1 24 Electrostatic Potential Energy. 2 3 4 Example as shown:

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)