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Prof. D. Wilton ECE Dept.

ECE 2317 Applied Electricity and Magnetism. Prof. D. Wilton ECE Dept. Notes 6. Notes prepared by the EM group, University of Houston. z. P ( x,y,z ). r. y. x. Review of Coordinate Systems. An understanding of coordinate systems is important for doing EM calculations. z.

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Prof. D. Wilton ECE Dept.

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  1. ECE 2317 Applied Electricity and Magnetism Prof. D. Wilton ECE Dept. Notes 6 Notes prepared by the EM group, University of Houston.

  2. z P (x,y,z) r y x Review of Coordinate Systems An understanding of coordinate systems is important for doing EM calculations.

  3. z P (x,y,z) r y x Rectangular Short hand: Some books: dy dS = dxdy dx dz dS = dxdz dS = dydz

  4. z B C dr r A r+dr y x Rectangular (cont.) Path Integral (need dr) Note on notation: dlis often used instead of dr

  5. Kinds of Integrals That Often Occur

  6. y z    x P (, , z) z . y  x Cylindrical .

  7. z . y y x Note: and depend on (x, y)  x Cylindrical (cont.) Unit Vectors Point in the direction of increasing coordinate

  8. y  x Cylindrical (cont.) Expressions for unit vectors Assume Then we have: Similarly, Hence

  9. Cylindrical (cont.) Summary of Results

  10. dS =  d d z  dz dS =  d dz  d d dS = d dz y x Cylindrical (cont.) Differentials Note: dS may be in three different forms

  11. d y y y  d z d dz   x x y x Cylindrical (cont.) Path Integrals First consider differential changes along any of the three coordinate directions:

  12. C y dr x Cylindrical (cont.) In general:

  13. z z  P (r, , ) P (r, , )   . . z r r y y   x x Note: 0 <  <  Note:  = r sin  Spherical

  14. z  P (r, , )  . z r y  Note:  = r sin  Spherical (cont.)

  15. z . y x Note: ,and depend on (x, y, z) Spherical (cont.) Unit Vectors Point in the direction of increasing coordinate

  16. z . y x Spherical (cont.) Transformation of Unit Vectors

  17. Spherical (cont.) Differentials z r sin d r d  d dS = r2 sin d d dr d y Note: dS may be in three different forms (only one is shown). The other two are: x dS = rdr d dS = rsin dr d

  18. dr z z d dr r y y x x Spherical (cont.) Path Integrals z dr r d y x

  19. Path Integral Parameterization C Typically, t is time, arc length, or angle

  20. Example P1 (4, 60, 1) [m] P2 (3, 180, -1) [m] (cylindrical coordinates) Given: Find d = distance between points d = 6.403 [m]

  21. z  y  x Example Derive Let

  22. z ( / 2) -   y L x  x Example (cont.) Hence

  23. Example Derive Let

  24. z z    y y x x Example (cont.) z   y x Result:

  25. Example Given: v = -3x10-8 (cos2 / r4) [C/m3] , 2 < r < 5 [m] Find Q z Solution: a y b x a = 2 [m], b = 5 [m]

  26. Example (cont.) Q = -5.655x10-8 [C]

  27. Find Q Example

  28. z E (x,y,z) . (0,1,0) y . B C (1,0,0) A x Example Find VAB y 1 x 1 VAB = -7/6[V]

  29. y B Find VAB x A 3 [m] C Example Scalar integrand; most conveniently performed in cylindrical coords.

  30. Example (cont.) VAB = 9/2[V]

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