Section 2 1 TRƯỜNG ĐIỆN TỪ

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


2.1
The Line Integral


2.1-3

The Line Integral
Work done in carrying a charge from A to B in an
electric field:
B
E1
A

1
l1
n

E2

2
l2

WAB   dWj
j1


2.1-4

dW j qE j cos  j l j 
qE j l j cos  j
qE j  l j
n

 WAB q  E j • l j
j1

n
WAB
VAB 
  E j • l j (Voltage between
q
A and B)
j1


2.1-5

In the limit n   ,
B
VAB   E • dl
A

= Line integral of E
from A to B.

C

E • dl = Line integral of E
around the closed
path C.


2.1-6

A

If
R
C

C
L
B

C

E • dl = 0 ,

B
then  E • dl
A
is independent of
the path from A to B
(conservative field)

ARBLA E • dl  ARB E • dl  BLA E • dl
 ARB E • dl – ALB E • dl 0

ARB E • dl  ALB E • dl


2.1-7

Ex. For F  yza x  zxa y  xya z , find
(1,2,3)

 F • dl

along the straight line paths (0, 0, 0)

from (0, 0, 0) to (1, 0, 0), from (1, 0, 0) to
(1, 2, 0) and then from (1, 2, 0) to (1, 2, 3).
z

(0, 0, 0)
(1, 0, 0)
x

(1, 2, 3)

y
(1, 2, 0)


2.1-8

From (0, 0, 0) to (1, 0, 0),
y z 0 ; dy dz 0
F 0 ,

(1,0,0)

(0,0,0) F • dl 0

From (1, 0, 0) to (1, 2, 0),
x 1, z 0 ; dx dz 0
F  ya z
dl dx a x  dy a y  dz a z dy a y
F • dl 0,

(1,2,0)

(1,0,0) F • dl 0


2.1-9

From (1, 2, 0) to (1, 2, 3),
x 1, y 2 ; dx dy 0
F 2za x  za y  2a z , dl dz a z
F • dl 2 dz ,


(1,2,3)

(1,2,3)

(1,2,0) 2 dz 6

(0,0,0) F • dl 0  0  6 6


2.1-10





In fact, F  d l  yza x  zxa y  xyaz

 dx ax  dy a y  dz az 

 yz dx  zx dy  xy dz
d  xyz 
1,2,3



0,0,0 

1,2,3

F  d l 

0,0,0 

1,2,3
0,0,0 

d  xyz   xyz 

12 3  0 0 0
6, independent of the path.