# Engineering Electromagnetics _Pt.1_

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```					Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Chapter 1 Electromagnetic Model
A field is a spatial distribution of a quantity（a scalar or
a vector）, which may or may not be a function of time.

A time-varying electric field is accompanied by a magnetic
field and vice versa.

（That is , time-varying electric and magnetic fields are
coupled resulting in an electromagnetic field.）

Under certain conditions, time-dependent electromagnetic
fields produce waves that radiate from the source.

1           Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

（Postulating the existence of electric and magnetic fields
and electromagnetic waves ）

Field → wave
↑
（Time-varying field）

In the transmitting unit ,when the length of the antenna is an
appreciable part of the carrier wavelength a non-uniform
current will flow along the open-ended antenna.

2          Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

This current radiates a time-varying electromagnetic field in space,
which propagates as an electromagnetic wave and induces
currents in other antennas at a distance. The message is then
detected in the receiving unit.

screen

RS
A      B

B
VS ( t )
AC                                 RL             e

E

Fig 1-1                          Fig 1-2

3     Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Force → Field → Wave
（Gravity , Electricity Magnetism）       （due to time-varying field）

E  mc2

Field：presence of energy

Wave：signaling or action-at-a-distance

4          Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Chapter 2 Vector Analysis
Vector Algebra
Vector representation
 
（1）A Vector       A  a AA
             
Where A A       and   aA   is a dimensionless unit vector

               A
specifying the direction of A , i.e. , a A 
A
                                      
（2）Equal vector A  B  a B B where         AB      and    aA  aB

Even though they may be displaced in space.

             Fig 2-1
A        B
5          Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

  
（1） A  B  C
C

B

A
Fig 2-2

   
（Ex） A B D  C                                          
                 D
C


B

A
Fig 2-3
6       Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
  
（2） A  B  C
                               
key： C arrowhead points to that of   A

           
B           C


A

Fig 2-4

7         Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Vector Multiplication
 
（1）Dot product A  B  AB cos  AB , especially

                                       
A  A  A  A  A2  A          A  A  A       
B
                        AB          
Key：the correlation of     A and                                  A
B cos AB
B

Fig 2-5

8             Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
    
（Ex） Find     AB  C

           
          C
      
Sol： A  B  C                      B  AB                 B

   
 A  B  A  B

A
2 2                     Fig 2-6
    A  B  2 A B cos AB

 A 2  B 2  2AB cos

 
From definition   ax ax 1

9            Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Note：Dot product （i.e., inner product）has two definitions

   N         
A  B   i  i  A  B cos AB
i 1

                   
B is projected onto A
or sum of product of their components on the same base.

  
A  B  a x A1  a y A2  a z A3 a x B1  a y B2  a z B3 
                                
i.e.
3                            
  Ai Bi  a x  a y  0..., etc. 
                         
i 1                             

10               Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

（2）Cross product

                                               
A  B  a n A  B sin  AB                
（read “ A cross B ”）
                                                                  
where   an   is a normal vector perpendicular to the plane containing A and B
                                           
From a right hand, “ A cross B ”means the gingers rotate from A to B through  AB


B
 AB
              
an             A
Fig 2-7

11            Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Note：
(1)Length evaluation
      12
C  C  C                From definition

az
       
ax ay  az
                                           
az ax  ay                                   ay
                         
ay az  ax                 ax
                          Fig 2-8
where     ax  ay  az

(2)Area evaluation
 
AB               
A  B  a A A  a A B cos AB  a A, B sin  AB 
                        

 a n A B sin  AB
           
where     a A  a A,  a n
12            Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
（3）Triple product
(i) Scalar triple product

              
A  B  C   A  a n BC sin  BC
  
Volume evaluation ： A  B  C

                         
Important identity      A  B  C  B  C  A   C  A  B

           
A           C      
      a BC
n
a CA
n
        Fig 2-9
B

13        Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Note：The normal vectors for each cross product in this
identity points to the interior of the volume.

(ii) Vector triple product

              
A  B  C   A  a n BC sin  BC

Note：The above vector manipulations do not involve the
concepts of coordinate system.

14           Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Orthogonal Coordinate System
Introduction
Z
（1）Cartesian Coordinate ：

P  ( X , Y , Z)
‧A right- handed system                   

0
                                              Y
Base vector a x a y a z
                      X
e.g., az  ax ay                        Fig 2-10

(i) Point P=（x , y , z）

15       Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

(ii) Position vector
                        
OP  a x x  a y y  a z z  x, y, z

(iii) Vector
                  
A  a x AX  a YAY  a ZAZ


（x
(iv) Vector field A , y , z）

Scalar field r（x , y , z）

16            Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

(v) Vector differentials or products

Vector differential line
                        
d  a d  ax dx  ay dy  az dz

Z
  
d  a d

            
  d
0                        Y
   ( x , y, z )

X
Fig 2-11

17                   Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Vector differential surface

             d
                                             an   dz
d s  a n ds
dy
For example ：                                        dx
Z   
d s  a x dx   a y dy 
                                                d        ds

Y
      
a n  a x  a y , and       ds  dxdy              X
Fig 2-12

i.e., the unit normal vector perpendicular
to the plane containing d s


18          Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Differential volume dv

For example ：

dv  a z d z   a x dx   a y dy
                        

                                       
 a z a n dxdydz whrer
                  an  ax ay 

                              
 
 dxdydz a z  a n  1

19               Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

（2） Cylindrical coordinates

Z
‧A right-handed system
P  (X, Y, Z)

                              
Base vectors a r , a  a z                              Z
0
r          Y
      
e.g., a z  a r  a               X       

(i) Point P  r,  , z                    Fig 2-13

(ii) Vector

                    
A  a r A r  a  A  a z A z

20     Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

(iii) Vector field A r,  , z 

Scalar field V r,  , z 
            
*If A  a r A r  a z A z , i.e., A  0
                      
then A is a position vector 

(iv) Vector differentials or products

Vector differential line (or length)
                                          
d  a  d  a r dr  a  rd  a z dz ,where   r,  , z 
↑
Metric coefficient for expressing vectors
(∵A vector consists of its length and direction.)

21           Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

                  
Vector differential surface                                 d   ax dx  ay dy  az dz
 
ds  an ds
rd
For example：                                                                       
              
d s  (a r dr)  (a  rd )                                    dz         d
Z
    
                                               d
 (a r  a  )(rdrd )

                                                                dr
 a z rdrd
r              Y
Differential volume dv                                   X 
Fig 2-14
For example：                                 
dv  (a z dz)((a r dr)  (a  rd ))
                                     
 (a z a n )rdrddz( where a n  a r  a   a z )

           
 rdrddz( a z  a n  a z  a z  1)
22               Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Note ： Vector representation in cylindrical coordinates
                   
A  a r Ar  a  A  a z A z
Involves the concept of metric coefficient since


a  is a has vector for angle , not for length

Therefore , A  should contain a metric coefficient ,

so that A  is the value of length

23          Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
（3）Spherical coordinates

Z
‧A right-hand system
P  ( R , ,  )
                              θ R
Base vector     a R , a , a                       
r            Y
                               X       
e.g. , a   a R  a 

(a) Point   P  ( R , ,  )                          Fig 2-15

                   
(b) Vector A  a R A R  a A  a  A

24       Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU


(c) Vector field A ( R , ,  )
dR
Scalar field   V( R , ,  )                  Z

R sin d          d
 
                   d
θ 
R          Rdθ
(d) Vector differentials or products
r
Y
Vector differential line (or length)

                                 X
d   a                                     Fig 2-16
                
 a R dR  a  Rd  a  R sin  d
 
where   ( R , ,  )

25        Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Vector differential surface
 
d s  a n ds
               
For example ：        d s  (a R dR )  (a  Rd )
                        
 a  RdRd , ( a   a R  a  )

Differential volume
           
For example ：         a  dA  d s
                               
dv  (a  R sin  d )  ((a R dR )  (a  Rd ))
 
 (a   a  )R 2 sin dRd d

 R 2 sin  dRd d

26            Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Note ： (1) R：the radius of 3-dimensional sphere

(2)Volume is not directional

(3)The angle Φ is cylindrical coordinate require a metric
coefficient r to convent “a differential angle change ”
d to a differential length change” i.e., rd

(4)Similar to (3), the metric coefficient corresponding to the
angles θ and Φ in spherical coordinates are R and
Rsinθ, respectively.

27           Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Let the metric coefficients     h 1, h 2, h 3 correspond to the space variables

(u1, u2, u3) in a general coordinate system. Especially,

u3
˙In Cartesian coordinates (u1, u2, u3)=(x, y, z)                       h 3du 3
h1=1, h2=1, h3=1
h 2du 2
h1du1
u2
˙ In Cylindrical coordinates (u1, u2, u3)=(r, Ø, z)
h1=1, h2=r, h3=1
u1
Fig 2-17

˙In Spherical coordinates (u1, u2, u3)=(R, θ, Ø)
h1=1, h2=R, h3=Rsinθ

28            Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

         
(6) Dot product of A and B in Cartesian coordinates
                                                    
A  B  (a x A x  a y A y  a z Az )  (a x Bx  a y By  a z Bz )

 A x  Bx  A y  B y  A z  Bz

                               
A  B  a A  A (a A  B cos AB  a A  B sin  AB )

 A  B cos AB
                                    
( a A  a A  1, a A  a A '  0( a a   a A ' ))

29            Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

     
(7) Cross product of A and B is Cartesian coordinates

                                                    
A  B  (a x A x  a y A y  a z Az )  (a x Bx  a y By  a z Bz )
                        
 a x ( Ay Bz  Az B y )  a y ( Az Bx  Ax By )

 a z ( Ax B y  Ay Bx )

           
aX     aY    aZ
 Ax      Ay    Az
Bx     By    Bz

30             Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

(8) A vector in Spherical coordinates

                         
A  a R AR  a A  a A ; a   direction           A   length

where    A  and A  should contain metric coefficients for

representing vector, since a vector consists of its length and direction.

(9) Unlike the Cartesian coordinates, in cylindrical coordinates and

Spherical coordinates, expressing a position vector is trivial , since

                    
a r A r  a z A z and a R A R are respectively position vector.

31           Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Coordinate Transforms
(1) Point coordinates

(a) (r,Φ, z) → (x, y, z)
x= rsinΦ                         Z

y= rsinΦ
P
z=z                               θ R

(b) (R, θ, Φ) → (x, y, z)                          r       Y

∵ r = Rsinθ               X
Fig 2-18
∴ x= (Rsinθ)cosΦ

y= (Rsinθ)sinΦ

z= Rcosθ
32       Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
(2) Vector (differential) transforms
                                            Z
d  a x dx  a ydy  a z dz
 
where           l ( x, y , z )                                
d

          
(a) (r,Φ, z) → (x, y, z)                                          d
0
Y
dx= cosΦdr – rsinΦdΦ
X              Fig 2-19
dy= sinΦdr + rcosΦdΦ

dz= dz      (Base on the corresponding point coordinates,
the vector differentials are taken.)

dx  cos          sin     0  dr 
 dy   sin 
                  cos      0 rd 
     
 dz   0
                    0       1  dz 
     

33           Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
                    
Similarly, Let A  a x Ax  a y A y  a z Az
                 
 a r A r  a  A  a z A z

We have
Ax  cos         sin    0  Ar 
 Ay    sin    cos      0 A 
                            
 Az   0
                   0      1  Az 
 

Note : Since a vector consists of its length and direction we have to
consider the metric coefficient for a vector differential, e.g., rdΦ.
Then we can extend the transform of a vector differential
to that of a vector.

34             Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Note : Another solution to the vector transform

(r, Φ, θ) → (x, y, z) at D.K.Cheng, PP31-32

By dot product techniques, since
                  
A  a r Ar  a A  a z Az
                 
 a xAx  a yA y  a zAz

We have
                                   
A x  A  a x  (a r A r  a  A   a z A z )  a x
                  
 (a r  a x ) A r  (a   a x ) A 
    
az  a x

To see the details, refer to D.K.Cheng

35               Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

(b) (R, θ, Ø) → (x, y, z)

From the corresponding point coordinates , we take the vector differential .

dx = sinθcosΦ dR + RcosθcosΦ dθ- RsinθsinΦ dΦ

dy = sinθsinΦ dR + RcosθsinΦ dθ- RsinθcosΦ dΦ

dz = cosθdR – Rsinθdθ

dx  sin  cos cos cos  sin    dR 
dy   sin  sin  cos sin  cos   Rd 
                                             
 dz   cos
                      0        1  R sin d 
           

36               Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

                   
Similarly, let    A  a R A R  a A  a  A
              
 a xAx  a yAy  a zAz

We have

 Ax  sin  cos        cos cos     sin    AR 
 Ay    sin  sin                             
                       cos sin    cos    A 

 Az   cos
                           0           1   A 
 

37            Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Note :

(1) In Cartesian coordinates (x, y, z), the base vector
  
a x , a y , a z are position-invariant; i.e., the directions

of these unit vectors are unchanged to represent
                    
a vector     A  a x Ax  a y A y  a z Az

(2)In cylindrical coordinates (r, Φ, z) ,the base vectors
       
a r and a  are varied with position; i.e., the directions

of the two unit vectors are dependent on the position

on which the represented vector is located.

38             Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
                                     Z
A  a r A r  a  A  a z A z         
az     
A
Therefore, the cylindrical coordinates can be
                       r              Y
easily applied to describe a position vector              X                    
                                                                      ar
i.e., A  a r A r  a z A z  
Fig 2-20
if A  0                                                                 
Z            aZ        
a

When     Ais not a position vector, the component                                           
                                                                                       A
a  A  plays an important role in representing    A              ar                 ar
Z
                                                              r
Y
,since   ar   expresses only the direction of a position   X                         
ar
vector projected onto the x-y plane for describing
                                                  Fig 2-21
the location of the vector   A
39              Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
See the top view of the cylindrical coordinate to show the above

point of view.

ie.,   a r is change from a location to another.


