Chapter 26
Capacitance and Dielectrics
(Cont.)
Dr. Jie Zou
PHY 1361
1
Outline
„
Combination of capacitors
„
„
„
„
„
Parallel combination
Series combination
Equivalence capacitance
Energy stored in a charged capacitor
Capacitors with dielectrics
„
„
„
Dielectric constant
Dielectric strength
Dielectric breakdown
Dr. Jie Zou
PHY 1361
2
Parallel combination of
capacitors
„
Properties:
The individual potential
difference across capacitors
connected in parallel are the
same and are equal to the
potential difference applied
across the combination.
(2) The total charge on capacitors
connected in parallel is the sum
of the charges on the individual
capacitors.
(3) Ceq = C1 + C2 + C3 + …
(1)
Ceq of a parallel combination
is greater than any of the
individual capacitances.
Dr. Jie Zou
PHY 1361
3
Series combination of
capacitors
„
Properties
The charges on capacitors
connected in series are the
same.
(2) The total potential difference
across any number of
capacitors connected in
series is the sum of the
potential differences across
the individual capacitors.
1
1
1
1
=
+
+
+ ...
(3)
Ceq C1 C2 C3
(1)
Ceq of a series combination is
always less than any individual
capacitance in the combination.
Dr. Jie Zou
PHY 1361
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Example 26.4 Equivalent
capacitance
„
Find the equivalent capacitance between a
and b for the combination of capacitors
shown in Figure (a). All capacitances are in
microfarads.
Dr. Jie Zou
PHY 1361
5
Energy stored in a charged
capacitor
„
The total work required to charge the
capacitor from q = 0 to some final
2
Q q
Q
1
Q
charge q = Q is W =
dq =
qdq =
∫
0
„
„
Dr. Jie Zou
C ∫0
2C
The work done in charging the capacitor
appears as the electric potential
energy U stored in the capacitor.
„
„
C
U = Q2/2C = (1/2)QΔV = (1/2)C(ΔV)2.
We can consider the energy stored in a
capacitor as being stored in the electric field
created between the plates as the capacitor is
charged.
Energy density = energy per unit
volume = uE = (1/2)ε0E2; uE ∝ E2.
PHY 1361
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Example 26.5 Rewiring two
charged capacitors
„
(a)
Two capacitors C1 and C2 (where C1>C2)
are charged to the same initial potential
difference ΔVi. The charged capacitors
are removed from the battery, and their
plates are connected with opposite
polarity as shown in Figure (a). The
switches S1 and S2 are then closed, as
shown in Figure (b).
„
„
(b)
Dr. Jie Zou
(A) Find the final potential difference ΔVf
between a and b after the switches are
closed.
(B) Find the total energy stored in the
capacitors before and after the switches are
closed.
PHY 1361
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Real world applications
Defibrillator
Dr. Jie Zou
PHY 1361
Camera flash unit
8
Capacitors with dielectrics
„
„
Dielectrics: A dielectric is
a nonconducting
material.
If a dielectric completely
fills the space between
the plates, the
capacitance increases by
a dimensionless factor κ,
the dielectric constant
of the material.
„
Dr. Jie Zou
C = κC0.
PHY 1361
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Dielectric strength and
dielectric breakdown
Dielectric strength: maximum
electric field.
„ If the magnitude of the
electric field in the dielectric
exceeds the dielectric
strength, then the insulating
properties beak down and the
dielectric begins to conduct.
„ Table 26.2: Dielectric
constants and dielectric
Dielectric breakdown in air strength of various materials
„
Dr. Jie Zou
PHY 1361
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Example 26.6 A paper-filled
capacitor
„
A parallel-plate capacitor has plates of
dimensions 2.0 cm by 3.0 cm separated
by a 1.0-mm thickness of paper (The
dielectric constant and dielectric
strength for paper are 3.7 and 16 x 106
V/m).
„
„
(A) Find its capacitance.
(B) What is the maximum charge that can
be placed on the capacitor?
Dr. Jie Zou
PHY 1361
11