Welcome Back

Exam returned Wed or Friday.





Problem 1 – We did it in class
Problem 2 - A Web Assign Problem
Problem 3 - Superposition
Quiz on Friday (Capacitors)
New WebAssign Posted … Wait until
Wednesday to try.
February 14, 2005
Capacitors
1
Chapter 25

Capacitors
Battery
February 14, 2005
Capacitors
2
Remember distributed
Charges
February 14, 2005
Capacitors
3
Infinite Metal Plates
+
+
+
+
+
+
+
+
-
February 14, 2005
Capacitors
4
Addemup
E=0
February 14, 2005
-

E
0
Capacitors
+
+
+
+
+
+
+
+
E=0
5
Not quite infinite …
February 14, 2005
Capacitors
6
Capacitor


Composed of two metal plates.
Each plate is charged




one positive
one negative
Stores Charge
Can store a LOT of charge and can be
dangerous!
February 14, 2005
Capacitors
7
A Simple Electric Circuit
February 14, 2005
Capacitors
8
Don’t ask questions because I
don’t know the answers!
Zn Metal
Cu Metal
Aqueous
Solution
of
February 14, 2005
Capacitors
9
What’s Next?

Zn(solid) Zn2+ +2e


Electrons Hang Around
Zn ion goes into the solution.
Cu2+(solution) +2e- Cu depositing on
Cu electrode
February 14, 2005
Capacitors
10
February 14, 2005
Capacitors
11
February 14, 2005
Capacitors
12
February 14, 2005
Capacitors
13
Gauss on Capacitors
Gauss
d
q
 E  dA  
Air or Vacuum
E
-Q
+Q
Area A
V=Potential Difference
0
 EA  0 A   EA  
Gaussian
Surface
Q
0
Q   0 EA
Q
(Q / A) 
E


0 A
0
0
February 14, 2005
Same result from other plate!
Capacitors
14
Two Charged Plates
(Neglect Fringing Fields)
d
Air or Vacuum
E
-Q
+Q
Symbol
Area A
V=Potential Difference
February 14, 2005
Capacitors
15
Note
d
Consider a +q charge
at the (-) plate.
Air or Vacuum
+
E
-Q
+Q
Work to do this is
W=Fd=qEd
also
W=q(Vf-Vi)=qV
Area A
V=Potential Difference
February 14, 2005
Move it to the (+)
plate
Therefore
Ed=V
E=V/d
Capacitors
16
Device



The Potential Difference
is APPLIED by a battery
or a circuit.
The charge q on the
capacitor is found to be
proportional to the
applied voltage.
The proportionality
constant is C and is
referred to as the
CAPACITANCE of the
device.
February 14, 2005
Capacitors
q
C
V
or
q  CV
DEFINITION
17
UNITS
Q coulomb
C 
 Farad
V
volt
February 14, 2005
Capacitors
18
Look again
q  A

E
0
V 0 A
q   0 AE   0 A 
V  CV
d
d
so
C
February 14, 2005
0 A
d
Capacitors
19
Continuing…

C
February 14, 2005
0 A
d

The capacitance of
a parallel plate
capacitor depends
only on the Area
and separation
between the plates.
C is dependent only
on the geometry of
the device!
Capacitors
20
Diversion on Capacitors
Two Metal Plates a Capacitor
Make.
February 14, 2005
Capacitors
22
More is better!
February 14, 2005
Capacitors
23
Implementation - Variable
February 14, 2005
Capacitors
24
How do you do that?
February 14, 2005
Capacitors
25
Roll it up, Scottie
February 14, 2005
Capacitors
26
Stacked Disks, etc.
February 14, 2005
Capacitors
27
Units of 0
Coulomb 2 Coulomb 2
 0  

2
Nm
m  Joule
Coulomb 2

m  Coulomb  Volt
Coulomb Farad


m  Volt
m
and
Joule
Volt 
Coulomb
 0  8.85 10 12 F / m  8.85 pF / m
February 14, 2005
Capacitors
pico
28
Simple Capacitor Circuits

Batteries



Apply potential differences
Capacitors
Wires



Wires are METALS.
Continuous strands of wire are all at the same
potential.
Separate strands of wire connected to circuit
elements may be at DIFFERENT potentials.
February 14, 2005
Capacitors
29
Size Matters!




