Transcript Document 7455956

Y.M. Hu, Associate Professor, Department of Applied Physics
Introduction to electrodynamics
Third Edition
評分標準:
1. 平時成績 (40 %)
2. 期中考 (30 %)
3. 期末考 (30 %)
David J. Griffiths
2016/5/23
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Y.M. Hu, Associate Professor, Department of Applied Physics
Contents
上學期
Ch1. 向量分析(Vector analysis)
Ch2. 靜電學(Electrostatics)
Ch3. 特殊技巧(Special techniques)
Ch4. 物質中的電場(Electric fields in matter)
Ch5. 靜磁學(Magnetostatics)
Ch6. 物質中的磁場(Magnetic fields in matter)
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Y.M. Hu, Associate Professor, Department of Applied Physics
Contents
下學期
Ch7. 電動力學(Electrodynamics)
Ch8. 守恆律(Conservation laws)
Ch9. 電磁波(Electromagnetic waves)
Ch10. 位勢與場(Potential and fields)
Ch11. 輻射(Radiation)
Ch12. 電動力學與相對論(Electrodynamics and relativity)
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Y.M. Hu, Associate Professor, Department of Applied Physics
Ch1. 向量分析(Vector analysis)—基本定義
在物理量中,向量是具有方向的量,例如:
direction
magnitude
位移(displacement),
速度(velocity),
加速度(acceleration),
力(force),
動量(momentum)
向量符號 (vector notation):
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A


boldface
arrow
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Y.M. Hu, Associate Professor, Department of Applied Physics
向量分量(Vector components)
A vector can be identified by specifying its three Cartesian
components:
分量(component)

  A x x̂  A y ŷ  A z ẑ
單位向量(unit vector)
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Y.M. Hu, Associate Professor, Department of Applied Physics
向量加減(Vector addition and subtraction)
# 向量相加(vector addition):
兩個向量相加產生第三個向量
右圖顯示向量相加是可互換的
(commutative)
# 向量相減(vector subtraction):
“減”等於加上一個向量的反向
  

  B  A  ( B)
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Y.M. Hu, Associate Professor, Department of Applied Physics
向量乘法(Vector multiplication)
Vector multiplication by a scalar:
The direction of the resulting vector is the same
as the direction of the original vector if a > 0.
The direction of the resulting vector is opposite
to the direction of the original vector if a < 0.
The magnitude of the resulting vector is the
product of the magnitude of the scalar and the
magnitude of the original vector.
Vector multiplication is distributive:

 

a (   B)  aA  aB
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Y.M. Hu, Associate Professor, Department of Applied Physics
向量之純量積(Scalar product)
Dot product (scalar product) of two vectors
 
  B  AB cos   純量(scalar)
 
  B  A x B x  A y B y  A z Bz
The dot product is commutative:
   
 B  B 
The dot product is distributive:
   
  
  ( B  C)  A  B    C
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Y.M. Hu, Associate Professor, Department of Applied Physics
向量之向量積(vector product)
Cross product (vector product) of two vectors
 
  B  AB sin  n̂
x̂
 
  B  Ax
Bx
ŷ
Ay
By
向量(vector)
ẑ
Az
Bz
方向: use right-hand rule
n̂
The cross product is not commutative:
 
 
  B  B  
The cross product is distributive:
   
  
  ( B  C)  A  B    C
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Y.M. Hu, Associate Professor, Department of Applied Physics
向量之向量積(vector product)
Parallelogram representation of the vector product
  
C  AB
y
Bsinθ

B
Area
θ

A
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C  ABsin 
x
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Y.M. Hu, Associate Professor, Department of Applied Physics
向量之三重純量積
Triple Scalar product
  
A  ( B  C)  A x ( ByCz  Bz Cy )  A y ( Bz Cx  Bx Cz )  A z ( Bx Cy  ByCx )
  
  
 B  (C  A)  C  ( A  B)
  
  
  
