Transcript 전 자 기 학

Chapter 5. Conductors, Dielectrics, and
Capacitance
1.
Current and Current Density
•
Current(A) : a rate of movement of charge passing a given
reference point (or crossing a reference plane).
dQ
dt
Current density : J (A/m 2 )
I  J N  S  J  S
I
I   J  dS
Q   v Sx
Q
x
  v S
  v Sv x
t
t
J x  vvx
I 
J  v v
목원대학교 전자정보통신공학부 전자기학
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2.
Continuity of Current
•
The principle of conservation of charge: charges can be
neither created nor destroyed.
dQi
: decrease of positive charge inside the closed surface
S
dt
 v
d
J

d
S

(


J
)
dv



dv


v
S
vol
vol t dv : constant surface
dt vol

(  J )v   v v
t
 v
I   J  dS  
(  J )  
t
•
The current, or charge per second, diverging from a small
closed surface per unit volume is equal to the time rate of
decrease of charges per unit volume at every point.
•
A numerical example: p. 123
목원대학교 전자정보통신공학부 전자기학
5-2
3.
Metallic Conductors
In the conductors , electrons : Q  e  F  eE
Free space : the electron w ould accelerate and continuous ly increase its velocity.
Crystallin e material : collisions with the thermally excited crystallin e lattice structure
 constant average velocity ( drift velo city v d )
v d   e E
 e : the mobility of electron (  e  0)
J  v v 
J  e e E
목원대학교 전자정보통신공학부 전자기학
 e  0 : the free - electron charge density
5-3
The point form of Ohm’s law
J  E
 : conductivi ty
  e e
Isotropic: same properties in every direction
Anisotropic: not isotropic
Resistivity: reciprocal of the conductivity
Superconductivity: the resistivity drops abruptly to zero at a few kelvin
Higher temperature→greater crystalline lattice vibration→lower drift
velocity →lower mobility →lower conductivity →higher resistivity
J , E : uniform
I   J  dS  JS
S
a
a
Vab    E  dL  E   dL  E  L ba E  L ab
b
V  EL
b
I
V
J   E  
S
L
V 
Ohm' s law : V  RI where R 
L
I
S
L
S
a
목원대학교 전자정보통신공학부 전자기학
Vab  b E  dL
R

I
 σE  dS
S
5-4
4.
Conductor Properties and Boundary Conditions
•
Suppose that there suddenly appear electrons in the interior of a
conductor→Electric fields by these electrons →The electrons
begin to accelerate away from each other →The electrons reach
the surface of the conductor
•
Good conductor: zero charge density within a conductor and a
surface charge density resides on the exterior surface

No charge, no electric field within a conducting material

•
Relate external fields to the charge on the surface of the conductor
The external electric field intensity is decomposed into tangential
component and normal component to the conductor surface.
Static condition: tangential one may be zero. If not, there will result
in a movement of electrons.
•
목원대학교 전자정보통신공학부 전자기학
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 Guass’s law: The electric flux leaving a small increment of surface
must be equal to the charge residing on that incremental surface.
• The flux must leave the surface normally!
• The flux density per square meter leaving the surface normally is equal
to the surface charge density per square meter D  
N
 E  dL  0

b
a

c
b

d
c

a
d
0
Rememberin g E  0 within th e conductor
1
1
Et w  E N , at b h  E N , at a h  0
2
2
h  0, keeping w small but finite : Et w  0 Et  0
목원대학교 전자정보통신공학부 전자기학
S
 D  dS  Q 
S
top

DN S  Q   S S
bottom

sides
Q
DN   S
5-6
Boundary conditions for the conductor-free space boundary in electrostatics
Dt  Et  0
DN   o E   S
Summary: p. 132
5. The Method of Images
•
The dipole field: the infinite plane at zero potential that
exists midway between the two charges.
Remove conducting
plane and locating a
negative charge (image)
목원대학교 전자정보통신공학부 전자기학
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6.
Semiconductors
•
Current carriers: electrons (conduction band), holes (valence band)
 e, e (electrons ),  e, h (holes)
•
•
  - e  e   h  h
Temperature↑: mobility↓, charge density ↑(more rapidly)
 Conductivity ↑
Doping
•
•
Donors: additional electrons, n-type
Acceptors: extra holes, p-type
목원대학교 전자정보통신공학부 전자기학
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7.
The Nature of Dielectric Materials
•
•
•
•
Bound charges: bound in place by atomic and molecular forces.
Only shift positions slightly in response to external fields.
Dielectric materials can store electric energy (a shift in the relative
positions of the internal, bound positive and negative charges
against the normal molecular and atomic forces)
Polar molecule: random dipole → alignment
Nonpolar molecule: dipole arrangement after a field is applied
p  Qd
Eq.(37) in Sec. 4.7
nv
p total   p i
n dipoles per unit volume
Random orientatio n of p i  p total  0
i 1
•
Define: Polarization as the dipole moment per unit volume
1 nv
P  lim
pi

