Download 2.3.4 一個局域電荷分佈的電位(The Potential of a Localized Charge

Ch2. 靜電學(Electrostatics)
2.1 電場(The Electric Field)
2.2 靜電場的發散與旋度(Divergence and Curl of Electrostatic Fields)
2.3 電位(Electric Potential)
2.4 靜電學中的功與能量(Work and Energy in Electrostatics)
2.5 導體(Conductors)
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.1.1 電場(The Electric Field)
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.1.2 庫倫定律(Coulomb’s Law)
1.
2.
3.
4.
5.
Charles Coulomb 1785 : fundamental law of electric force
between two stationary charged particles. The force is:
inversely proportional to square of the distance between the
particles,
directed along the line joining the particles,
proportional to the product of the two charges and
attractive if particles have charges of opposite sign and
repulsive if charges have same sign.
r
r̂
Q1
Q
2
Q1Q2
F12  ke 2 rˆ
r
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.1.3.1 電場(The Electric Field)
Source point


'
r
 
  E( r ) 

r
Field point
 
E( r ) 
1
40
n
qi
ˆ

2 i
i 1  i

 '
1 n
ˆ i
1 n
1 n
q i (  2 )   q i (i )   q i ( r  ri )

40 i 1
i
0 i 1
0 i 1
 ' ' n
 ' ' n
  qi( r  ri )d   qi  ( r  ri )d   qi  q
n
The far right term
V i 1
i 1
V
i 1
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.1.3.2 電場(The Electric Field)
Example : Electric dipole
  
E  E1  E2
E1  E2  ke
E  2 E1 cos   2ke
q
q

k
e
r2
y2  a2
q
a
2qa

k
e
( y 2  a 2 ) ( y 2  a 2 )1/ 2
( y 2  a 2 )3 / 2
Because y >> a, we neglect a2 and write
2qa
E  ke 3
y
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.1.4.1 連續電荷分佈(Continuous Charge Distributions)
'
( r ) '
L 2 ˆdl
'
( r )
'
ˆ

da
S 2
Source : a line charge
 
E( r ) 
1
40
Source : a surface charge
 
E( r ) 
1
40
Source : a volume charge
 
E( r ) 
'
1
( r )
'
ˆ

d

40 V  2
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.1.4.2 連續電荷分佈(Continuous Charge Distributions)
Example :
A rod of length l has a uniform charge per unit length λand a total charge Q.
Calculate the electric field at a point P along the axis of the rod at a distance
a from one end. Note that λ= Q/l.
Uniform positive charge per unit length 
dq  dx
dq
dx
dE  ke 2  ke 2
x
x
E
a
a
  a dx
dx
1 a  k  ( 1  1 )  keQ
ke 2  ke 
 ke [ ]a
e
2
a
a a
a (  a )
x
x
x
If a >> 
E
k eQ
a2
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.1.4.3 連續電荷分佈(Continuous Charge Distributions)
Example : Electric Field of a Uniform Ring of Charge
dE  k e
dEx  dE cos   (ke
Ex  
dq
r2
dq x
ke x
)

dq
2
2
2 3/ 2
r r (x  a )
ke x
ke x
ke x
dq

dq

Q
2
2 3/ 2
2
2 3/ 2 
2
2 3/ 2
(x  a )
(x  a )
(x  a )
If x >> a
keQ
Ex  2
x
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.1.4.4 連續電荷分佈(Continuous Charge Distributions)
Example : Electric Field of a Uniformly Charged Disk
dq  2rdr
ke x
ke x
dq

(2rdr )
2
2 3/ 2
2
2 3/ 2
(x  r )
(x  r )
dEx 
E  ke x 
R
0
R
2rdr
2
2 3 / 2
d (r 2 )
2
2 3 / 2  ke x  ( x  r )
0
(x  r )
( x 2  r 2 ) 1 / 2 R
x
x
 ke x [
]0  2ke (  2
)
2 1/ 2
 1/ 2
x (x  R )
If R >> x
E  2ke 

