PHYS 2421 - Fields and WavesInstructor: Jorge A. López

Office: PSCI 209 A, Phone: 747-7528

Textbook: University Physics 11e, Young and Freedman

24.1 Capacitors and capacitance

24.2 Capacitors in series and parallel

24.3 Energy storage in capacitors

24.4 Dielectrics

00 0

d dq

dW F dl dq E dl dq dA

Energy needed to charge a capacitor:

In terms of energy stored in the capacitor:

2 2

0 0 00 02 2

q Q q Q

q q

q d Q d QW dq d qdq

A A A C

221 1

2 2 2

QU CV QV

C Valid for all capacitors

Energy density (per unit volume) for parallel plate capacitor

2 2 2/ 2 1 1

2 2 2Volume

Q U Q C CV QVU u

C Ad Ad Ad

In terms of E (V=Ed and C=0A/d)

202 2 2 2 2

0

/1 1 1 1

2 2 2 2

A dCV Cu E d E d E

Ad Ad Ad

22

0

1 1

2 2

CVu E

Ad

Hmwk: Problems 24.25 and 24.27 (11th or 12th Eds.)

Step 1: Charge and disconnect

Charge? 0 1 8 120 960F V CQ C V

Energy stored?0

1 1960 120 0.058

2 2C V JInitialU Q V

Step 2: Connect to an uncharged capacitor

Voltage? 0

0 1 2 1 2

1 2

96080

8 4

CV

F F

QQ Q Q CV C V V

C C

Charges:

1 1 8 80F V=640 CQ CV

2 2 4 80F V=320 CQ C V Energy:

1 2

1 1 1640 0 80 0.038

2 2 2C+32 C V JFinalU QV Q V

Energy is lost as heat in moving the charges

Electric field needed?

2

0

12

0

5

1

2

2 2 1

8.85 10

4.75 10

3 2 2

Volume

J

Volume 1m C /Nm

V/m

Uu E

UE

Hmwk: Probl. 24.30 (11th or 12th Eds.)

Soln: u=5.63x10-7 J/m3

Summary of Section 24.3

Homework for Section 24.3:

Problems 24.25, 24.27 and 24.30 (11th or 12th Eds.)

221 1

2 2 2

QU CV QV

C Energy stored in a capacitor:

Energy density:2

2

0

1 1

2 2

CVu E

Ad

Consider a capacitor with Q charges, it produces an E

200 0 0

1

2,

VE u E

d

Now, charge

it to same Q

Since medium polarizes, the net field inside is smaller

0 0 ,Inside Pol Inside DielectricVE E E E E

d

Now consider the capacitor with a polarizable medium (dielectric)

0

1

2

2

Dielectric insideu Eand its energy

is smaller

Can charge to same Q with less energy (smaller V)

Capacitance increases when using a dielectric

0Dielectric Dielectric

Dielectric

Q QV V C C

V V Since

0 0 0, , ,Q C V E , , ,Dielectric Dielectric DielectricQ C V E

The ratio

0

1DielectricCK

C , in terms of V: 0

Dielectric

VK

V

Capacitance increases when using a dielectric

Dielectric

Dielectric

Dielectric

V V

C C

E E

0K 0

An example…

Area = 2000 cm2, d=1 cm, V0=3000 V

1

12 10

0 0 2

2 108.85 10 1.77 10

1 10

2mF

m

AC

d

10 7

0 0 1.77 10 3000 5.31 10F V CQ C V

500 2

30003 10

10

VV/m

1 m

VE

d

Capacitance?

Charge?

Electric Field?

Now insert a dielectric . . .

Now insert dielectric,

voltage drops to V=1000 V

7105.31 10

5.31 101000

CF

V

QC

V

10

10

0

5.31 103.00

1.77 10

F

F

CK

C

12 11

0 3 8.85 10 2.66 10 2 2C /NmK

10

0 1.77 10 FC 75.31 10 CQ

5

0 3 10 V/mE

Capacitance?

Dielectric constant?

Permittivity

Electric Field?5

50 3 101 10

V/mV/m

3

EE

K

Again, V < V0, E < E0 and C > C0

10

0 1.77 10 FC 75.31 10 CQ

5

0 3 10 V/mE

Energy stored?

105.31 10 FC

51 10 V/mE

1000 VV 3000 VV

75.31 10 CQ

221 1

2 2 2

QU CV QV

C

2

0 0 0

10

4

1

2

11.77 10 3000

2

7.97 10

F V

F

U C V

2

10

4

1

2

15.31 10 1000

2

2.66 10

F V

F

U CV

Again, U0 > U

Homework for Section 24.4

(11th or 12th Eds.) :

Problem 24.43 a) and b) only

Problem 24.45

Problem 24.46

Solution: a) V remains unchanged, K=1.8

b) V=2 Volts

c) E0=999 N/C

E=999 N/C since the power supply was

connected during insertion of slab

Summary of Section 24.4

Capacitance increases when using a dielectric

0 0

0

1Dielectric

Dielectric Dielectric

C V EK

C V E

Dielectric

Dielectric

Dielectric

V V

C C

E E

0K

Homework for Section 24.4 (11th or 12th Eds.) :

Problem 24.43 a) and b) only, 24.45 and 24.46

Summary of Chapter 24

PHYS 2421 - Fields and WavesInstructor: Jorge A. López

Office: PSCI 209 A, Phone: 747-7528

Textbook: University Physics 11e, Young and Freedman

24.1 Capacitors and capacitance

24.2 Capacitors in series and parallel

24.3 Energy storage in capacitors

24.4 Dielectrics