偏心          A
Z                                                                
a a r                                a          ar
Rsinθ
對準圓心
                        r
θ
R            a               Z
-

Z                                            Y
r
Y
                                                                  
X                                                                 A  a r A r  a  A
^^^^ 軸向大小與方向
Fig 2-22                                      X
Fig 2-23

40                  Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
The expression of vector transform (r,ψ,z) → (x,y,z) is a general

form for the point transform (r,ψ,z) → (x,y,z) since a point in a coordinate

system can be regarded as a position vector.

A x  cos        sin         0  A r 
                                    
 A y    sin 
          cos          0  A  

A   0                          1 A Z 
 z                   0           
                                         
A point vector     a x x  a y y  a zz  A  a xAx  a yA y  a zAz
          
 a rAr  a zAz             ( A   0)

A x   x  cos              sin    0 A r 
 
 A y    y    sin 
                    cos     0  0 
 
A  z   0                            1  Az
 z                          0         

41                 Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

x  r cos 

y  r sin       ie., the point coordinates     (r,ψ,z)→(x,y,z)
zz

Therefore the vector representation in (r,ψ,z) is relevant to it’s location.

If a vector     A   is treated as a position vector, its representation is

changed since a r A r is changed to express a distinct radial component.

42            Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

(3) All the argument in this Note(2) hold for vector representation in

spherical coordinates.

A point p in (r,ψ,z) → (x,y,z) ←─→ A vector A in (r,ψ,z) → (x,y,z)

or (R,θ,ψ) → (x,y,z)                           or (R,θ,ψ) → (x,y,z)

                                 
This is because    A  a R A R  a A  a  A , where a R means the unit

vector to indicate the position vector of the location of A

43           Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

                                                   
If   A   is a position vector, we have                  A  a R A R ( A   A   0)

Z                    
a a r                               
A  a R AR  a A  a A

R            
θ                    a
r
Y
X        

Fig 2-24

44                Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

When the vector     A   is translated and treated as a position vector, it’s vector

representations in cylindrical coordinates and in spherical coordinates

are changed.

However the vector representation for         A in Cartesians coordinates are invariant

, and the vector   A   always has no change in its direction and magnitude
Z                              
in any coordinate system.                          A  a R A R  a A  a  A
   
A  a R AR

Y
X

Fig 2-25

45              Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Physical meaning of curl of a vector field A

Fig 2-26

Quiz #2.
1. Give an interpretation of the curl of a vector field

B and illustrate its meaning in detail. (60%)
2. Compute the divergence of the curl of a vector field
                   
B (i.e., (   (  B) ) and show your result. (40%)

46             Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Vector Calculus
•Gradient of a scalar field let        ( 1 ,  2 ,  3 ) be a scalar function

of space coordinates        ( 1 ,  2 ,  3 ) and it may be constant along

certain lines or surface.

Consider the space rate of change of                ( 1 ,  2 ,  3 ) in a specified

direction, e.g., the direction of          d  , is a directional derivative.

47               Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Note：

P3


d                             P2
α
1  d

dn
P1

 ( 1 ,  2 ,  3 )  1           Fig 2-27
for some  1 ,  2 , and  3

48                Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Illustrate the meaning of gradient of Φ and grad           
     
where      ax     ay     az       in Cartesian coordinates.
x      y      z

<Ans>

The directional directive

d   d     d  d dn   d                        
    al      al        al   cos   al grad  al 
d  al d l   d      dn d       d 

     d               
           an al      
      d                
 d   
 cos  a 
                an al 
                        
l
dn

d                     d  d
dn                      dn      dn

49              Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

From the above, we see that

d           
d
                   
 d  grad  a l  d  grad  d   1

Total derivatives in Cartesian coordinates

              
d   dx 
 x         dy      dz 
         y       z   
        
 a x dx  a y dy  a z dz  2
        
 a x
 x       ay     az
              y      z    
                       
d

50            Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Compare (1) with (2), we have

     
grad  a x     ay     az     
x      y      z

      
where          a x
ˆ      ay     az 
 x      y     z 


     

Note： A  B C  A B  C
    
 
  
                不滿足結合律
scalar         scalar

                          
∴   grad d   d  grad d  a   d a 
               

51     Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
A vector field

• Divergence of         A

div A   Fig 2-28

                                  
A(X, Y, Z)                         A(X, Y, Z)

q                                                          
A(X 0 , Y0 , Z 0 )
(X 0 , Y0 , Z0 )                   q (X 0 , Y0 , Z0 )

Fig 2-29                                   Fig 2-30
52     Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Gauss’s divergence theorem

        
   Adv   A  ds
v           
Fig 2-31

Helmholtz theorem：

A vector field is determined if both its divergence and its curl

are specified everywhere.

53               Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Illustrate the meaning of divergence of        A and
       
A   a x
 x  a y y  a z z 
Y
                       
 a x A x  a y A y  a z A z 
                  
dz                     dy
A x   A y A z                                                    SL
                                                     SF         ST               SB      Ax
x    y    z                              Ax                                  Ax        dx
                    x
SR         X 0 , Y0 , Z 0
X
<Ans>                                                         S TB dx
Consider the special case：                     Z
Fig 2-32
S F flux at x 0 : A x dydz
A x 
S B flux at x 0  dx :  A x 
           dx dydz  A x x 0  dx dydz
        x           

a A  x  dx a ds
x    x       0   x

                                 A x             
 A x x 0  dx   A x x 0          x x 0 dx 
                                  x              
54                   Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Outward flux over          SF and SB
A x
A x x 0  dx dydz  A x x 0 dydz           x x0   dxdydz
x
Similarly, we have the outward flux over               S T and S B

A y
y y0   dxdydz and outward flux over          S L and S R
y
A z
z  z 0 dxdydz
z
Therefore, the net outward flux at point              (x 0 , y0 , z0 )

 A x A y A z                                
dxdydz    A dv   ax          a z a x A x  a y A y  a z A z dv
         
                                         x  ay
 x
        y      z  
 dv                            y      z 

            
 
        
        v

55                 Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU


It follows that we can define the divergence of A

 
       
divA    A  lim
ˆ
v0    v
where

 A x A y Az           
A                  and A ( x 0 , y 0 , z 0 ) is continuous and differentiable.
x   y   z

 
d  A  d s

56              Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Fig 2-33

point (or an object)

Fig 2-34

57    Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Gauss’s Divergence theorem

           
   A  dv   A  ds
v               

The divergence theorem is an important identity in vector analysis.
It converts a volume integral of the divergence of a vector to a
enclosed surface integral of the vector, and vice versa.

     
flux :  A  Ads

Fig 2-35
58             Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

• Curl of a vector field

Curl： A( x, y, z)

Ax
Ax        dy
y
Ax
Ax
Ax        dy
regarded as                                   y
or
Ax
2       dy
2dy                              y
Ax
dy
1dy                   y
Fig 2-36                          Ax

59               Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU


Divergence: A( x, y, z)

Ax
Fig 2-37                     Ax        dx
x
dz
dy
dx

 Ax
 0)
y

Fig 2-38

60   Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Az
Az        dz
z       Ay
Ay      dy
y
Ax
Ax                        Ax         dx
Fig 2-39                                              x
Ay
Az
Ay
(Ay      )dS XZ  AydS XZ
y

                
ax           ay   az
                                
Illustrate the meaning of curl of     A and       A 
x           y   z
Ax           Ay   Az

61            Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Ay
Fig 2-40
Ay       dx
x
Ay          Y
Ax
<pf> Consider the special case on                                                                Ax        dy
(X 0 , Y0 )                y
Ax
the x-y plane, we see that
Ax
Ay                              Ax        dy
Ay      dx            Xo                     y
  Ay  Ax          
  curlA z a z
x                                          X
az           ˆ
 x  y 
                                                       (X 0 , Y0 , Z0 )
         
             a X  aY  aZ
Z     (  A) Z

Similarly, on the y-z plane
Ay
                    dx Y
aZ                x                                    Ax
  Az  Ay                                                                                     dy
  curlA x a x
y
ax           ˆ
 y  z                                                                                    (X 0 , Y0 )
Ax
         dy
Ay    (X 0 , Y0 )                              y
and on the z-x plane                           dx                                                   X
x                                       
- aZ
  Ax  Az         
ay          curlA y a y
ˆ                                              Ay      Ay            
z  x 
aZ      (     ) ( ) Z a z
                                                             x       x 
Z
Fig 2-41
62                Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
    
ax ay az
                                                        
curl A  a x curl A x  a y curl A y  a z curl A z
                
∴                                                                 
x y z
                                  Ax Ay Az
curl A    A
ˆ

Note：
      Ay  Az  
1.   A   a x 
Y

 z    y  

            
  Az  Ax  
 ay x    z    
                                                           X

  Ax  Ay  
 y    x  
 az                                               Z
            

Fig 2-42

63              Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Illustrate Stokes’s theorem
Consider the line integral along path a b c d on the x-y plane.
                                    
A  a x A x  a y A y and d   a x dx  a y dy

Path ab：Axdx
Y                  Ax
Ax        dy
      Ay                                                      y
Path bc：  Ay      dx dy
       x                                      d                c

      Ax                                A                              Ay
Path cd：   Ax                                                              Ay        dy
          dy dx
                            y                               x
       y                              Y a A                 b
x                       X
         Ax          
 a x  Ax 
          dy    a x dx 

X                X  dx
       y                                        Ay
Ay 

   

dy
Path da：  Aydy  a y Ay   a y dy                                   x

Fig 2-45
          Ay Ax            
  Ad                    A dS xy
Sxy  x   y  Sxy
abcd
         
64       Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Similarly, in the 3 dimensional case, we have circulation

                
 A  d      Ads
C          S

 
ds  a n ds
         right-hand rule
az
∵intuitively for example,
  
 az  ax  ay
ay            
                 (a x Ax)  (a y Ay)                                           Fig 2-47
ax                              
 (Ax  Ay)a z                               C
Fig 2-46                                               where the surface S is
bounded by the contour C

65            Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

We may define the curl of     A

       
curlA    A  lim
ˆ
S0 S

1   
a n  A  d
C

max

Where S is the area enclose by the contour     C

Since A is a vector point function, the value of line integral
 
 A  d  depends on the determination of the contour
C
C

curl A is a measure of strength of a vortex source.

66             Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Contour C
Orienting the contour C
in the such a way that
the circulation is a maximum.


S                          (Curl A ) z
Fig 2-48

Curl( A )


(Curl A ) y

(Curl A) x

Fig 2-49
67           Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
(5) TWO NULL IDENTITIES

(1)     V   0

It means that gradient (of a scalar field) is curl-free or invitational.

(zero net circulation)

<pf>

By Stokes’s Theorem

     V ds   Vdl   dV  0
S                   C          C

     V   0

68            Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU



(2)     A  0  
It means that gradient (of a scalar field) is curl-free or invitational.
(zero net circulation)
<pf>
By Divergence Theorem
                                               
V     Adv  S   Ads  S1   Aan1 ds  S2   Aan2 ds
        
  A  d   A  d  0
C1         C2

(∵ C1 and C 2 traverse the same path in opposite directions )

                               V
a n1              C1
S1             C2 S
2     
a n2
Fig 2-50                   outward normal vector
(for evaluating outward flux)
69             Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Note：

(1) Divergence Theorem：

Outward flux of a vector field

         
i.e.    V
  Adv   A  ds
S

(2) Stokes’s Theorem：

Net circulation a vector field

         
i.e.          Ads   A  d 
S              C

70               Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Divergence theorem

For example,

 v
E 
0
                   1                    Q
  E  d s     Edv            v dv 
S            V        0   V               0

71            Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Stokes’s theorem

For example,

E  0
           
  E d      E ds  0
C          S

72       Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Chap3. Static Electric Fields

Virtual displacement
( Gauss s law )
Force                    Field          Energy ( work )
( Coulomb s law )       ( Field intensity )


Electric displacementD        Potential V ( Electric dipoles )
( Electric flux density )    ( Scalar field )           

( P polarization vector in material media )       ( J in a good conductor )

73             Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Cheng’s approad for static electric fields：
Two postulates ：（in different form）

 
（1）   E  v     0 （Gauss’s law）

       
（2）   E  0 （ E is irrotational i.e. it is conservation）

The postulate 1 can be derived later.

74        Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Two postulate of E filed in free space
 
1.   E  v
0


         F
where     E  lim
q 0
q

 v ：volume charge intensity of free charges.
q is small enough not to disturb the distribution of source charges.
 0 ：permittivity of free space.

75          Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

+

+                 -
-Q                       +
-                        -
r=a
+
+Q +           +
+        +    r=b
+

-                        -
+                 -               +

+
( Electric displacement )

76              Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

＊ Definition of Electric field intensity


         F
E  lim
q 0
( V m or        N
C )
q

where the force is measured in newtons（N）and charge q in Coulombs （C）

77             Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

NOTE：
1. The test charge q , of course , cannot be zero in practice；
as a matter of fact , it cannot be less than the charge on
e 1
an electron ’e’ , where.602  10 19      （C）

2. Also , the test charge is small enough not to disturb the charge
distribution of the source

78           Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Coulomb’s law
                     q 1q 2
In 1785 , Coulomb found that                    F12  q 2 E12  a r
12
40 r122

1
Where the permittivity of free space                     0         109         ( Fm )
36

r12a r
12

q1
             q2

r12                       F12

Field point charge              Test point charge

Fig.3-2

NOTE：
                                                           q1 
F12  q1q2                1
F12  2              and      E12               ar
,
r12                                40 r12
2  12

79                Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
3-2 Gauss’s Law

Consider electric field intensity                      E    going thru
 
a differential surface dS  a n dS and the corresponding flux.

