Chapter 24 Capacitance, dielectrics and electric energy storage

1
Ch 26.4 Energy Stored in a Capacitor charging
a capacitor
Switch goes shut. ? Battery establishes E-field
in wires, charge builds up on the
capacitor. Chemical energy in battery goes down
and electrical potential energy stored in
capacitor goes up. Electrical potential energy
results from charge separation across capacitors
plates
2
Ch 26.4 Energy Stored in a Capacitor charging
a capacitor
To charge the capacitor, an external agent
(battery) must do work to separate the
charges. Step by step, the battery grabs a
small amount of charge dq off one capacitor plate
and move it to the other. At first, this requires
no work, because the uncharged capacitor has no
electric field to resist the movement of
charge. But, once dq has been transferred, the
capacitor starts to develop a potential
difference. Now, to move more charge from one
plate to the other, the battery must do some
amount of work dW, to overcome the rising
potential difference across the plates. As more
and more charge is transferred, the work required
to transfer the same amount of charge, dq, goes
up.
3
Ch 26.4 Energy Stored in a Capacitor charging
a capacitor
Suppose q is the amount of charge on the
capacitor at some instant during the charging
process. At the same instant, the potential
difference across the capacitor is ?V q/C The
amount of work necessary to transfer dq amount of
charge across this potential difference is dW
?Vdq (q/C)dq
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Ch 26.4 Energy Stored in a Capacitor
Thus, the total amount of work required to charge
the capacitor from q 0 to a final charge of q
Q is
But, in an isolated system with no
non-conservative forces, total mechanical energy
must be conserved. Therefore, the work done to
charge the capacitor must equal the change in the
systems potential energy.
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Ch 26.4 Energy Stored in a Capacitor
Energy stored in a charged capacitor
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Ch 26.4 Energy Stored in a Capacitor
-Its not obvious, but the potential energy
stored in the capacitor actually resides in its
electric field. -This implies we should be able
to solve the density of the energy stored in the
field (J/m3). -For a parallel plate capacitor, we
already know -and, its capacitance is
just -Substituting these into the purple
equation,
-Dividing by the volume in between the plates of
the capacitor (VAd), we get
Energy per unit volume in a capacitor (J/m3)
7
Ch 26.4 Energy Stored in a Capacitor
-We dont attempt it here, but it can be shown
that this result is valid for any electric field!
Energy per unit volume in an electric field.
-In a very real sense, electric fields carry
energy.
8
EG 26.4 Rewiring two Charged Capacitors

• Two capacitors, C1 and C2 (C1 gt C2), are charged
  to the same initial potential difference, ?Vi.
  The charged capacitors are removed from the
  battery, and their plates are connected with
  opposite polarity, as shown. The switches, S1
  and S2, are then closed.
• Find the final potential difference ?Vf between a
  and b after the switches are closed.
• Find the total energy stored in the capacitors
  before and after the switches are closed and
  determine the ratio of the final energy to the
  initial energy.

9
Ch 26.5 Capacitors with dielectrics
A dielectric is something you stick in between
the plates of a capacitor to change (increase)
its capacitance. The term comes from the fact
that, at the atomic level, such materials can be
polarized into arrays of dipoles.
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Ch 26.5 Capacitors with dielectrics
Common dielectric materials are wax, paper, oil,
polymers, fluid (electrolyte), etc. Dielectrics
are insulators. Since you stick the dielectric
material into the region of the capacitors
E-field, it changes how good the capacitor is
at establishing the E-field ? ?V? for the same
amount of Q on the capacitor plate.
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What happens?
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Ch 26.5 Capacitors with dielectrics
Consider parallel-plate capacitor where ?V0
Q0/C0 Assume no battery is connected ? Q cant
change When you stick a dielectric in between the
plates ?
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Ch 26.5 Capacitors with dielectrics
Consider parallel-plate capacitor where ?V0
Q0/C0 Assume no battery is connected ? Q cant
change When you stick a dielectric in between the
plates ?
-where ? is a dimensionless constant called the
dielectric constant
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Ch 26.5 Capacitors with dielectrics
-Q on the capacitor does not change -Therefore
-the capacitance is changed by a factor of
?. -as ? goes up, C goes up.
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Ch 26.5 Capacitors with dielectrics
-For a parallel plate capacitor

• To make capacitance ?
• -decrease d
• -increase A
• -increase ?

