1

24 Electrostatic Potential Energy

2

24-1Electrostatic Potential Energy

3

4

5

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

6

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:

7

ji ij

ji

r

qkq

2

1

a

kq

a

kq

a

kq 222 2

2

22

2

1

a

kq

a

kq

a

kq 222 2

2

22

2

1

qqq ji

aarij 2,

Example with:

8

ji ij

ji

r

qkq

2

1

a

kq

a

kq

a

kq 222

2

a

kq

a

kq

a

kq 222

2

qqq ji

aarij 2,

Example with:

a

kq

a

kq

a

kq

a

kq

a

kq

2

22

22

22222

9

ji ij

ji

r

qkq

2

1

qqq ji

aarij 2,

Example with:

2

1

1

22

2

22

222

a

kq

a

kq

a

kq

242

2

1

4 22

a

kq

a

kq

10

24-2Capacitance

11

SI Unit: farad [F] = C/V

Capacitance: Charge Storage per Volt Applied

12

Real Parallel-Plate Capacitor

Note:

• Uniform Field

• Fringing

13

Rolled Parallel-Plate Capacitor (Can Shape)

14

Parallel Plate Capacitor

o

E

E is nearly uniform

o

dEdV

V

QC

A

od / d

Ao

15

Cylindrical Capacitor

r

16

rLEEA 2

r

rkE

12

inkQ4

LQin

Lk4

17

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

18

24-3The Storage of Electrical

Energy

19

Work done in Charging a Capacitor

= (Q)(Vavg)

20

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

21

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

22

24-4Capacitors, Batteries, and

Circuits

23

Capacitors in “Parallel” Arrangement

CVQ

""12 VVVV BA

eqBA QQQ

BA QQ VCVCVC eqBA

eqBA CCC

Ex. FFFCeq 18126

24

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

25

equivalent value?

26

24-5Dielectrics

27

DielectricConstant K

• reduces E and V (E = Eo/K)

• C = KCo

• C = Capacitance with Dielectric

• Co = “Empty” Capacitor

28

Ex. K’s

• vacuum: 1 exactly

• air: 1.00059

• paper: 3.7

• water: 80

• barium titanate: 1200

• potassium tantalate niobate (0 °C): 34,000

29

Supercapacitors

• porous structure

• surface areas much greater

• charge separation distance < 1 nm

• very high capacitance

30

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.

31

Problems

32

33

34

35

36

37

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