( Assume that there is no charge outside the enclose hypothetical surface S )
 
E  a rE

E
 
dS  a n dS

ar
d                                          Fig.3-4

+q
P

A point charge located at P
S

Fig.3-3
80           Dr. Gao-Wei Chang
OptoelectronicSystemsLab., Dept. of Mechatronic Tech., NTNU
       
d  E  dS  Ea r  dSa n
q
            d cos 
40 r 2
E
q
         d
40

A partial area of sphere

dS cos 
where d         2
is the solid angle involvingd cos 

       
and  is the angle between a n and a r ,

（ i.e. dScos  is called the effective area ）

81              Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

The total flux going thru the entire enclose surface S

q         q           q
   d  S        d       Sd        4
40      40        40

 
d  E  dS

E


dS                
an

Fig.3-5

82         Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Consider the point charge is located at P’ outside the closed surface S.

To evaluate flux resulting from a point source , we introduce the concept

of solid angle.
( Assume that there is no charge inside the closed hypothetical surface S )

dS2

                    E2
r2           2
dS'2

dS1'          
E1
d
1
+q                                               S
dS1
P'

Fig.3-6

83                    Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
       
d  E1  dS1  E 2 dS2
'
dS1                             dS'2
   
q  a r dS1 a r dS2 
       2  2   1 d1  d 2 
1            2
0
40  r1
          r2    40
              
a dS           a dS
( ∵ d1  r 2 1  d 2   r 2 2  d )
1                         2

r1             r2
( That d1  d ,              d 2  d )

∴ 0
0   , q is located outside S
 
｛
∴  EdS 
S
q , q is located inside S
0
……( 1 )

84              Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

NOTE：
Any charged object may be regarded as a collection of an infinite

number of point charges.


ar


ar                  dS2

dS1
d

Fig.3-7

85         Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

R sin 

dS

R

d

Fig.3-8

86            Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

dS  Rd 2R sin   2R 2 sin d

∴ The area of a sphere：
                                  
S  0 2R sin d  2R
2                 2

0
sin d

 2R 2  cos 

0
 4R 2

∴ Steradian of a sphere is define as
S
2
 4
R

In general , we define steradian
S

R2
87                Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Suppose a collection of point charges q 1 , q 2 ,…. q n are distributed

inside enclosed surface S.
The Eq(1) can be rewritten as
  1 n
SEdS    q i
0 i 1

Or for a charged object with volume density    . Eq(1) can be rewritten as
  1
SEdS   v v dv ……（2）
0

Where  v is volume charge density and  0 is permittivity of free space .

88          Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

In view of the Gauss’s divergence theorem , Eq(2) becomes

      v                            q
SEdS  v  dv  v  Edv        ( 
0
)
0

 v
∴ E     （point form of Gauss’s law）
0

（Usually we don’t use the differential form since the derivative does not

exist at boundary points or discontinuous points.）

NOTE：
A Gauss’s surface is a hypothetical surface over which Gauss’s law

is applied and it is needed for the integral form of the law .

89                Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Ex：Electric field intensity due to a point charged .
     
E  ERa R

q          
R  Ra R

Fig.3-9

Since a point charge has no preferred direction , its electric field must be

everywhere radial and has the same intensity at all points on the spherical

surface .（i.e. the Gauss’s surface）

Due to the fact that ( electric force lines do not intersect with each other )

90             Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

（The choice of Gauss’s surface is very important to simplifying the

integration in Gauss’s law）
                         q
SEdS  Sa R E R   a R dS 
0
or
E R SdS  E R 4R 2  
q                    q
=>    ER 
0                 40 R 2
Therefore
                  q
E  a RER  a R
40 R 2

91                Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
NOTE：
Electric field intensity of an isolated point charge at an arbitrary

location P .

 
R  R0
             
R0            R

Fig.3-10

92             Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
            q
E P  a qP                2
40 R  R 0

Where the unit vector a qP drawn from q to P

 
      RR
a qP    0
R  R0

R 0 is the position vector of q .

and R is the position vector of field point P .

Thus , we have
 
       qR  R 0 
EP           3
40 R  R o

93   Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
NOTE：
            
qa r
1. ∵      E           f q is linear
40 r 2

                                                          
∴     f  a1q1  a 2 q 2   a1f q1   a 2 f q 2   a1E1  a 2 E 2

2. A single（point）charge → Continuous charge distribution
（given charge distribution） （linear , planar , spherical , disk ,….）

94              Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Gauss’s law
  v v dv
SEdS  
0

Gauss’s law is particularly useful in determing the E-field of charge
distributions with some symmetry conditions , such that the normal component of

E is constant over an enclosed hypothetical surface（called a Gauss’s surface）.

95         Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU


Ex：Determine E of an infinitely long straight line charge .

     
                      E  ERa R
dS  a R dS
     
E  ERa R
R
l

Fig.3-12                       L

Fig.3-11

96   Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Sol：
Applying Gauss’s law ,

  q
SEdS  
0

Where S is a Gaussian surface

L                                 l
E R 2RL   l     =>        E  a R ER  a R
0                                 20 R

97              Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Ex：An infinite planar charge


az
A                          
E  a zEz
S
S
+    +     +    +     +    +          L

    
E  a z E z
Fig.3-13

98             Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Sol：
Applying Gauss’s law , we have

  q
SEdS  
0

where S is Gaussian surface

                                    S  A
=> E Z a Z  a Z A  E Z  a Z  a Z A 
0

=>  EZ  S
2 0
 

﹛
aZ S       z0
               20
∴ E=
  S z  0
aZ
20
99             Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Ex：A parallel charged plate

    
EZ  a zEz
A
S
S
+    +   +   +      +   +

E i  0 ( 相互抵消 )
+    +   +   +      +   +

Fig.3-14

100           Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Sol：
Applying Gauss’s law , we have
  q
SEdS  
0

Where S is a Gaussian surface

                 S A
=>   a Z E Z  a Z A 
0
                                d
∴ EZ  a Z S         ,        z >
0                         2
       S              z<
d
E Z  a Z          ,
0                       2
                        
d
z
d
EZ  0              ,
2     2

101             Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Ex：A spherical cloud of electrons with a volume charge density

 v   0 for 0  R  b （both  0 and b are positive）

     
E  ERa R

Sout
-           -
-           -               -
Sin

-
-           -

Fig.3-15

102                      Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Sol：            
To determine E , we consider the following two cases：

( 1 ) 0R b

r           dr

Fig.3-16

103     Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Constructing a Gaussian surface Sin inside the spherical cloud , we have ,
from Gauss’s law ,

         4 R
0
3

3
Sin EdS            o

 4 3 
 0  R 
        2             3    
=> E R a R  4R a R 
0

                
=>    E   o Ra R  E R a R
3 0

104           Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

(2) R>b

Constructing a Gaussian surface Sout outside the spherical cloud ,
it follows that from Gauss’s law ,

4 3
 
 0  b 
3     E a  4R 2 a
         
Sout EdS 
0
R R          R

          0 b3 
=> E  Ea R           aR    , R>b
3 0 R 2

105          Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Ex：A uniform charged disk of radius b that carries a surface charge
intensity  S

z

P                 Ep

L

b

y
r

 d
rddr  dS
x

Fig.3-17

106            Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Although the disk has circular symmetry , we cannot visualize a

hypothetical surface around it over which the normal component E

has a constant magnitude ; hence Gauss’s law is not useful for the

solution of this problem .
To solve this problem efficiently , we introduce the concept if electrical

potential.
Q
（pending until V               is introduced）
40 R

107           Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Sol：
The electrical potential at the point P(0,0,z) referring to the point
at infinity is
s 2  b        r
V        0 0 2 2 12 drd
40        z  r 
s
 2  0 z  r  2 dz 2  r 2 
1 b 2       1
                      2

40       2


s
2 0
 z  r  
2   2
1 b
2

0
s
2 0

z  r  2  z
2   2
1
    , z0

( i.e. If z >0 and z < 0 )
 

a z s 1  zz 2  b 2  2
1

﹛
,z>0
             V          20
∴ E  V  a z           =
z            
20

 a z s 1  zz 2  b 2  2
1
   ,z<0

108          Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

NOTE：                
For determining E ,

（1）it is simplest to apply Gauss’s law if a symmetrical

Gaussian surface enclosing the chargse can be found

over which the normal component of the field is constant .

（2）it is simpler to find V (a scalar) first , and then obtain E

from  V , if a proper Gaussian surface is not found .

109          Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

‧Principle of conservation of electric charge

（1）Electric charge is conserved ; that is , it can neither be created nor

be destroyed .

（This is a law of nature and cannot be derived from other principles

or relations）

110          Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

（2）Electric charges can move from one place to another and can be

algebraic sum of the positive and negative charges in a closed（isolated）

system remains unchanged .

（This principle must be satisfied at all times and under any circum stances）

（Energy stored or Work doned does not depend on the different paths with

the same starting and end points）

111          Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
‧Principle of conservation of energy in a static electric field .

（1）Analogous to the concept of potential energy in mechanics ,

the electric field E is conservative or irrotational .
          
∴   E  0（∵ cEd l  0（By Stokes’s thm））
         
 Fd l  cqEd l  0

（2）There exists a scalar field V s.t. E  V
（∵ a null identity   V   0 ）

NOTE：
Postulating the conservation of energy in a static electric field

is similar to postulating that in a gravitational field .

112               Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
3-3 Electric potential
Consider the charge in energy（of a static electric field）due to the

movement of a unit positive test point charge q along the direction of

a differential displacement vector d l


dl
+          
                                                    Fe
dWe  Fe  d l           +                    qt
qf
Fig.3-18

113          Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
 
Where the sign “ - ” means the sign of dW is opposite to that of（ Fe d l ）

 
As Fe d l is positive（“+”）, the energy stored in the field is released

（or decreased）and thus dWe is negative .

 
On the other hand , as Fd l is negative（“ - ”）, the energy stored is

increased（i.e. external work is needed）and thus dW is positive .

114         Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

NOTE：                                   
1. If d l is along the direction of Fe , the mechanical work Wm
is positive .

2. Principle of energy conservation：
Wm  We  const
 
=> dWm  dWe  0 =>            dWe  dWm  Fe d l

3. Principle of charge conservation：
+q                        +q     -q
+                        +       -

Isolated point charge   Induced charges from dielectric

115              Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Electric potential（cont’d）

In moving a unit charge from point P1 to point P2 , in an electric field E

, the external work Wm must be done against the field and the energy

stored in the electric field .

              
We   P            Fe d l   P qEd l
P2              2P

1               1

116        Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Many paths may be followed in going from P1 to P2 , due to the principle

of conservation of energy in electric field . Let’s define the difference of

potentials at P1 and P2
We      P  
V12  V2  V1       P Ed l
2

q         1

Usually the zero potential point is taken at infinity . The potential at P2

（or any point P）is denoted by
                         
V  V2   
P2
Ed l （or   dV  Ed l ） or   E  V

       
（This implies   E  0 ∴ E is conservative）

117         Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU


Ex：A test point charge q 0 is moved from infinity to the position vector    r

    
qa r
E
4 0 r 2
         q0
+          r

+q             dl

Fig.3-19

118              Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Sol：
               
q       ar (      
V    Ed l 
r

 r 2  dl  a l )
r

40

Where   l  lr, , 
r
q         dr  q             q
 r 2  4 (1)r  4 r
r
V                           1

40                0           0

 
( a r  a l  1 and dl=-dr )

119             Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

V

 “+”
E
Fig.3-20

∴ The electric potential V of a point at a distance r from a point charge

q referred to that at infinity.
q
V
40 r

120         Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Electric potential due to charge
distribution
The electrical potential at r due to a system of n discrete charges

q1,q2,、、、,qn located at     r1' , r2' ,、、、, rn' is by superposition ,
the sum of the potentials due to the individual charges：
y

 
q1      r  r1
1 n qk
V         '
40 k 1 r rk

r1                              
r                    r  r2

distance
r2
q2
x

Fig.3-21
121             Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Ex：An electric dipole：

It consists of equal and opposite point charges +q and –q separated

by a small distance d

az
P

R+
 
         R
R  aRR
R-
+q
      
a dd  d
d << R            d
  cos 
-q           2
Fig.3-22

122              Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

The potential V at P：
q  1      1 
V                   ….（1）
40  R  R  

    
For   d  R , we write

d                        d
R   R  cos     and   R   R  cos 
2                        2

123         Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Form Eq（1）, we have
 
q        d cos         qd cos            1             P  aR
V                                                     
40                     40 R 2                        40 R 2
2                               2
 d                          d        
R   cos  
2
1      cos  
2                          2R        
           
a R  a d  a R a d cos   cos                   1

 a R  cos 
     1 
V  2 
    R 

      
where P  qd is called the electric dipole moment

           V  V         qd                   
E  V  a R     a             a R 2 cos   a  sin 
R      R 40 R 3

124              Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Pr, , 
r1
+q +
r
d

r2
-
-2q

+q +

Fig.3-23

125        Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
The electrical potential at P（by superposition）
1  q  2q q 
V        
r         
40  1    r    r2 

Assume d << r , using the approximation method similar to that for

an electric dipole , we have
1 2qd 2 3 cos 2   1
V      3 
40 r         2
1                         1
V  3               E  V  4
r                          r
For electric multiples ,
1                     1
V              and     E  n2
r n 1                r
where n represents the number of independent displacements between

any two opposite charges
126          Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
＊ Electric potential due to continuous charge distribution

（1）For a line charge distribution,
1     ldl'       (i.e. dq)
V      
40 L r
'

length

（2）For a surface charge distribution,
1     s ds '      dq
V      
40 S r
'

surface
（3）For a volume charge distribution,

1     v dv'         dq
40 v r
V            '

volume

127           Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Material media in static electric field
consists of atoms

In general，we classify materials，according to their electrical properties

（or energy bands of atoms），into three types：

（1）Conductors

conductor band
valence band

Fig.3-24

128             Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
（2）semiconductors

E  hf
Energy gap
( typically 1ev )

Fig.3-25
（3）insulators ( or dielectric )

Energy gap
( >>1ev )

Bounded charges
( no currents )
129
Fig.3-26         Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

It is found that                          
In good conductors the conduction current J takes place as an external

electric field E is applied
        
J  ee E ( Conduction current )
conductivity

In terms of the band theory of solids , we find that there are allowed energy

bands for electrons , each band consisting of many closely spaced discrete

energy states

( between these energy bands there may be forbidden regions or gaps where
non-electrons of the solid atom can reside )

130             Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

（1）Conductors have an upper energy band partially filled with electrons

or an upper pair of overlapping bands that are partially filled so that

the electrons in those bands can move from one to another with only

a small charge in energy.