- Only limited by dielectric strength of the
dielectric
16
Example values of dielectric constant
Dielectric strength is the maximum field in the
dielectric before breakdown. (a spark or flow of
charge)
17
EG 26.5 Energy stored before and after
A parallel-plate capacitor is charged with a
battery to a charge of Q0. The battery is then
removed, and a slab of material that has a
dielectric constant ? is inserted between the
plates. Identify the system as the capacitor and
the dielectric. Find the energy stored in the
system before and after the dielectric is
inserted.
18
EG 26.4 Rewiring two Charged Capacitors
A parallel-plate capacitor is charged with a
battery to a charge of Q0. The battery is then
removed, and a slab of material that has a
dielectric constant ? is inserted between the
plates. Identify the system as the capacitor and
the dielectric. Find the energy stored in the
system before and after the dielectric is
inserted.
Before
After
Where did the energy go?
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Ch 26.6 Electric Dipole in an Electric Field
The combination of two equal charges of opposite
sign, q and q, separated by a distance
2a Every dipole can be characterized by its
dipole moment. - vector which points from q
to q -magnitude p 2aq
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Ch 26.6 Electric Dipole in an Electric Field
What happens when we pop this baby in an external
E-field?
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Ch 26.6 Electric Dipole in an Electric Field
What happens when we pop this baby in an external
E-field?
-external field exerts FqE on each charge -net
torque about the dipoles center -dipole rotates
to align with the field
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Ch 26.6 Electric Dipole in an Electric Field
What happens when we pop this baby in an external
E-field?
-external field exerts FqE on each charge -net
torque about the dipoles center -dipole rotates
to align with the field
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Ch 26.6 Electric Dipole in an Electric Field
but,
and
Thus
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Ch 26.6 Electric Dipole in an Electric Field
but,
and
Thus
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Ch 26.6 Electric Dipole in an Electric Field

• The dipole and the external field are a system
• -electric force is an internal conservative force
• we can describe its work using a potential energy
• In other words, different configurations of the
  dipole-field system have different potential
  energies.

26
Ch 26.6 Electric Dipole in an Electric Field
As the dipole aligns with the field, the systems
potential energy goes down.
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Ch 26.6 Electric Dipole in an Electric Field
-Work must be done to un-align the dipole from
the field. -in an isolated system, the work
input must correspond to an increase in potential
energy.
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Ch 26.6 Electric Dipole in an Electric Field
-Work must be done to un-align the dipole from
the field. -in an isolated system, the work
input must correspond to an increase in potential
energy. W ?K ?U
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Ch 26.6 Electric Dipole in an Electric Field
-To rotate the dipole through some small angle
d?, an amount dW of work must be done.
but,
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Ch 26.6 Electric Dipole in an Electric Field
-To rotate the dipole through some small angle
d?, an amount dW of work must be done.
but,
-so, to rotate the dipole from ?i to ?f, the
change in potential energy is
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Ch 26.6 Electric Dipole in an Electric Field
Lets define the zero potential energy as being
when the dipole is at ? 90,
when
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Ch 26.6 Electric Dipole in an Electric Field
Lets define the zero potential energy as being
when the dipole is at ? 90,
when
Well use this reference energy as an anchor
point. At any time, we can write the systems
instantaneous potential energy, U, with respect
to the zero-point potential energy.
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Ch 26.6 Electric Dipole in an Electric Field
Lets define the zero potential energy as being
when the dipole is at ? 90,
when
Well use this reference energy as an anchor
point. At any time, we can write the systems
instantaneous potential energy, U, with respect
to the zero-point potential energy.
But, we already know
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Ch 26.6 Electric Dipole in an Electric Field
Lets define the zero potential energy as being
when the dipole is at ? 90,
when
Well use this reference energy as an anchor
point. At any time, we can write the systems
instantaneous potential energy, U, with respect
to the zero-point potential energy.
But, we already know
35
EG 26.6 The Water Molecule

• A water molecule has an electric dipole moment of
  6.3x10-30 Cm. A sample contains 1021 water
  molecules. All of the dipoles are oriented in
  the direction of an external E-field, which has a
  magnitude of 2.5x105 N/C.
• How much work is required to rotate all the
  dipoles from this orientation (? 0) to one in
  all the dipoles are perpendicular to the external
  field (? 90)?

36
EG 26.6 The Water Molecule

• A water molecule has an electric dipole moment of
  6.3x10-30 Cm. A sample contains 1021 water
  molecules. All of the dipoles are oriented in
  the direction of an external E-field, which has a
  magnitude of 2.5x105 N/C.
• How much work is required to rotate all the
  dipoles from this orientation (? 0) to one in
  all the dipoles are perpendicular to the external
  field (? 90)?

37
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?
38
Example P26.21

• Four capacitors are connected as shown in Figure
  P26.21.
• Find the equivalent capacitance between points a
  and b.
• Calculate the charge on each capacitor if ?Vab
  15.0 V.

39
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.
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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?
. Therefore, the
,
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Example P26.43
Determine (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.
42
Example P26.59
A 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.