Section 2 1 TRƯỜNG ĐIỆN TỪ

No Slide Title Slide Presentations for ECE 329, Introduction to Electromagnetic Fields, to supplement “Elements of Engineering Electromagnetics, Sixth Edition” by Nannapaneni Narayana Rao Edward C Jor[.]

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

The Line Integral

1 2

A l1

l2

B

The Line Integral

Work done in carrying a charge from A to B in an

electric field:

j1 n



 WAB q E j • l j

j1

n



VAB  WAB

n

 (Voltage between

A and B)

 cos  

cos

qE l q





 E  l

In the limit , n  

V AB A B E • dl

= Line integral of E

from A to B.

E • dl

C

 = Line integral of E

around the closed

path C.

L

If E • dl = 0 ,C

then AB E • dl

is independent of

the path from A to B

(conservative field)

E • dl

ARBLA

ARB E • dl – ALB E • dl 0

E • dl

ARB

(1, 0, 0)

z

x

y

(1, 2, 3)

(1, 2, 0) (0, 0, 0)

Ex For find F  yza x  zxa y  xya z ,

F • dl

(1,2,3) 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)

From (0, 0, 0) to (1, 0, 0),

From (1, 0, 0) to (1, 2, 0),

y z 0 ; dy dz 0

(0,0,0)

(1,0,0)



x 1, z 0 ; dx dz 0

F  ya z

(1,0,0) (1,2,0)



From (1, 2, 0) to (1, 2, 3),

x 1, y 2 ; dx dy0

F 2za x  za y  2az , dl dz a z

(1,2,0)

(1,2,3)



(0,0,0)

(1,2,3)



In fact ,  

 

d yz zx xy

dx dy dz

yz dx zx dy xy dz

d xyz







 

 

 

 

    

0,0,0

1 2 3 0 0 0

6, independent of the path.

     