v  0 v
i 1
목원대학교 전자정보통신공학부 전자기학
5-9
Assume nonpolar molecules
Apply an E  movement of positive and negative charges
1
Charges within d cos  of the surface S cross the surface
2
n molecules/ m 3 : Q b  nQd cos S  nQd  S
Q b  P  S
The net total charge which
crosses the elemental
surface
The net increase in the bound charge within the closed surface
Qb   P  dS
S
(*Resemblance to Gauss’s law)
목원대학교 전자정보통신공학부 전자기학
5-10
•
Generalize the definition of electric flux density
QT    o E  dS where QT  Qb  Q
S
QT : total enclosed charge, Qb : bound charge, Q : free charge
Q  QT  Qb   ( o E  P)  dS
S
D   oE  P
Qb    b dv
v
  P  b
QT    T dv
   o E  T
Q    v dv
  D  v
v
v
•
Isotropic material: linear relationship between E and P
P   e o E
D   o E   e  o E  (  e  1) o E   o  R E  E
( R   e  1: reletive permitivit y or dielectric constant )
목원대학교 전자정보통신공학부 전자기학
   o  R : permittivi ty
5-11
8.
Boundary Conditions for Perfect Dielectric Materials
 E  dL  0
h  0
Etan 1w  Etan 2 w  0
E tan 1  E tan 2
Dtan 1
1
 D  dS  Q
S
 E tan 1  E tan 2 
DN 1S  DN 2 S  Q   S S
Perfect dielectrcs :  S  0
Dtan 2
2
or
Dtan 1  1

Dtan 2  2
DN 1  DN 2   S
DN 1  DN 2
 1 E N1   2 E N 2
목원대학교 전자정보통신공학부 전자기학
5-12
D N 1  D1 cos 1  D2 cos  2  D N 2
Dtan 1
D sin 1  1
 1

or  2 D1 sin 1   1 D2 sin  2
Dtan 2 D2 sin  2  2

tan 1  1

if  1   2 , 1   2
tan  2  2
목원대학교 전자정보통신공학부 전자기학
2
D2  D1

cos 2 1   2
 1

 sin 2 1

E 2  E1

sin 2 1   1
2

 cos 2 1

2
5-13
The boundary conditions at the interface between a conductor and a dielectric
1. D and E are both zero inside the conductor
2. The tangentia l E and D field components must both be zero to satisfy  E  dL  0 and D  E
3. Gauss' s law
 D  dS  Q,
S
DN   S , E N   S / 
Dt  Et  0
DN  E N   S
How any charge introduced within a conductor arrives at the surface
(surface charge)
 v



   E   v ,   D   v
t
t

t
  v
  v

 D  
, v  
  v   o exp(  t )
 t
 t

J  E ,   J  
목원대학교 전자정보통신공학부 전자기학

: relaxation time

5-14
9.
Capacitance
Q at M 2 and - Q at M1 : total charge  0
Surface charge, normal electric field,
equipotential surface
Electric Flux from M 2 to M1
Define : capacitanc e C 
Q
Vo
E  dS

C
  E  dL
S


Charge density  Electric flux density  Electric field intensity  Potential difference 
The ratio is constant
The capacitance is a function only of the physical dimensions of the system
of conductors and of the permittivity of the homogeneous dielectric.
목원대학교 전자정보통신공학부 전자기학
5-15
E
S
a z and D   S a z

Vo   
lower
upper
Q  S S
E  dL   
0
d
0
S

a z  dza z    S dz  S d
d 


C
Q S

d
Vo
The total energy stored in the capacitor
1
1 S d  S2
1  S2
1 S  S2 d 2
2
WE   E dv     2 dzdS 
Sd 
vol
0
0
2
2
2 
2 d 2

1
1
1 Q2
2
WE  CVo  QVo 
2
2
2 C
목원대학교 전자정보통신공학부 전자기학
5-16
10. Several Capacitance Examples
A coaxial capacitor (inner radius a, outer radius b)
L
L
a
E

a

S
2
2
a 


b
a
L
Vo   
d   L ln  b  L ln
b 2
2
2 a
D  dS  Q 2LD   L L D 
C
2L
ln( b / a)
A spherical capacitor (inner radius a, outer radius b)
a
a
Q
Q
Q  1
Q 1 1
Er 
V


dr






  
o
b 4r 2
4  r  b 4  a b 
4r 2
C
Q
4

Vo 1 1

a b
Capacitance of an isolated spherical conductor
C  4a
목원대학교 전자정보통신공학부 전자기학
5-17
Coating this sphere with a different dielectric material
Dr 
Q
Q
Q
Er 
(a  r  r1 ) Er 
(r1  r )
2
2
4r
41r
4o r 2
r1
Q

4o r 2
Va  V   
C
a
Q
r1
41r 2
dr  
dr 
Q
4o r1

Q 1 1 Q 1 1 1 1 
  
   

41  a r1  4   1  a r1   o r1 
4
1 1 1 1
   
1  a r1   o r1
Multiple dielectrics
D1  D2   S
 1 E1   2 E2   S
Vo 
C

목원대학교 전자정보통신공학부 전자기학
E1 
S

E2  S
1
2
S

d1  S d 2 Q   S S
1
2
S S

d1  S d 2
1
2
Q

Vo  S
1
d1
d
 2
1S  2 S

1
1
1

C1 C 2
5-18
11. Capacitance of a Two-Wire Line
zero reference at R10 and R20
V 
V
Choose R10  R20 ( x  0 plane)
R
L
  L R20
ln 10 V 
ln
2 R1
2
R2
R  
R R
 L  R10
 ln
 ln 20   L ln 10 2
2  R1
R2  2 R20 R1
L
 L ( x  a) 2  y 2
( x  a) 2  y 2
V
ln

ln
2
( x  a) 2  y 2 4 ( x  a) 2  y 2
목원대학교 전자정보통신공학부 전자기학
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