Vector Analysis

http://optics.hanyang.ac.kr/~shsong 송석호 (물리학과)

Field and Wave Electromagnetics, David K. Cheng Reviews on

(Week 1) 2. Vector Analysis

3. Static Electric Fields

(Week 2) 4. Solution of Electrostatic Problems 5. Steady Electric Currents

(Week 3) 6. Static Magnetic Fields

7. Time-varying Fields: Faraday’s Law Introduction to Electromagnetics, 3^rd Edition, David J. Griffiths

(Week 4-5) 7. Electrodynamics: Maxwell’s Equations (Week 6) 8. Conservation Laws

(Week 7-8) 9. Electromagnetic Waves (Week 9-10) 10. Potentials and Fields (Week 11-12) 11. Radiation

(Week 13-14) 12. Electrodynamics and Relativity

Chapter 2.

Vector Analysis

한양대학교, 전기공학과 정진욱

2-4. Orthogonal coordinate systems

• Cartesian, cylindrical, spherical coordinates

• In 3D space, the three families of surface are described by

u₁

=const, u

₂

=const and u

₃

=const

• In Cartesian coordinate system – u

₁

= x, u

₂

= y and u

₃

= z

1 1 1

, ,

y z x z

0

x y z

d dx dy dz

d  dxdydz

  

  

  

  



x y z y z x z x y

x y

x y z

x y z

a a = a a a = a a a = a a a a a a a

OP a a a

l a a a

  

Cylindrical coordinates

•

u₁

, u

₂

, u

₃

= ( r, 

, z )

, ,

r z z r z r

dl

r

dr rd dz d rdrd dz

  





 

  

  



z

a a = a a a = a a a = a

a a a

r

• Differential volume element

d   rdrd dz 

Spherical Coordinates

•

u₁

, u

₂

, u

₃

= ( R, 

,

 )

2

, ,

sin sin

     

 

  

   

  

  



a a = a a a = a a a = a

l a a a

R R R

d dR R Rd R d

d R dRd d

sin cos sin sin cos

x R y R z R

 

 









2

sin

d   R  dRd d  

Generalized Orthogonal Coordinate

• Base vectors

• Displacement vector

• Differential volume

• Differential area

1 2 3

1 1 2 2 3 3

, ,

u u u

u

A

u u

A

u u

A

u

  

a a a

A a a a

     

     

1 1 1 2 2 2 3 3 3

2 2 3

1 1 2 2 3 3

u u u

d h du h du h du

dl h du h du h du

  

  

l a a a

1 2 3 1 2 3

dv h h h du du du 

1 2 3 2 3 2 1 3 1 3

,

,

d

n

ds

ds h h du du ds h h du du



 

s a

h₁ h₂ h₃

x, y, z 1 1 1

r, , z 1 r 1

R, ,  1 R R sin

Metric coefficients

2-6. Gradient of a Scalar Field

•

Gradient :

the vector that represents both the magnitude and the direction of the maximum space rate of increase of a scalar.

 

1 1 2 2 3 3

( , , ),

i

i i i i i

i i i i i i

u

i i i

d du h du dl d

u h u

d h du h du h du

h u

 

  

 

 

      

 

    



  



l

l a

1 2 3

1 2 3

1 2 3

1 1 2 2 3 3

( , , ) ( , , )

In general orthogonal coordinates ( , , )

u u u

u u u x y z

V V

x y z

u u u

h u h u h u



    

          

    

          

x y z

a a a

a a a

2-7. Divergence of a Vector Field

• Flux lines :

representation of field variations graphically by directed field lines.

• Magnitude of the field at a point :

either depicted by the density or by the length of the directed lines in the vicinity of the point

•

Divergence at a point:

the net outward flux of A per unit volume as the volume about the point tends to zero

lim

0 ^s

d

div

^{ }



  



 ^{A s}

A   A 

The flow of an incompressible fluid

  : Equal by definition 

Net outward flux indicates the presence of a source

 Flow source

 Div A is a measure of

the strength of the flow source

Divergence of a Vector Field

•

 

0

0

0 0 0

0 0 0 0 0 0

( ,

lim

On the front face

( , , )

2

( , , ) ( , , )

2 2

 

 

  

           

  

             



 

  



       



A s

A A s A s

A s A s A a

 





s

s

front back right left top bottom

front front front x x

front

x

x x

x y

d

d d

d y z A x x y z y z

A

x x

A x y z A x y z

x

 

0 0

0 0 0

0 0

, )

0 0 0

0 0 0 0 0 0

( , , )

( , ,

higher-order terms

( , , )

2

( , , ) ( , , ) higher-order terms

2 2

H.O.T.



