100 學年上學期高二物 理
黃信健
3 Gauss’ Law
How wide is a
lightning strike ?
3-1 A New Look at Coulomb’s Law
• Using Gauss’s law to take advantage of special
symmetry situations
• Gaussian surfaces
• 高斯面上各點電場與面 內總電荷相關
3-2 Flux (
通量 / 流 量 )
• For a fluid
A v
vA A
v
( cos ) cos
m/s × m2 = m3/s
3-3 Flux of an Electric Field
• For a flat surface
• For a arbitrary (asymmetric) surface
A E
EA A
E
( cos ) cos
E A E dA
• An
arbitrary Gaussian surface
Ex.3-1 A cylindrical Gaussian surface
0
0
EA EA
A d E
A d E
A d E
A d E
c b
a
Ex.3-2 A nonuniform electric field and a Gaussian cube
3 .0 ˆˆ 4 .0 ˆ
(3 .0 ˆˆˆ 4 .0 ) ( )
r
E x i j
d A d A i
E d A
x i j d A i
2
(3.0 ˆˆˆ 4.0 ) ( )
[(3.0 )( ) ˆˆˆˆ (4.0)( ) )
(3.0 0) 3.0
3.0 3.0 9.0 36 /
r E dA
xi j dAi
x dA i i dA j i
xdA xdA
dA dA N m C
2
left face: ˆ
3.0 1.0 3.0 12 /
l
dA dAi
dA dA N m C
2
(3.0 ˆˆˆ 4.0 ) ( )
[(3.0 )( ) ˆˆˆˆ (4.0)( )( )]
(0 4.0 ) 4.0 16 /
t xi j dAj
x dA i j dA j j
dA dA N m C
3-4 Gauss’ Law
• Flux
enclosed charge
q enc
A d E
0 0
3-5 Gauss’ Law and Coulomb’ Law
• From G.L. to C.L.
2 0
2 0
0
0 0
4 1
) 4
(
r E q
q r
E
q dA
E
q EdA
A d E
3-6 A Charged Isolated Conductor
• 同號電荷相斥
• 導體內部無電場
• The external electric field
qenc
A d
E
0
0 0
,
E A
EA
EA A
q enc
3-7 Applying Gauss’ Law:
Cylindrical Symmetry
E r
h rh
E
rh A
h q enc
0 0
2 2
2
= ,
Ex.3-5 A lightning strike
m r E
6
) 10
3 ( 2
10 1
2
6 0
3 0
3-8 Applying Gauss’ Law:
Planar Symmetry
0
0 ( ) 2
E A
EA EA
EdA A
d E
• Two Conducting Plates
E 2 1
0 0
ex.3-6
Two‖nonconducting sheets
3-9 Applying Gauss’
Law: Spherical Symmetry
) (
0
) 4 (
1
2 0
R r
E
R r r
E q
• The Shell Theorems
A spherical distribution
) 4 (
1
2 0
R r r
E q
Ex.3-7 The electric field vs. r
C r N
E q
C e
Ze q
/ 10
3.0 4 =
1
10 264
. 1 )
79 (
21 2
0
17
R r q r
E q
R q r
q R
r q
q
4 ) 4 (
1 3 43
4
3 0
2 0
3 3 3
3
0 0
0
0 0
0
1
e n c e n c
S V
e n c
E d A q E d A q
E d A E d V
E d V q d V
E
3-10 Gauss’ Law in Differential Form
Total charge
Charge density Divergence
散度
