Section 3 3 TRƯỜNG ĐIỆN TỪ

3.3-1

Slide Presentations for ECE 329,
Introduction to Electromagnetic Fields,
to supplement “Elements of Engineering
Electromagnetics, Sixth Edition”
by

Nannapaneni Narayana Rao

Edward C. Jordan Professor of Electrical and Computer Engineering
University of Illinois at Urbana-Champaign, Urbana, Illinois, USA
Distinguished Amrita Professor of Engineering
Amrita Vishwa Vidyapeetham, Coimbatore, Tamil Nadu, India


3.3
Curl and Divergence


3.3-3

Maxwell’s Equations in Differential Form
B
× E = 
t
D
× H = J 
t

Curl

ax
∂
× Α 
∂x
Ax

 D 

 B 
ay
∂
∂y
Ay

az
∂
∂z
Az

A y A z

A
x
Divergence  A =


x
y
z

3.3-4

Basic definition of curl
Lim  C A d l 
× A =
an
S  0  S 

 max
 × A is the maximum value of circulation of A per

unit area in the limit that the area shrinks to the point.
Direction of  × A is the direction of the normal
vector to the area in the limit that the area shrinks
to the point, and in the right-hand sense.


3.3-5

Curl Meter
is a device to probe the field for studying the curl of the
field. It responds to the circulation of the field.


3.3-6


3.3-7

a
 2x
for 0  x 
 v0 a az
2
v 
2x 
a
 v0  2 
 az for  x  a
a 
2
 

ax

ay

az


× v 
x
0


y
0


z
vz

   × vy

vz

ay
x

a

negative
for
0

x


2

 positive for a  x  a

2

 2v0
  a a y

 2v0 a y

 a


3.3-8

Basic definition of divergence
Lim
  A=
v  0

 A d S
v

is the outward flux of A per unit volume in the limit that
the volume shrinks to the point.

Divergence meter
is a device to probe the field for studying the divergence
of the field. It responds to the closed surface integral of
the vector field.


3.3-9

Example:

At the point (1, 1, 0)
(a)

 x  1

2

 y  1 a y
Divergence positive

(c) x a y

y

1

ax

Divergence zero
(b)

x

z
x

y

1

1

z
x

y

1

1

Divergence negative
z

1

y


3.3-10

Two Useful Theorems:
Stokes’ theorem

 A  d l =  × A   d S
C

S

Divergence theorem

 A  d S =    A  dv
S

V

A useful identity

   × A 


3.3-11

ax

ay

az


× Α 
x
Ax


y
Ay


z
Az





× A =
  × A x    × A  y    × A z
x
y
z

x


x
Ax


y

y
Ay


z

0
z
Az