A Random Access Memory stores
information on small capacitors which are
either charged (bit=1) or uncharged
(bit=0).
Voltage across one of these capacitors ie
either zero or the power source voltage
(5.3 volts in this example).
Typical capacitance is 55 fF (femto=10-15)
Question: How many electrons are stored
on one of these capacitors in the +1
state?
February 14, 2005
Capacitors
30
Small is better in the IC world!
q CV (55 1015 F )(5.3V )
6
n 


1
.
8

10
electrons
19
e
e
1.6  10 C
February 14, 2005
Capacitors
31
TWO Types of Connections
SERIES
PARALLEL
February 14, 2005
Capacitors
32
Parallel Connection
q1  C1V  C1V
V
C1
C2
C3
q2  C2V
q3  C3V
QE  q1  q2  q3
V
CEquivalent=CE
February 14, 2005
QE  V (C1  C2  C3 )
therefore
C E  C1  C2  C3
Capacitors
33
Series Connection
q
V
-q
C1
q
-q
C2
The charge on each
capacitor is the same !
February 14, 2005
Capacitors
34
Series Connection Continued
V1
q
V
C1
V2
-q
q
-q
C2
V  V1  V2
q
q
q


C C1 C 2
or
1
1
1


C C1 C 2
February 14, 2005
Capacitors
35
For Bunches of Capacitors
Series
1
1

C
i Ci
Parallel
C   Ci
i
February 14, 2005
Capacitors
36
February 14, 2005
Capacitors
37
Example
C1
C2
C1=12.0 uf
C2= 5.3 uf
C3= 4.5 ud
(12+5.3)pf
V
C3
February 14, 2005
Capacitors
38
More on the Big C
E=0A/d
+dq
+q
February 14, 2005
-q
• We move a charge dq
from the (-) plate to
the (+) one.
• The (-) plate becomes
more (-)
• The (+) plate
becomes more (+).
• dW=Fd=dq x E x d
Capacitors
39
So….
dW  dq  Ed
Gauss
 q 1
E

0 A 0
q 1
dW 
d  dq
A 0
Q
d
q2 d Q q2 1 Q
W U  
qdq 
|
|
A 0
2 A 0 0 2 ( A 0 ) 0
0
d
or
Q 2 C 2V 2 1
U

 CV 2
2C
2C
2
February 14, 2005
Capacitors
40
Not All Capacitors are Created
Equal
• Parallel
Plate
• Cylindrical
• Spherical
February 14, 2005
Capacitors
41
Spherical Capacitor
Gauss
q
 E  dA  
4r E 
2
0
q
0
q
E (r ) 
2
4r  0
surprise ???
February 14, 2005
Capacitors
42
Calculate Potential Difference V
positive. plate
Eds

V
negative. plate
q 1
V  
 2 dr
40  r 
b
a
(-) sign because E and ds are in OPPOSITE directions.
February 14, 2005
Capacitors
43
Continuing…
q
b
dr
q
1
V

( )
2

40 a r
40 r
q 1 1
q ba
V
  


40  a b  40  ab 
q
ab
C   40
V
ba
Lost (-) sign due to switch of limits.
February 14, 2005
Capacitors
44
Real Materials

Consist of atoms or molecules bonded
together.
Some atoms and molecules do not
have dipole moments when isolated.
Some do.

Two types to consider:




Polar
Non-Polar
February 14, 2005
Capacitors
45
Polar Molecule
E
February 14, 2005
Capacitors
46
Polar Materials
February 14, 2005
Capacitors
47
February 14, 2005
Capacitors
48
Apply an Electric Field
Some LOCAL ordering
February 14, 2005
Large Scale Ordering
Capacitors
49
Adding things up..
-
E +
Net effect REDUCES the field
February 14, 2005
Capacitors
50
Non-Polar Material
February 14, 2005
Capacitors
51
Non-Polar Material
 net
E
0
Electric Field
in the dielectric
is reduced
Effective Charge is
REDUCED
February 14, 2005
Capacitors
52
Effect of Capacitor Material
Dielectric
 net
E
0
Electric Field
in the dielectric
is reduced
Effective Charge is
REDUCED
February 14, 2005
Capacitors
53
We can measure the C of a
capacitor (later)
C0 = Vacuum or air Value
C = With dielectric in place
Definition
C=kC0
February 14, 2005
Capacitors
54
Dielectric Constant
C
k
C0
February 14, 2005
Capacitors
55
How to Check
C0
V0
Charge to V0 and then disconnect from
the battery.
Q
V
Connect the two together
C0 will lose some charge to the capacitor with the dielectric.
We14,can
voltmeter (later).
February
2005measure V with aCapacitors
56
Checking the idea..
q0  C0V0
V
q1  C0V
q2  CV
q0  q1  q2
C0V0  C0V  CV
 V0 
C  C0   1
V

February
Note: 14,
When
2005
two Capacitors are Capacitors
the same (No dielectric), then V=V0/2.
57
Some k values
Dielectric
Strength
1
Breakdown
KV/mm
3
Polystyrene
2.6
24
Paper
3.5
16
Pyrex
4.7
14
Strontium
February
14, 2005
Titanate
310
8
Material
Air
Capacitors
58
Messing with Capacitor
+
+
-
-
+
+
-
-
The battery means that the
potential difference across
the capacitor remains constant.
V
For this case, we insert the
dielectric but hold the voltage
constant,
q=CV
V
since C  kC0
qk kC0V
Remember – We hold V
constant with the battery.
February 14, 2005
THE EXTRA CHARGE COMES
FROM THE BATTERY!
Capacitors
59
WHERE IS THIS NEW
CHARGE?
Hang on … we will get there.
But there is more capacity so there
is more charge for the same
applied voltage
Another Case




