Download lecture 16 capacitors

LECTURE 16
CAPACITORS
Instructor: Kazumi Tolich
Lecture 16
2
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Reading chapter 20.5 & 20.6
¤ Capacitance
¤ Parallel-plate
capacitor
¤ The storage of electrical energy
¤ Dielectrics
Capacitors
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A capacitor is a device consisting of two
conductors: one conductor with charge 𝑄, and
the other with charge −𝑄.
The symbol for a capacitor in a circuit
diagram is
The capacitance of the capacitor is defined as
𝑄
𝐶=
𝑉
Quiz: 1
4
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A capacitor of capacitance 𝐶 holds a charge 𝑄 when the potential
difference across the plates is 𝑉 . If the charge on the plates is
doubled to 2𝑄 , what is the capacitance of the capacitor now?
Quiz: 16-1 answer
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𝐶
The capacitance is a function of geometry only. The geometry did not change, so the
capacitance did not change.
You might think that capacitors depend on charge and potential difference by looking at the
'
definition of capacitance: 𝐶 = . However, the charge a capacitor store is proportional to the
(
potential difference. And that proportionality constant is the capacitance, 𝑉 = 𝐶𝑄.
The slope 𝐶 is independent of the value of 𝑄.
Parallel plate capacitors/Demo: 1
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The electric field between the plates is uniform and given by
𝐸=
¨
𝑄
𝜖+ 𝐴
The potential difference between the plates is given by
∆𝑉 = 𝐸𝑑 =
¨
The capacitance of a parallel-plate capacitor is
𝐶=
¨
𝑄𝑑
𝜖+ 𝐴
𝜖+ 𝐴
𝑑
Demo: variable capacitors (tuning capacitor used in AM radios)
Quiz: 2
7
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Suppose a parallel plate capacitor is fully charged by a battery and
then disconnected from the battery. The positive plate has a charge
+ 𝑞 and the negative plate has a charge −𝑞. The plate separation is
doubled from its initial separation. How did the potential between the
plates change?
A.
B.
C.
Remained the same.
Doubled
Halved
Quiz: 16-2 answer
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Doubled
∆𝑉 = 𝐸𝑑
Since the capacitor is isolated, the
amount of charge does not
change.
So, 𝐸 is constant.
Dielectrics
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If we place a non-conducting dielectric material between the
plates of a capacitor, molecules within can be polarized.
The capacitance with a dielectric installed is increased and
given by
𝐶 = 𝜅𝐶+
where 𝐶+ is the capacitance without any dielectric, and 𝜅 is the
dielectric constant.
¤ The dielectric constant is a property of the material.
¤ 𝜅 of air is very close to 1 (close to vacuum).
Quiz: 3
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A charge 𝑄 is placed on a plate of an isolated parallel-plate
capacitor, and −𝑄 is placed on the other plate. The voltage across
the plates is 𝑉. Now a dielectric material with a dielectric constant
𝜅 > 1 is inserted between the plates without touching them. Has the
potential difference between the plates increased, decreased, or
stayed the same?
A.
B.
C.
Increased
Decreased
Stayed the same
Quiz: 16-3 answer/Demo: 2
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Decreased.
Since the capacitor is isolated, the amount of charge on the capacitor
does not change.
The capacitance increases to 𝐶 = 𝜅𝐶+ .
¨
𝐶 = , so the voltage must decrease.
¨
𝑉=
¨
¨
'
(
(4
5
Quiz: 4
12
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A parallel-plate capacitor is connected to an ideal battery. A charge
𝑄 is on a plate, and −𝑄 is on the other plate. Now a dielectric
material with a dielectric constant 𝜅 > 1 is inserted between the
plates without touching them. Has the amount of charge stored on the
plate with a positive charge increased, decreased, or stayed the
same?
A.
B.
C.
Increased
Decreased
Stayed the same
Quiz: 16-4 answer
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increased.
Since the capacitor is connected to a battery, the potential difference
across the capacitor remains constant, the voltage of the battery.
The capacitance increases to 𝐶 = 𝜅𝐶+ .
¨
𝐶 = , so the charge must increase.
¨
¨
'
(
Example: 1
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A parallel-plate capacitor is
constructed with circular plates of
radius r = 0.056 m. The plates are
separated by d = 0.25 mm, and the
space between the plates is filled with
a dielectric with dielectric constant κ.
When the charge on the capacitor is
Q = 1.2 µC, the potential difference
between the plates is V = 750 V. Find
the value of κ.
Dielectric breakdown/Demo: 3
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Dielectric breakdown occurs when a material is ionized in very high electric fields
and becomes a conductor.
The magnitude of the E field for which dielectric breakdown occurs in a material is
called the dielectric strength of that material.
The dielectric strength of air is ~3×106 V/m.
¤
¤
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In air, the existing ions are accelerated in the electric field, collides with air molecules, ionizing them.
The dielectric discharge occurs more often during a dry day because moisture in the air can conduct the
charge away before breakdown occurs.
Demo: Jacob’s ladder
¤
15,000 volts applied to two long vertical electrodes forms a spark at the bottom.
Electrostatic field energy/Demo: 4
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The energy stored in a capacitor is the total amount of energy required to
charge the plates,
1
1 𝑄7 1 7
𝑈 = 𝑄𝑉 =
= 𝐶𝑉
2
2 𝐶
2
¨
The density of this energy stored in the electric filed is
𝑢9 =
¨
electric energy 1
= 𝜖+ 𝐸 7
volume
2
Demo: capacitor energy
¤
Three capacitors with a total capacitance of 4.5 mF are charged to 400 volts.
¤
𝑈 = 𝐶𝑉 7 =
H
H
7
7
4.5 mF 400 V
7
= 360 J
Quiz: 5
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Consider a parallel-plate capacitor whose plates are given equal
and opposite charges and are separated by a distance 𝑑. Suppose
the plates are then pulled apart until they are separated by a
distance 2𝑑. What is the ratio of the final to initial electrical energies,
stored in the capacitor, 𝑈R ⁄𝑈S ?
Quiz: 16-5 answer
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2
The charge on the plates does not
change, so the electric field and
energy density do not change:
𝑢9 =
¨
UVUWXYSW UZUY[\
]^V_`U
=
H
𝜖+ 𝐸 7
7
The volume the electric field
occupies doubles so the stored
energy doubles.
Applications of capacitors
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Camera flash
Defibrillator
Capacitor microphone
¤ A diaphragm acts as one plate of a parallelplate capacitor.
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Stud finder
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¤
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A stud finder has a capacitor with its plates
arranged side by side.
When it is moved over a stud, the capacitance
increases since the stud acts as a dielectric material.