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Prof. D. Wilton ECE Dept.

ECE 2317 Applied Electricity and Magnetism. Prof. D. Wilton ECE Dept. Notes 9. Notes prepared by the EM group, University of Houston. E. q. Electric Flux Density. Define:. “flux density vector”. Analogy with Current Flux Density. The same current I passes through every

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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 9 Notes prepared by the EM group, University of Houston.

  2. E q Electric Flux Density Define: “flux density vector”

  3. Analogy with Current Flux Density The same current I passes through every sphere concentric with the source, hence J r I current flux density vector due to a point source of current Note: if I is negative, flux density vector points towardsI

  4. Current Flux Through Surface J I S

  5. D q S Electric Flux Through Surface

  6. Example z Find the flux from a point charge going out through a spherical surface. D q y S x (We want the flux going out)

  7. Spherical Surface (cont.)

  8. 3D Flux Plot for a Point Charge

  9. D S Flux Plot (3D) Rules: 1) Flux lines are in direction of D 2) S = small area perpendicular to the flux vector NS = # flux lines through S

  10. D l0 Flux Plot (2D) Rules: 1) Flux lines are in direction of D 2) L = small length perpendicular to the flux vector NL = # flux lines through L Note: We can construct a 3D problem by extending the contour in the z direction by one meter to create a surface.

  11. Example Draw flux plot for a line charge y  Nflines x Hence l0[C/m]

  12. y Choose Nf = 16 x l0[C/m] Example (cont.) Note: If Nf = 16, then each flux line represents l0/ 16[C/m]

  13. S Flux Property • The flux through a surface is proportional to the number of flux lines in the flux plot that cross the surface (3D) or contour (2D). • Flux lines begin on positive charges (or infinity) and end on negative charges (or infinity) NS : flux lines Through S

  14. NS: flux lines Through S D S  D  NS : # flux lines S S D S   S Flux Property (Proof)

  15. D   S Flux Property Proof (cont.) Also, (from the definition of a flux plot) Hence Therefore,

  16. y S x Example l0 = 1 [C/m] Nf = 16 Find z = 1 [m] for surface S

  17. CV D dr Equipotential Surfaces (Contours) D  CV Proof: On CV : CV: (V = constant )

  18. CV D Equipotential Surfaces (cont.) 2D flux plot Assume a constant voltage difference V between adjacent equipotential lines in a 2D flux plot. Theorem: shape of the “curvilinear squares” is preserved throughout the plot. “curvilinear square”

  19. CV D W B L A Equipotential Surfaces (cont.) Proof: Along flux line, E is parallel to dr Hence, Or

  20. CV D W B L A Equipotential Surfaces (cont.) Also, so Hence,

  21. y D x l0 Example Line charge

  22. y r = (x, y) R1 R2 x -l0 l0 h h Example Flux plot for two line charges

  23. line charges of opposite sign flux lines - - - - - - - - - - - equipotential lines

  24. line charges of same sign

  25. - + Example Find the flux through the red surface indicated on the figure (z = 1 m) Counting flux lines:

  26. - + Example

  27. Example Software for calculating cross-sectional view of 3D flux plot for two point charges: http://www.xmission.com/~locutus/astro2-old/ElectricField/ElectricField.html

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