Capacitor Circuits
ii
i i
CC
ParallelCC
Series11
Thunk some more …
C1 C2
V
C3
C1=12.0 fC2= 5.3 fC3= 4.5 d
(12+5.3)pf
So….
2222
0
2
0
2
0 0
0
00
21
22
)(
122
1
1
CVCVC
CQU
ordA
qAdqqdq
AdUW
dqdAqdW
AqE
GaussEddqdW
Q
Sorta like (1/2)mv2
DIELECTRIC
Polar Materials (Water)
Apply an Electric Field
Some LOCAL ordering Larger Scale Ordering
Adding things up..
- +Net effect REDUCES the field
Non-Polar Material
Non-Polar Material
Effective Charge isREDUCED
We can measure the C of a capacitor (later)
C0 = Vacuum or air Value
C = With dielectric in place
C=C0
(we show this later)
How to Check This
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).
Checking the idea..
V
00
0
000
210
2
01
000
1 CVVCC
CVVCVCqqq
CVqVCqVCq
Note: When two Capacitors are the same (No dielectric), then V=V0/2.
Messing with Capacitors
+
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 C0
qC0V
THE EXTRA CHARGE COMES FROM THE BATTERY!
Remember – We hold V constant with the battery.
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.
No Battery
+
-
+
-
q0
q
q0 =C0Vo
When the dielectric is inserted, no chargeis added so the charge must be the same.
0
0000
0
VV
orVCqVCq
VCq
V0
V
A Closer Look at this stuff..Consider this capacitor.No dielectric experience.Applied Voltage via a battery.
C0
00
00
00
VdAVCq
dAC
++++++++++++
------------------
V0
q
-q
Remove the Battery
++++++++++++
------------------
V0
q
-q
The Voltage across thecapacitor remains V0
q remains the same aswell.
The capacitor is (charged),
Slip in a DielectricAlmost, but not quite, filling the space
++++++++++++
------------------
V0
q
-q
- - - - - - - -
+ + + + + +
-q’
+q’
E0
E
E’ from inducedcharges
Gaussian Surface
000
0
....
AqE
qd
gapsmallin
AE
A little sheet from the past..
+++
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Aq
AqE
AqE
dialectricsheet
sheet
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Some more sheet…
AqqEnet
soAqE
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0
00
0arg
'
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A Few slides backNo Battery
+
-
+
-
q0
q
q=C0Vo
When the dielectric is inserted, no chargeis added so the charge must be the same.
0
0000
0
VV
orVCqVCq
VCq
V0
V
From this last equation
0
00
00
0
1
EE
EE
VVthus
dEVEdV
and
VV
Add Dielectric to Capacitor
• Original Structure
• Disconnect Battery
• Slip in Dielectric
+
-
Vo
+
-
+
-
V0
Note: Charge on plate does not change!
SUMMARY OF RESULTS
0
0
0
EE
CC
VV
APPLICATION OF GAUSS’ LAW
qqq
andAqE
EAqqE
AqE
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'
0
0
0
00
New Gauss for Dielectrics
0
0
sometimes
qd freeAE