16-1基础物理学讲义Conductor and Dielectrics

substances silver copper water
fused quartz
conductivity (W 1 m1) 6.2×107 5.8×107 2×10-4 10-17
The reciprocal of the conductivity σ is called the
resistivity
0A
d
Example Find the capacitance of a cylindrical capacitor.
Solution: Assume the radius of the inner conducting wire is a and the radius of the out cylindrical shell is b. Let L be the length of the capacitor. Q and -Q be the charges on the inner and outer conductors respectively. The electric field between two conductors is
Van de Graaf generator
107 volts
The charge density on the surface of a conductor
V1
1
40
q r
V2
1
40
Q R
1 2
q 4 r2
:
Q 4 R2
R r
1
E~ ~
0
3×10 6 N/C
The surface charge density as well as the electric field is inversely proportional to the radius of curvature.
C Q 20L .
V ln b a
Suppose that battery does work and move dq from one plate to another
dW Vdq q dq C
W dW
1 q2 1 CV 2 U
2C 2
u U
1 C(V )2 10 A( V )2
○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+
substitutional impurity
interstitial impurity
vacancies dislocations
○+ ○+ ○+ ○+ ○+ ○+
○+ ○+ ○+ ○+
○+
○+ ○+ ○+ ○+ ○+ ○+
○+ ○+ ○+ ○+ ○+ ○+
○+ ○+ ○+ ○+ ○+ ○+
○+ ○+ ○+ ○+ ○+ ○+
○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+
S
surface of the
cavity is zero.
The charge density on each point of the inner surface is zero.
Electrostatic shielding
E=0
Faraday’s ice pail
electroscope
Ein = 0 also means that the conductor is an equipotential body.
B
VB VA Ein dl 0
A Ein = 0
A
B
V 0 V C.
The field near the surface of the conductor
E
0
I V GV, G R1 conductance R
unit: W1 or siemens (S)
negative differential
conductance
(resistance)
G=dI/dV
(R=dV/dI)
Nonlinear voltage-current(V-A) relation of a tunnel diode
microfarads (μF) , picofarads (pF)
1F 106 F, 1pF 1012F
Capacitance of two conductor A, B
CAB
qA VA VB
In a parallel-plate capacitor
E
Ignore the edge effect
Fx dx dW B dU
dW B Vdq dqV d CV 2 2dU
Fx dx dU
Fx
U x
V
U 1 C(V )2 1 0 A (V )2
2
2x
Fx
0A
2x2
(V
)2
Assignment: 16.1, 16.3, 16.6,
16.3 Conductivity and Ohm’s law
copper is n 8.41028 / m2 . Estimate the drift
velocity of the free electrons.
Solution:
d
j ne
2.4106 A/ m2 8.41028 / m31.61019 C
1.8104 m/ sec.
Nonohmic conductors
E
Surface
?
On microscopic scale, is it a real step function?
A conductor with a cavity
S-a closed surface just around
the cavity The total
E dS 0 charge on the
○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+ ○+
a periodic arrangement of positive ions
lattice
When a crystal is formed, we have energy bands
Overlapping
e. g.,Alkaline earth metals Mg (magnesium) 3s2
A complete filled band does not contribute current when the electric field is applied.
j ne2 E E
m
Ohm’s law in differential from
where is called electric conductivity.
unit : W -1m-1 .
The electric conductivity for some typical substances
When no external electric filed is present, the electrons move in all directions, no net current results.
If an external field is applied, a drift motion appears, and an net electric current results.
16.1 Uniform conductor in an electrostatic field
electrostatic equilibrium requires Ein = 0, E// = 0(on the surface) B
VB VA E// dl 0 A
the surface of a conductor is an equipotential surface
Chapter 16 Conductor and Dielectrics Plenty of free electrons are the characteristic of a conductor.
The atoms consisting conductor usually have unfilled shell, e.g. 11Na:1s22s22p63s.
conductors metals electrolytes plasmas
charge carries electrons ions ions and electrons
Free electron model of a metallic conductor
free electrons (electron gas)
thermal motion
j n(e) υ
drift velocity of electrons
υ
υ (kBT ) υ d (E )
υ
d
(E)
eE m
where rmsis the relaxation time. Then the
electric current density is
R
When R and R’ approach to the infinity and the difference R’-R keeps constant d, we have
C
4 0
R(R d
d)
1 d
0 4R2
R d
result for parallel-plate capacitor
1.
E V l
j I S
I V
Sl
l
S
E
j
I
Iwk.baidu.com
V l
S
V
l
S
V R
resistance
V
Ohm’s law
The origin of the resistance a perfect crystal lattice --- no resistance
a. imperfections of crystals impurities
R
VR VR E dl E dr
R
R
q
4 0
1 R
1 R
C
VR
q VR
40
RR R R
C
VR
q VR
40
RR R R
It can be considered as capacitors combined in series
1 1 1
C 40 R 40 R
R
1 1 1
C 40 R 40 R
Er
1
2 0
r
1
2 0
Q. rL
coaxial cable
The potential difference between the two conductors is then
a
V Va Vb Erdr
b
Q b dr Q ln b ,
20L a r 20L a
and the capacitance is
0
VU VL
E dl Ed d 0
upper plate
C
q V
A d
0A
d
0
lower plate
Example 16.1 Capacitance of concentric
spherical shells of radii R and R'.
Er
1
40
q r2
R r R
R
corona discharge lightning rod
16.2 Capacitance capacitor
for a isolated conductor
C q V
capacitance of a sphere or shell
V 1 q,
40 R
C
q V
40
R
units
coulombs/volt (C/V) farads (F)
2
2 d
Ad
Ad
Ad
0
2
V d
2
1
2
0
E
2
* Forces on plates
q = const. Fext dx Fxdx dU
Fx
U x
q
For parallel plate capacitor
U 1 q2 x q2 2C 2ε 0 A
Fx
U x
q
1
20 A
q2
V = const.
vacancy edge dislocation
b. thermodynamic oscillations of the ions
Example Assume the current density in a copper wire is 2.4A/mm2 , the free electron density in the