1
24 Electrostatic Potential Energy
2
3
ji ij
ji
r
qkq
2
1
DA
AD
AD
DA
CA
AC
AC
CA
BA
AB
AB
BA
r
qkq
r
qkq
r
qkq
r
qkq
r
qkq
r
qkq
2
1
DC
CD
CD
DC
DB
BD
BD
DB
CB
BC
BC
CB
r
qkq
r
qkq
r
qkq
r
qkq
r
qkq
r
qkq
2
1
4
ji ij
ji
r
qkq
2
1
a
kq
a
kq
224
22
aarij 2,
Example as shown:
242
24
22
a
kq
a
kq
5
SI Unit: farad [F] = C/V
Capacitance: Charge Storage per Volt Applied
6
Real Parallel-Plate Capacitor
Note:
• Uniform Field
• Fringing
7
Parallel Plate Capacitor
o
E
E is nearly uniform
o
dEdV
V
QC
A
od / d
Ao
8
Cylindrical Capacitor
r
9
rLEEA 2
r
rkE
12
inkQ4
LQin
Lk4
10
rkE
12
2
1
R
REdrV 2
1
12
R
Rdrr
k 2
1ln2
R
Rrk
1
2ln2R
RkV
V
QC
LQ
V
L 12ln2 RRk
L
12ln2 RRk
L
12ln
2
RR
LC o
11
Work done in Charging a Capacitor
= (Q)(Vavg)
12
Q = VC Vavg = ½ Q/C
Work = (Q) x (½ Q/C)= ½ Q2/C
= area under curve
22
2
1
2
1
2
1CVQV
C
QU
13
Ad
CV
volume
energyue
221
Energy Density Inside a Capacitor
Ad
dCE 2221
Ad
dE
d
A o22
21
221 Eu oe
Ex: Lab Capacitor, C = 1F, V = 6V, vol.=2x10-5 m3.
SI Unit: [J/m3]
36335
221
/10/000,900102
0.60.1mJmJ
mx
VFue
31033
8
/105.3104
104.1mJx
mx
Jxugasoline
about 35,000 higher than capacitor
14
Capacitors in “Parallel” Arrangement
CVQ
""12 VVVV BA
eqBA QQQ
BA QQ VCVCVC eqBA
eqBA CCC
Ex. FFFCeq 18126
15
eqBABA C
Q
C
Q
C
QVV
Capacitors in “Series” Arrangement
C
QV eqBA QQQ
Q = 0eqBA CCC
111
12
1
6
11
eqCEx.
FCeq 4
16
equivalent value?
17
DielectricConstant K
• reduces E and V (E = Eo/K)
• C = KCo
• C = Capacitance with Dielectric
• Co = “Empty” Capacitor
18
Ex. K’s
• vacuum: 1 exactly
• air: 1.00059
• paper: 3.7
• water: 80
• barium titanate: 1200
• potassium tantalate niobate (0 °C): 34,000
19
Supercapacitors
• porous structure
• surface areas much greater
• charge separation distance < 1 nm
• very high capacitance
20
Batteries
slow/special charging
limited # cycles with decreasing utility
short life
high energy density
poor low temp. performance
Capacitors
simple/fast charging
over 500,000 cycles at 100%
10 to 12 year life
low energy density
good low temp. perf.
21
Summary
• Electrostatic potential energy
• Capacitance: field, energy, voltage, charge
• Capacitors in circuits
• Dielectrics
• /
22
23
24
25
26
27
28
29
ji ij
ji
r
qkq
2
1
a
kq
a
kq
a
kq
a
kq
a
kq
a
kq 222222
222
1
a
kq
a
kq
a
kq
a
kq
a
kq
a
kq 222222
222
1
qqq ji
aarij 2,
Example with:
30
Rolled Parallel-Plate Capacitor (Can Shape)