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February 14, 2005 Capacitors 1
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 7
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 9
Don’t ask questions because I don’t know the answers!
Zn Metal Cu Metal
AqueousSolution
of
February 14, 2005 Capacitors 10
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 14
Gauss on Capacitors
d
Air or Vacuum
Area A
- Q +QE
V=Potential Difference
GaussianSurface
000
0
0
0
)/(
0
AQ
A
QE
EAQ
QEAAEA
qd
Gauss
AE
Same result from other plate!
February 14, 2005 Capacitors 15
Two Charged Plates(Neglect Fringing Fields)
d
Air or Vacuum
Area A
- Q +QE
V=Potential Difference
Symbol
February 14, 2005 Capacitors 16
Noted
Air or Vacuum
Area A
- Q +QE
V=Potential Difference
+
Consider a +q chargeat the (-) plate.
Move it to the (+)plate
Work to do this isW=Fd=qEdalsoW=q(Vf-Vi)=qV
ThereforeEd=VE=V/d
February 14, 2005 Capacitors 17
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.
CVq
orV
qC
DEFINITION
February 14, 2005 Capacitors 20
Continuing…
d
AC 0
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!
February 14, 2005 Capacitors 28
Units of 0
mpFmF
andm
Farad
Voltm
CoulombVoltCoulombm
Coulomb
Joulem
Coulomb
Nm
Coulomb
/85.8/1085.8 120
2
2
2
2
0
pico
Coulomb
JouleVolt
February 14, 2005 Capacitors 29
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 30
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 31
Small is better in the IC world!
electronsC
VF
e
CV
e
qn 6
19
15
108.1106.1
)3.5)(1055(
February 14, 2005 Capacitors 33
Parallel Connection
V
VCEquivalent=CE
C1 C2 C3
321
321
321
33
22
111
)(
CCCC
therefore
CCCVQ
qqqQ
VCq
VCq
VCVCq
E
E
E
February 14, 2005 Capacitors 34
Series Connection
V C1 C2
q -q q -q
The charge on eachcapacitor is the same !
February 14, 2005 Capacitors 35
Series Connection Continued
21
21
21
111
CCC
or
C
q
C
q
C
q
VVV
VC1 C2
q -q q -q
V1 V2
February 14, 2005 Capacitors 39
More on the Big C
• 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+q -q
E=0A/d
+dq
February 14, 2005 Capacitors 40
So….
2222
00
2
00
2
0 0
0
00
2
1
22
|)(
1
2|
2
1
1
CVC
VC
C
QU
ord
Aq
A
dqqdq
A
dUW
dqdA
qdW
A
qE
Gauss
EddqdW
QQQ
February 14, 2005 Capacitors 41
Not All Capacitors are Created Equal
• Parallel Plate
• Cylindrical• Spherical
February 14, 2005 Capacitors 42
Spherical Capacitor
???
4)(
4
02
0
2
0
surprise
r
qrE
qEr
qd
Gauss
AE
February 14, 2005 Capacitors 43
Calculate Potential Difference V
drr
qV
EdsV
a
b
platepositive
platenegative
20
.
.
1
4
(-) sign because E and ds are in OPPOSITE directions.
February 14, 2005 Capacitors 44
Continuing…
ab
ab
V
qC
ab
abq
ba
qV
r
q
r
drqV
b
a
0
00
02
0
4
4
11
4
)1
(44
Lost (-) sign due to switch of limits.
February 14, 2005 Capacitors 45
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 52
Non-Polar Material
Effective Charge isREDUCED
Electric Fieldin the dielectric
is reduced0
netE
February 14, 2005 Capacitors 53
Effect of Capacitor MaterialDielectric
Effective Charge isREDUCED
Electric Fieldin the dielectric
is reduced0
netE
February 14, 2005 Capacitors 54
We can measure the C of a capacitor (later)
C0 = Vacuum or air Value
C = With dielectric in place
C=C0
Definition
February 14, 2005 Capacitors 56
How to Check
Charge to V0 and then disconnect fromthe battery.C0 V0
Connect the two togetherV
C0 will lose some charge to the capacitor with the dielectric.We can measure V with a voltmeter (later).
