Section 2 2 TRƯỜNG ĐIỆN TỪ

Slide 1 Slide Presentations for ECE 329, Introduction to Electromagnetic Fields, to supplement “Elements of Engineering Electromagnetics, Sixth Edition” by Nannapaneni Narayana Rao Edward C Jordan Pro[.]

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

Slide Presentations for ECE 329,

Introduction to Electromagnetic Fields,

to supplement “Elements of Engineering Electromagnetics, Sixth

Edition”

2.2 The Surface Integral

B

an

S

B an

S

The Surface Integral Flux of a vector crossing a surface:

Flux = (B)(S)

Flux = 0

Flux (B cos  ) S

B S cos 

B • S a n

B • S

B

an

S



Bj

anj

j

S

S j

Flux   j

j1

n



 B j • S j

j1

n



In the limit n   ,

Flux,  = B • dSS

= Surface integral of B over S.

z

y x

2

2

B • dS

S

 = Surface integral of B

over the closed surface S.

D2.4

(a) x 0, a n ax

A 0, A • dS 0

A • dS 0



A • dS

 x y 

x

 

2, 2

x y

x

x



z

y x

2

2

2

A • dS 2 dy dz

A • dS y02 z02 2 dy dz 8



A • dS

(b)

z

x

y

2

2

A • dS x02 z02 x dx dz 4



A • dS

(c)

x y

y

y

x





A  S

(d) From (c),

A • dS x dx dz

A • dS x02 z0 2–x x dx dz



 x(2 – x) dx02

4

3

A • dS

3

z

x

y

2

2

x + z = 2