 A  (C  B)  C  ( B  A)  B  (A  C)
Ax
Ay
Az
 Bx
Cx
By
Cy
Bz
Cz
純量(scalar)
The dot and the cross may be interchanged :
  
  
A  (B  C)  (A  B)  C
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Y.M. Hu, Associate Professor, Department of Applied Physics
向量之三重純量積
Parallelepiped representation of triple scalar product
 
  

A  ( B  C)  Volume of parallelepiped defined by A , B , and C
z

C
 
BC

A

B
y
x
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Y.M. Hu, Associate Professor, Department of Applied Physics
向量之三重向量積
Triple vector product
z
 
BC

A

B
x
  
A  ( B  C)
y

C
  
A  ( B  C)
  


A  ( B  C)  xB  yC
   
 
 
A  [A  ( B  C)]  0  xA  B  yA  C
 
 
x  zA  C y  zA  B
  
   
A  ( B  C)  z( BA  C  CA  B)
   
 ( BA  C  CA  B)
BAC-CAB rule
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Y.M. Hu, Associate Professor, Department of Applied Physics
向量之三重向量積
  
   
Proof : z = 1 in A  ( B  C)  z( BA  C  CA  B)
Let us denote
B̂  Ĉ  cos 
Ĉ  Â  cos
[Â  ( B̂  Ĉ)]2  Â2 ( B̂  Ĉ)2  [Â  ( B̂  Ĉ)]2
 1  cos2   [Â  ( B̂  Ĉ)]2
  B̂  cos 
(Â  B̂)2  Â2 B̂2  (Â  B̂)2
( B̂  Ĉ)2  B̂2Ĉ2  ( B̂  Ĉ)2
 z 2 [( Â  Ĉ) 2  (Â  B̂) 2  2(Â  B̂)( Â  Ĉ)B̂  Ĉ]
 z 2 (cos2   cos2   2 cos  cos  cos  )
[Â  ( B̂  Ĉ)]2  1  cos2   z 2 (cos2   cos2   2 cos  cos cos  )
The volume is symmetric in αβ,γ
For the special case
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x̂  (x̂  ŷ)  ŷ
z2 = 1
z=±1
z 1
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Y.M. Hu, Associate Professor, Department of Applied Physics
向量之三重向量積
  
  
  
A  (B  C)  B(A  C)  C(A  B)
But
  
  
(A  B)  C   C  (A  B)
  
  
  A(B  C)  B(A  C)
  
  
 B(A  C)  A(B  C)
  
  
  
A  (B  C)  B(A  C)  C(A  B)
     
  
(A  B)  C  B(A  C)  A(B  C)
  
  
A  (B  C)  (A  B)  C
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Y.M. Hu, Associate Professor, Department of Applied Physics
向量之多重向量積
當一乘積式中含有兩個以上(含)的  符號出現時,都可將
其減化至每項最多只剩一個.
  
  
  
例如: A  (B  C)  B(A  C)  C(A  B)
又如:
   
   
   
(A  B)  (C  D)  (A  C)( B  D)  (A  D)( B  C)
   
   
   
A  (B  (C  D))  B(A  (C  D))  (A  B)(C  D)
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Y.M. Hu, Associate Professor, Department of Applied Physics
位置向量與位移向量
位置向量

r  x x̂  y ŷ  z ẑ
ẑ

r
r̂
z
ŷ
x
x̂
y
位移向量
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
r  r  x 2  y2  z2

r x x̂  y ŷ  z ẑ
r̂  
r
x 2  y2  z2

d l  dx x̂  dy ŷ  dz ẑ
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Y.M. Hu, Associate Professor, Department of Applied Physics
距離向量
Source point

r'
  
  r  r'


 
  r  r'
Field point

r
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

ˆ  

 
r  r'
 
r  r'
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Y.M. Hu, Associate Professor, Department of Applied Physics
向量轉換(Vector transformation)
一個向量的分量(components)是與所選取的座標
系統(the coordinate system)有關.
對於同一個向量,不同的座標系統會產生不同的分
量,此過程稱之為向量轉換(vector transformation).
適當選取座標系統並使用向量轉換,可以有效改善
電動力學(electrodynamics)中問題的複雜度(the
complexity).
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Y.M. Hu, Associate Professor, Department of Applied Physics
向量轉換(Vector transformation)
A y '  cos A y  sin A z
In matrix notation
A z '   sin A y  cos A z
 A y '   cos  sin   A y 

