Physics 212 Lecture 7, Slide 1

1
Physics 212 Lecture 7
Today's Concept Conductors and Capacitors How
are charges distributed on conductors? What is a
capacitor and how can we calculate C?
Extra 1
Preflight 1
Preflight 13
Extra 2
Preflight 16
Preflight 4
Music
Extra 3
Preflight 10
Calculation
Preflight 7
Example
2
Things you identified as difficulties

• Adding conductor between the plates of a
  capacitor
• We have prepared exercises to understand this
  example

2) Capacitors and Calculations What is a
Capacitor? Calculation of Capacitance ? We will
calculate the capacitance of a cylindrical
capacitor
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Preflight 1
Most confident
Least confident
4
Preflight 4
Most confident
Least confident
5
Preflight 7
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Least confident
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Example Problem
Two parallel plates of area A separated by a
distance d carry equal an opposite charge Q0. An
uncharged conducting plate having thickness t is
slipped midway between the plates. How does the
voltage between the plates change?
Q0
d
-Q0
d
t
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As in Pre-Lecture 7
First figure out DV without conductor
Q0
Integrate from the bottom plate to the top plate
DV Ed
d
-Q0
so
Also, since
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Demo
-Q0
-Q0
Q0
Q0
V0
V1

• How does V1 compare to V0
• V1 gt V0
• V1 V0
• V1 lt V0

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After the conductor is inserted, the charge on
the plates is Q1. Compare this to Q0.
Q0

• Q1 lt Q0
• Q1 Q0
• Q1 gt Q0

d
-Q0
Q1
We haven't touched the plates so Q cant change.
d
t
-Q1
10
Q0
d
t
-Q0
What is the total charge induced on the bottom
surface of the conductor?

• Q0
• -Q0
• 0
• Its ve but the magnitude is not known
• Its ve but the magnitude is not known

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Q0
E
E 0
E
-Q0
12
Now figure out DV with conductor
Q0
Again, integrate from the bottom plate to the top
plate DV E(d-t)
t
d
-Q0
so
So to make DV the same as before you have to make
Q bigger
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Preflight 13
Two parallel plates carry equal and opposite
charge Q0. The potential difference between the
two plates is measured to be V0. An uncharged
conducting plate (green) is slipped into the
space between the plates without touching either
one. The charge on the plates is adjusted to a
new value Q1 such that the potential difference
between the two plates remains the same as
before.
Most confident
Least confident
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Now figure out new capacitance
Q0
t
d
DV E(d-t)
-Q0
We just showed
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Preflight 16
Two parallel plates carry equal and opposite
charge Q0. The potential difference between the
two plates is measured to be V0. An uncharged
conducting plate (green) is slipped into the
space between the plates without touching either
one. The charge on the plates is adjusted to a
new value Q1 such that the potential difference
between the two plates remains the same as
before.
Most confident
Least confident
16
Now figure out V out as a function of distance
from the bottom conductor. Choose V0 to be at
the top conductor
Q0
E
r
r
E 0
V
r
-Q0
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Preflight 10
Suppose the electric field is zero in a certain
region of space. Which of the following
statements best describes the electric potential
in this region?
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Calculation
cross-section
a4
a3

• A capacitor is constructed from two conducting
  cylindrical shells of radii a1, a2, a3, and a4
  and length L (L gtgt ai).
• What is the capacitance C of this capacitor ?

a2
a1
metal
Q
metal

• Conceptual Analysis

But what is Q and what is V.. They are not given??

• Important Point C is a property of the
  object!! (concentric cylinders here)
• Assume some Q (i.e., Q on one conductor and Q
  on the other)
• These charges create E field in region between
  conductors
• This E field determines a potential difference V
  between the conductors
• V should be proportional to Q the ratio Q/V is
  the capacitance.

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Calculation
cross-section
a4
a3

• A capacitor is constructed from two conducting
  cylindrical shells of radii a1, a2, a3, and a4
  and length L (L gtgt ai).
• What is the capacitance C of this capacitor ?

a2
a1
metal
Q
metal

• Strategic Analysis
• Put Q on outer shell and Q on inner shell
• Cylindrical symmetry Use Gauss Law to calculate
  E everywhere
• Integrate E to get V
• Take ratio Q/V should get expression only using
  geometric parameters (ai,L)

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Calculation
cross-section
a4
a3

• A capacitor is constructed from two conducting
  cylindrical shells of radii a1, a2, a3, and a4
  and length L (L gtgt ai).
• What is the capacitance C of this capacitor ?

Q
a2
a1
-Q
metal
metal
Why?
21
Calculation
cross-section
a4
a3

• A capacitor is constructed from two conducting
  cylindrical shells of radii a1, a2, a3, and a4
  and length L (L gtgt ai).
• What is the capacitance C of this capacitor ?

Q
a2
a1
-Q
r gt a4 E(r) 0
metal
metal
Where is Q on outer conductor located? (A) at
ra4 (B) at ra3 (C) both surfaces
(D) throughout shell
Why?
We know that E 0 in conductor (between a3 and
a4)
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Calculation
cross-section
a4
a3

• A capacitor is constructed from two conducting
  cylindrical shells of radii a1, a2, a3, and a4
  and length L (L gtgt ai).
• What is the capacitance C of this capacitor ?

Q
a2
a1
-Q
-
r gt a4 E(r) 0
metal
Where is -Q on inner conductor located? (A) at
ra2 (B) at ra1 (C) both surfaces
(D) throughout shell
Why?
We know that E 0 in conductor (between a1 and
a2)
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Calculation
cross-section
a4

• A capacitor is constructed from two conducting
  cylindrical shells of radii a1, a2, a3, and a4
  and length L (L gtgt ai).
• What is the capacitance C of this capacitor ?

a3
Q
a2
a1
-Q
-
r gt a4 E(r) 0
metal
Why?
24
Calculation
cross-section
a4

• A capacitor is constructed from two conducting
  cylindrical shells of radii a1, a2, a3, and a4
  and length L (L gtgt ai).
• What is the capacitance C of this capacitor ?

a3
Q
a2
a1
-Q
-
r gt a4 E(r) 0
a2 lt r lt a3
metal
r lt a2 E(r) 0 since Qenclosed 0

• What is V?
• The potential difference between the conductors

What is the sign of Vouter - Vinner? (A)
Vouter-Vinner lt 0 (B) Vouter-Vinner 0
(C) Vouter-Vinner gt 0
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Calculation
cross-section
a4

• A capacitor is constructed from two conducting
  cylindrical shells of radii a1, a2, a3, and a4
  and length L (L gtgt ai).
• What is the capacitance C of this capacitor ?

a3
Q
a2
a1
-Q
-
r gt a4 E(r) 0
r lt a2 E(r) 0
a2 lt r lt a3
metal
V proportional to Q, as promised
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Music

• Who is the Artist?
• Professor Longhair
• Johnny Adams
• Henry Butler
• Dr. John
• Allen Toussaint

Its FAT TUESDAY ! All New Orleans greats!!
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Extra 1
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Extra 2
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Extra 3