Section 5 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


5.1
Gradient, Laplacian, and
the Potential Functions


5.1-3

Gradient and the Potential Functions
ax

× A 
x
Ax

ay

y

Ay

az

z
Az




   × A    × A x    × A  y    × A  z
x
y
z

x
 
x
Ax
0


y

y
Ay


z


z
Az


5.1-4

Since   B 0,
B can be expressed as the curl of a vector.
Thus

B =  A

A is known as the magnetic vector potential.
Then


 × E =   × A 
t
A
  ×
t


5.1-5

A 

×  E +
 0
t 


A
E+
 
t
A
E =   
t
is known as the electric scalar potential.

 


  ax
 ay
+ az  
y
z 
 x




ax 
ay 
az
x
y
z
is the gradient of 

5.1-6

ax

 ×  
x
 x
ax

 x

x
0

ay

y
 y

ay

y

y

az

z


z

az

z
 z


5.1-7

Basic definition of 

Q  x  dx, y  dy, z  dz 

  d
dl



d    d l

P  x, y , z 

From this, we get

d
 
an
dn

   Maximum rate of increase of 
an  direction of the maximum rate of increase, which
occurs normal to the constant  surface.


5.1-8

B
× E = 
t

D 
× H = J +

t 

D = 
B = 0
B = × A
A
E =   
t

and using   A =  
t
2



2    2 

t
 Potential function
2

A
equations
2
 A   2   J
t


5.1-9

2

    

Laplacian of scalar

2 A =    A    ×  × A Laplacian of vector
In Cartesian coordinates,
2

2

2

  
 2  2  2
x

y
z
2

2 A = 2 Ax ax  2 Ay a y  2 Az az