  
 
 A z '    sin  cos   A z 
The rotation considered here leaves
the x axis untouched.
0
0  A x 
 A x'   1

 
 
 A y '    0 cos  sin   A y 
 A   0  sin  cos   A 
 z 
 z'  
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Y.M. Hu, Associate Professor, Department of Applied Physics
向量轉換(Vector transformation)
Coordinate transformation resulting from a rotation
around an arbitrary axis can be written as:
or, more compactly, with x denoted as 1, y as 2, z as 3:
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Y.M. Hu, Associate Professor, Department of Applied Physics
向量轉換(Vector transformation)
The rotation matrix R is an example of a unitary transformation:
one that does not change the magnitude of the object on
which it operates:
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Y.M. Hu, Associate Professor, Department of Applied Physics
向量轉換(Vector transformation)
Vectors are first-rank tensors (張量), having three independent
components that can be represented by a column matrix.
A second-rank tensor with nine independent components is a
quantity which transforms with two factors of R:
An nth-rank tensor has n indices and 3n components, and
transforms with n factors of R.
Scalars are zero-rank tensors.
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Y.M. Hu, Associate Professor, Department of Applied Physics
全微分與偏微分
當f = f (x)
全微分
df
df  ( )dx
dx
當f = f (x,y,z)
f
f
f
df  ( )dx  ( )dy  ( )dz
x
y
z
偏微分
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Y.M. Hu, Associate Professor, Department of Applied Physics
梯度(Gradient)
f
f
f
df  ( )dx  ( )dy  ( )dz
x
y
z
f
f
f
 ( x̂ 
ŷ  ẑ)  (dxx̂  dyŷ  dzẑ)
x
y
z




 ( x̂ 
ŷ  ẑ)f  d l
x
y
z
The definition of the gradient of a scalar function



f  ( x̂ 
ŷ  ẑ)f
x
y
z
Gradient operator “acting on” a
scalar field produces a vector field
The “del” or gradient operator   (
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


x̂ 
ŷ  ẑ)
x
y
z
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Y.M. Hu, Associate Professor, Department of Applied Physics
梯度(Gradient)的幾何意義
如果 f 為高度的函數,則 f 代表從低到高的最大斜率變化
f
f(x,y)
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Y.M. Hu, Associate Professor, Department of Applied Physics
散度(Divergence)



  ( x̂ 
ŷ  ẑ)
x
y
z

v  v x x̂  v y ŷ  v z ẑ




  v  ( x̂ 
ŷ  ẑ)  ( v x x̂  v y ŷ  v z ẑ)
x
y
z
v x v y v z



 g( x, y, z)
x
y
z
The operation of divergence produces a new scalar field
g(x,y,z) which is related to the density of a scalar quantity
such as charge, mass, etc.
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Y.M. Hu, Associate Professor, Department of Applied Physics
散度(Divergence)的幾何意義
Divergence is a measure of how fast the field lines stretch and/or
spread out.
電荷q的密度
 
E 
0

 v  0
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
 v  0
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Y.M. Hu, Associate Professor, Department of Applied Physics
旋度(Curl)



  ( x̂ 
ŷ  ẑ)
x
y
z

v  v x x̂  v y ŷ  v z ẑ
x̂
ŷ
ẑ

  v   / x  / y  / z
vx
vy
vz
v y v x
v z v y
v x v z
 x̂(

)  ŷ(

)  ẑ(

)
y
z
z
x
x
y
The operation of curl produces a new vector field which is
related to the density of a vector quantity such as current.
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Y.M. Hu, Associate Professor, Department of Applied Physics
旋度(Curl)的幾何意義
Curl is a measure of how fast the field lines of a vector field
twist or bend in a direction set by a right-hand-rule
電流I的密度


