Section 3 2 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[.]

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

3.2 Gauss’ Laws and the Continuity Equation

GAUSS’ LAW FOR THE ELECTRIC FIELD

D • dS 

S

z

(x, y, z) y

x

z

y x

x x x x x

y y y y y

z z z z z

x y z







   











0

0

Lim

x

y

z

x y z

x y z



 

 

 

  



  

 

+Δ

0

0

0

Lim

x x x x x

y y y y y

z z z z z x

y

z

x y z





 

 

 

  

D x

x 

D y

y 

D z

z

Longitudinal derivatives

of the components of D

  

 • D 

Divergence of D = 

Ex Given that

Find D everywhere.

  0 for – a  x  a

0 otherwise







Noting that  = (x) and hence D = D(x), we set



y  0 and



z  0, so that

 • D D x

x 

D y

y 

D z

z 

D x

x

0

x=–a x=0 x=a

• •

• •

• •

• •

• •

• •

• •

• •

• •

• •

• •

• •

• •

• •

• •

• •

• •

• •

• •

• •

• •

• •

which also means that D has only an

x-component Proceeding further, we have

where C is the constant of integration

Evaluating the integral graphically, we have the following:

 • D =  gives

D x

x (x)

D x – x  x dx  C

–a 0 a x

 0

(x) dx

– 

x



2 0a

From symmetry considerations, the fields on the two sides of the charge distribution must

be equal in magnitude and opposite in

direction Hence,

C = – 0a

– 0a

D 

–0a a x for x  –a

0x a x for – a  x  a

0a a x for x  a











B • dS = 0 = 0 dvV

S



• B 0

GAUSS’ LAW FOR THE MAGNETIC FIELD

From analogy

Solenoidal property of magnetic field lines Provides test for physical realizability of a given vector field as a magnetic field.

D • dS = V dv

S



• D 









 • B 0

LAW OF CONSERVATION OF CHARGE

J • dS  ddt V dv 0

S



• J  t( ) 0







aaaa

• J  t 0

 ContinuityEquation

(4) is, however, not independent of (1), and (3) can be derived from (2) with the aid of (5).

 E – B

t

 H J  D

t

 • D 

 • B 0

(1)

(2) (3) (4) (5)

 • J  

t 0