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).
I
dQ dt
Current density
I
J N
S
J
:
J
(A/m
S
2 )
Q
I J x
v
S
x
Q
t v v x
v
S
x
t
v
Sv x
J
v
v
I
S
목원대학교 전자정보통신공학부 전자기학 5-1
2.
• •
Continuity of Current
The principle of conservation of charge: charges can be neither created nor destroyed.
I
S
J
d
S
dQ i dt
: decrease of positive charge inside the closed surface
S
J
d
S
vol (
J
)
dv
d dt
vol
v dv
vol
t v
(
J
)
v
t v
v
(
J
)
t v dv
: constant surface 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
e
E
Free space : the electron w ould accelerate and continuous ly increase its velocity.
Crystallin e material : collisions with the thermally constant
v
d
e
average velocity
E
e ( drift velo : the mobility of city
v
electron )
d
( e excited crystallin 0 ) e lattice structure
J
v
v
J
e
e
E
e
0 : the free electron charge density 목원대학교 전자정보통신공학부 전자기학 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 목원대학교 전자정보통신공학부 전자기학
J
,
E
: uniform
I
S
J
d
S
JS V ab V
b a EL
E
d
L
J
I S
E
E b
a d
L
E
L
ba
V L
V
E
L
ab
L
S I
Ohm' s law :
V
RI
where
R
L
S R
V ab I
S σ b a
E E
d
L
d
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 5-5
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
S
E
d
L
0
a b
Rememberin g
E
b c
c
d
0 within th
d a
0 e conductor
E t
h
w
E N
, at b 1
h
E N
, at a 2 0 , keeping
w
small 1 2
h
0 but finite :
E t
w
0
E t
0 목원대학교 전자정보통신공학부 전자기학
S
D
d
S
D N
S
Q
top
Q
bottom
S
S D N
sides
S
Q
5-6
Boundary conditions for the conductor-free space boundary in electrostatics
D t
E t
0
D N
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.
목원대학교 전자정보통신공학부 전자기학 Remove conducting plane and locating a negative charge (image) 5-7
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 5-8
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
Q
d
Eq.(37) in Sec.
4.7
p
total
n
v i
1
p
i n
dipoles per unit volume Random orientatio n of
p
i
p
total 0 Define: Polarization as the dipole moment per unit volume
P
lim
v
0 1
v i n
v
1
p
i
목원대학교 전자정보통신공학부 전자기학 5-9
Assume nonpolar molecules Apply an
E
movement of positive and negative charges Charges within 1 2
d
cos of the surface
S
cross the surface
n
molecules/ m 3 : Q
b
nQd
cos
S
nQ
d
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
Q b
S
P
d
S
(
*Resemblance to Gauss
’
s law
) 목원대학교 전자정보통신공학부 전자기학 5-10
• Generalize the definition of electric flux density
Q T
S
o
E
d
S
where
Q T
Q b
Q Q T
: total enclosed charge,
Q b Q
Q T
Q b
S
(
o
E
: bound charge,
Q
P
)
d
S
: free charge
D
o
E
P
Q b
v
b dv Q T Q
v
T dv
v
v dv
P
o
E
b
T
D
v
• Isotropic material: linear relationship between E and P
P
e
o
E D
(
R
E
o
e
e
o
E
1 : reletive (
e
1 )
o
E
permitivit
o
R
E
y or dielectric
E
constant ) 목원대학교 전자정보통신공학부 전자기학
o
R
: permittivi ty 5-11
8.
