( = 6.50) is
slipped between the plates. What is the potential energy of the capacitorslab both before and after the slab is put into place.
2006
Physics 2112
Fundamentals of Physics
Chapter 25
24
Molecular View of a Dielectric
Dipoles, permanent or induced, tend to align with an electric field, producing an
internal electric field in the opposite direction.
2006
Physics 2112
Fundamentals of Physics
Chapter 25
25
Surface Charge of a Dielectric
Electrically neutral
atoms in a dielectric
material.
The atoms acquire an
induced dipole moment in
an electric field, and align
with the field direction.
The aligned atoms induce
charges on the surfaces of
the dielectric, resulting in
an electric field in the
opposite direction of the
original field.
The surface charge induced on the surface of the dielectric weakens the
original electric field between the plates.
2006
Physics 2112
Fundamentals of Physics
Chapter 25
26
Surface Charge of a Dielectric
The surface charge induced on the surface of the dielectric weakens the original electric
field between the plates.
E
polystyrene:
dry air:
2006
Dielectric Constant = 2.6
Dielectric Constant = 1.0005
Physics 2112
E0
Dielectric Strength = 24 kV/mm
Dielectric Strength = 3 kV/mm
Fundamentals of Physics
Chapter 25
27
Dielectrics and Gauss Law
0
0
E dA
qenc
E0
q
0
qenc
(qenc includes only the charge on the plates)
E
E
E dA
E
A
q
q
0 A
In a region filled with a material with dielectric constant k, all electrostatic equations
containing e0 are modified by replacing 0 by k 0.
2006
Physics 2112
Fundamentals of Physics
Chapter 25
28
Example 6
A= 115 cm2 d=1.24 m V0=85.5 V
b = 0.780 cm k = 2.61 (inserted after
battery disconnected)
a) Capacitance C0 before slab is
inserted
b) Free charge appearing on plates
c) E0 in gap between plate and slab
d) E1 in slab
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Physics 2112
Fundamentals of Physics
Chapter 25
29
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