  B  0 J
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Y.M. Hu, Associate Professor, Department of Applied Physics
Six product rules
Two for gradients
 (fg )  fg  gf
 







 (A  B)  A  (  B)  B  (  A)  (A   )B  (B   )A
Two for divergences



  (fA)  f (  A)  A  (f )
 




  (A  B)  B  (  A)  A  (  B)
Two for curls



  (fA)  f (  A)  A  (f )
 




 


  (A  B)  (B   )A  (A   )B  A(  B)  B(  A)
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Y.M. Hu, Associate Professor, Department of Applied Physics
Three quotient rules
f
gf  fg
( ) 
g
g2



A
g(  A)  A  (g)
( ) 
g
g2



A
g(  A)  A  (g)
( ) 
g
g2
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Y.M. Hu, Associate Professor, Department of Applied Physics
Second derivatives
The gradient T is a vector
1. Divergence of gradient :
  (T)  ( x̂



T
T
T
 ŷ  ẑ )  ( x̂
 ŷ
 ẑ )
x
y
z
x
y
z
 2T  2T  2T
2
2
2
 2  2  2  ( 2  2  2 )T   2 T
x
y
z
x
y z
 2 T : the Laplacian of T  scalar
Laplacian can operate on scalar or vector functions
2
 v  ( 2 v x )x̂  ( 2 v y ) ŷ  ( 2 vz )ẑ
2. Curl of gradient :
  (T )  0
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Y.M. Hu, Associate Professor, Department of Applied Physics
Second derivatives

The divergence   v is a scalar
1. Gradient of divergence :

 (  v) seldom occurs in physical applications

The curl   v is a vector
1. Divergence of curl :

  (  v)  0
2. Curl of curl :


2
  (  v)   (  v)   v
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Y.M. Hu, Associate Professor, Department of Applied Physics
線積分(Line integrals or path integrals)
b

 
P v dl
along a prescribed path P
a
通常,線積分的值與所選定的路徑是有關
If a = b

 
 v dl
Line integral around curve C
drawn in the vector field is
called the “circulation”
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Y.M. Hu, Associate Professor, Department of Applied Physics
梯度定理(The gradient theorem)
b


P ( T )  d l  T ( b)  T (a )
a
Integral is taken along any curve P that has end-points a and b
Fundamental theorem of calculus
for a function of one variable
b
df ( x )
a dx dx  f (b)  f (a )
For the electric potential, the line integral of a gradient is
independent of the path taken from a to b. The gradient
theorem ensures this. Such fields are said to be ‘conservative
fields’
b
b

 
 E  d l    (V)  d l  V(a )  V(b)
a
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a
36
Y.M. Hu, Associate Professor, Department of Applied Physics
發散定理(The divergence theorem)
亦稱為高斯定理(Gauss’s theorem)或格林定理(Green’s theorem)

 
 (  v)d   v  da

S
一個場的發散值為某個物理量(例如:電荷)的密度,此物理量是該
場的來源.因此,定理的左式代表在體積V之內場源的總合.
定理的右式代表該場越過邊界表面S(所包圍的體積為V)的淨向
外通量(net outward flux).
 (faucets

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within the volume )   (flow out through the surface )
S
37
Y.M. Hu, Associate Professor, Department of Applied Physics
發散定理(The divergence theorem)

Example: point charge q that is a source of electric field E

 
 (  E)d   E  da

S


q
  (  E)d   ( )d 
0
0


  q
  E  da 
0
S
Gauss’s Law
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Y.M. Hu, Associate Professor, Department of Applied Physics
史托克定理(The Stokes’ theorem)
 
 
 (  v)  da   v  d l
S
C
定理左式代表面積S(由封閉曲線C所包圍)之內環流(circulation)


源的總合.
例如: v 改為磁場 B 時,則環流源為穿過表面S的電流

密度 J . 定理右式代表環繞曲線C的磁場的淨環流.


