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[.]

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 1.3 Cylindrical and Spherical Coordinate Systems 1.3-3 Cylindrical and Spherical Coordinate Systems Cylindrical (r, , z) z Spherical (r, , ) z az a 90Þ x x  r y z ar a  r ar y  x Only az is uniform All three unit ar and a are nonuniform vectors are nonuniform 90Þ a y 1.3-4 = r cos  = r sin  =z x y z D1.7 (a) x = r sin  cos  y = r sin  sin  z = r cos  (2, 5/6, 3) in cylindrical coordinates z  x 5/6 y x 2 cos 5 –    2 y 2 sin 5 61  z 3 1.3-5 (b) (4, 4 3, – 1) in cylindrical coordinates z 4/3 y x x 4 cos 4 –    12 4 y 4 sin 4 –  z – 1.3-6 (c) (4, 2 3,  6) in spherical coordinates z 2/3 x y /6 2  x 4 sin cos 3    2  y 4 sin sin     4    z 4 cos –  (d)  1.3-7  8,  4,  in spherical coordinates z /4 x /3 y   x  sin cos 1       y  sin sin  3       z  cos 2   1.3-8 Conversion of vectors between coordinate systems a rc  cos  sin  a x     a  a  –sin  cos       y      a z    a z   a rs  sin  cos  sin  sin     a  cos  cos  cos  sin     cos   a    – sin  a az ay  ax arc a az  ars arc a cos  a x    – sin  a y    a z  1.3-9 P1.18 A = ar at (2, /6, 2) B = a at (1, /3, 0) C = i at (3, /4, 3/2) A z /6  x /2 y   A sin a y  cos a z 6  ay  az 2  1 4 1.3-10 z  y B x  1 4 z C /4 3/2 x   B sin a x – cos a z 6  ax – az 2 y C a x 1.3-11 1  1  3 (a) A  B  a y  az    ax   az  2  2    1  (b) A  C  a y  az   ax 2   0 1.3-12 1  (c) B  C  ax  a  a  z x 2     (d) A B • C C • A B = – 3 –