2 0
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.1.4.5 連續電荷分佈(Continuous Charge Distributions)
Example : Electric Field of an Infinite Plane of Charge
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.1.4.6 連續電荷分佈(Continuous Charge Distributions)
Example : Electric Field Between Two Oppositely Charged Parallel Plates
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.1.4.7 連續電荷分佈(Continuous Charge Distributions)
Example 2.1
Find the electric field a distance z above the midpoint of a straight line segment
of length 2L, which carries a uniform line charge .

dE 

r
1 dx
1 dx
( 2 ) cos ẑ 
( 2 ) cos ẑ
40 r
40 r
1 dx
2
( 2 ) cos ẑ
40 r
dx
z
cos   
r
z
z2  x 2
2L
1
E
40

L
2z
2z
x
0 (z 2  x 2 )3/ 2 dx  40 [ z 2 z 2  x 2 ]
0
L
1
2L
40 z z 2  L2
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.1.4.8 連續電荷分佈(Continuous Charge Distributions)
Problem 2.4
Find the electric field a distance z above the center of a square loop (side a)
carrying uniform line charge .
z
Remember example 2.1
L
a
2
a 2
z z ( )
2
2
a
Field of one edge is :
1
E
40
For four sides :

1
E4
40
a
a 2 2
a 2
z  ( ) z  2( )
2
2
2
1
2 L
E
40 z z 2  L2
a
a
a
z 2  ( ) 2 z 2  2( ) 2
2
2
cos ẑ 
1
40
4az
a 2
a 2
2
[z  ( ) ] z  2( )
2
2
2
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
ẑ
2.2.1.1 場線(Field Lines)
E
+q
Point away from
positive charge
E
-q
Point towards
negative charge
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.2.1.2 場線(Field Lines)
場線規則
1.
2.
3.
4.
5.
6.
Field lines always begin on positive charges and end on negative
charges. (In doing so, they may leave and re-enter the picture frame).
The (net) number of lines exiting or entering a charge is proportional
to the charge magnitude.
The density of lines (number of lines per unit area through a surface
perpendicular to the lines) is proportional to the field strength there.
At any point, the direction of a field line (its tangent) is the direction of
the electric field vector E at that point.
Field lines, representing the total E field, never cross one another.
Field lines very close to a point charge are distributed symmetrically.
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.2.1.3 場線(Field Lines)
+q
-q
+q
+q
•Electric field lines are not material objects
•Finite number of lines can be misleading
•The electric field is continuous and exists at every point
•The electric field is three dimensional
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.2.1.4 通量(Flux)
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.2.1.5 通量(Flux)
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.2.1.6 通量(Flux)
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.2.1.7 高斯定律(Gauss’s Law)
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.2.1.8 高斯定律(Gauss’s Law)
Divergence theorem :
 

 E  da   (  E)d
S
V
Qenc   d
V
  Q enc

 E   E  da 
  d
0

S
V 0
 
E 
0
Gauss’s law in differential form
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.2.1.9 高斯定律(Gauss’s Law)
Problem 2.9
Suppose the electric field in some region is found to be
coordinates (k is some constant).

E  kr 3r̂
, in spherical
(a). Find the charge density .
(b). Find the total charge contained in a sphere of radius R, centered at the origin.

