Section 1 3 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

1.3 Cylindrical and Spherical

Coordinate Systems

Cylindrical (r, , z) Spherical (r,  ,  )

Only az is uniform All three unit

ar and a are vectors are

a



x

x y

y r

z

z a r

az

90Þ

z

r

y

ar

90Þ

x

a

a





Cylindrical and Spherical Coordinate Systems

x = r cos  x = r sin  cos 

D1.7 (a) (2, 5/6, 3) in cylindrical coordinates



x 2 cos 5 6 – 3

y 2 sin 5 61





z 3

x

z

3

5/6

x 4 cos 4 3 – 2

y 4 sin 4 3 – 2 3





z – 1

(4, 4 3, –1) in cylindrical coordinates

1 4

x

y

4/3

z

x 4 sin 2



y4 sin 2



z 4 cos 















9  3  4 4

(4, 2 3,  6) in spherical coordinates

2/3

x

y

z

/6 4

x  8 sin 



y  8 sin 



z  8 cos 



















1  3  4  8

/4

/3

z

y x

8

Conversion of vectors between coordinate systems

arc

a

az











































ax

ay

az





















ars

a

a















































ax

ay

az





















ax arc

az a ay



az

ars

arc

a

a



P1.18 A = ar at (2, /6, 2)

A sin 



6 az

1

2 ay  32 az 1

x

y

/2



/6

B sin 



6 az

1

2 ax – 32 az

1

C ax

C

x

y

z

/4 3

3/2

x

y z

B

1 3



(b)

3 4

       



0

  



(d) A B • C C • A B

=

0 1

2

3 2 1

– 3

4

1 2

