http://optics.hanyang.ac.kr/~shsong
송석호 (물리학과)
Introduction to Electrodynamics, David J. Griffiths
Review:
1.
2.
3.
4.
5.
Vector analysis
Electrostatics
Special techniques
Electric fields in mater
Magnetostatics
6.
7.
8.
9.
Magnetic fields in matter
Electrodynamics
Conservation laws
Electromagnetic waves
10. Potentials and fields
11. Radiation
12. Electrodynamics and relativity
Introduction to Electrodynamics, David J. Griffiths
Electromagnetics ?
•
History of Electromagnetics
BASIC EQUATIONS OF ELECTRODYNAMICS in SI units
Maxwell's Equations
Lorentz force law
Auxiliary Fields
Potentials
Review: Chapter 1. Vector Analysis
1.1.4 Position, Displacement, and Separation Vectors
Position vector:
Infinitesimal displacement vector:
Separation vector from source point to field point:
1.2.3 The Del Operator:

: a vector operator, not a vector.
(gradient)
 Gradient represents both the magnitude and the direction
of the maximum rate of increase of a scalar function.
(divergence)
(curl)
1.2.2 Gradient
 What’s the physical meaning of the Gradient:
 Gradient is a vector that points in the direction of maximum increase of a function.
Its magnitude gives the slope (rate of increase) along this maximal direction.
 Gradient represents both the magnitude and the direction of the maximum rate of
increase of a scalar function.
Gradient of Separation distance
•
Magnitude of separation vector:
( x, y , z )
r
r  r  ( x  x) 2  ( y  y) 2  ( z  z) 2
r
1/2
1
 ( x  x) 2  ( y  y) 2  ( z  z ) 2 
r
 1
 1
 1
1
    ax    a y    az  
x  r 
y  r 
z  r 
r
a ( x  x)  a y ( y  y)  a z ( z  z )
ar
r
=  x




3/2
r3
r2
( x  x) 2  ( y  y) 2  ( z  z) 2 
a
1
     2r : in field coordinates
r
r
ar
1

     2 : in source coordinates
r
r

r'
( x, y, z )
1.2.4 The Divergence div A   A
Ax Ay Az
 A 


x
y
z
: scalar, a measure of how much the vector A
spread out (diverges) from the point in question
: positive (negative if the arrows pointed in) divergence
: zero divergence
: positive divergence
1.2.5 The Curl
curl A  rot A    A
: vector, a measure of how much the vector A
curl (rotate) around the point in question.
Zero curl :
Non-zero curl :
1.2.7 Second Derivatives
 The curl of the gradient of any scalar field is identically zero!
 If a vector is curl-free, then it can be expressed as the gradient of a scalar field
E  0
E  V
 The divergence of the curl of any vector field is identically zero!
 If a vector is divergence-free, then it can be expressed as the curl of a vector field
B  0
B  A
Laplacian and Laplace equation
Laplacian = “the divergence of the gradient of ”
 2  
 

   V
V
V 
 V  V   a x  a y
 a z  a x
 ay
 az







x
y
z
x
y
z



2
2
2



V
V
V
 2V  2  2  2
x
y
z
2
Laplace equation:
 2V  0
Poisson equation:

V 
0
2
Useful product rules
Triple Products
(BAC-CAB rule)
Product Rules
Second Derivatives
[Appendix A] Vector Calculus in Curvilinear Coordinates
(Orthogonal) Curvilinear Coordinates: (u , v, w)
Gradient Theorem
Gradient in arbitrary curvilinear coordinates.
 Fundamental theorem for gradients
f
g
h
x, y, z
1
1
1
s, , z
1
r
1
r, , 
1
r
r sin
Divergence in Curvilinear Coordinates:
The divergence of A in curvilinear coordinates is defined by
f
g
h
x, y, z
1
1
1
s, , z
1
r
1
r, , 
1
r
r sin
 Divergence theorem
 It converts a volume integral to a closed surface integral, and vice versa.
Divergence Theorem
For a very small differential volume element  j bounded by a surface s j
   A  j  j   s
j
A  ds     A  j 

sj
A  ds
 j
N

N

lim      A  j  j   lim    A  ds 
s
  0
 j 1
   0  j 1 j

N

lim      A  j  j      Ad
V
  0
 j 1

N

lim    A  ds    A  ds
s
S
  0
 j 1 j



V
  Ad   A  ds
S
Curl in Curvilinear Coordinates:
The curl of A in curvilinear coordinates is defined by
f
g
h
x, y, z
1
1
1
s, , z
1
r
1
r, , 
1
r
r sin
 Stokes’ theorem
 It converts a volume integral to a closed surface integral, and vice versa.
Stokes’s Theorem
    A   ds  
S
C
   A  j  ds j   C
N
lim
s j  0
   A 
j 1
j
A  dl
A  dl
j
 s j      A   ds
S
 A  dl   A  dl
  c
  C

s j  0
j

j 1 
N
lim
Laplacian in Curvilinear Coordinates:
Laplacian = “the divergence of the gradient of ”  2  
 Gradient of t
 Divergence of A
(Ex)
Laplace equation:  2V  0
Poisson equation:  2V  

0
f
g
h
x, y, z
1
1
1
s, , z
1
r
1
r, , 
1
r
r sin
1.5 The Dirac Delta Function
Consider the divergence of E:
(divergence in terms of r)
Since the r-dependence is contained in
r = r - r',

 r 
 How do we solve the divergence?    2 
r 
The Divergence of
Consider the vector function directed radially:
Let’s apply the divergence theorem to this function:
Does this mean that the divergence theorem is false? What's going on here?
 The divergence theorem MUST BE right since it’s a fundamental theorem.
 The source of the problem is the point r = 0, where v blows up!
(
) vanishes everywhere except r = 0, its integral must be 4.
 The entire contribution of
must be coming from the point r = 0!
 No ordinary function behaves like that.
 It's zero except at the source location, yet its integral is finite!
 It’s called the Dirac delta function.
 It is, in fact, central to the whole theory of electrodynamics.
1.5.3 The Three-Dimensional Dirac Delta Function
Generalize the delta function to three dimensions:
with its volume integral is 1:
 As in the one-dimensional case, integration with  picks out the value f at r = 0.
 The divergence of
is zero everywhere except at the origin.
 The integral of
over any volume containing the origin is a constant (= 4)
More generally,
or
Since
Dirac Delta Function and Divergence of E
E 
 (r )
 This is Gauss's law in differential form
0
1.6 The Theory of Vector Fields
1.6.1 The Helmholtz Theorem
Maxwell reduced the entire theory of electrodynamics to four differential equations,
specifying respectively the divergence and the curl of E and B.
 The Helmholtz theorem guarantees that
the field, E or B is uniquely determined by its divergence and curl.
For example, in electrostatics
(V: Scalar potential)
In magnetostatics,
(A: Vector potential)
1.6.2 Potentials
Note the two null identities
 the curl of the gradient of any scalar field is identically zero:    V   0
 The divergence of the curl of any vector field is identically zero:      A   0
  F  0  If the curl of a vector field (F) vanishes (everywhere),
F can be written as the gradient of a scalar potential (V) 
F  V
(The minas sign is purely conventional.)
 F  0  If the divergence of a vector field (F) vanishes (everywhere),
F can be written as the curl of a vector potential (A) 
F   A
For all cases, any vector field can be written as


F  V    A