Boundary Conditions for Perfect Dielectric Materials
h
0
S
D
d
S
Q
L
0
E
tan 1
w
E
tan 2
w
0
E
tan 1
E
tan 2
D
tan 1 1
E
tan 1
E
tan 2
D
tan 2 2 or
D
tan 1
D
tan 2 1 2
D N
1
S
D N
2
S
Q
S
S D N
1
D N
2
S
Perfect dielectrcs :
S
0
D N
1
D N
2 1
E N
1 2
E N
2 목원대학교 전자정보통신공학부 전자기학 5-12
D N
1
D
1 cos 1
D
2 cos 2
D N
2
D
tan 1
D
tan 2
D
1 tan tan
D
2 1 2 sin sin 1 2 1 2 if 1 1 2 or 2 , 2 1
D
1 sin 2 1 1
D
2 sin 2 목원대학교 전자정보통신공학부 전자기학
D
2
D
1 cos 2 1 2 1 2 sin 2 1
E
2
E
1 sin 2 1 1 2 2 cos 2 1 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
d
L
0 and
D
E
3.
Gauss' s law
S
D
d
S
Q
,
D N
S
,
E N
S
/
D t
E t
0
D N
E N
S
How any charge introduced within a conductor arrives at the surface (surface charge)
J
E
,
D
J
t
t v v
,
v
E
t v
t v
,
v
o
D
exp(
t v
t
) 목원대학교 전자정보통신공학부 전자기학 : relaxation time 5-14
9.
Capacitance
Q
at
M
2 and -
Q
at
M
1 : total charge 0 Surface charge, normal electric field, equipotential surface Electric Flux from
M
2 to
M
1 Define : capacitanc e
C
Q V o C
S
E
E
d
S
d
L
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
V o Q
S
S S
a
z
and
D
S
a
z
lower upper
E
d
L
d
0
S
a
z
dz
a
z
d
0
S dz
S d C
Q V o
S d
The total energy stored in the capacitor
W E
1 2 vol
E
2
dv
1 2 0
S
0
d
S
2 2
dzdS
1 2
S
2
Sd
1 2
S d
S
2 2
d
2
W E
1 2
CV o
2 1 2
QV o
1 2
Q
2
C
목원대학교 전자정보통신공학부 전자기학 5-16
10. Several Capacitance Examples
A coaxial capacitor (inner radius
a,
outer radius
b
)
S V o
D
d
S
b a
2
Q
2
LD L
d
2
L
L L
D
ln
a b
2
L
a
2
L
ln
b a
E
2
L
a
C
2
L
ln(
b
/
a
) A spherical capacitor (inner radius
a,
outer radius
b
)
E r
Q
4
r
2
V o
b a Q
4
r
2
dr
Q
4 1
r b a
Q
4 1
a
1
b C
Q V o
4 1
a
1
b
Capacitance of an isolated spherical conductor
C
4
a
목원대학교 전자정보통신공학부 전자기학 5-17
Coating this sphere with a different dielectric material
D r
Q
4
r
2
E r
Q
4 1
r
2 (
a
r
r
1 )
E r
Q
4
o r
2 (
r
1
r
)
V a C
V
r
1
Q
4
o r
2 1 1 1
a
4 1
r
1 1
o r
1
dr
r
1
a Q
4 1
r
2
dr
Q
4
o r
1
Q
4 1 1
a
1
r
1
Q
4 1 1 1
a
1
r
1 1
o r
1 Multiple dielectrics
D
1
D
2
S
1
E
1 2
E
2
S E
1 1
S E
2 2
S V o
S
1
d
1
S
2
d
2
Q
S S
목원대학교 전자정보통신공학부 전자기학
C
Q V o
1
d
1 1
S
d
2 2
S
1
S d
1
S
S
S
2
d
2 1 1
C
1 1
C
2 5-18
11. Capacitance of a Two-Wire Line
zero reference at
R
10 and
R
20
V
V
2
L
ln 2
L
ln
R
10
V
R
1
R
10
R
1 ln 2
L R
20
R
2 ln
R
20
R
2 2
L
ln
R
10
R
2
R
20
R
1 Choose
R
10
R
20 (
x
0 plane)
V
2
L
ln (
x
(
x
a
)
a
) 2 2
y
2
y
2 4
L
ln (
x
(
x
a
) 2
a
) 2
y y
2 2 목원대학교 전자정보통신공학부 전자기학 5-19