EMLAB

1

5. Conductors and dielectrics

EMLAB

2Contents

1. Current and current density

2. Continuity of current

3. Metallic conductors

4. Conductor properties and boundary conditions

5. The method of images

6. Semiconductors

7. Dielectric materials

8. Boundary conditions for dielectric materials

EMLAB

3

Current and voltage

EMLAB

4

EMLAB

55.1 Current and current density

dt

dQI

I

S

dII aJSJ

nJ ˆS

I

n

• Current is electric charges in motion, and is defined as the rate of movement of charges passing a given reference plane.

• In the above figure, current can be measured by counting charges passing through surface S in a unit time.

S

• In field theory, the interest is usu-ally in event occurring at a point rather than within some large region.

•For this purpose, current density measured at a point is used, which is current divided by the area.

I

J

S

Current

Current density

EMLAB

6Current density from velocity and charge density

tv

S

vJ

S

IS

t

QI

tSVQ

,

volume)(

Charges with density ρ

With known charge density and velocity, cur-rent density can be calculated.

EMLAB

7Continuity equation : Kirchhoff ’s current law

Q I

J

tSq nJ ˆ

Charges going out through dS.

nFor steady state, charges do not accumulate at any nodes, thus ρ become constant.

.currentSteady;0

t

J

t

dt

dddt

dd

dddtdQ

VVVC

VC

J

JaJ

aJ

dS

t

J t

dQI

nn

differential form

integral form

Kirchhoff ’s current law

EMLAB

8

+

-

Electron energy level

- -

- -

- -

1 atom

Electrons in an isolated atom

Tightly bound electron

Energy levels and the radii of the electron orbit are quantized and have discrete values. For each energy level, two electrons are accommodated at most.

-

-

More freely moving electron

EMLAB

9

+-

+-

+-

+-

+-

+-

+-

+-

+-

+-

+-

+-

+-

+-

+-

+-

Atoms in a solid are arranged in a lattice structure. The electrons are attracted by the nuclei. The amount of attractions differs for various material.

Electrons in a solid

Freely moving elec-tron

Tightly bound elec-tron

-

Electron energy level

To accommodate lots of electrons, the discrete energy levels are broad-ened.

extE

External E-field

EMLAB

10

-

Energy level of insula-tor atoms

+-

+-

+-

+-

+-

+-

+-

+-

+-

+-

+-

+-

+-

+-

+-

+-

-

Energy level of con-ductor atom

+-

+-

+-

+-

+-

+-

+-

+-

Insulator and conductorInsulator atoms Conductor atoms

Occupied energy level

Empty energy level

External E-field External E-field

EMLAB

11Movement of electrons in a conductor

EMLAB

12

EMLAB

13

+-

+-

+-

+-

+-

+-

+-

+-

+-

+-

+-

+-

+-

+-

+-

+-

E

EEF eq

Ev e

EE ee nene ))((

vJ

EJ

: Electric conductivity

; Ohm’s law

Electron flow in metal : Ohm’s law

• n: Electron density (number of electrons per unit vol-ume.

• μ : mobility

EMLAB

14

J

AB

BA

A

B

A

B

B

ABA dddV LJ

rJ

rErE

BABABA LS

IL

JV

S

lRIRVBA ,

Example : calculation of resistance

SSd

d

d

d

I

VR

aE

rE

aJ

rE

S

AB

S

EMLAB

15Conductivities of materials

EMLAB

16

E

-q1+q1

Conductor

1. Tangential component of an external E-field causes a positive charge (+q) to move in the di-rection of the field. A negative charge (-q) moves in the opposite direction.

2. The movement of the surface charge compensates the tangential electric field of the external field on the surface, thus there is no tangential electric field on the surface of a conductor.

3. The uncompensated field component is a normal electric field whose value is proportional to the surface charge density.