We charge the capacitor to a
voltage V0.
We disconnect the battery.
We slip a dielectric in between the
two plates.
We look at the voltage across the
capacitor to see what happens.
February 14, 2005
Capacitors
61
Case II – No Battery
q=C0Vo
+
q0
V0
-
+
qk  kC0V
qk
V
-
When the dielectric is inserted, no charge
is added so the charge must be the same.
q0  C0V0  qk  kC0V
or
V
February 14, 2005
Capacitors
V0
k
62
Another Way to Think About
This



There is an original charge q on the
capacitor.
If you slide the dielectric into the capacitor,
you are adding no additional STORED
charge. Just moving some charge around in
the dielectric material.
If you short the capacitors with your fingers,
only the original charge on the capacitor can
burn your fingers to a crisp!
February 14, 2005
Capacitors
63
q0
February 14, 2005
Capacitors
64
A Reminder of days past

Q
E 
0 0 A
V
E
d
equating :
Qd
V
0 A
Q
C  (definition )
V
0 A
C
d
February 14, 2005
Capacitors
65
A Closer Look at this stuff..
q
Consider this capacitor.
No dielectric.
Applied Voltage via a battery.
++++++++++++
V0
C0
-q
------------------
C0 
0 A
d
q  C0V0 
February 14, 2005
Capacitors
0 A
d
V0
66
Remove the Battery
q
++++++++++++
V0
The Voltage across the
capacitor remains V0
q remains the same as
well.
-q
February 14, 2005
-----------------The capacitor is fat (charged),
dumb and happy.
Capacitors
67
Slip in a Dielectric
Almost, but not quite, filling the space
Gaussian Surface
q
V0
-q
++++++++++++
- - - - - - - -
-q’
+ + + + + +
+q’
------------------
E0
E
E’ from induced
charges
in..small..gap
q
 E  dA 
0
February 14, 2005
Capacitors
q   
  
E0 
0 A  0 
68
A little sheet from the past..
-q’
+q’
- -q
-
+
q+
+
Esheet

q'


2 0 2 0 A
q'
0
February 14, 2005
2xEsheet
0
q'
Esheet / dialectric  2 

2 0 A  0 A
Capacitors
69
Some more sheet…
Edielectricch arg e
q
E 0
0 A
 q'

0 A
(original field)
so
q  q'
Ein dielectric 
0 A
material
February 14, 2005
Capacitors
70
A Few slides back
Case II – No Battery
+
q0
V0
-
+
q=C0Vo
When the dielectric is inserted, no charge
is added so the charge must be the same.
qk
V
-
qk  kC0V
q0  C0V0  qk  kC0V
or
V0
February 14, 2005
V

Capacitors
k
71
From this last equation
V
V0
k
and
V  Ed
V0  E0 d
thus
V 1 E
 
V0 k E0
E
February 14, 2005
Capacitors
E0
k
72
A Bit more…..
 q 


0 A 
V0
E0

k 

V
E  q  q' 


 0 A 
therefore
q
q  q' 
k
February 14, 2005
Capacitors
73
Important Result
(We already know)
• Electric Field is
Reduced by the
presence of the
material .
• The material reduces
the field by a factor k.
E
E0
k
k is the DIELECTRIC CONSTANT
of the material
February 14, 2005
Capacitors
74
Another look
Vo
+
-
February 14, 2005
Parallel  Plate
0 A
C0 
d
 0 AV0
Q0  C0V0 
d
Electric  Field
V0
E0 
d
Q0  0V0
0 

Capacitors
A
d
75
Add Dielectric to Capacitor
Vo
• Original Structure
+
-
+
• Disconnect Battery
V0
+
• Slip in Dielectric
Note: Charge on plate does not change!
February 14, 2005
Capacitors
76
What happens?
o +
o -
i 
i 
E0
V0 1
E

k
d k
and
V0
V  Ed 
k
Potential Difference is REDUCED
by insertion of dielectric.
Q
Q
C 
 kC0
V V0 / k 
Charge on plate is Unchanged!
Capacitance increases by a factor of k
February 14, 2005
Capacitors
as we showed previously
77
SUMMARY OF RESULTS
V
V0
E
E0
k
C  kC0
February 14, 2005
k
Capacitors
78
APPLICATION OF GAUSS’ LAW
q
E0 
0 A
q  q ' E0
E

0 A
k
E
q
k 0 A
and
q  q' 
February 14, 2005
Capacitors
q
k
79
New Gauss for Dielectrics

A
d

E
k

q free
0
sometimes
  k 0
February 14, 2005
Capacitors
80
The Insertion Process With A
Battery
Vo
+
F
--------
-
February 14, 2005
Capacitors
81