Short Version : 23. Electrostatic Energy & Capacitors
23.1. Electrostatic Energy
Electrostatic Energy = work done to assemble the charge configuration of a system.
Reference ( 0 energy): when all component charges are widely separated.
Bringing q1 in place takes no work.
2 2 1 2W q V rBringing in q2 takes
1 1 0 0W q V
12
qq k
a
3 3 1 3 2 3W q V V r rBringing in q3 takes
1 23
q qq k k
a a
Total electrostatic energy1 2 2U W W W 1 2 2 3 3 1
kq q q q q q
a
23.2. Capacitors
Capacitor: pair of conductors carrying equal but opposite charges.
Usage: store electrical energy
Parallel-Plate Capacitor:2 conducting plates of area A separated by a small distance d .
Plates are initially neutral.
They’re charged by connecting to a battery.
Charge transfer plates are equal but oppositely charged.
Large A, small d E 0 outside.
0
ˆinside
E zFar from the edges0
ˆQ
A z ˆinsideV d E z
0
Qd
A
Capacitance
0
QV d
AParallel-plate capacitor: 0 AQ V
d
CV
0 ACd
C = Q / V = capacitance
Parallel-plate capacitor
See Probs 41 & 42
CC
V farad F
Practical capacitor ~ F ( 106 F) or pF ( 1012 F )
0
C d
A
/F m 1dV dQ
C
Charging / Discharging
dQC
dV
Energy Stored in Capacitors
When potential difference between capacitor plates is V,
work required to move charge dQ from to + plate is
Work required to charge the capacitor from 0 to V is
0
VW C V dV 21
2CV = U = energy stored in capacitor
Note:
In a “charged” capacitor, Q is the charge on the + plate.
The total charge of the capacitor is always zero.
plate
plate
dW dQ d
E r V C dV plate platedQ V V E dr < 0
2
2
Q
C 1
2QV
Example 23.1. Parallel-Plate Capacitor
A capacitor consists of two circular metal plates of radius R = 12 cm, separated by
d = 5.0 mm. Find
(a) Its capacitance,
(b) the charge on the plates, and
(c) the stored energy when the capacitor is connected to a 12-V battery.
0
AC
d
100.8 10 F
22
9 3
12 101
4 9 10 / 5.0 10
m
Vm C m
80 pF
(a)
(b)
(c)
Q CV 80 12pF V 960 pC
21
2U CV 21
80 122
pF V 5760 pJ
5.76 nJ
CF
V
Practical Capacitors
Inexpensive capacitors:
Thin plastic sandwiched between aluminum foils
& rolled into cylinder.
Electrolytic capacitors (large capacitance):
Insulating layer developed by electrolysis.
Capacitors in IC circuits (small capacitance):
Alternating conductive & insulating layers.
Dielectrics
Dielectrics: insulators containing molecular dipoles but no free charges.
Dielectric layer lowers V between
capacitor plates by factor 1/ ( > 1).
0
A
d
QC
V
= dielectric constant
Molecular dipoles aligned by E0 .
Dipole fields oppose E0.Net field reduced to E = E0 / .
Hence V = V0 / .Q is unchanged, so C = C0 .
: 2 ~ 10 mostly
Working voltage V = Max safe potential < Ebkd d
Example 23.2. Which Capacitor?
A 100-F capacitor has a working voltage of 20 V,
while a 1.0-F capacitor is rated at 300 V.
Which can store more charge? More energy?
100 FQ CV 6100 10 20F V 32 10 C 2 mC
61 1 10 300FQ F V
43 10 C 0.3mC
2100
1
2FU CV 12 20
2mC V
1
2QV 20 mJ
1
10.3 300
2FU mC V 45mJ
Connecting Capacitors
Two ways to connect 2 electronic components: parallel & series
Parallel: Same V for both components
Q CV 1 2Q Q 1 2C V C V
1 2C C C
Series: Same I (Q) for both components
QV
C 1 2V V
1 2
Q Q
C C
1 2
1 1 1
C C C
1 2C C or C
1 2C C or C
Bursts of Power
San Francisco’s BART train:
KE of deceleration stored as EE in ultracapacitor.
Stored EE is used to accelerate train.
Capacitors deliver higher energy
much more quickly than batteries.
Flash light:
Battery charges capacitor,
which then discharges to give flash.
Other examples:
Defibrillator, controlled nuclear fusion, amusement park rides, hybrid cars, …
23.4. Energy in the Electric Field
Charging a capacitor rearranges charges energy stored in E
Energy density = energy per unit volume
Parallel-plate capacitor:2
2
QU
C
2
02
QAd
E
Uu
A dEnergy density :
2
202
Q
A
2
02
20
1
2E
0
E
is universal2
0
1
2Eu E 3/Eu J m
2
0
1
2 U dVE
Example 23.4. A Thunderstorm
Typical electric fields in thunderstorms average around 105 V/m.
Consider a cylindrical thundercloud with height 10 km and diameter 20 km,
and assume a uniform electric field of 1105 V/m.
Find the electric energy contained in this storm.
111.39 10 J
20
1
2Eu E
EU u V 2 25 3 39 2 2
1 110 / 10 10 10 10
2 4 9 10 /V m m m
Nm C
139 GJ
~ 1400 gallons of gasoline.
Example 23.5. A Shrinking Sphere
A sphere of radius R1 carries charge Q distributed uniformly over its surface.
How much work does it take to compress the sphere to a smaller radius R2 ?
2ˆ
Qkr
E r2
0
1
2 U dVE
1
2
22
1 1
2
R
Rk Q dr
r 1
2
22
2
14
8
R
R
Qk r dr
k r
2
2 1
1 1 1
2k Q
R R
2 10U for R R Work need be done to shrink sphere
2 1U U U 2 1
2 20
14
2 R Rr dr
E
Extra energy stored here