Current I that is a source of B
 
 
 (  B)  da   B  d l
S
C

 

 (  B)  da   0 J  da  0Ienclosed
S
Ampere’s Law
S
 
 B  d l  0Ienclosed
C
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Y.M. Hu, Associate Professor, Department of Applied Physics
史托克定理(The Stokes’ theorem)
 
 
 (  v)  da   v  d l
S
C
The surface S having the boundary line C
is arbitrary. We should therefore choose a
surface that allows the easiest valuation of
the integral.
For a closed surface, the integrals are zero
because the boundary line then shrinks
down to a point
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40
Y.M. Hu, Associate Professor, Department of Applied Physics
史托克定理(The Stokes’ theorem)
 
 
 (  v)  da   v  d l
S
C
The velocity field on the left has a certain amount of vorticity.
Tornadoes develop from such flow fields. Although drawn
inaccurately, the resulting flow field on the right would have
exactly the same amount of vorticity.
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41
Y.M. Hu, Associate Professor, Department of Applied Physics
笛卡兒直角座標(Cartesian Coordinates)
ẑ

r  x x̂  y ŷ  z ẑ

r

d l  dx x̂  dy ŷ  dz ẑ
r̂
z
ŷ
x
x̂
y
The unit vectors do not change direction from point to point
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Y.M. Hu, Associate Professor, Department of Applied Physics
球形極座標(Spherical Polar Coordinates)
x  r sin  cos 
y  r sin sin 
z  r cos 
0    2
0 r 
0 

A  Ar r̂  A ˆ  A ˆ
r̂  sin  cos  x̂  sin  sin  ŷ  cos  ẑ
ˆ  cos  cos  x̂  cos  sin  ŷ  sin  ẑ
ˆ   sin  x̂  cos  ŷ
The unit vectors change direction from point to point
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43
Y.M. Hu, Associate Professor, Department of Applied Physics
球形極座標(Spherical Polar Coordinates)

A  x x̂  y ŷ  z ẑ  r sin  cos  x̂  r sin  sin  ŷ  r cos  ẑ



A / 
A / 
A / r
ˆ
ˆ
 
 
r̂  
A / 
A / 
A / r

A / r  sin  cos  x̂  sin  sin  ŷ  cos  ẑ

2
A / r  sin 2  cos 2   sin 2  sin 2   cos 2   1
 r̂  sin  cos  x̂  sin  sin  ŷ  cos  ẑ
其他依此得出!
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44
Y.M. Hu, Associate Professor, Department of Applied Physics
球形極座標(Spherical Polar Coordinates)
Homework : Prove that
x̂  sin  cos  r̂  cos  cos  ˆ  sin  ˆ
ŷ  sin  sin  r̂  cos  sin  ˆ  cos  ˆ
ẑ  cos  r̂  sin  ˆ
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Y.M. Hu, Associate Professor, Department of Applied Physics
球形座標下的體積與面積單元(Volume and area
elements in spherical coordinates)

d l  dlr r̂  dl ˆ  dl ˆ  dr r̂  rd ˆ  r sin d ˆ
dr
r
rsind
rd
r

d
r
d
rsin
Surface element
(see P.41)
r is constant

da1  r̂dldl  r 2 sin ddr̂
 is constant = /2

da 2  ˆ dl r dl  rdrdˆ
Infinitesimal volume element
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d  dl r dl dl r 2 sin drdd
46
Y.M. Hu, Associate Professor, Department of Applied Physics
球形座標下的梯度、散度、旋度與拉普拉斯
T ˆ 1 T ˆ 1 T
T  r̂