1  2
1
3
   0  E   0 2 (r  kr )   0 2 k (5r 4 )  5 0 kr 2
r r
r
R
Q enc   d   (5 0 kr 2 )( 4r 2 dr )  40 kR 5
V
or
Qenc
0
 
  0  E  da   0 (kR 3 )( 4R 2 )  40 kR 5
S
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.2.2 電場的發散(The Divergence of E)
 
E( r ) 
Source point


'
r

r
Field point

E 
'
1
( r )
'
ˆ

d

40 V  2
  
  r  r'
ˆ
3 
  ( 2 )  4 ()



1
1 
3 
4

(
r

r
'
)

(
r
'
)
d

'

( r )

40
0

  1
Q enc
V (  E)d  S E  da  0 V d  0
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.2.3.1 高斯定律的應用(Application of Gauss’s Law)
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.2.3.2 高斯定律的應用(Application of Gauss’s Law)
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.2.3.3 高斯定律的應用(Application of Gauss’s Law)
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.2.3.4 高斯定律的應用(Application of Gauss’s Law)
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.2.3.5 高斯定律的應用(Application of Gauss’s Law)
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.2.3.6 高斯定律的應用(Application of Gauss’s Law)
l
Example 2.3
  ks
s
Find the electric field inside this cylinder
  1
S E  da  0 Qenc
S
2
Q enc   d   (ks' )(s' ds' ddz)  2kl  s' ds'  kls 3
3
0
 
 



 E  da   E  da   E da  E  da  E 2sl
2
S

2
E 2sl 
kls 3
3 0

1
E
ks 2ŝ
3 0
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.2.4.1 電場的旋度(The Curl of E)

E
Consider a point charge at the origin :
z
If we calculate the line integral of this field
rb
 
 E  dl
b
q
y
ra
x
In spherical coordinates :
 
E  dl 
 
 E  dl 
b
a
1 q
r̂
2
40 r
1
40
a
d l  dr r̂  rd ˆ  r sin d ˆ
1 q
dr
2
40 r
q
1 q b
1 q q
a r 2 dr  40 r r  40 ( ra  rb )
a
b
r
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.2.4.2 電場的旋度(The Curl of E)
 
 E  dl 
b
a
1
40
q
1 q
1 q q
a r 2 dr  40 r r  40 ( ra  rb )
a
rb
b
For the integral around a closed path (ra = rb) :
史托克定理(The Stokes’ theorem)
 
 E  dl  0
 
 
 (  v)  da   v  d l
S
C
 
 
 (  E)  da   E  d l  0
S

E  0
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.3.1 電位的介紹(Introduction to Potential)
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.3.1 電位的介紹(Introduction to Potential)
Since we will only ever calculate differences in potential, the
value of the potential at the reference point O can be anything as
long as it is finite. Therefore, we can simply define the electric
potential at a point P(x,y,z) as:
Normally, the reference point is at infinity where V=0. In fact, V
is not physical (charges can’t feel it directly). It only becomes
physical once you take its gradient to form E.
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.3.1 電位的介紹(Introduction to Potential)
V obeys the superposition principle
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.3.1 電位的介紹(Introduction to Potential)
Equipotential surface
surface
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.3.2 關於電位的重點
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.3.2 關於電位的重點
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.3.2 關於電位的重點
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.3.2 關於電位的重點
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.3.3 Poisson’s Equation and Laplace’s Equation
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.3.4 一個局域電荷分佈的電位(The Potential of a
Localized Charge Distribution)
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.3.4 一個局域電荷分佈的電位(The Potential of a
Localized Charge Distribution)
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.3.4 一個局域電荷分佈的電位(The Potential of a
Localized Charge Distribution)
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.3.4 一個局域電荷分佈的電位(The Potential of a
Localized Charge Distribution)
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.3.4 一個局域電荷分佈的電位(The Potential of a
Localized Charge Distribution)
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.3.4 一個局域電荷分佈的電位(The Potential of a
Localized Charge Distribution)
E=0 inside conductor in equilibrium. Any net charge
resides on the surface of the conductor. Electric field
immediately outside the conductor is perpendicular
to the surface of the conductor.
Consider the two points A and B on the surface of the
charged conductor. Along a surface path connecting
these points E is always perpendicular to ds.
V is constant everywhere on the
surface of a charged conductor in
equilibrium. Since electric field inside
the charged conductor in equilibrium
is zero, the potential is constant
everywhere inside the conductor and
equal to its value at the surface.
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.3.5 總結; 靜電的邊界條件(Summary; Electrostatic
Boundary Conditions)
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.3.5 總結; 靜電的邊界條件(Summary; Electrostatic
Boundary Conditions)
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.3.5 總結; 靜電的邊界條件(Summary; Electrostatic
Boundary Conditions)
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.3.5 總結; 靜電的邊界條件(Summary; Electrostatic
Boundary Conditions)
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.3.5 總結; 靜電的邊界條件(Summary; Electrostatic
Boundary Conditions)
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.3.5 總結; 靜電的邊界條件(Summary; Electrostatic
Boundary Conditions)
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.4.1 移動一電荷所做的功(The Work Done to Move
a Charge)
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.4.2 一個點電荷分佈的能量(The Energy of a Point
Charge Distribution)
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.4.3 一個點電荷分佈的能量(The Energy of a Point
Charge Distribution)
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.4.4 一個點電荷分佈的能量(The Energy of a Point
Charge Distribution)
Problem 2.31
-q
(a). V
+q
(b)
W1  0
1