（2）In semiconductors , the energy gap of the forbidden regions is relatively

small and small amounts of external energy may be sufficient to excite

the electrons in the filled upper band to jump into the next band , causing

conductor.

131          Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

（3）Insulators or dielectrics are materials with a completely filled upper band

, so conduction could not normally occur because of the exisrence of a

large energy gap to the next higer band.

132         Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Polarization of a dielectric material
                                Polarization
E                  ( due to exertion of Columbs s force )
No E-field applied
-                     -
-        -           -             -
-q          +q
Electron -       +        -   -             +        -
cloud                                                                 -           +
-        -           -              -
-                     -                                      d
Unpolarized atom         Polarized atom             Electron cloud       Positive mucleus

Fig.3-27

133                 Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

To analyze the macroscopic effect of induced dipoles , we define a

polarization vector P

electric dipole moment

nv

         Pk
P  lim k 1      ( Induced bound charges appear in pair
v o
v               i.e. electric dipoles )

Where n is the number of the induced dipoles per unit volume and

the numerator represents the vector sum of the induced dipole moments

contained in a very small volume v

134          Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
 
The electric potential corresponding to dP  Pdv


aR
                               +q +              
dP  a R P  a R dv                                   R
dV            
40 R  2
40 R 2
                              d
1 P  aR
V
40
v R 2 dv                       -q -

Fig.3-28

Where the R is the distance from the elemental volume dv to a fixed field point

135           Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

The effects of the induced
electric dipoles
（1）Equivalent polarization surface charge density ps

The bounded charge distributed over a specified surface S

 
d cos                     S  a n S

+         +        +       +

                            d                
d'                                             E
-        -         -        -

a n  d'
Fig.3-29

136            Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

v
 
q b  nqSd '  nqd cos S  nqdS                     n
                                              Pk
 P  S  P  a n S                     P  lim k 1
v 0
v
q b  
 P  a n c / m 2 
Polarization vector
=> ps 
S
dq b      q
( or ps          lim b )
dS S0 S
                                  
（ That is ,       q b  P  a n S or    dq b  P  a n dS  PdS ）

137            Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
（2）Equivalent polarization valume charge density pv

The net charge remaining within the volume V is bounded by S
（ surface charge q b , remaining charge  q b ）
 
q b  Sdq b  SP  a n dS
             
q r  q b    P  dS  v P dv ( by Divergence thm )
S
              
 vpv dv                 ( i.e.   dq b    P  a n dS    P  dS )
S          S
 
(   q b  P  a n S )
               
pv  P ……（1） （ 即   P   pv ）

（∵ The total charge of the dielectric after polarization must remain zero）

138           Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Total charge    S ps dS  v pv dv
              
 SP  a n dS  vPdv  0
              
( i.e SP  a n dS  vPdv )

139           Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Electric flux density and dielectric
constant
The electric field intensity due to a given source distribution volume density of

free charges  v in a dielectric .

Dielectric

+         + -    +       -   +          -   +   -   + -     -

+         + -    +                              -   + -     -

+         + -    +                              -   + -     -

+         + -    +                              -   + -     -

Fig.3-30
140            Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

 1
E   v   pv 
0

 
Using Eq(1) , we have  0 E  P  v 

Now we define a new fundamental field quantity , the electric flux density
                   
or electric displacement P such that D   0 E  P ….（2）


D  v ….（3）

141         Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

                        
or   vDdv  vv dv  q f  sD  dS ….（4）

Eq(4) another form of Gauss’s law , states that the total outward flux

of the electric displacement over any closed surface is equal to the

total free charge enclosed in the surface.

 pv  P
By Gauss’s law , E   v   pv 
 
=> 0 E  P  v

0

142            Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

P

                
E                EP
+            -     +              -   +                -
+                                                      -
+            -     +              -   +                -
+                                                      -
+            -     +              -   +                -
+                                                      -
                       
+            -     +              -   +                -   E decreased to become E i  E  E P
+                                                      -
+            -     +              -   +                -
+                                                      -
+            -     +             -   +                -
Ei

induced electric field
Dielectric
Fig.3-33

143             Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
 v

                       
E0                     E

medium
Fig.3-32

                        
D unchanged（∵ including P  q b d ）

144          Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
When the dielectric properties of the medium are linear and isotropic we have
                                      
P  0 Xe E             （conductor： J  E ）current density

 Xe 0 E
    
insulator： D  E electric displacement

Where X e is a constant called electric susceptibility .

From Eq(2)
                             
D  0 1  Xe E  0  r E  E

（for conductors ,  r  1 ∴ X e  0 ）（∵ P  0 ）


Where     r  1 X e         is called relative permittivity or dielectric constant
o
and  is the absolute permittivity .（often called simply permittivity）

145           Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Boundary conditions for electrostatic
fields
To investigate the relations of the field quantities at an interface

between two media .

（1）A conductor-free space interface                                              E1t

free space ( or dielectric1 )                                      h
W              d

a                            +
+
+
+                       c
+
conductor ( or dielectric2 )
b
Fig.3-34

146                       Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Inside a conductor , the field exerting a force on the charges and making

them move away from one another , until all the charges reach the surface

in such a way that .  v  0 and E  0

Under static conditions , the E field on a conductor surface is everywhere

normal to the surface , In other words , the surface of a conductor is an

equipotential surface under static condition .

To see this , let’s construct a small path abcd , as shown in Fig.3-34 ,

where the width ad  cb  W , and the height ab  dc  h sides

ad and bc are parallel to the interface .

147          Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Since E is conservation , we have
                                            
 Ed l  E t W as h  0                （∵   E  0）
abcd

∴ Et  0


That is the tangential component of     E   on a conductor surface

is zero under static conditions .

To evaluate     E n . We construct a Gaussian surface in the form of
a thin pillbox as shown in Fig.3-35 .

148           Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
free space ( or dielectric1 )
S
h

conductor ( or dielectric2 )

Fig.3-35

Using Gauss’s law , we have
            S S
SEdS  E n S  
0
S
or E n 
0

149            Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

（2）A dielectric1-dielectric2 interface

In Fig.3-34 since E is conservative , we have
                    
abcd Ed l  E1  W  E 2  W  E1t  W  E 2 t  W  0
     
           
         
∴ E1t  E 2 t ,   a n  E1  E 2  0

150         Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU


This states that the tangential component of an E field is
continuous across an interface .


E1 t

E1 n
W
a                      b
1
h
2

d                      c   E2t
W               
E2n
Fig.3-36

151           Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

To evaluate E n , we apply another form of Gauss’s law

           
a n 2 D1n  a t 2 D 2 n

D1
an2       
S2  a n 2 S

S                                    dielectric1
dielectric2
 
S1  a n1S
a n1    
D2
Fig.3-37

152         Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

               
SDdS  D1a n 2  D2 a n1 S
         
 a n 2 D1  D2 S
  S S
    
∴ a n 2  D1  D2   S
or

D 2 n  D 1n   S

                    
a n 2 a n 2 D1n   a n 2 a n 2 D 2 n    S =>   D 2 n  D 1n   S

153            Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Capacitances and capacitors
（Capacitances：due to the property of equipotential of a conductor）

A conductor in a static electric field is an equipotential body（due to

overlapping of the conduction band and valence band of its atom）and

that charges deposited on a conductor will distribute themselves on

its surface in such a way the electric field inside vanishes .

154           Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

The potential of the surface of an isolated conductor is directly

proportional to the total charge on it , sine

（1)     E  V

（increasing the potential V by a factor of K increases E        by K）

（2）The boundary condition at a conductor-free space interface
  S
E  an
0

（as a result ,  S（or the total charge Q）increases by k when E
increases by a factor of k）

155           Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Therefore , the radio Q       remains uncharged . For the isolated
V
conducting body , we define capacitance


Q
C
V

Of considerable importance in practice is the capacitor（or condenser）,

which consists of two conductors separated by free space or dielectric media .

The capacitance of a capacitor is a physical property of the two conductor

system . It depends on the capacitor and on the permittivity of the medium .

156         Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Ex1：Determine the capacitance of a parallel-plate capacitor

y

+   +       + …………… +       S
y1  d
                        area A
E


dl
y0  0
-   -       - …………… -

Fig.3-38

157       Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Sol：
 
∵ V10   y Ed l
y1

0

d     
  0 E  dya y
 S   
  0   a y dya y
d

      
 0 
d
S y   S
      
0 0 0

Q Q     Q     A
∴    C             
V V10 Q  d    d
A

158    Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Ex2：Determine the capacitance of a cylindrical capacitor


l         
a
E

b

Gaussian surface S

Fig.3-39

159        Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Sol：
By Gauss’s law , we have

（neglecting the fringing effect of the field near the edges of the conductors）

 
 E  dS  S  2b  l
                  
  a r E r  a r dS  a r E r a r 2rl  S  2b  l
  S b
∴ E  ar
r

        b     S b      
Vba  a Ed l  a a r         a r dr
b

r
r
b
1             b
 S  ln r  S b  ln
        a                a

160            Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Electrostatic problems.
Field problems associated with electric charges at rest.

Charges in motion that constitute current
flow .
(Problems of current flow in a conductive medium are governed by Ohm’s law.)
Question: How about the problems of current flow in a good insulator?

161           Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Two types of electric current:
(Caused by the motion of electric charges)

(1) Convection current
(2) Conduction current
Convection current：
The result of hydrodynamic motion involving a mass transport, are
not governed by Ohm’s law）:
(1) Electron beams in a cathode-ray tube
(2) Violent motions of charged particles in a thunderstorm..

162           Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Conduction current：

As an external electric field is applied on a conductor, an organized motion of
conduction (or valence) electrons, which may wander from one atom to another in a
random manner is produced.

The conduction electrons collide with the atoms in the course of their motion,
dissipating part of their kinetic energy as heat：thermal radiation. This phenomenon
manifests itself as a damping force or resistance, to current flow.

163           Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
                              
The redation mbetween conduction current density J（and electric field intensity E mgive
us a point form of Ohm’s law), i.e.,
    
J  E

Where  is a macroscopic constitutive parameter of the medium called conductivity .
    

( In a dielectric (or an insulator) the electric displacementis D given by D  E

Where  is called permittivity.)
Recall that       A good condutor                      A good insulator

Conduction
                           band
J
Energy
Valence                           gap >>
                          band
1ev

E

Fig.4  3

164        Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU


The same I J x, y  but
magnetic field B effects,
i.e. different

Charge carrier

Fig.4  4

165             Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Convection current:

Point charge(s) moving with a velocity v in free space.

                    Free space
qv
Fig.4  5

Conduction current
Metallic conductor is filled with free electrons.

Under the influence of E conduction electrons collide with atoms and consequently
conduction current produces.

Fig.4  6
166           Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Current density and Ohm’s law
1.Convection current:
Consider the steady motion of one kind of charge carriers, each of charge q, across an elem
of surface ΔS with a velocity

through the surface s

The amount of charge passing                                        
S  an  s
Where N is the number of charge carriers
per unit volume and the vector quantity                                          
u
 
s = a n s
Q          
From Eq(1), we have I                  v  u  s
t

Fig.4  7

167        Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

In field theory, we are usually interested in events occurring at a point rather than
within some large region, and we shall introduce the concept of current density


J  v  2

(where  v  Nq , is free charge per unit volume. )
 
I  J  s
so that

168              Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

          
ue    e E (m / s )  5                            
E  ue  ue  E

Where  e is the electron mobility measured in ( m 2 / v  s )

( This is because conduction currents are the result of the drift motion of charge carriers
under the influence of applied electric field intensity.)