4. With zero tangential electric field, the conductor surface can be assumed to be equi-potential.

0

Sn

t

ˆ

0

nE

E

-q1

Conductor

EE in

Electric field on a conductor due to external field

normal component

tangential component

EMLAB

17Charges on a conductor

1. In equilibrium, there is no charge in the interior of a conductor due to repulsive forces between like charges.

2. The charges are bound on the surface of a conduc-tor.

3. The electric field in the interior of a conductor is zero.

4. The electric field emerges on the positive charges and sinks on negative charges.

5. On the surface, tangential component of electric field becomes zero. If non-zero component exist, it induces electric current flow which generates heats on it.

0inE

EMLAB

18Image method

• If a conductor is placed near the charge q1,

the shape of electric field lines changes due to the induced charges on the conductor.

• The charges on the conductor redistribute themselves until the tangential electric field on the surface becomes zero.

•If we use simple Coulomb’s law to solve the problem, charges on the conductors as well as the charge q1 should be taken into account. As

the surface charges are unknown, this approach is difficult.

• Instead, if we place an imaginary charge whose value is the negative of the original charge at the opposite position of the q1, the

tangential electric field simply becomes zero, which solves the problem.

+q1

Perfect electric conductor

0ˆ0tan nEEn

-q1

0ˆ0tan nEE

Image charge

+q1

- - - -

EMLAB

19

+q1

도체

0ˆ0tan nEE

n

-q1

0ˆ0tan nEE

Image charge

+q1

0

zya

ˆa2

4

q)0x(

zy)ax(

zˆyˆ)ax(ˆ

4

q

zy)ax(

zˆyˆ)ax(ˆ

4

q

tan

2/32220

1

2/32220

1

2/32220

1

E

xE

zyx

zyxE

z

x)0,0,a(

)0,0,a(

)0,0,a(

• The electric field due to a point charge is in-fluenced by a nearby PEC whose charge distri-bution is changed. In this case, an image charge method is useful in that the charges on the PEC need not be taken into account.

•As shown in the figure on the right side, the presence of an image charge satisfies the boundary condition imposed on the PEC sur-face, on which tangential electric field be-comes zero.

• This method is validated by the uniqueness theorem which states that the solution that sat-isfy a given boundary condition and differen-tial equation is unique.

z

x

Example : a point charge above a PEC plane

EMLAB

20

molecule

Dielectric material

The charges in the molecules force the molecules aligned so that externally applied electric field be decreased.

EMLAB

21Dielectric material

S

0

S0 ˆ

zE

d

zx

S

0E -+

-+

S S

pE

inESˆ zD Sˆ zD

• D (electric flux density) is related with free charges, so D is the same despite of the dielectric material.

• But the strength of electric field is changed by the induced dipoles inside.

EED )1( e0

(1) No material (2) With dielectric material

inine

einine

inein

ineppin

EEED

EEEE

EEE

EEEEE

)1(

1)1(

,

000

00

0

0

0E

EMLAB

22

+q

-q

)cosr,sinr,0( P

)2/d,0,0(Q

)2/d,0,0(Q

)d,0,0(d

d

.momentdipole;1

drp qqN

iii

θ

z

Electric dipole

sinˆcos2ˆ4

cos

4

ˆ

4

cos

4

4

11

444

30

20

20

20

0000

θrE

rp

r

qdV

rr

qd

r

PQPQq

PQPQ

PQPQq

PQPQ

q

PQ

q

PQ

qV

EMLAB

23

S

d

xS

0E -+

-+

S S

pE

inE0E

0

S0 ˆ

zE

r

S

0

zE

-+

-+

Electric field in dielectric material

Induced dipole 에 의해 물질 내부 전기장 세기 줄어듦 . 도체 양단의 전압을 측정하면 전압이 줄어듦 .

EMLAB

24

-+ -+

+q1-

+

-+ -

+

- +

-+

-+

-+

-+

-+

-+

-+

-+

-+

-+

-+

-+

-+

- +

-+-

+ -+

- +

-+

-+

-+

-+ -

+

-+

-+ -

+

-+-+

-+

-+

-+

-+

-+

-+

-+ -

+-+

-+

-+

-+

-+

-+

-+

Gauss’ law in Dielectric material

PED

aDaPE

aPaE

in

SS

in

SS

in

dd

dd

0

free0

freeboundfreetotal0

q

qqqQ

-+

- +

DipoleLength : d

d

SSS

S

ddNdNqdqNVq

ddSdV

aPapan

an

ˆ)()(

ˆ

bound

V

ineEP 0

Induced dipole

p

EMLAB

25Relative permittivity

EMLAB

26Boundary conditions

(1) Boundary condition on tangential electric field component

Tangential boundary condition can be derived from the result of line integrals on a closed path.