Q
February 14, 2005 Capacitors 57
Checking the idea..
V
100
000
210
2
01
000
V
VCC
CVVCVC
qqq
CVq
VCq
VCq
Note: When two Capacitors are the same (No dielectric), then V=V0/2.
February 14, 2005 Capacitors 58
Some values
Material
Dielectric
Strength
Breakdown
KV/mm
Air 1 3
Polystyrene 2.6 24
Paper 3.5 16
Pyrex 4.7 14
Strontium Titanate
310 8
February 14, 2005 Capacitors 59
Messing with Capacitor
+
V-
+
V-
+
-
+
-
The battery means that thepotential difference acrossthe capacitor remains constant.
For this case, we insert the dielectric but hold the voltage constant,
q=CV
since C kC0
qk kC0V
THE EXTRA CHARGE COMES FROM THE BATTERY!
Remember – We hold V constant with the battery.
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
February 14, 2005 Capacitors 61
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 62
Case II – No Battery
+
-
+
-
q0
q
q=C0Vo
When the dielectric is inserted, no chargeis added so the charge must be the same.
0
0000
0
VV
or
VCqVCq
VCq
V0
V
February 14, 2005 Capacitors 63
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 64
q0
February 14, 2005 Capacitors 65
A Reminder of days past
d
AC
definitionV
QC
A
QdV
equatingd
VE
A
QE
0
0
00
)(
:
February 14, 2005 Capacitors 66
A Closer Look at this stuff..Consider this capacitor.No dielectric.Applied Voltage via a battery.
00
00
00
Vd
AVCq
d
AC
C0
++++++++++++
------------------
V0
q
-q
February 14, 2005 Capacitors 67
Remove the Battery
++++++++++++
------------------
V0
q
-q
The Voltage across thecapacitor remains V0
q remains the same aswell.
The capacitor is fat (charged),dumb and happy.
February 14, 2005 Capacitors 68
Slip in a DielectricAlmost, but not quite, filling the space
++++++++++++
------------------
V0
q
-q
- - - - - - - -
+ + + + + +
-q’
+q’
E0
E
E’ from inducedcharges
Gaussian Surface
000
0
....
A
qE
qd
gapsmallin
AE
February 14, 2005 Capacitors 69
A little sheet from the past..
+++
---q-q
-q’ +q’
A
q
A
qE
A
qE
dialectricsheet
sheet
00/
00
'
2
'2
2
'
2
0 2xEsheet 0
February 14, 2005 Capacitors 70
Some more sheet…
A
qqE
so
A
qE
A
qE echdielectric
0material dielectricin
00
0arg
'
'
(original field)
February 14, 2005 Capacitors 71
A Few slides backCase II – No Battery
+
-
+
-
q0
q
q=C0Vo
When the dielectric is inserted, no chargeis added so the charge must be the same.
0
0000
0
VV
or
VCqVCq
VCq
V0
V
February 14, 2005 Capacitors 74
Important Result (We already know)
• Electric Field is Reduced by the presence of the material .
• The material reduces the field by a factor .
0EE
is the DIELECTRIC CONSTANTof the material
February 14, 2005 Capacitors 75
Another look
+
-
Vo
d
V
A
Qd
VE
FieldElectricd
AVVCQ
d
AC
PlateParallel
0000
00
00000
00
February 14, 2005 Capacitors 76
Add Dielectric to Capacitor
• Original Structure
• Disconnect Battery
• Slip in Dielectric
+
-
Vo
+
-
+
-
V0
Note: Charge on plate does not change!
February 14, 2005 Capacitors 77
What happens?
0
00 1
VEdV
andd
VEE
+
-
ii
oo
Potential Difference is REDUCEDby insertion of dielectric.
00 /
CV
Q
V
QC
Charge on plate is Unchanged!
Capacitance increases by a factor of as we showed previously