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Chapter 9 Conductors and Dielectrics in Electrostatic Field
§9-3 Dielectrics §9-3 Dielectrics 电介质电介质
§9-4 Gauss’ Law in Dielectric §9-4 Gauss’ Law in Dielectric 有电介质时的高斯有电介质时的高斯定律 定律 Electric Displacement Electric Displacement 电位移电位移
§9-5 Energy in Electric Field §9-5 Energy in Electric Field 电电场的能量场的能量
§9-2 Capacitance §9-2 Capacitance 电容器电容器
§9-1 Conductors §9-1 Conductors 导体 导体 Elecrostatic Induction Elecrostatic Induction 静电感应静电感应
Conductor:
§ 9-1 Conductors and § 9-1 Conductors and Electrostatic Induction Electrostatic Induction
There are many free electrons in it.
can move in the conductor randomly
ee e
e eee
e
A
B
1. The phenomena of the electrostatic induction
The charges of an insulated
conductor are redistributed
because of external E-field.
No external E-field
The process of electrostatic induction of a conductor
0E
external E-field is supplied--the electrons start to move.
The process of electrostatic induction of a conductor
The process of electrostatic induction of a conductor
0E
external
induced 'E
0E
The process of electrostatic induction of a conductor
0E
external
induced 'E
0E
The process of electrostatic induction of a conductor
0E
external
induced 'E
0E
The process of electrostatic induction of a conductor
0E
externalinduced 'E
0E
The process of electrostatic induction of a conductor
0E
external
induced 'E
0E
The process of electrostatic induction of a conductor
0E
externalinduced 'E
0E
The process of electrostatic induction of a conductor
0E
external
induced 'E
0E
The process of electrostatic induction of a conductor
0E
external
induced 'E
0E
The process of electrostatic induction of a conductor
0E
external
induced 'E
0E
The process of electrostatic induction of a conductor
0E
external
induced 'E
0E
The process of electrostatic induction of a conductor
E
静电平衡状态
0E
'E
0'0 EEE
(2). Electrostatic equilibrium conditions ：
(1). Electrostatic equilibrium state ： There is no any charge moving along a definite direction macroscopically inside the conductor or on the surface of the conductor.
2. Electrostatic equilibrium
The distribution of charges does not change with time.
The E-field equals zero everywhere inside the conductor.
The conductor is an equipotential body.The conductor surface is an equipotential surface.
The E-field at the surface of the conductor is perpendicular to the surface.
E=0
E
The behavior of E-field
The behavior of E-potential
Electrostatic field influences conductor: --electrostatic induction --make the charges in conductor redistribution.
Conductor influences electrostatic field : -- make the electrostatic field redistribution.
Example
The field is uniform before the metal sphere is put in.
E
The field is no longer uniform after the metal sphere is put in.
+++
++++
E
S--any Gaussian surface inside conductor P
S
S
SdE 0 0
内Siq
P--any point inside conductor, S--infinite small ，No excess charge inside conductor.
(1). Entire conductor ：
3. The charge distribution on conductor at the electrostatic equilibrium state.
Prove ： i
i
S
qSdE0
1
Use
No excess charges inside the conductor. They are found only on the conductor surface.
Q
(2). A conductor with a cavity ： Assume charged Q
No charge in the cavity:
Prove:
Draw a Gaussian surface S surrounding the cavity tightly.
No charge inside and internal surface of the conductor.
All the excess charges distribute on the outside surface of conductor.
Q
SSdE 0
0内S
iqi.e.,
--No net charge inside S
S
Question ？ Are there any equal magnitude and opposite sign charges on the internal surface of the conductor ？
0E
Inside the conductor
Not at all.
q
q
There are charges q in the cavity ：
On the inner surface of the conductor ：
--induction charges -q
On the external surface of the conductor ：The original charges Q of the conductor
+ induction charges +q
3.The relation about the charge distribution on the conductor surface and the its radius of curvature.
Rr
Q qR
r
Example:
Two conducting spheres of different radii connected by a long conducting wire.
They are equipotential.
R
QU R
04
1
00
2
0 4
4
4
1
rr
r
r
r
r
r
qU
R
R R
0
2
4
4
0 RR
rR
q
Q rR UU
In a qualitative way, for a conductor of arbitrary shape, the charge density distribution on its surface is inverse proportional with its radius of curvature.
rR
R
r
---
---
- -
- -
4. The relation about the E-field just outside the conductor surface and the charge density on the conductor surface.
.pE
sdE
sE
s
Sqi
i 00
11
0
E
is set up by all charges in the space (on and outside the conductor).
