Download Dielectric material

1
5. Conductors and dielectrics
EMLAB
Contents
2
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
5
5.1 Current and current density
S
Current
I
dQ
dt
n̂
J
I
nˆ
S
I
Current density
•
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.
J
I
S
I  J  S  I   J  da
S
• In field theory, the interest is
usually 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.
EMLAB
Current density from velocity and charge density
6
With known charge density and velocity,
current density can be calculated.

S
vt
Charges with
density ρ
Q  V ( volume)  S  t
Q
I
I 
 S  , J 
 v
t
S
EMLAB
Continuity equation : Kirchhoff ’s current law
7
J
Kirchhoff ’s current law
I
Q
J  
Charges going
out through dS.
dS
n̂

 0 ; Steady current .
t
For steady state, charges do not accumulate at
any nodes, thus ρ become constant.
q  J  nˆ St


Q    J  da dt  d   d
V
C

d

J

d
a



d




J
d



C
V
V t d
dt V
  J  

t

J  
t
differential form
I
n
n

dQ
t
integral form
EMLAB
8
Electrons in an isolated atom
Electron
energy
level
1 atom
-
+
-
-
-
-
-
-
-
Tightly bound
electron
-
More freely moving 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.
EMLAB
9
Electrons in a solid
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.
Freely moving
electron
+
Eext
External E-field
+
+
Tightly bound
electron
+
+
-
+
-
-
-
+
+
+
-
+
-
-
-
+
+
+
-
+
-
-
-
+
+
-
Electron
energy
level
-
-
-
To accommodate lots of electrons,
the discrete energy levels are
broadened.
EMLAB
10
Insulator and conductor
Insulator atoms
+
+
Conductor atoms
+
-
+
-
+
-
+
-
+
-
+
-
External E-field
+
-
+
-
+
-
+
-
-
-
External E-field
+
+
-
-
+
+
-
-
+
+
-
-
+
+
-
-
+
+
-
-
+
+
-
-
Empty energy level
-
Energy level of
insulator atoms
Occupied energy level
-
Energy level of
conductor atom
EMLAB
Movement of electrons in a conductor
11
EMLAB
12
EMLAB
Electron flow in metal : Ohm’s law
13
F  qE  eE
+
+
-
+
+
+
-
+
-
-
+
-
E
v   eE
+
-
+
-
-
+
-
-
+
-
+
-
+
+
-
+
Jv
-
+
-
• n: Electron density (number
of electrons per unit volume.
-
• μ : mobility
 (ne)(  e )E  neeE
J E
; Ohm’s law
 : Electric conductivity
EMLAB
14
Example : calculation of resistance
B
A
S
J
B
A
VBA    E  dr   E  dr  
A
VBA 
J

LBA 
B

V
R 
I
B
I
L
 S BA
VBA  IR , R 
A
B
J

 dr 
J

 L BA
A
S
l
S

E  dr



 J  da   E  da

S
E  dr

S
EMLAB
Conductivities of materials
15
EMLAB
16
Electric field on a conductor due to external field
tangential component
Et  0
E
E n  nˆ
normal component
+q1
-q1
S
0
-q1
Ein  E
Conductor
Conductor
1. Tangential component of an external E-field causes a positive charge (+q) to move in the
direction 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.
EMLAB
17
Charges 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
conductor.
Ein  0
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.
EMLAB
18
Image method
+q1
• 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.
Etan  0  E  nˆ  0
n̂
-
-
-
-
Perfect electric conductor
•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
Etan  0  E  nˆ  0
-q1
Image charge
EMLAB
19
Example : a point charge above a PEC plane
• The electric field due to a point charge is
influenced by a nearby PEC whose charge
distribution 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
surface, on which tangential electric field
becomes zero.
• This method is validated by the uniqueness
theorem which states that the solution that
satisfy a given boundary condition and
differential equation is unique.
E
+q1
n̂
Etan  0  E  nˆ  0
도체
+q1
 E( x  0) 
 E tan  0
x̂
(a ,0,0)
ẑ


x̂
ẑ
q1 xˆ ( x  a )  yˆ y  zˆ z
40 ( x  a ) 2  y 2  z 2 3 / 2
q
xˆ ( x  a )  yˆ y  zˆ z
 1
40 ( x  a ) 2  y 2  z 2 3 / 2

(a ,0,0)
Etan  0  E  nˆ  0

q1
 2a xˆ
 2
40 a  y 2  z 2


3/ 2
-q1
(a ,0,0)
Image charge
EMLAB
Dielectric material
20
molecule
The charges in the molecules force the molecules aligned so
that externally applied electric field be decreased.
EMLAB
21
Dielectric material
ẑ
(1) No material
D  zˆ S
 S
E0
(2) With dielectric material
x̂
 S
d
D  zˆ S Ein
E0
+
+
Ep
 S
E0   zˆ
 S
S
0
Ein  E0  E p , E p    e Ein
Ein  E0   e Ein
(1   e )Ein  E0  Ein 
D  E  0 (1  e )E
E0
1  e
 D   0 E0   0 (1   e )Ein   Ein
•
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.
EMLAB
22
Electric dipole
ẑ
P (0, r sin , r cos )
θ
Q (0, 0, d / 2)
+q
d
-q
Q (0, 0,  d / 2)
d (0, 0, d)
N
p   qiri  qd ; dipole moment .
i 1
 1
1 
q  PQ  PQ 

 