              

  

   



      

    

 

 



 

A s A s A a

A s

z

back back back x x

back

x

x x

x y z

x

x y z front back

d y z A x x y z y z

A

x x

A x y z A x y z

x d A

x 0)

  x y z

0 0 0

( , , )

x y z

s

x y z

A A A

d x y z

x y z

   

           



^{A s}



• Following the same procedure for 4 faces

x

A

y z

A A

x y z

  

   

  

 A

Divergence of a Vector Field

• In general orthogonal curvilinear coordinates (u

₁,u₂,u₃

)



2 3 1

 

1 3 2

 

1 2 3



1 2 3 1 2 3

1 h h A h h A h h A

h h h u u u

    

  A          

Divergence Theorem

V

  d  

S

 d

 ^A   ^{A s}

 It converts a volume integral to a closed surface integral, and vice versa.

2-9. Curl of a Vector Field

C

 dl

 ^A



0

lim 1

_n

s C

dl

 

s  

  ^{A }   ^a    ^A  

: Circulation of a vector field A around contour C caused by a vortex source

y x y x

z z

x y z

A A A A

A A

y z z x x y

 

         

 A a     a     a     

x y z

x y z

x y z

A A A

  

     a a a A

In general orthogonal curvilinear coordinates

1 2 3

1 2 3

1 2 3 1 2 3

1 1 2 2 3 3

1

u u u

h h h

h h h u u u

h A h A h A

  

    

a a a

A

Laplace equation

Laplacian = “the divergence of the gradient of ”    

²



2

2 2 2

2

2 2 2

V V V

V V

x y z x y z

V V V

V x y z

         

                      

  

   

  

x y z x y z

a a a a a a

 

Laplacian in orthogonal curvilinear coordinates (u₁,u₂,u₃)

 

     

2 2 3 1 3 1 2

1 2 3 1 1 1 2 2 2 3 3 3

2 3 1 1 3 2 1 2 3

1 2 3 1 2 3

1

1

h h V h h V h h V

V V

h h h u h u u h u u h u

h h A h h A h h A

h h h u u u

             

                            

    

 

A

         





Laplace equation:

Poisson equation:

2

2

0

0 V

V 



 

  

2-10. Stokes’s Theorem

 

S

   d 

C

 dl

 ^{A s}   ^A

The surface integral of the curl of a vector field over open surface

Is equal to the closed line integral of the vector along the contour bounding the surface.

 It converts a surface integral of the curl of a vector to a line integral, and vice versa.

Note! Divergence Theorem

V

  d  

S

 d

 ^A   ^{A s}

It converts a volume integral to a closed surface integral, and vice versa.

2-11. Two Null Identities

(I) The curl of the gradient of any scalar field is identically zero.

(ex)

If a vector is curl-free,

then it can be expressed as the gradient of a scalar field.

  ^{V 0}

  

  V E

  E 0

 

⁰

S  V d  C   V dl dV   V dl



^s

 

(II) The divergence of the curl of any vector field is identically zero.

  ⁰

   A 

   

   

1 2 1 2

1 2 0

V S

n n

S S C C

d d

ds ds d d



     

          

 

   

A A s

A a A a A  A 



   

(ex) If a vector is divergenceless,

then it can be expressed as the curl of another vector field.

   B 0

 

B A

Field Classification and Helmholtz’s Theorem

• A field is Solenoidal and irrotational if

• Solenoidal but not irrotational if

• Irrotational but not solenoidal if

• Neither solenoidal nor irrotational if

•

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

are specified everywhere.



Helmholtz’s theorem

0, 0 (static electric field in a charge-free region)

  F  F

0, 0 (A steady magnetic field in a current-carrying conductor)

  

F

 

F

0, 0 (A static electric field in a charged region)

  F    F

0, 0,

(An electric field in a charged medium with a time-varying magnetic field)

  

F

 

F

 A general vector function F can be written

as the sum of the gradient of a scalar function and the curl of a vector function

Some useful vector formulas

     

     

 

 

 

     

 

 

2

0 0

V V V

V

  

  

  

    

 

    

    

    

     

     

 

  

A B C B C A C A B A B C = B A C - C A B

A A A

A A A

A B B A A B

A A A

A

  

 

  

  





Chapter.3

Static Electric Fields

한양대학교 전기공학과

정진욱

Coulomb’s law

• The experimental law of Coulomb (1785)

– http://navercast.naver.com/contents.nhn?contents_id=4647

1 2

 2

F q q a _R k r

9 2 2

9 10 N m /C

  

k

Electrostatics in Free Space

• Electric field density : the force per unit charge (very small)

The two fundamental postulates of electrostatics in free space.