Table the electron motilities for some conductors

Conductor                Copper                Aluminum                 Silver

e                     3.2                    1.4                    5.2
 10 3                10 4                 10 3

169            Dr. Gao-Wei Chang
(Unit in m / v  s )
2
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

6
5.2

4
3.2

2                                   electron mobility

0.4
0
copper   aluminum   silver
Tab.4  1
electron mobility    3.2       0.4       5.2

Each in   10 3  m 2 / v  s
                 
From Eqs (4) and (5) , we have J    e  e E  E (Point form of Ohm’s law)

Where the negative quantity  e   N e is the volume charge density of the
drifting electrons and the conductivity     e  e (A/vm or Siemens per meter(s/m))

170             Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

For semiconductors, conductivity     depends on the concentration and mobility of
both electrons and holes：

    i i    e  e   n  n
i

mobility
concentration

where the subscript h denotes hole.
 e  Ne   Nq

171           Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Equation of continuityand kirchhoff’s
current law

Conservation of charge：

Electric charges may not be created or destroyed (just transferred
from one place to another)

172        Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Continuity of current：
If a net current I flows across an enclosing surface（封閉面） out of the bounded
volume V, the net charge Q in V must decrease at a rate that equals current

       dQ    d
 I   J  ds   i     v dv
S           dt   dt V
Qo

By Divergence theorem, we have
Qi
            v
V  J  dv  V t dv
For arbitrary choice of V, it follows that

    
  J   v  1
t                                          Fig.4  8
173    Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
 Qi  Qo  0     Qo  Qi
dQo   dQ
That is , I      o
dt    dt
Outward current

Qi       Qo
dQ
I             Qi  Qo  Q  Q  0
dt

Fig.4  9
This point relationship derived from the principle of conservation   of charge is
called the equation of continuity (of current).

For steady electric currents charge density does not vary
 v                                              
with time,       0   v is fixed.）Equation (1) becomes   J  0
（
t
Thus, steady currents are divergenceless or solenoidal.

174             Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Over any enclosed surface, we have, by Divergence theorem,

 
 J  ds  0
S

Which can be written as       Ij
j    0 2

Equation (2) is an expression of kerchief’s current law.

<Ex>consider that charges introduced（引進） in the interior（內部） of a conductor
will move to the conductor surface and redistribute（重新分佈） themselves in such a

way as to make  v  0 and E  0 inside under equilibrium（平衡，均勢） conditions.
Please calculate the time it tables to reach（取得，抓到） an equilibrium.

175         Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
<Pf> :
From Ohm’s law, the equation of continuity becomes

      v          
 J            E
t

Where is the conductivity of the conductor

In a simple medium, Gauss law

From the above eqs, we have   E 
v


176           Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

 v 
 0
t                                               
   J    E   v   v
        t

It can be readily obtained that  v   0 e  ( /  )t (c/m3).

Where  0 is the initial charge density at t=0.

For a good conductor such as copper,   5.8  10 ( s / m)
7

,ε≒  0（like vacuum no electric dipole） 8.85  10 12 ( F / m)


constant    /   1.53  10 sec ,a very short time indeed.
19

177             Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Power dissipation and joule’s law

Under the influence of an electric field E , conduction electrons conductor under go a drift
motion macroscopically, and they collide with atoms on lattice sites. (Energy is thus transmitted
from the electric field to the atoms in thermal vibration.)

                                                       
The work ΔW done by E in moving a charge q a distance  is w  Fe     qE 


W       dl         
Which corresponds to a power        P  lim       qE     q  E u
t 0 t        dt

                                        
F  l                       F  l      dl
W                         lim           qE
t                    t 0  t           dt

178           Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

ＳＷ
1
0 
LC
Ｃ                 Ｌ

                                        
Energy storage in E field                 Energy storage inB field
Ｒ
Power dissipation
Fig.4  10

The total power delivered to all the charge carriers in a volume is dv

179             Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

                 
dP   dP  E( Ni qiui )dv  E  J  dv
i
i                 i

P  
and         E  J (w)
v

Is thus the point form of a power density under steady-current conditions.
For a given volume V, the total electric power converted into heat is

 
P   E  J  dv                   Which is known as Joule’s law.
V

180           Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

<Ex> Show that in a conductor of a constant cross section
, we have P=I2R(w)
<Pf>：              
Where d  is measured in the direction of
 
 dv  ds  d 


ds

                                                         d
 P   E  J  dv   Edl   Jds  V  I
V            L        S

Fig.4  11
Since V=RI , we have        P=I2R(w)

181   Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

density.
Basic quantity：

Current density vector

182       Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Governing eqs
                                                             
 Jds  0

(1)   J  0 or                                                        J 2n  a2n  J 2n
S

means J is divergence less.
      
J 1n  a1n  J 1n

At an interface between two different conductors,
Fig.4  12
J 1n  J 2 n ( A / m 2 )
2
                                             
(2)   ( J /  )  0          (∵    E  0     and J  E )                          J1t       J 2t
1                                                1                           
or    
C
Jdl  0
Fig.4  13
1        2

at on interface between two different conductors
J 1t       J 2t        J 1t  1
                   
1         2     or   J 2t  2
183      Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Resistance calculations
      
V  LEdl  LEdl
R=                   
I    Jds  Eds
S            S

<Ex>
Derive the voltage-current relationship (i.e. resistance) of a piece of homogenous material
of conductivity σ, length l, and uniform cross section S, as shown below.

184           Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

<Sol>：
The potential difference
or voltage between terminals 1 and 2 is

－
0    
V12  E           V12  
V 12
Ed        ＋
                          
J                
                                                                   a
Where E  a E and   a                                        E       

Ｓ
０                    2
The total current is
１
 
I   Jds  J  S    E  S
S
V12    
∴      R         ()
I    S                               Fig.4  14
185   Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Chap 5 Static Magnetic Fields in Free
Space

186       Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Field and Wave Electromagnetics
due to
Force exertion                 Field existance
(   )
          
Maxwell's Equations

(action - at - a - distance)
due to
Energy t
ransfer                    Wave motion
(    )
      
Poy ting's thm

(or Energy flow)

187                Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Static Magnetic Field in the Free Space

＊ A magnetic field can be cause by

（1）a permanent magnet（like the magnetized lodestone）

（2）moving charges

（3）a current flow

188   Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Static Magnetic Field in the Free Space

I

I

(a)                                  (b)

189                Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Static Magnetic Field in the Free Space

A magnetic field can be characterized by a so-called magnetic flux density
                                   
B ,which is defined in terms of     Fm experienced by a moving charge q ,i.e.,

      
Fm  qu  B    (N)

190            Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
5-2 Static Magnetic Field in the Free Space

                                                       
where   u   (m/s) is the velocity of the moving charge and      B
2
is measured in webers per         square meter (Wb/ m ) or teslas (T).

teslas = 104 Gauss


（Here , B has been not yet defined）

191           Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Static Magnetic Field in the Free Space

＊ Lorentz’s force equation

When a test charge q is placed in an electric field E

and it is also in motion in a magnetic B

the total electromagnetic force on it is

                
F  Fe  Fm  q (E  u  B)       ……(1)

which is called Lorentz’s force equation

192          Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Static Magnetic Field in the Free Space

Magnetic force is a kind of transverse force

analogous to the electric force （or Coulomb’s force）

193            Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Static Magnetic Field in the Free Space

Note：
                                 
（1） B and u are perpendicular to       Fm This phenomenon is

found by Oersted.    Magnetic force Fm is a transverse

force found by Oersted.         Specifically Fm is perpendicular
     
to both   B and u

194            Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Static Magnetic Field in the Free Space

（2）

195           Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Magnetic Force between Two Charges
＊ Law of magnetic force between two moving point charges
（Popovic ,Introductory Engineering Electromagnetics,1971）

The magnetic force   Fm12 exerted by a charge q 1 on the other
charge q 2 is found by indirect experiments, involving steady

current system, to be

           
         (q2u2 )  (q1u1  ar12 )
Fm12  km             2
……(2)
r12
    
  km (q1u1  ar12 )
B
r2
12
196           Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Magnetic Force between Two Charges

(the permeability of a vacuum) in MKSA system, and

     
r12  a r12  r12 is the vector length from   q 1 to q 2

（Equation(1) may be compared with Coulomb’s law in electrostatic fields.）

                                         
r12  a r12  r12                          B
q1
q2                                      
                                          u1                     u2
u1                     
B          F                                       Fm12
u 2 m12               Test moving                Field moving
charge q1                   charge q 2

197            Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Magnetic Force between Two Charges

＊ The concept of magnetic field


magnetic flux density    B from Eqs(1) and (2) we can see that

 
           0 q1u1  ar12                   
Fm12  q2u2  
 4    2
  q2 u 2  B

     r12                 
 
 0 q1u1  ar12
where         B
4     2
r12
is the definition of the flux density vector produced by a point
               
charge   q 1 moving with a velocity     u1          q 1 u 1：field moving charge

q 2 u 2：test moving charge
198            Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Magnetic Force between Two Charges

 0I                0 Id sin                                               
s                                   0 I  sin d s
B                                                   B  2 dB 
2 0 r 2
dB 
2a                 4   r2                             0

sin   sin     
a
r  s2  a 2       ,
s2  a 2

          0I      ad s
B  2  dB 
0     2 0 s 2  a 2
              3
2


0I        s            I
                        0
2a  (s 2  a 2 ) 12   2a
                 0

199        Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Magnetic Force between Two Charges

Note：
If a test charge    q t resides at a fixed location outside the current-carrying
conductor, it will be acted on neither by the electric force nor by the

magnetic force.

（1）The moving charges inside the conductor are compensated so that there

is no appreciable electric field outside (nor in side) the conductor.
（2）Since charge q t is stationary according to the law of magnetic force

between two moving point charges the magnetic force on q t is also zero.

（In metallic conductors, charge carriers are called conduction electrons.

In a good insulator (or dielectric), induced charges are called bound charges.）
200            Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Magnetic Force between Two Loops
＊Ampere’s law of force

Consider two idealized complete circuits                      C1

C 1 and C 2 , consisting of two very thin                I1        dl1                            C2
R  r2  r1

dl2
conducting loops(wires) carrying filamentary                                                             I2
r1

r2
currents I1 and I 2 respectively.
O

In Ampere’s extensive experiments, he found that in free space,
              
            ( I 2 d  2 )  ( I1d 1  aR )    
c2 c1
F12  0
4                      R2
F12 ( I 2 ) is linear
 
where R  a R R
201             Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Magnetic Force between Two Loops

This equation is referred to Ampere’s law of force and it constitutes

the foundation of magnetostatics.
（Usually, the magnetic force due to two moving charge acting on them

is very relatively small. For example, in a conductor, charges forming

steady current are moved by both electric force and magnetic force.

However, the magnetic force is much small than the electric force.

As a result, the average drift velocity   v     is governed by the electric field

intensity E , i.e.,）
                 
J  E (i.e., J  E )
    
v  E , where     is the mobility of the charges.        (see popovic )
202            Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Magnetic Force between Two Loops

Suppose n charges      q 1、 2 , …….., q moving with velocities
q
n

           
u 1 , u 2 …….u n 。

According to the superposition principle, the total magnetic force
                                                    
Fm on a test charge     q t moving with a velocity u t

 
           0   n  qi ui  arit   
Fm  qt  ut   
 4                  
 ……(3)
 i 1     rit2       
 
 qt ut  B
203            Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Magnetic Force between Two Loops


Where   a rit are with vectors directed ( at the time instant considered )
from charges q 1、 2 , …….., q n toward the test charge
q                                            qt

 
 0          n     q i u i  a rit
B
4
i 1         r 2
……(4)
it

     
and    rit  a rit  rit

204     Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Biot-Savart Law
＊The magnetic field of steady electric current：

The Biot-Savart law (for conduction current)

Suppose that the number of free charges per unit volume of a conductor is N。

Then，inside a small volume v ，there are       Nv charges moving with the

same velocity   v ，since v   is supposed to by very small。

Equation (4) becomes

 0          
Nq  u  ar
B
4
 r 2 V
V
                 ……(5)
    J a
 0  2 r V
4 V r
205            Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Biot-Savart Law

               
where     J  Nq u   V  u represents the current density vector

at point inside the volume element         v and  V is volume charge density。

If V is assumed to be “ physically small ” the magnetic flux density due

to the steady current in the conductor is given by

 
 0         J  ar
B
4    
V     r 2
dV ……(6)

206             Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Biot-Savart Law

where
   is the unit vector directed from the volume element           dv
r

towards the fixed point at which B is being determined.

v'

V


J                      

ar  r  ar  r
qt
Position of a testing point

a small charge q t in the field of charges moving inside a conductor.

207                 Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Biot-Savart Law

Filamentary current I

＊ Biot-Savart law (cont’d)
 
r  ar  r
J

 
 0          J  ar
B       
Field point
dV ……(6)
4     V     r 2

In practice ，the current is very often flowing thru thin conducting wires.
                 
Suppose the cross-section area of the wire is S and then V  S  d 

where d  is the vector differential element of the wire Equation (6) yields
 
  0 I d   ar
B         
4 C r 2
208                      Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Biot-Savart Law

where the current element

                    
I  d   J  a n  S  d 
      
 J  S  d  
                                                   
 J  dV                                              S
       
d      r  ar  r
 
and      S  a n  S                                                             
qtut
Conductor current I( conduction current in a flamentary conductor )
(or thin wrie )

This important formula is known as the Biot-Savart law.
209              Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Biot-Savart Law

 
Note ：          0           J  ar
B
4      V ' r 2 dV
   
0 J  ar S  d  

4 C         r2
   
0 J  S d   ar

4 C         r2
 
 0 I d   ar
         
4 C r 2
 
 0 I d   ar  r

4 C   r3

210         Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Biot-Savart Law

【EX】

Find the magnetic flux density          B at a point located at a distance

r from the current-carrying straight wire ，as shown below 。

<sol>

The distance vector from the source element             dz'

to the field point p is
               
R  ar  r  (  a z  z ' )
                             
d  R  az dz  ar r  az z  a rdz

211             Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Biot-Savart Law

By applying Biot-Savart law ，we have
  0 I                                                                             z
L       rdz '
B  a
4    
z' r 
L                3
2    2 2                                               L

      0 I         L                                                               Id l  az  dz '
 a                                             source element

2r L2  r 2                                               z'
p
 
 I  d   aR 
（∵ dB  0 
 a z  dz '                r  ar  r
4  R2                                                                                            Field point

 
 I  d   aR 
 0   R3       
4                                                                    -L
                
1
R  Z ' R 2       2 2
）         212                Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Biot-Savart Law

【EX】
Find the magnetic flux density at center of a planar square loop ,
with side W carrying a direct current I.
z
<sol>
From the preceding example, we have
O
I
      I          W/2             2 2 0 I                      W
B  4a z 0                       az
2  W   W  2  W  2      W
     
 2  2  2

B的方向和迴路中電流的方向遵循右手規則
(homework or exercise)
213           Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Biot-Savart Law

【EX】

Find the magnetic flux density at a point on the axis of a circular loop

of radius b that carries a direct current I.
P(0,0,z)
               R 
<sol>                                                                  a zz
d 
b
I                             y
We apply Biot-Savart law to the circular loop

                                          x
d   abd 
         
R  azz  arb
 R  z   2
 b2   
1/ 2

214    Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Biot-Savart Law

                                                                    
Recall that R is the vector from the source element                             d  to the field point P.
                                               
d   R  a  bd   a z z  a r b  a r bzd   a z b 2 d 

(differential length vector  distance vector in free space)


We need only consider the                    a z component of this cross product since the

a r component is canceled due to cylindrical symmetry.