r
r 
r sin  

v 
v
1


2
[sin  (v r r )  r (v sin )  r
]
2
r sin 
r


r̂
1


v  2
r sin  r
vr
rˆ


rv 
r sin ˆ


r sin v 
1
 2 T

T
1  2T
  T  2
[sin  (r
)  (sin  ) 
]
2
r sin 
r
r

 sin  
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Y.M. Hu, Associate Professor, Department of Applied Physics
圓柱座標(Cylindrical Coordinates)
x  s cos 
y  s sin 
zz
0s
0    2
 z 
ŝ  cos  x̂  sin  ŷ
ˆ   sin  x̂  cos  ŷ
ẑ  ẑ
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48
Y.M. Hu, Associate Professor, Department of Applied Physics
圓柱座標(Cylindrical Coordinates)

A  x x̂  y ŷ  z ẑ  s cos  x̂  s sin  ŷ  z ẑ



A / z
A / s
A / 
ˆ
ẑ  
ŝ  
 
A / z
A / s
A / 

A / s  cos  x̂  sin  ŷ

2
A / s  cos 2   sin 2   1
 ŝ  cos  x̂  sin  ŷ
其他依此得出!
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Y.M. Hu, Associate Professor, Department of Applied Physics
圓柱座標下的體積與面積單元(Volume and area
elements in cylindrical coordinates)

d l  dls ŝ  dl ˆ  dlz ẑ  ds ŝ  sd ˆ  dz ẑ
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Y.M. Hu, Associate Professor, Department of Applied Physics
圓柱座標下的梯度、散度、旋度與拉普拉斯
T ˆ 1 T
T
T  ŝ

 ẑ
s
s 
z
1 v v z
 1
v 
(sv s ) 

s s
s  z
ŝ sˆ
ẑ


 1 
v 
s s  z
v s sv  v z
1  T
1  2T  2T
  T 
(s )  2 2  2
s s s
s 
z
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51
Y.M. Hu, Associate Professor, Department of Applied Physics
The Dirac Delta Function – The Divergence of r̂ / r 2
The example of a point charge:
然而當我們計算電場的發散(divergence)值時,
But the divergence theorem says:

 
 (  v)d   v  da

S
Our analysis goes wrong at r = 0 because of the 1/r2 term which
becomes singular (we are not treating that point correctly).
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52
Y.M. Hu, Associate Professor, Department of Applied Physics
The One-Dimensional Dirac Delta Function
 0,
( x )  
,
and
if x  0
if x = 0

 (x )dx  1

The one-dimensional Dirac delta-function is named after the Cambridge
physicist Paul Dirac who invented it in 1927 while investigating
quantum mechanics. The delta-function is an example of what
mathematicians call a generalized function, or distribution: it is not welldefined at x = 0, but its integral is nevertheless well-defined.


 f (x )(x )dx  f (0)  (x )dx  f (0)

 f ( x ) ( x  x

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

0
)dx  f ( x 0 )
53
Y.M. Hu, Associate Professor, Department of Applied Physics
The Three-Dimensional Dirac Delta Function
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54
Y.M. Hu, Associate Professor, Department of Applied Physics
Classification of fields

Class I fields :   F  0

F  f
and

F  0
everywhere in a region
 2f  0
Laplace’s equation
Electrostatic fields in charge-free medium
Magnetic fields in current-free medium

Class II fields :   F  0

F  f
and

 F  0
everywhere in a region
 2f  0
Poisson’s equation
Electrostatic fields in a charged region
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55
Y.M. Hu, Associate Professor, Department of Applied Physics
Classification of fields


everywhere in a region
Class III fields :   F  0 and   F  0
 



Coulomb’s gauge   A  0
 F  J
F   A


2
Poisson’s vector equation
 A  J
Magnetic fields within a current-carrying conductor

Class IV fields :   F  0
  
FGH
and

 F  0

G  Class

H  Class
everywhere in a region
III
II


G   A

H  f
Hydrodynamic fields in a compressible medium
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