Section 2 5 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


2.5
Gauss’ Laws


2.5-3

Gauss’ Law for the Electric Field

 C

3
S D  dS V  dv  m3 m , or C 
D

•
V

S

dS

Displacement flux emanating from a closed surface S =
charge contained in the volume bounded by S = charge
enclosed by S.


2.5-4

Gauss’ Law for the Magnetic Field
S B • dS = 0
B

S

Magnetic flux emanating from a closed
surface S = 0.

dS


2.5-5

P2.21 Finding displacement emanating from a surface
enclosing charge



(a)   x, y, z  0 3  x  y  z
2

2

2



Surface of cube bounded by

x 1, y 1, and z 1

 D  d S   dv
  
S

V

1

1

1

x  1 y  1 z  1
1

8 0 

0 3  x2  y2  z 2  dx dy dz

  3  x
1

1

x 0 y 0 z 0

1 1 1

8 0  3    
3 3 3

16 0

2

 y2  z 2  dx dy dz


2.5-6

(b)   x, y, z  0  x y z 
Surface of the volume x > 0, y > 0, z > 0, and (x2 + y2 + z2) < 1.

 D  d S   dv
S

V

1



1 x2

1 x2  y2

 

0 xyz dx dy dz

1 x2

xy 1  x2  y2  dx dy

x 0 y 0

z 0


 0
2

 

0

2

 xy
x y
xy 
x0  2  2  4  dx
y 0


 0
4

1

3
3
5
3
5 
x

x

x

x

x

2
x


x
dx


x0 

2


0

4

0  x2 x4 x6 
 x 3 1 5
x0  2  x  2 x  dx  4  4  4  12 
0



0
48

1

x 0 y 0
1

2

3

2

4

1 x2

1

1

1


2.5-7

P2.23

z
dS4

1

dS1

dS2

x

y


dS3

S B • dS
  B • dS +  B • dS2
S1

S2

 S B • dS3  S B • dS 4
3
4
0


2.5-8

  B  d S1
S1

  B  d S2 
S2
1

 





z 0 x 0

B0

 B  dS   B  dS
 ya  xa    dx dz a 
3

S3
x

y

y 0

 0 0
1

 





z 0 x 0

B0 2


2

B0 x dx dz

2

B0 
Absolute value =
Wb
2

S4

4

y