 0 I        2            b 2 d                   0 Ib2
B                az                           az                       Teslas
4      0
z   2
b    
2 3/ 2
2z  b
2

2 3/ 2

215                Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

The Curl and Divergence of B

＊ The curl and divergence of        B
 
 0 J  a r
Recall   B
4 v r 2 dv
          
whereB  B(r) J  J (r)


and   r(x,y,z) is the position vector of the field point p

( x, y, z)
V 
          
J         r  ar  r

P(x,y,z)
216             Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
The Curl and Divergence of B


ar      1
Since          
r2      r
  0        1
We have B     r
4 v
J   dv              ……(1)

From the vector identity

J   1  1       
      J    J ……(2)
r   r      r

it follows that

 0         J
B     r
4 v
   dv
 
217           Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
The Curl and Divergence of B

In space change rate

                 J
B  ( 0
4      V' r dv') ……(3)
Let’s define the vector magnetic potential

 0        J
A
4   V' r dv' ……(4)
Where the source coordinates are primed.

            
Therefore          B    (  A)  0


∴   B  0 for determining B
218           Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
The Curl and Divergence of B

Furthermore, suppose the stationary current I in a thin wire It appears that the

vector magnetic potential                        
 0         J
A
4    V' r dv'

 I d 
 0 
4 L ' r

According to Helmholtz’s theorem, a vector field is determined if both its

divergence and its curl are specified everywhere Therefore, we need to
                 
further evaluate   B for determining B
219            Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
The Curl and Divergence of B

Note：
Since it is found that

                J
B  ( 0
4    V ' r dv')

We thus define the rector magnetic potential

 0         J
A
4   V ' r dv'

Where the source coordinates are primed.

220           Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
The Curl and Divergence of B

            
Therefore     B    (  A )  0 (due to the null identity)


or     B  0 (any where)
Source free


B  0


That is, B is rotational.                                    
B  0
dS


∴   B  0 anywhere

221            Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Ampere’s Circuital Law
Ampere’s circuital law

                 
    A    A   A
2

                           2 2 2
  ax     ay     az              2 2 2
2

x      y      z            x y z

 2V  2V  2V
V 2  2  2
2

x   y   z

                 2  2  2 
 2V   a x  a y  a z  a x  a y  a z V   2  2  2 V
 x
          y    z  x
        y    z   x y z 
             

222        Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Ampere’s Circuital Law

Consider
                             
  B      A    A   A … (1)
2

from vector identities ,
where               2         2         2
 A  a x  Ax  a y  Ay  a z  Az
2

                  
and               A  ax Ax  a y Ay  az Az
1. The first term on the right hand side of Eq . (1)
                      
 0              Jdv'  
  A 
0         J 
                      dv'
v'  r  
                                              … (2)
4                   4  v '  r  
 

223           Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Ampere’s Circuital Law

From vector identities ,

 J   1 1         
   J       J  … (3a)
r
         r r
or        
J     ' 1
     J    
r                                     … (3b)
           r
                                   
(  r  a x  x  x '  a y  y  y '  a z  z  z '
,


r   x  x '    y  y '    z  z '
2             2            2

1
2    and  ' is the

differentiation with respect to the source coordinate  x' , y ' , z.' )

224          Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Ampere’s Circuital Law
From vector identities ,
                              
J                            J
r
 ' 1 1 ' 
     
 '    J        J     
r
… (4)
          r r               

( for any source element the field point is specified )

225             Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Ampere’s Circuital Law

Eqs . (3b) and (4) yields
           
J      ' J  1 '  
          J 
r                             … (5)
       r r        


(Since   J  0 is a necessary condition for static
'

magnetic fields (i.e. , steady current) . )

Eqs . (2) and (5) yields

      0      '  J 

  A           dv'
r                  … (6)
4  v '      

226         Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Ampere’s Circuital Law

From Divergence theorem , it follows that

     0  J  
  A     ds '             … (7)
4  s ' r



where S ' encloses the volume V ' .

Since J is the volume current density ( i.e. ,all
currents are enclosed inside S ' ) its normal
component is always equal to zero .

 
 J  ds ' 0

227           Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Ampere’s Circuital Law

Equation (7) becomes

   A  0     …(8)

This is a general form for Coulomb condition
. 
 A  0
2. For the second term on the right side of Eq . (1) ,


 0      2 J       0              2 1
 A    v'   r dv'  4         v' J    r dv'
2
… (9)
4        
                               

228            Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Ampere’s Circuital Law

If the field point P is not located inside the volume V '
                                        
1

( i.e. , r   x  x'   y  y '   z  z ') ,       0
2            2            2 2

we have
1
2    0
r

1   2   1 
( 2   
1
        r       0                          for   r  0)
r
2
r r        r  r 
1
For    2    0 , the field point must be located inside
r
the volume V ' and it is infinitely close to a source
element .

229                 Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Ampere’s Circuital Law
 r  0 , if r is located inside the volume V ' i.e. ,
the field point is infinitely close to a source point
( i.e. ,        ,      ,        ).

x  x'       y  y'     z  z'
Therefore , Eq . (9) becomes

 0           2 1                    '2  1 
 A       v ' J    dv'  0  v '0 J    r dv'
2

4               r     4                  
                             
 0 J                  1      J            ' 1  
4 v '0                          s '0   r ds '
 2 A             '   '  dv'  0
r      4              

( ∵ Jis constant , as V '  0)
      
( By Divergence theorem , and      r'  r   , but   r '  r   )

230              Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Ampere’s Circuital Law

we have
                
 0 J         ar '  ds '             
4 s '0
2 A                            ( ar '  ar )
r '2

0 J
4 s '0
      d ' ( d '  ds' )
r '2

 0 J    … (10)

231             Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Ampere’s Circuital Law

From Eqs . (1) and (10) ,

We have                    
  B   0 J … (11)

which is called Ampere’s circuital law . ( or
simply called Ampere’s law . )

232           Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Ampere’s Circuital Law

Also , by Stokes’s theorem ,

                
   Bds   Bdl  0  Jds '  0 I
s              c          s
… (12)

where the surface S is enclosed by the contour C .

          
In addition , from Eq . (10) ,     A   0  J
2

is called a vector Poisson’s equation .

233               Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Ampere’s Circuital Law

In Cartesian coordinates ,
  2                        
 A  ax   Ax  a y  2 Ay  az  2 Az
2

where

 2 Ax    0  J x   … (11a)

 2 Ay   0  J y    … (11b)

and                 2 Az    0  J z   … (11c)

234            Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Ampere’s Circuital Law

Hence , the solution for Eq . (11a) is

0    J
Ax         x  dv'
4 v ' r

∴Consistently ,


 0       J
A          dv'
4   v' r

235          Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Ampere’s Circuital Law

Note :

In a static electric field

 v
E 
0

( Point form only valids
v
for the point having          )
0

236             Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Ampere’s Circuital Law

In a static magnetic field

        
  B  0  J

( Point form only valids

for the point having  0  J )

237             Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Chap 6 Magnetic Dipole &
Behavior of Magnetic Materials

238       Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Magnetic Dipole
＊ Magnetic Dipole

Let’s evaluate the magnetic flux density at a distance point of

a small circular loop of radius b that carries a current I ( a

magnetic dipole ) .

239          Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Magnetic Dipole

We choose a spherical coordinate system s.t. the field point

p( R, , ) is located in the yz plane for convenience
2
z                                   
p ( R , , )
2

R
                          Φ=π/2

R1


y
p' '
'             p'
dl'
x

240                      Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Magnetic Dipole

It is intended to find the vector magnetic


 0 I      dl                       1     l dl
' R                     4 0  R
A                  analogous to
4
L

                                            
and then      B   A           is determined analogous to E  V
 
 0 I       dl  aR1
(∵      B
4     L' R12 )

241          Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Magnetic Dipole

From the top view of the small loop
                 
dl '  (-ax sin  'ay cos ' )dl'
                
or    dl '  (-ax sin  'ay cos ' )bd '
     
dl '  ax

'
b
p   
ay

dl '

'     '


ax
242               Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Magnetic Dipole

      I          2   b sin  ' d '
A  a x 0
4     0            R1


     0 Ib 2 sin  ' d '
2  2
or        A  a       
R1

( the a y component is canceled due to the source element I dl '
is symmetric to the y axis )

The law of cosine gives

R12  R 2  b 2  2bR cos  '
or     R12  R 2  b 2  2bR sin  sin  '
( left as an exercise )

243                  Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Magnetic Dipole

1
1   1   b  2
2b                        2
  1  2  sin  sin  ' 
R1 R 
   R   R              


1
1  1  2b                         2
1   b              
or                 1  sin  sin  '                     1  sin  sin  ' 
R1 R   R                                  R R                

b2
( assuming
R 2  b 2
i.e.
2
 1 )
R

      0 Ib 2      b
2R  2
∴            A  a        (1  sin  sin  ' ) sin  ' d '
R

244                  Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Magnetic Dipole

   0 Ib 2
A  a    2
sin 
4R
as a result

        0 Ib          2

B   A      3
(a R 2 cos  a sin  )
4R

245       Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Magnetic Dipole

Note：The vector magnetic potential can be rearranged as

 
  0 m  aR
A
4R 2


法線方向是        az   方向

                    
where    m  a z Ib  a z IS  a z m
2
is defined as the

magnetic dipole moment

246            Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Magnetic Dipole

also ,We can rewrite the magnetic flux density vector as

 0m                 
B       a R 2 cos  a sin  
4R 3
Comparing with the similar expressions for the electric dipole

in static fields , we have
 
P  aR
V
and
4 0 R 2
                                        
E   V 
P
a R 2 cos  a sin  
4 0 R   3

                   
where    p  qd  qda z  p  a z    is the electric dipole

and the magnetic dipole are also similar.

247            Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Magnetic Dipole

+

X

-

Magnetic dipole
Electric dipole

248         Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Magnetization and Equivalent Current
Density
＊ Magnetization and Equivalent Current Density  J mv , J ms 
Suppose the orbiting electrons in a material cause circulating

currents and form microscopic magnetic dipoles .

The application of an external magnetic field causes both an

induced magnetic moment due to a change in the orbital

motion of electrons and an alignment of the magnetic dipole
Nv

moments of the spinning electrons
            mk
M  lim      k 1
Let’s define a magnetization vector         v '0     v'

249              Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Magnetization and Equivalent Current Density


where m k is the magnetic dipole moment of an atom and

N stands for the number of atoms per unit volume .

250           Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Magnetization and Equivalent Current Density

   
since dm  Mdv' , we have
 
 0 M  aR        
dA                dv
4 R   2

0         1
    M   ' ( )dv'
4          R
1    1 
(∵        ' ( )  2 aR        )
R   R
       0                          1
v'
∴ A  dA 
4               v' M   '  dv '
 R
( where v' is the volume of the magnetized material )
               
       M ' 0
'
M
 0 '
4 v '
dv           '  ( )dv '
4 v R                         R
251             Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Magnetization and Equivalent Current Density


(  From the vector identities ,  '  M   1  '  M   '  1   M
               
 
R R                    
                       R

1 '           1
   M  M  '  
R                R

                          
J
 mv                    J ms
0 '  M      0               M  an'

4 v ' R dv' 4           S ' R ds'

252           Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Magnetization and Equivalent Current Density

Define the equivalent magnetization volume current
density vector

  '                                               
J mv    M        ( analogous to  pv          P )

and the magnetization surface current density

                                             
J ms  M  a n '     ( analogous to     ps    P  an )

( For notational simplicity, we omit the primes )

Consider                                         Bi 
J mv     M   
 
 0
253            Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Magnetization and Equivalent Current Density

       
( compared to Ampere’s law   B   0 J in free space )

                
where the internal flux density B is produced by M


In addition , we see that due to the free current density J


 Be  
   J
 
 0

where B e denotes the external magnetic flux density

254            Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Magnetization and Equivalent Current Density

Thus , the resultant magnetic flux density in the presence of

a magnetized is changed by an amount B i ；i.e.