1C

τEτE

τEτEsE

ˆˆ

)wˆwˆ(d0V

21

C

21

1

unit vector tangential to the surface

t2t1 EE

(2) Boundary condition on normal component of electric field

Boundary condition on normal component can be obtained from the result of surface integrals on a closed surface.

density) charge surface(ˆˆ

0ˆ

,0 If

ˆ)ˆˆ(

12

side curved

side curved12

S

S VV

h

h

h

ShhS

ddda

nDnD

τD

τDnDnD

DD

Sn11n22Sn1n2 EEDDS

n

Unit vector normal to the surface

S

h

Medium #1

Medium #2

Medium #1

Medium #2

τ

w

EMLAB

27

E

-q1+q1

Conductor

1. Tangential component of an external E-field causes a positive charge (+q) to move in the di-rection of the field. A negative charge (-q) moves in the opposite direction.

2. The movement of the surface charge compensates the tangential electric field of the external field on the surface, thus there is no tangential electric field on the surface of a conductor.

3. The uncompensated field component is a normal electric field whose value is proportional to the surface charge density.

4. With zero tangential electric field, the conductor surface can be assumed to be equi-potential.

0

Sn

t

ˆ

0

nE

E

-q1

Conductor

EE in

Example – conductor surface

normal component

tangential component

EMLAB

28

1E

2E21

21

: component normal

:component tangential

nn

tt

DD

EE

1

2

12

2

2

11

21

2

112

1211

222

2222

1112221n12n21n2n

11221t2t

cossinE

cossincossin

coscos

sinsin

EEEEE

EEEEDD

EEEE1

2

Surface charge density of dielectric interface can not be infinite.

Example – dielectric interface

EMLAB

29Example – dielectric interface

0

z

D

z

D

y

D

x

D zzyxD

Sz

z

D

CD

The normal component of D is equal to the surface charge density.

x

yz

Szz DD 21

Szz EE 2211

11

22

1122

0dddEdEdV SS

zzd

rE

SSdQ ssS )ˆ)(ˆ( zzaD

1

1

2

21

12

2

dd

S

dd

S

V

QC

SS

s

Capacitance :

EMLAB

30

0C drE 0 E

Static electric field : Conservative property

정전기장에 의한 potential difference VAB 는 시작점과 끝점이 고정된 경우 , 적분

경로와 상관없이 동일한 값을 갖는다 .

21 CC

BA ddV rErE

AAV

BBV

1C

2C

3C

021

CC

Cddd rErErE

EMLAB

31

n

CndrE

C

drE

nC

임의의 닫혀진 경로에 대한 선적분은 매우 작은 폐곡선의 선적분의 합으로 분해할 수 있다 .

Stokes’ theorem

aErE ddSC

• 벡터함수를 임의의 닫혀진 경로에 대한 선적분을 하는 경우 그

경로로 둘러싸인 면에 대한 면적분으로 바꿀 수 있다 .

• 이 때 피적분 함수는 원래 함수의 ‘ curl’ 로 바꿔야 한다 .

EMLAB

32Line integral over an infinitesimally small closed path

xy

yEy

yEyx

xEx

xE

dxy

yEdyx

xEdxy

yEdyx

xE

dxEdyEdxEdyEd

xxyy

xyxy

xyxyCn

2222

2222

1

4

4

3

3

2

2

1

1

4

4

3

3

2

2

1rE

S

xy dyxy

E

x

EaE

yx

d

y

E

x

EnC

S

xyz

rEE

0Lim)(

),,( zyx y

x

z

yxz ˆ)( E

zyxE ˆˆˆ zyx EEE

y

E

x

E

x

E

z

E

z

E

y

E xyzxyz zyxE ˆˆˆ