Section 5 1 TRƯỜNG ĐIỆN TỪ

5 1 Gradient, Laplacian, and the Potential Functions Slide Presentations for ECE 329, Introduction to Electromagnetic Fields, to supplement “Elements of Engineering Electromagnetics, Sixth Edition” by[.]

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

Gradient and the Potential Functions

0







a a a

× A



Since   B  0,

B can be expressed as the curl of a vector.

Thus

A is known as the magnetic vector potential

Then

t

t













A

×



A

E +   



is known as the electric scalar potential.

0

t

A

× E +   

   



 

t

 A

E =    



is the gradient of 

     

0

x y z

x y z

 

  



×

 

Basic definition of

From this, we get





d

dl



Maximum rate of increase of

  

 , , 

P x y z

Q x dx y dy z dz   

d      d l

n

d dn





direction of the maximum rate of increase, which occurs normal to the constant surface.

n



a 

Potential function equations

2 2

2

t



t





× E =



t



  

 



D

× H = J +

D =



 

t



  



B = 0

B = × A

A

E =









and using

t

 





A =

 

2 2

2

t







 





Laplacian of scalar

Laplacian of vector

In Cartesian coordinates,

2

     

