100 學年上學期高二物理 黃信健. 3 gauss ’ law how wide is a lightning strike ?...
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100 學年上學期高二物理
黃信健
3 Gauss’ Law
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
AvvAAv cos)cos(
m/s × m2 = m3/s
3-3 Flux of an Electric Field• For a flat surface
• For a arbitrary (asymmetric) surface
AEEAAE
cos)cos(
AdEAE
• An arbitrary Gaussian surface
Ex.3-1 A cylindrical Gaussian surface
00
EAEA
AdEAdEAdE
AdE
cba
Ex.3-2 A nonuniform electric field and a Gaussian cube
ˆ ˆ3.0 4.0
ˆ
ˆ ˆ ˆ(3.0 4.0 ) ( )
r
E xi j
dA dAi
E dA
xi j dAi
2
ˆ ˆ ˆ(3.0 4.0 ) ( )
ˆ ˆ ˆ ˆ[(3.0 )( ) (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
ˆleft face:
3.0 1.0 3.0 12 /l
dA dAi
dA dA N m C
2
ˆ ˆ ˆ(3.0 4.0 ) ( )
ˆ ˆ ˆ ˆ[(3.0 )( ) (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
encq
AdE
0
0
3-5 Gauss’ Law and Coulomb’ Law• From G.L. to C.L.
20
20
0
00
4
1
)4(
r
qE
qrE
qdAE
qEdAAdE
3-6 A Charged Isolated Conductor • 同號電荷相斥• 導體內部無電場
• The external electric field
encqAdE
0
00
,
EAEA
EAAqenc
3-7 Applying Gauss’ Law: Cylindrical Symmetry
rE
hrhE
rhAhqenc
0
0
2
2
2= ,
Ex.3-5 A lightning strike
m
Er
6
)103(2
101
2
60
3
0
3-8 Applying Gauss’ Law: Planar Symmetry
00 2
)(
EAEAEA
EdA AdE
• Two Conducting Plates
E 2 1
0 0
ex.3-6 Two‖nonconducting sheets
3-9 Applying Gauss’ Law: Spherical Symmetry
)( 0
)( 4
12
0
RrE
Rrr
qE
• The Shell Theorems
A spherical distribution
)( 4
12
0
Rrr
qE
Ex.3-7 The electric field vs. r
CNr
qE
CeZeq
/103.0= 4
1
10264.1)79(
212
0
17
rR
q
r
qE
R
rqq
R
r
q
q
)4
(4
13434
30
20
3
3
3
3
0 0
0
0 0
0
1
enc
enc
S V
enc
E dA q
qE dA
E dA EdV
qEdV dV
E
3-10 Gauss’ Law in Differential Form
Total charge
Charge density
Divergence散度