                        
  B    Be  Bi    0 J  J mv 

Note：The application of an external magnetic field causes both

(1) an induced magnetic moment in a magnetic material

(2) an alignment of the internal dipole moment and

255           Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Magnetization and Equivalent Current Density

Fig . The induced magnetic dipole moment are partially aligned


along dl by an Externally applied magnetic field

magnetic dipole moment
     
m  Ids
                                         
dl

256           Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Magnetic Field Intensity and Relative
Permeability

＊ Magnetic Field Intensity and Relative Permeability

since the magnetic flux density   B   in the magnetic material
1                           
can be express by          B  J  J mv  J    M
0

257             Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Magnetic Field Intensity and Relative Permeability


We define the magnetic field intensity   H


     B    
H         M
0
                  A
Thus ,   H  J         (         m2 )

( another form of Ampere’s law )

where    J is the volume density of free current

258              Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Magnetic Field Intensity and Relative Permeability

According to Stokes’s theorem , we have
         
S
(  H )ds   J  ds
S

 
  Hdl  I
C

where C is the contour bounding the surface S and I is

the total free current passing thru S .

259         Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Magnetic Field Intensity and Relative Permeability

 H          
ds


H
C
S

The above formula holds in a nonmagnetic as well as a

magnetic medium

When the magnetic properties of the medium are linear and

isotropic, the magnetization vector
      
M  xm H
260       Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Magnetic Field Intensity and Relative Permeability

where   x m is a dimensionless quantity called magnetic

susceptibility

                                               
Therefore , B       0 ( H  M )   0 (1  x m ) H   0  r H  H

where the dimensionless quantity  r  1  x m  
0
is called the relative permeability of he medium and      ( H/m
)
is known as the ( absolute ) permeability.

261            Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Magnetic Field Intensity and Relative Permeability

Note：

xm :   magnetic susceptibility
x e : electric susceptibility

 r : relative permittivity               r :   relative permeability

 r  1  xe                                r  1  xm

   0 r                                    0  r

262            Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Magnetic Field Intensity and Relative Permeability

Note：
1. Electric dipole     vs     Magnetic dipole


S

+


d                                              X

-
I

                           
Electric moment    P  qd   Magnetic moment m  IS
263            Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Magnetic Field Intensity and Relative Permeability

Electric charge        vs       X (no magnetic charge)

          
P   0 xm E
+        -        +         -
+        -        +         -
+        -        +         -
+        -        +         -

Ed
        
Ei  E0  Ed

264          Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Magnetic Field Intensity and Relative Permeability


2. The magnetic field intensity H is introduced as the

basic quantity of the fields, the generalized Ampere’s law

 
 H  J

holds across any media

265           Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Magnetic Field Intensity and Relative Permeability

3. Recall that the potential due to the polarized dielectric
 
1       P  aR
V
4 0 V ' R 2 dv'

 
where R  aR R         is the distance vector from   dv' to
a fixed field point.


 1  aR
Since '    2
 R R

266             Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Magnetic Field Intensity and Relative Permeability

1             1
4 0 V '
it follows that
V             P  '  dv'
R

1             P        1    
        V ' ' dv' V ' 'Pdv'
 R
4 0                    R       
           
     
( By the vector identity ' f  A  f  ' A  A  ' f      )
 ps                        pv
                              
1        P  a 'n       1           'P dv'
4 0 S ' R                    V ' R
                    ds'
4 0

267           Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Magnetic Field Intensity and Relative Permeability


where a' n is the outward normal from the surface element
 
ds '  a ' n ds' of the dielectric

+     -
S’
+       -

+       -                 V’

Dielectric
268          Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Magnetic Field Intensity and Relative Permeability

Therefore , the electric potential due to a polarized dielectric

1            pv            1           ps
V
4 0   V'    R
dv'
4 0   
S'    R
ds'


where    pv    'P ( polarized volume charge density )

 
and     ps     P  an   ( polarized surface charge density )
 1
∵     E   v   pv  in the dielectric
0
            
   0 E    v  P
         
  v   0 E  P   D

269                 Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Magnetic Field Intensity and Relative Permeability

  
where   D   0 E  P is called electric displacement vector .

          
since   P   0 xe E     , we have

                                   
D   0 E   0 x e E   0 1  x e E
    
  0  r E  E

270           Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Magnetic Field Intensity and Relative Permeability

4.             Polarization & Magnetization
Polarization                       q          p

P
Thru V
 pv ,  ps

Magnetization
I         m

M
Thru A
J ms , J mv

271                   Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Behavior of Magnetic Materials

Magnetic Materials can be roughly classified into three main groups in accordance
with their  r values.
(1)Diamagnetism if  r  1 (  m  0 and  m  0 ) (The word “dia” in Greek mean
“across”)

(2)Para magnetism, if  r  1 (  m  0 and   m  0   ) (The word “Para” in Greek mean
“along”)

(3)Ferro magnetism, if  r  1 (    m  1 )

272             Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Behavior of Magnetic Materials

nucleus


+
az

B     
electron in orbital                                      ds
motion
I

B is produced due to current loop I
     
electron in spinning motion   (or magnetic dipole moment m  Id s )

273            Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Behavior of Magnetic Materials


m

+
       -
I
Ue                                                    
（a）                    （b）                      ms
（c）

274        Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Behavior of Magnetic Materials

(1) Diamagnetic materials:

As      B appl  0,   ( ie, no external magnetic field )

morb  mspin  0 ( for an atom )

as                is applied, B int  B appl      and B int  B appl
B appl
This is because the induced magnetic moment always apposes the applied field

according to Len’s law of electromagnetic induction. As a result, the magnetic flux
density is reduced.
The effect is equivalent to that of a negative magnetization ( ie,  m  0 ) and it

is usually very small. For diamagnetic materials,copper,lead,……etc.

 m is of the order of -10.
5

m                                         275              Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Behavior of Magnetic Materials

(2) Para magnetism materials:
As B appl  0, morb  mspin is small

as B appl is applied, morb  mspin is aligned in the direction of the applied

field.
s.t. B int  B appl and B int  B appl

However, the alignment process is impeded by the forces of random thermal
vibrations; as a result the paramagnetic effect is temperature dependent in contrast
to that of diamagnetic materials.For paramagnetic materials, e.g, aluminum,
tungsten,…..etc.  m is usually of the order of -10-5.

5
276      Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Behavior of Magnetic Materials

(3) Ferro magnetic materials:

As    B appl  0,

m spin  m orb
As   B appl   is applied
B int  B appl

Due to the postulate of magnetized domains proposed by Weiss in 1907 (Called
Weiss’ domains)

277           Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Behavior of Magnetic Materials

(a)                            (b)
Fig. schematic of an unmagnified (a) paramagnetic and (b) ferromagnetic material.
The arrows qualitatively show atom magnetic moments.

278           Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Behavior of Magnetic Materials

15   16
These domains, each containing about 10 or 10 atoms and usually having
the linear dimension of about 105 m, are fully magnetized in the sense
that they contain aligned magnetic dipoles resulting from spinning
electrons even in the absence of an applied magnetic field.
There are strong coupling forces between the magnetic dipole
moments of the atoms in a domain, holding the dipole moments in
parallel.
atoms thick called a domain wall.

279             Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Behavior of Magnetic Materials

( a)                        (b)                             (c)
Applied magnetic field
Fig. (a) unmagnetized
(b) magnetic-domains translated
(c) magnetic-domains rotated ferromagnetic materials

Above a certain temperature, called the curie temperature, the thermal vibrations
completely prevent the parallel alignment of molecule magnetic moments, and
ferromagnetic materials become paramagnetic. This temperature is 770 C for iron

280             Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Behavior of Magnetic Materials

H                       H
( a)                 (b)                     (c)

281        Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Behavior of Magnetic Materials


B

b
Br c

d         a           g               
0                           H

f
e
Hysteresis loop in B-H plane for ferromagnetic materials

282            Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Behavior of Magnetic Materials

(4)Anti ferromagnetic materials
As   B appl  0, m spin  m orb

As B appl is applied

B int  B appl

(5)Ferromagnetic materials
As Bappl  0,
m spin  m orb
As B appl is applied

B int  B appl
283      Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Behavior of Magnetic Materials

(a)

(b)

(c)

Fig. Schematic atomic spin structures for
(a) ferromagnetic,
(b) antiferromagnetic,
and (c) ferrimagnetic materials

284           Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Behavior of Magnetic Materials

Note（1）

1.For convection currents, ( which does not satisfy ohm’s law)the amount of
moving charges

               
q  Nq (ut )  s   vu  s  t
 
l                                        
S
q                  
     J  s ( J   v u )
t
 
 I  J  s                                                     
l

285           Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Behavior of Magnetic Materials

2.For conduction currents, (which leads to ohm’s law and KCL)

the volume current density vector
                       
J   N i qi u i    i u i
i             i

where more then one kind of charge carriers qi drifting with different velocities vi .

                    
For metallic conductors, we write the drift velocity. u e    e E Where    e is the

electron mobility measured in (m2/V.S)

                 
 J    e  e E  E where  e   Ne and conductivity     e  e

286           Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Behavior of Magnetic Materials

3.Equation of continuity

       dQ    d
I   J ds          v dv
dt    dt V
S 
 
flowing outward current
negative decreasing net charge rate ( where Q is locally existing charge)

By Divergence thm,
              d                                d
              
 Jds V J dv   dt V  v dv
S
J    v
dt

287           Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Behavior of Magnetic Materials

Note（2）

.static magnetic field

Fm  (qu)  B ………….Lorentz Force equ.


A point charge moving at a const. speed v go thru a magnetic field B Magnetic

force acting on the moving charge due to the convection current is negligible

288           Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Behavior of Magnetic Materials

2. law of magnetic force between two moving point charges
          
(q2u2 )  (q1u1  ar12 )
Fm12  km                     2           ……..testing charge in motion
r12                  (convection current)
0
where           km 
4
According to Lorentz’s force equ.
    
0      q1u1  ar
B                             12
is called magnetic flux density
4             r12
2

Ampere’s law of force {for conduction current (specifically filamentary
currents)}

0          ( I d  )  ( I1d  1  aR)
F 12         2 2                             where R  aR  R
4   c 2 c1            R   2

289           Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Behavior of Magnetic Materials

3.Biot-Savart law:

( B due to large amount of charges in motion)specifically free charges per unit volume of a

conductor is relative to a specified location (or a field point )

0     N q u  ar
B
4
 r 2 v
v

0   J  ar
       r 2 v
4 v

Where       J  N q u  v u
  0          d J  ar
B
4        c r 2 dv

290            Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Behavior of Magnetic Materials

4.Point charge

q    compared to
N q dv   v dv(   d)

compared to
qu                   ( N q dv) u   v u dv  J dv  J  S  d   Id 

291            Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Behavior of Magnetic Materials

    N q u  ar
5.Since B 0             v
4 v    r 2

(    r  ar  r         is the distance vector from the source element

N q u or             J to a specified field point)
 
 0                                          ( I dl )  ar
or B  0
( J  dv)  a r
we have       B
4   v r 2                      4    
L        r   2
(Biot-Savart law)

analogous to

1      (  v dv)  ar          1     (   d)  a r
E
4 0 v       r2
or E 
4 0 L      r2
   For source element (a point charge in motion)

0 q u  a r                                                     
B                                         0 N q u  ar
B                v                as v  0
4
or
r2                                4     r 2

292                    Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Behavior of Magnetic Materials

By superposition, we obtain the Biot-Savart law as formulated above
Also, for the source element (a point charge) in a static electric field

1  q ar                    1 N q ar
E                   or   E               v as
4 0 r 2                 4 0 r 2                    v  0

Again, by superposition        E

due to a volume distribution of charge or it due to a line charge is obtained

as show above, respectively.

293           Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Behavior of Magnetic Materials

6.The vector magnetic potential

           J                          0          I d 
A 0
4     V 1 r dv          or        A
4      V  r
analogous to the electric potential

1          v                         1          v d 
V
4 0   
V    r
dv     or   V
4 0    
L       r
q
     for a point charge            V
4 0 r
By superposition, we obtain the above eq.
for a volume charge distribution or for a line charge.

294                Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Chap 7 Time-Varying Fields and
Maxwell’s Equations

295       Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
induction                             X                X
 
dw  awdw   X
  d
B  as
ds
From Lorentz’s force eq., the magnetic force
l
exerted on the conductor shown in the figure is       X                                X

                                                                          Area Σ
expressed as
Fm  qu  B  qEi                     X
  
dl  al dl   X               X
enclosed by L

                                                        A moving conductor in a
where E is denoted as impressed electric field intensity.          magnetic field
i
                dw          d           dw             d
 Ei  u  B  ( aw           )  (  )  (aw            )  (as  )
dt      ds              dt              ds
dw d                            d           dw        
(            )  ( aw  as )  (            )(        )( al )
dt ds                          l  dw          dt
       d                                  
 Ei  
dt
or      Ei  dl   t  B  ds
L                       
296               Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
By Stokes’s thm, for an arbitrary area 

                             B 
 Ei  dl   (  Ei )  ds  ( t )  ds
L                            

      B where the subscript i is omitted is called Faraday’s law
We see that    E  
t
of EM induction.

In addition, the voltage across the terminals a and b of the conductor

b  d
Vab    ( Ei  dl ) 
a              dt
can be applied to that of a coil with N turns.

Nd    di
Vab      L
dt    dt
297           Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

where the polarities of the voltage are plus and minus on the terminals a and b,

respectively and the inductance L is defined as

Nd

L
di

298       Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Magnetic Forces and Torques
Hall effect

Consider a uniform magnetic field
                                                                                   
B  a z B0

B  a z B0 and a uniform direct current
flows in the y-direction:                                                                                       
J  ay J0
                                                                     z

J  ay J 0  Nqu                                                                     y

d
o
Vn
where N is the number of charge carriers per                                                  x

unit volume, moving with a velocity v , and                                b

J  ay J0

q is the charge on each charge carrier.
 
B  a z B0

It can be observed that
(1)The magnetic force tends to move the charge carriers in the positive x-direction,
creating a transverse electric field.

