Engineering Electromagnetics _Pt.1_

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Engineering Electromagnetics _Pt.1_ Powered By Docstoc
					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

     Vector Addition and Subtraction
            
      (1) A  B  C
            A head-to-tail rule           
                                          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
              Y:the radius of 2-dimensional circle



            (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:

        Gradient: a scalar field



                                      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
      Where grad        an or      grad        an
                       dn                      dn      dn

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

     From the above, we see that

       d           
           grad  a l
       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 
                                                                
                                            
                            Ad s  divAdv
                                
                            Ad s
             
    Ads   Adv
           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
                                            Ads
                                    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

‧ Faraday Experiment in 1837
                                     +

                   +                 -
                            -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 
                          r        ( steradian )


                                   
 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



Steradian or solid angle
                                        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

                       -       -                           Radius b
                                                   -
                                   -           -



                                   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

     redistributed under the influence of an electromagnetic field ; but the

     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



Ex:A linear electric quadru-pole


                                                       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

    Chap. 4 Steady Electric Currents
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
                        different lead todifferent
                        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

Governing equation for steady current
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
            About Magnetic Field
        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
               
    Thad is, B is rotational.
                                                                        
                                                                      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

   It follow that
                                                       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.
    Between adjacent domains there is a transition region about 100
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
   Faraday’s law of electromagnetic EM
                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
     In addition,since    dwmg 
                                   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
                                   of flux linkage)
                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
                                                                            will link with loop
  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

                 Faraday`s law of electromagnetic induction


          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
                Adjust:林裕軒 2005/10/20


                             333        Dr. Gao-Wei Chang

				
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