Department of Semiconductor Systems Engineering SoYoung Kim

Engineering Electromagnetics- 1 Lecture 8: Electric Dipole

SoYoung Kim

ksyoung@skku.edu

Department of Semiconductor Systems Engineering

College of Information and Communication Engineering

Sungkyunkwan University

Department of Semiconductor Systems Engineering SoYoung Kim

Outline

Electric Potential Review

Electric Dipole

Energy

Energy in Field

Department of Semiconductor Systems Engineering SoYoung Kim

Review

Department of Semiconductor Systems Engineering SoYoung Kim

Department of Semiconductor Systems Engineering SoYoung Kim

Department of Semiconductor Systems Engineering SoYoung Kim

Electric Dipole: Potential

Electric dipole: two point charges with opposite signs seprated by a small distance

Dipole moment p:

Electric potential due to an electric dipole

+Q

-Q

d d

In terms of dipole moment,

Dipole located at origin Dipole located at r’

2 1

1 2 1 2

2

2 1 2 1

2 2

1 1

4 4

If : ~ cos , ~

cos cos

4 4

o o

o o

r rQ QV

r r r r

r d r r d r r r

Q d pV

r r

32

( ') or ( )

4 4 '

r

o o

V Vr

p a p r rr

r r

p dQ

Department of Semiconductor Systems Engineering SoYoung Kim

Find the electric field

?E

Eplot

Department of Semiconductor Systems Engineering SoYoung Kim

Electric Dipole: Electric Field

From the definition of electric potential

Comparison with monopole

For monopole

Electric field varies inversely as r2

Potential varies inversely as r

For dipole

Electric field varies inversely as r3

Potential varies inversely as r2

3 3

1 cos sin

2 4r r

o o

V V Q d Q dV

r r r r

E a a a a

3(2 cos sin )

4E a a

r

o

p

r

Department of Semiconductor Systems Engineering SoYoung Kim

Electric Flux Line and Equipotential Surface

Electric flux line Lines whose direction at any point is same as the direction

of the electric field at the point

Lines to which electric flux density D is tangential at every

point

Equipotential surface Surface on which the potential is the same

No work is done in moving a charge on the equipotential surface

2D version is called equipotential lines

Equipotential lines and electric flux lines are always normal

Department of Semiconductor Systems Engineering SoYoung Kim

Energy in Electrostatic Field: Point Charge

Assume you move charges Q1, Q2, Q3 to positions P1, P2, P3

If you move the charges in the order of Q1, Q2, Q3, total work done is:

If you move in reverse order:

By adding the two:

In general,

Assume Vij is potential at point Pi due to charge Qj in this analysis

[J]

1 2 3

2 21 3 31 320 ( )

EW W W W

Q V Q V V

3 2 1

2 23 1 12 130 ( )

EW W W W

Q V Q V V

1 12 13 2 21 23 3 31 32

1 1 2 2 3 3

1 1 2 2 3 3

2 ( ) ( ) ( )

1( )

2

E

E

W Q V V Q V V Q V V

Q V Q V Q V

W Q V Q V Q V

1

1

2

n

E k k

k

W Q V

Solve Ex. 4.14

Department of Semiconductor Systems Engineering SoYoung Kim

Energy in Electrostatic Field: Distributed Charge

For distributed charge

1 (line charge)

2

1 (surface charge)

2

1 (volum e charge)

2

1( )

2D D

E LL

E SS

E vv

E vv

W V dl

W V dS

W V dv

W V dv

Department of Semiconductor Systems Engineering SoYoung Kim

Energy and Energy Density in Electrostatic Field

Electrostatic energy

Electrostatic energy density

2

2 31 1 [J/m ] or

2 2 2

E

E o E Ev

o

dW Dw E W w dv

dv

D E

U sing vector identity ( ) ( )

1 1 ( ) ( )

2 2

Applying divergence theorem ,

1 1 ( ) ( )

2 2

First term on R H S w ill becom e zero

A A A A A A

D D

D S D

Ev v

Ev

V V V V V V

W V dv V dv

W V d V dv

2

1 1 ( ) ( )

2 2

1 1

2 2

D D E

D E

Ev v

E ov v

W V dv dv

W dv E dv

[J]

Department of Semiconductor Systems Engineering SoYoung Kim

Ex. 4.15

From Gauss’s law, we obtained

3 3

0 0

2

0 0

3

2 20 0 0

2

0 0 0

3 3

33 3 6

r r

r

r R r R r

r r rR R

R Rr R V E dr dr

r r

r R

R rV E dr E dr E dr dr dr R r

r