40
qi
1 q
q
q
 r  4 ( a  2a  a )
ij
0
q
1

( 2 
)
-q
40a
2
q2
1

( 2 
)
40a
2
1
q2
q2
1  q2
(
 )
W2 
(
) W3 
40 2a a
40 a
q2
1
W4 
( 2 
)
40a
2
2q 2
1
Wtotal 
( 2 
)
40a
2
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.4.5 一個連續電荷分佈的能量(The Energy of a
Continuous Charge Distribution)
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.4.6 一個連續電荷分佈的能量(The Energy of a
Continuous Charge Distribution)

1
W   Vd
   0  E
2

0
W   (  E )Vd 
2


 
 f (  A)d    A  (f )d   fA  da
V
V
S

  0
 
0
2
W  [  E  (V)d   VE  da ]  [  E d   VE  da ]
2 V
2 V
S
S
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.4.7 一個連續電荷分佈的能量(The Energy of a
Continuous Charge Distribution)
The integral above contains self-energy terms and interaction
energy terms. The former are always positive (and generally
large), whereas the latter can be positive or negative.
Example: Consider two point-like charges q1 and q2 at some
non-zero separation. Calculate the interaction energy using
the summation form and either of the integral forms above
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.4.8 一個連續電荷分佈的能量(The Energy of a
Continuous Charge Distribution)
interaction energy terms
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.5.1 導體的基本性質
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.5.2 導體的基本性質
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.5.3 導體的基本性質
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.5.4 導體的基本性質
(a).
q
R 
4R 2
q
a 
4a 2
q
b 
4b 2
b
a
R
0
 
1 q
1 q
(b). V(0)    E  d l    (
)dr   (0)dr   (
)dr   (0)dr
2
2
40 r
40 r


b
a
R
0

(c).
1 q q q
(   )
40 b R a
b  0
a
R
0
1 q
V(0)    (0)dr   (
)dr   (0)dr
2
40 r

a
R

1 q q
(  )
40 R a
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.5.5 在一導體上的表面電荷與力(Surface Charge
and the Force on a Conductor)
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.5.6 在一導體上的表面電荷與力(Surface Charge
and the Force on a Conductor)



E above  E ext 
n̂
2 0



E below  E ext 
n̂
2 0



1 
E ext  (E above  E below )  E average
2
The force per unit area on a conductor :


1 2
f  E average 
 n̂
2 0
The electrostatic pressure :
0 2
P E
2
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.5.7 電容器(Capacitors)
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.5.8 電容器(Capacitors)
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.5.9 電容器(Capacitors)
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.5.10 電容器(Capacitors)
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.5.11 電容器(Capacitors)
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.5.12 電容器(Capacitors)
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung
2.5.13 電容器(Capacitors)
Y.M. Hu, Associate Professor, Department of Applied Physics, National University of Kaohsiung