                                                                           
                                               Fm 　is　transverse　to　B
(i.e. Fm  qu  B ,   the same direction as that of   J   ,                                       )

299                       Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Magnetic Forces and Torques

(2)This will continue until the transverse field is sufficient to stop the drift of the charge
carriers.
In the steady state, the net force on the charge carrier is zero:
                                    
Eh  u  B  0         or       E h  u  B
                    
( Fm  Fe  0  qu  B  qEh  0)

This is known as the Hall effect, and E h is called the Hall field.
                        
Eh  (a y u0 )  az B0  axu0 B0
A transverse potential (denoted as Vh and called Hall voltage) appears across the sides
of the material.

Thus, we have
 
0         0           
Vh    Eh dl    a x u0 B0  a x dx  u0 B0 d for electron carriers.
d           d
300            Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Magnetic Forces and Torques

The Hall effect can be used for measuring the magnetic field and determining the sign
of the predominant charge carriers (distinguishing an n-type from a p-type
semiconductor).

Forces on current-carrying conductors
       
                      dl2  (dl1  aR12 )
Recall that     F12  0 I 2 I1  
4         c2 c1            2
R12                   
dl1              
a R12 R12  R12
It is an inverse-square relationship and should                I1                       
be compared with Coulomb’s law of force                                                dl 2
between two stationary charges.
I2

301           Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Magnetic Forces and Torques

Torques on current-carrying Conductors

Consider a small circular loop of radius b and carrying a current I in a uniform

magnetic field of flux density B

B//

dl1 dl sin 

B                       
          dl
o                        x
I
              T
dl2

(b)
(a )

                                         
It is convenient to resolve B        into B perpendicular and B// parallel to the plane of
 
the loop, i.e. . B  B  B
   //

302               Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Magnetic Forces and Torques

                                                                            
Obviously,  B tends to expand the loop but exerts no net force to move the loop and B//
produces a torque that tends to rotate the loop about the x-axis in such a way as to align the

magnetic field (due to I) with the external B// field.

       
The differential torque produced by dF1 and dF2 is
                         
dT  a x (dF )2b sin   a x ( Idl sin   B// )2b sin 
 
dT  a x 2 Ib2 B// sin 2 d
                                           
(  dF  dF1  dF2              and         dl  dl1  dl2  bd )

303           Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Magnetic Forces and Torques

The total torque acting on the loop is then
                                                                        
T   dT  a x 2 Ib B//  sin d  a x I (b ) B//  a y I (b )  a z B//  m  B
2         2               2                 2
0
             
(  sin 2 d         )
0             2
      
where    a y  is nthe unit vector of the surface of the loop,
a
                                                                          
m  B  m  ( B  B// )  m  B//       and       m  a y I (b2 )  an I (b2 )  an I  s

  
Therefore, we have           T  m  B　　  m)
(N
The principle of operation of direct-current (d-c) motors is based on this equation.

304               Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Magnetic Forces and Torques

Forces and torques in terms of stored magnetic energy

The principle of virtual displacement is an alternative method of finding magnetic
forces and torques. Let’s explore it in the following two cases.
(1)System of circuits with constant flux linkages:
 
The mechanical work F  dl done bythe system is at the expense of a decrease
in the stored magnetic energy, where F denotes the force under the constant-
flux condition.
                         
Thus    F  dl  dwmg  wmg  dl

That is,   F  wmg
If the ckt is constrained to rotate about an axis, say the z axis, the mechanical work
wm
done by the system will be (T ) z d and (T ) z         　　  m)
(N


305             Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Magnetic Forces and Torques

(2)System of circuits with constant currents

Since dws   I k d k is the energy supplied by the system source
k

( dwk  vk ik dt  ik d k 　 dws   dwk   I k d k for ik  I k )

k         k

We have       dws  dw  dwmg
1            1
2
 I k dk  dws
2
                                          
We have      dw  FI  dl  dwmg  (wmg )  dl or             FI  wmg ( N )

If the ckt is constrained to rotate about the z-axis, the z-component of the torque
acting on the ckt is
wm
(TI ) z           ( N  m)


306             Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Magnetic Forces and Torques
(1) In electrostatic field
 dwm                            
                                 dwe  Fe dl
 Fe dl  dwe
 
 qEdl
 qdv
 dwm  dwe  0
( wm  we  cons tan t )

External work   Electrostatic
energy

(2)      p  v i

307                Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Boundary Conditions for Magnetic Fields
(1)The normal components of magnetic field                                         
B1n
B1

by divergence theorem, since (  B  0) by divergence thm
      
                                                                      an 2  an

v
  Bdv   B  ds  B1n  (an s)  B2n  (an s)  0
s
                       
                                                1   B1 t                    B2t h  0
Assuming B1n   an B1n and B2 n  an B2 n
2
        
We have                                                                 a n1   a n
        
                                                                           B2 n      B2
(an B1n )( an s)  (an B2 n )( an s)  0  B2 n  B1n
                             
(Note that we may assume B1n  an B1n        and B2n  an B2n

As a result, B2 n   B1n following the directions we assume. )

Therefore, the normal component of B is continuous across an the interface.
                                               
For linear media, B1  1 H 1 and B2   2 H 2 , we have 1 H 1n   2 H 2 n
308           Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Boundary Conditions for Magnetic Fields

(2)The tangential components of magnetostatic field
      
an 2  an
                                                                                   
Since (  H  J ) by Stokes’s thm                                                
H1

                                                                   H 1n

 (  H )ds   Hdl   Jds  I
s                 c         s
b
a
H 1t
1     h  0    
H 2t
                                                   2
abcda dl  H1t  w  H 2t  w  J sn w
c                        d
H                                                                                 w

                                                               
H 2 H 2n
 ( a x H 1t )(  a x w)  ( a x H 2t )( a x w)                                                 
ax

                    
 H1t  H 2t  J sn ( A / m) or          an 2  ( H 1  H 2 )  J s

where an 2 is the outward unit normal from medium 2 at the interface.

Thus, the tangential component of H is discontinuous across an interface where a
free surface current exists. (However, when the conductivities of both media are
finite, currents are defined by volume current densities and free surface current do
not exist on the interface.)
309              Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Inductances and Inductors
Consider two neighboring closed loops,                             S2

C1 and C2 bounding surfaces S1 and S2,
respectively.                                          S1
 
12   B1ds2 ( Wb)                                                                 C2
s2


From Biot-Savart laws (determining B1 due to
the filamentary current I1)                                 I1
C1

                                                      Two magnetically
    I    dl 'a R                                                  coupled loops

B1  0 1 
4 c ' R 2
Note: L↑ means energy
increasing stored in a magnetic
N         N                    
L12  2 12  12  2
I1     I1   I1           s2
B1  ds2   field.

    N           
L11  11  1
I1   I1   
s1
B1  ds1 ( H )

310   Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Inductances and Inductors

<Ex>Find the self-inductance of a closely
wound toroidal coil.                                 I

Sol：
     2                                                           b
c
Bdl   (a B )( a rd )  2rB   0 NI
0                                                                 h

r     a
 NI                                               dr
 B  0
2r
        0 NI            0 NIh b dr  0 NIh b
   Bds   (a
2 a r
)(a hdr)                     ln
s       s    2r                             2      a

 N  0 N 2 h b
L            ln             (H )
I  I   2        a
311      Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Inductances and Inductors

Cause :        Capacitance: (involving the concept of
 1Q  2Q     charge storage)
Effect :
 1V  2V

I1   Cause :
Inductance: (involving the concept
1I1  2 I1
Effect :
11  21

312             Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Inductances and Inductors

Loosely
wound
L small

Tightly
wound
L large

Tightly
Loosely           neighboring
neighboring           C large
C small

313            Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Magnetic Energy
Consider a single closed loop with a self-inductance L1 in which the current i1
increases from zero to I1. At the same time, an electromotive force (emf) is induced to
oppose the current charge.

The work that must be done to overcome this induced emf is
I1          1
w1   v1i1dt  L1  i1di1      L1 I12
0           2
di1
where v1  L1     is the voltage across the inductance.
dt

Obviously, this work required is stored as magnetic energy.

314               Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Magnetic Energy

Now consider two closed loops, C1 and C2 carrying currents i1 and i2, which are
initially zero and are to be increased to I1 and I2, respectively.
        
Note : dw f  Fe dl  qEdl  qdv  dwe
 dwe  qdv
Consider　q  cv, 　We　have　dwe  cvdv
1 2
 we   cv 　　  ( Assume　we  0 　initially)
2
For　constsnt , 　ch arg e　case, 　dwe  qdv                       C1                1

or for constant voltage case
i1
v1
dwe  vdq
In the constant voltage case,
i
dwe    dq
p       v     vi
dt    dt                   315           Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Magnetic Energy

To find the amount of work required, we consider the following three cases:

Case1:

Keeping i2  0 and increasing i from zero to I 1
1
The work required in loop C1
C1       12
w1 
1
L1 I12                     v1     i1       11
2

C2
The current i1 linking with magnetic flux 1  11  12

316          Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Magnetic Energy

Case2:

Keeping i1 at I 1 and increasing i2 from zero to I 2

Because of mutual coupling, some of the magnetic flux due to                          i2
C1 giving rise to an induced emf that must be overcome
di2
by a voltage v21  L21       in order to keep i1 constant at its value I1.
dt

This work involved is

I2
w21   v21 I1dt  L21 I1  di2  L21 I1 I 2
0

317             Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Magnetic Energy
Case3:
At the same time, a work w22 must be done in loop C2 in order to counteract the induced
emf is increased from o to I2.

1                                                   i1  I
w22  L2 I 2
2

2                                                                            v2
The total amount of work done in raising               v21
the currents in loops C1 and C2 from 0 to I1
and I2, respectively.

1                    1        1 2 2
wm  w1  w21  w22  L1 I1  L21 I1 I 2  L2 I 2    L jk I j I k
2                    2

2                    2        2 j 1 k 1
1 2 2
wm    L jk I j I k 　　 )
(J
2 j 1 k 1
For a current I following in a single inductor with inductance L, the
stored magnetic energy is                 1 2
wm        LI 　　　)
(J
2
318            Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Wave Equations
In free space, we have


  E  0.......( M 1)


       H
  E  0            M
.......( 2)
t
                                
  H  0.......( M 3)        (B  0 H )


      E
  H  0           M
.......( 4)
t

319           Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Wave Equations

From (M2) and (M4), we have

                                                                E
2
  (  E )    0 (  H ) or                      (  E )    0 0
t                                                      t 2

According to the vector identities, we see

                                
  (  E )  (  E )   E
2
(   E  0)

                   E
2
That is        (  E )   E    0 0 2
2

t
                       
          E
2                   1  E
2
It follows       E   0 0 2
2
2E  2         .......( W 1)
t
or
 t
1
where the real number
  (  0 0 )        2

320           Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Wave Equations


In a similar fashion, we can obtain the same Eq(W1). Let’s use the notation U   to
       
stand for E or H .
It appears that

   1 2 
 U  2 2 U .......( W 2)
2
which is called wave eq.
 t

                                 
Assume           U ( x, y, z, t )  axU x  a yU y  azU z

For one-dimensional cases, one kind of the sols to Eq(W2) is

U ( x, t )  U m sin(kx   (t ))

321           Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Wave Equations
                                                                 
U ( x, y, z, t )  a u  a u  a u
x       x   y   y      z   z     U  a  u a  u a  u
2

x
2

x   y
2

y   z
2

z

U  U ( x, t )  U sin( kx   (t ))
X                      m

U(x,t)

Um

X                               X

-Um

322         Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Wave Equations

2
V (t )  Vm sin(t )            
T
U (t )  U m sin(kx   (t ))
1
               3 10 (m / s )
8

0   0

Vm

t
T

-Vm

323           Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Maxwell’s Equations

Electrostatic
model                                  Electromagnetic
(modified)           model (due to time-
varying field) :
(modified)
Magnetostatic
Maxwell`s eqs.
model

Equation of continuity

324              Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Maxwell’s Equations

Electrostatic model


 E  0

 
E 
   0


     B
 E  
t

This means that a changing magnetic field induces an electric field.

325        Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Maxwell’s Equations

Magnetostatic model


 B  0
    
  B  J

               
  (  B )  0     J
0

 
 J    0
t
V

326         Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Maxwell’s Equations

Q Q  0
0      i
Qi
Q     Q     
O
         dv
i

t     t    t
V

QO                    
t
 I   Jds     Jdv
Z       V                                  v

     
J      V
Q
t
O

327           Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Maxwell’s Equations

                                   
    B   0           H   0    J

       
  J  (  D)  0
t


 D
  (J     )0
t

                    D
  (  B )  0    (  ( J     ))
t
0

328             Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Maxwell’s Equations

    
D  E 0

              v
    H   0    J 
t

          D 
    H      J      
     t 

  D
 H  J 
t

           E
 H  J 
t
0

329              Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Maxwell’s Equations


     B
 E  
t

  D
 H  J 
t

  D  v


 B  0

330           Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Maxwell’s Equations

          d
C E  d  
dt

               dD
C H  d  I  
S dt
 ds


 D  ds  Q
s


 B  ds  0
s

331            Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Reference
1.張國維老師上課手稿

2.楊國輝、廖淑慧。民89。應用電磁學。臺北市：五南。

3.徐在新、宓子宏。民83。從法拉第到麥克斯韋。新竹市：凡異。

4.David K. Cheng(1993), Fundamentals of Engineering Electromagnetics, Addison Wesley.

5.王奕淳、張友福、張毓華、郭志成、林漢璿、廖家成上課作業。

332          Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Acknowledgement

Thanks to 王奕淳、林漢璿等人 For typing the
lecture notes