 PQ PQ  40  PQ  PQ 
q  PQ  PQ  qd cos 
p  rˆ




2
2
40 
r
40 r
40 r 2

qd cos 
 E  V 
rˆ 2 cos   θˆ sin 
40 r 3
V 
q
q
q


40 PQ 40 PQ 40


EMLAB
23
Electric field in dielectric material
 S
E0
x̂
 S
d
Ein
E0
+
+
+
Ep
 S
 S
S
0
E   ẑ
E0   zˆ
+
S
 0 r
Induced dipole에 의해 물질 내부 전
기장 세기 줄어듦. 도체 양단의 전
압을 측정하면 전압이 줄어듦.
EMLAB
Gauss’ law in Dielectric material
24
 E
0
in
 da  Q total  q free  q bound  q free   P  da
S
S
+
   0Ein  P   da  q free   D  da
S
S
+
 D   0Ein  P
+
+
+
+
+
p
Length : d
P   0  e Ein
+
+
+
+
+
+
+
+q1
+
+
+
+
Dipole
+
+
+
+
+
Induced
dipole
d
V
V  Sd   d nˆ  da
S
qbound  (V ) N (q)    Nqd nˆ  da    N p  da    P  da
S
S
S
EMLAB
Relative permittivity
25
EMLAB
26
Boundary 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.
w
C1
E1t  E2 t
τ̂
unit vector tangential to
the surface
Medium #2
V  0    E  ds  (E1  τˆ w  E2  τˆ w )
C1
Medium #1
Unit vector normal
to the surface
 E1  τˆ  E2  τˆ
(2) Boundary condition on normal component of electric field
n̂
S
Medium #2
h
S
Boundary condition on normal component can be obtained from
the result of surface integrals on a closed surface.
D2n  D1n  S  2E2n  1E1n  S
 D  da    D d    d
S
Medium #1
V
V
(D 2  nˆ  D1  nˆ )S  Dcurved side  τˆ h  hS
If h  0,
Dcurved side  τˆ h  0
 D 2  nˆ  D1  nˆ  h   S (surface charge density)
EMLAB
Example – conductor surface
tangential component
27
Et  0
E
E n  nˆ
normal component
+q1
-q1
S
0
-q1
Ein  E
Conductor
Conductor
1. Tangential component of an external E-field causes a positive charge (+q) to move in the
direction 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.
EMLAB
Example – dielectric interface
28
Surface charge density of dielectric
interface can not be infinite.
2
1
2
1
E1
E2
tangential component : Et1  Et 2
normal component
: Dn1  Dn 2
E t 2  E t1  E 2 sin 2  E1 sin 1
Dn 2  Dn1   2E n 2  1E n1   2E 2 cos 2  1E1 cos 1
E2 
E2 sin 2   E2 cos 2 
2
2

E1 sin 1 
2


  1 E1 cos 1 
 2

2
 E1
 
sin 1   1  cos 2 1
 2 
2
EMLAB
2
Example – dielectric interface
D 
Dx Dy Dz Dz



0
x
y
z
z
29
The normal component of D
is equal to the surface
charge density.
 Dz  C
 Dz   S
ẑ
ŷ
 D1z  D2 z   S
x̂
1E1z   2 E2 z   S
Capacitance :
0
V   E  dr  E2 z d 2  E1z d1 
d
S

d 2  S d1
2
1
Q   D  da  ( zˆ )(   s zˆ ) S   s S
S
C 
Q
sS
S


d 2 d1
V S d  S d

2
1
2
1
2
1
EMLAB
Static electric field : Conservative property
30
VB
C3
VA
B
C2
A
VBA    E  dr    E  dr
C1
C1
C2
정전기장에 의한 potential difference VAB는 시작점과 끝점이 고정된 경우, 적분
경로와 상관없이 동일한 값을 갖는다.
 E  dr   E  dr   E  dr  0
C
C1
 E  dr  0
C
C2
E  0
EMLAB
Stokes’ theorem
31
 E  dr    E  da
C
•
S
벡터함수를 임의의 닫혀진 경로에 대한 선적분을 하는 경우 그 경
로로 둘러싸인 면에 대한 면적분으로 바꿀 수 있다.
•
이 때 피적분 함수는 원래 함수의 ‘curl’로 바꿔야 한다.
 E  dr
C
  E  dr
n
Cn
Cn
임의의 닫혀진 경로에 대한 선적분은 매우 작은 폐곡선의 선적분의 합으로 분해할
수 있다.
EMLAB
Line integral over an infinitesimally small closed path
E  E x xˆ  E y yˆ  E z zˆ
32
ẑ
( x, y , z )
ŷ
x̂
2
3
4
1
2
3
4
 E  dr   E y dy   Ex dx   E y dy   Ex dx
Cn
1
2
3
4
1
x 
y 
x 
y 




  E y  x  dy   E x  y  dx   E y  x  dy   E x  y  dx
1
2
3
4
2 
2 
2 
2 




x 
x 
y 
y 
 
 


  E y  x    E y  x   y   E x  y    E x  y   x
2 
2 
2 
2 


 
 
 E E 
  y  x  xy     E  da
S
y 
 x
(  E)  zˆxy
E  dr

 E y E x 
Cn
 (  E) z  

 Lim
y  S 0 xy
 x
 E E   E E   E E 
  E  xˆ  z  y   yˆ  x  z   zˆ  y  x 
z   z
x   x
y 
 y
EMLAB