0





   E 

 

lim 0 V/m

q ^ q

 F

E

0 0

1

V V S

d

  

_d d Q

 

    

 ^

^E

  

^{E s}

  E 0  

_C

^{E   dl 0 } Kirchhoff’s voltage law

 Gauss’s law

Static electric field is irrotational!

Static E is Conservative !!

• Scalar line integral of E is independent of the path; it depends only on the end points.

 

0 0

S C

d d

    ^E  ^{E  s}    ^{E  l} 

1 2

2 1

1 1 2 2

2 2

1 1 1 2

0

!!

C C

P P

P C P C

P P

P C P C

d d

d d

d d

 

 



 

 

 

E l E l

E l E l

E l E l

 

 

 

 

Electrical potential

• From the null identity,

• Scalar quantities are easy to handle than vector quantities.

• If we can determine V more easily, then E can be found by a gradient operation.

• Work done from point P

₁

to point P

₂

 potential difference (Electric potential)

• Relative direction of E and increasing V.

0 V

     E E

 

 

2

1

2

1

2 1

J/C or V V

P P

P P

W d

q

V V d

 

  





E l E l





3-6. Conductors in static electric field

0, 0





 E 

• Inside a conductor ( under static conditions)

• Boundary conditions at a conductor-free space interface

0 0

0 0

s s

n n

S

t t

abcda

E d E S S

E d E w

 

 

     

    





E s E l









 Shielding from outside electric fields

Under static conditions,

 The E field on a conductor surface is everywhere normal to the surface.

 The surface of a conductor is an equipotential surface under static conditions.

3-7. Dielectrics in static E field

• Insulators ( or dielectrics) – Bound charges

– The induced electric dipoles will modify the electric field both inside and outside the dielectric material

+

 0

applied

E

E_applied0

+ p_induced



²



1

lim

0

C/m

n

k k













 



 ^p

P

Polarization vector P

:

Average volume density of electric dipole moment

S







  

  



P n P

Polarization charge densities (bound-charge densities)

 













P n P







charge on

Polarizati

^S

0 when 

_

   P 

S

 P  n

₁

 P



  ^P

S

 P  n

₂

 P   n

₁

 



n

₁

n

₂

=-n

₁

Physical meaning of polarization charge

n S

 P  n

₁

 P



n S

 P  n

₂

  P

 0

when 

_

   P 

 P









 0



 P

n P 

S

 n 

P 

S





Physical meaning of polarization charge

 

 

 

 

 

0

0

0

2 0

3

C/m C/m

p

free

 





 







 





   

  

   

    



  



  



E E P

E P

D E P

D

 



 

























e R

R

e e

e



























1

1

0

0 0

0 0

E D

E E

E D

E P

Gauss's law inside dielectric with no surface charge

 P



p

 charge, on

Polarizati 

Relative permittivity (dielectric constant) Permittivity (dielectric constant)

electric susceptibility

3-8. Electric Flux Density and Dielectric Constant

: Generalized Gauss's law

S d



Q



^{D s}

^



: not at all!

E P  

_e



₀

applied field ? induced polarization ?

오류 1 E를 인가했더니



_e



_eP가 유도 ?

E D  

오류 2 E는 인가된 전기장, D는 유도된 전기장 ?

: not at all!

E와 D는 서로 다른 물리량.

E = E_applied + E_{by dipoles}

D=D_applied + D_{by dipoles}

Common misunderstanding on E & D

3-9. Boundary conditions

• Tangential component of E

• Normal component of D

1 2

0 E

_t

E

_t

   E 

2

1 2

0

D_n D _n



_s

(C/m )

   

D

 

1 2 1 2

0 _{t t} 0

abcda

dl d d E dw E dw

         



^{E E w E w}





1 n2 2 n1



n2



1 2



V S

D ds S S d S h

   

   ^D

 

  ^{D a} ^D ^a   ^a  ^D ^D  



  

V  d



lim

0

s h h

 



 



Summary

• Electrostatic case

Charge density

Electric Field Potential

2 0

1 ˆ

4 ^V dV

R



 ^





^R

E

/

0

, 0

 

 

  E

E



V    ^{E } d ^l

  V E

0

1 4

^V

V dV

R





^

 

2

/

0

V  

  