Download 24 Electrostatic Potential Energy

24 Electrostatic Potential
Energy
1
2
kqi q j
1

2 i  j rij
1  kqAqB kqB q A kqA qC kqC q A kqAqD kqD q A 

 





2  rAB
rBA
rAC
rCA
rAD
rDA 
1  kqB qC kqC qB kqB qD kqD qB kqC qD kqD qC
 





2  rBC
rCB
rBD
rDB
rCD
rDC



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Example as shown:
rij  a, 2a
kqi q j
1

2 i  j rij
kq2
kq2
4
2
a
2a

kq 
2  kq

4 2
4 

a 
a
2
2
2

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Capacitance: Charge Storage per Volt Applied
SI Unit: farad [F] = C/V
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Real Parallel-Plate
Capacitor
Note:
• Uniform Field
• Fringing
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Parallel Plate Capacitor
E is nearly uniform

E
o
d
V  Ed 
o
o A
Q A
C


V d /  o
d
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Cylindrical Capacitor
r
8
EA  E2rL  4kQin  4kL
1
E  2k
r
r
Qin  L
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1
E  2k
r
Q  L
V 
R2
R1
1
Edr   2k dr  2k ln r R2
R1
R1
r
 R2 
V  2k ln  
 R1 
R2
Q L
L
L


C

2k ln R2 R1 
2k ln R2 R1 
V
V
2o L
C
ln R2 R1 
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Work done in Charging a Capacitor
= (Q)(Vavg)
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Q = VC
Vavg = ½ Q/C
Work = (Q) x (½ Q/C)= ½ Q2/C
= area under curve
1 Q2 1
1
2
U
 QV  CV
2 C 2
2
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Energy Density Inside a Capacitor
energy 12 CV 2 12 CE 2 d 2
A o
ue 



volume
Ad
Ad
d
ue   o E
1
2
2
1
2
E 2d 2
Ad
SI Unit: [J/m3]
Ex: Lab Capacitor, C = 1F, V = 6V, vol.=2x10-5 m3.
ue 
1
2
1.0 F 6.0V 
2
5
3
 900,000 J / m  10 J / m
3
6
3
2 x10 m
8
1.4 x10 J
10
3
u gasoline 

3
.
5
x
10
J
/
m
3 3
4 x10 m
about 35,000 higher than capacitor
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Capacitors in “Parallel” Arrangement
VA  VB  12V "V "
QA  QB  Qeq
Q  CV
QA  QB  C AV  C BV  CeqV
C A  C B    Ceq
Ex.
Ceq  6F  12F  18F
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Capacitors in “Series” Arrangement
QA  QB  Qeq
Q
V
C

Q=0 


VA  VB 
Q
CA

Q
CB

Q
Ceq
1
1
1

 
C A CB
Ceq
1
1 1
 
Ex.
Ceq 6 12
Ceq  4 F
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equivalent value?
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Dielectric
Constant K
•
•
•
•
reduces E and V (E = Eo/K)
C = KCo
C = Capacitance with Dielectric
Co = “Empty” Capacitor
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Ex. K’s
•
•
•
•
•
•
vacuum: 1 exactly
air: 1.00059
paper: 3.7
water: 80
barium titanate: 1200
potassium tantalate niobate (0 °C): 34,000
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Supercapacitors
•
•
•
•
porous structure
surface areas much greater
charge separation distance < 1 nm
very high capacitance
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Batteries
slow/special
charging
Capacitors
simple/fast
charging
limited # cycles with
decreasing utility
 over 500,000
cycles at 100%
short life
 10 to 12 year life
 high energy
density
low energy density
poor low temp.
performance
 good low temp.
perf.
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Summary
•
•
•
•
•
Electrostatic potential energy
Capacitance: field, energy, voltage, charge
Capacitors in circuits
Dielectrics
/
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Example with:
kqi q j
1

2 i  j rij
qi  q j  q
rij  a, 2a
1  kq2 kq2 kq2 kq2 kq2 kq2 

 





2 a
a
a
a 
2a
2a
1  kq2 kq2 kq2 kq2 kq2 kq2 

 





2 a
a
a
a 
2a
2a
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Rolled Parallel-Plate Capacitor (Can Shape)
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