5. Application of electrostatic induction.
(1).Tip discharge Lighting rod( 避雷针 )
(2).Electrostatic shielding
++++ ++++
++--
Electrostatic generator
candle
Electrostatic generator and electric wind
A conductor shell can shield the external field
0E
A conductor shell that is connected with the ground can shield the influence of the fields between inside and outside the conductor.
A
A
[Example 1]A neutral conductor sphere with radius R is put on the side of a point charge +q . Assume the distance between the spherical center and the point charge is d. Calculate: The E-field and potential at point 0 set up by the induced charges on the sphere. If the sphere is connected with the ground, how much is the net charge on the sphere?
qd
R0
'q 'q
The examples about electrostatic induction
qd
R0
'q 'q
x
Solution
The total field at 0 = the field set up by q + the field set up by ±q
Assume the induction charges are±q
=0 ！！
EEE q
0
qEE
'
0
)(
4 20
id
q
id
q 2
04
'q 'q
qd
R0 x
the potential at 0 set up by ±q :
'
04
''
q R
dqU
0
d
qUq
04
'0 UUU q d
q
04
the potential at 0 set up by q :
the sphere is connected with ground.
R
q
d
qU
0
1
00 44
0q
d
Rq 1
Assume net charge q1 is left on the sphere.
the potential U0 at 0 = Uq + Uq1
qd
R0
1q
[Example 2] Two large parallel plates with the area S carry charge Qa and QB respectively.
QAQB
Find ： The charge and field distribution.
solutionAssume the charge surface density areσ1,σ2,σ3 andσ4 on the four surfaces .
σ1 σ2
σ3 σ4
Draw a Gaussian surface1 2 3 4
S
SSsdES
320
1
0P
4321 EEEEE p
0
41
Ⅰ Ⅱ Ⅲ
32
at point P :
and AQS )( 21
BQS )( 43
S
QQ BA
241
S
QQ BA
232
1 2 3 4
Ⅰ Ⅱ Ⅲ
The field distribution:
E1 E1E2 E2
E3 E3
E4 E4
4321 EEEEE I
0
4321
2
E=0 inside the plates
outside the plates:
4321 EEEEE II
direction: point to left
point to right
4321 EEEEE III point to right
2 3
Ⅰ Ⅱ Ⅲ
0
3
0
2
22
ES
q
0
EⅠ = E Ⅲ= 0
0
E
discussion Two plates carry equal magnitude and opposite sign charge
QQQ AB
041
S
Q 32
the charges distribute on the inner surface only.
S
qE
ES
qE
III
I
00
4
0
1
00
4
0
1
22
022
1 4
Ⅰ Ⅱ Ⅲ
Charges distribute on the exterior surface only.
S
Q 41
032
Two plates carry equal magnitude and same sign charge
QQQ AB
1r 2r
3r[Example 3] conductor sphere with radius r1 carries ＋ q and conductor spherical shell with inner and exterior radii r2 and r3 carries ＋ Q.
Calculate the E distribution, the potential of sphere and shell U1 and U2, potential difference U△
Connect sphere and shell with a wire, find E, U1 and U2 , U △ =?.
If the shell is connected with ground, find E, U1 , U
2, U =△ ? If the sphere is connected with ground, find the ch
arge distribution. U2=?
solution ： the field distribution:
0: 11 Err
rr
qE
rrr
ˆ4
:
20
2
21
0332 Errr
rr
QqE
rr
ˆ4
:
20
4
3
1r 2r
3rq
q
sphere potential:
1
1 rrdEU
3
2
12
02
0 44 r
r
rdr
r
Qqdr
r
q
32104
1
r
r
q
r
q
3
2
142 r
r
rrdErdE
3
2 rrdEU
3
4rrdE
3
204r
drr
304 r
21 UUU
210
11
4 rr
q
The shell potential:
Potential difference:
0321 EEE
Connect sphere and shell with a wire,
rr
QqE ˆ
4 20
4
3
421 rrdEUU
drr
Qqr
3
204 304 r
All charges are on the exterior surface of the shell.
The shell is connected with ground,
rr
qE ˆ
4 20
2
0431 EEE
2
11
r
rrdEU 2
12
r
rrdE
210
11
4 rr
q
121 UUUU
U2 ＝ 0 ， no any charge on the exterior surface of the shell.
The sphere is connected with ground,
3020101 4
'
4
'
4
'
r
r
q
r
qU
0
Qrrrrrr
rrq
122313
12'
'q
'q
U1=0. Assume sphere charges q' ， then the inner surface of shell charges -q' ， its exterior surface charges （ Q ＋ q' ）
As r3 r1 < r3 r2 , q'<0
The potential of shell:
1323210
12
302 44
'
rrrrrr
rrQ
r
QqU
