99 學年上學期高二物理

99 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  d A 

• 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























) (

) 0

. ˆ 4

0 . 3 (

0 . 4 0

. 3

i dA j

i x

A d E

i dA A

d

j i

x E

r



 





 



 

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 j

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

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

q

A d

E  

 ^{ }



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

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



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 4 3

4

3 0

2 0

3 3 3

3









 



 



 