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1. ECE 2317 Applied Electricity and Magnetism Spring 2014 Prof. David R. Jackson ECE Dept. Notes 9
2. Flux Density E q From the Coulomb law: Define: “flux density vector” We then have
3. Flux Through Surface D q S Define flux through a surface: In this picture, flux is the flux crossing the surface in the upward sense. The electric flux through a surface is analogous to the current flowing through a surface. Top view The total current (amps) through a surface is the “flux of the current density.”
4. Flux Through Surface Analogy with electric current A small electrode in a conducting medium spews out current equally in all directions. I S Top view
5. Example z Find the flux from a point charge going out through a spherical surface. D r q y S x (We want the flux going out.)
6. Example (cont.) We will see later that his has to be true from Gauss’s law.
7. Water Analogy z qw y x Nf streamlines Each electric flux line is like a stream of water. Water streams Each stream of water carries qw/ Nf [liters/s]. Water nozzle qw=flow rate [liters/s] Note that the total flow rate through the surface is qw.
8. Water Analogy (cont.) Here is a real “flux fountain” (Wortham fountain, on Allen Parkway).
9. Flux in 2D Problems D C l The flux is now the flux per meter in the z direction. We can also think of the flux through a surface S that is the contour C extruded one meter in the z direction. l S 1 [m] C
10. Flux Plot (2D) 1) Lines are in direction of D 2) y D x l0 Rules: L= small length perpendicular to the flux vector NL = # flux lines through L Rule #2 tells us that a region with a stronger electric field will have flux lines that are closer together. Flux lines
11. Example y  Nflines x Draw flux plot for a line charge Rule 2: Hence l0[C/m]
12. Example (cont.) y  Nflines x This result implies that the number of flux lines coming out of the line charge is fixed, and flux lines are thus never created or destroyed. This is actually a consequence of Gauss’s law. (This is discussed later.) l0[C/m]
13. Example (cont.) y Choose Nf= 16 l0[C/m] x Note: Flux lines come out of positive charges and end on negative charges. Flux line can also terminate at infinity. Note: If Nf= 16, then each flux line represents l0/ 16 [C/m]
14. Flux Property C The flux (per meter) lthrough a contour is proportional to the number of flux lines that cross the contour. NC is the number of flux lines through C. Please see the Appendix for a proof. Note: In 3D, we would have that the total flux through a surface is proportional to the number of flux lines crossing the surface.
15. Example y C x l0 = 1 [C/m] Nf = 16 Graphically evaluate
16. Equipotential Contours y D l0 x The equipotential contour CV is a contour on which the potential is constant. Line charge example Flux lines = -1 [V] Equipotential contours CV = 0 [V] = 1[V]
17. Equipotential Contours (cont.) Property: D  CV CV The flux line are always perpendicular to the equipotential contours. D (proof on next slide) CV: (= constant )
18. Equipotential Contours (cont.) Proof of perpendicular property: Proof: Two nearby points on an equipotential contour are considered. On CV : CV D B r A The r vector is tangent (parallel) to the contour CV.
19. Method of Curvilinear Squares 2D flux plot Assume a constant voltage difference Vbetween adjacent equipotential lines in a 2D flux plot. CV D - + B A Note: Along a flux line, the voltage always decreases as we go in the direction of the flux line. “Curvilinear square” If we integrate along the flux line, E is parallel to dr. Note: It is called a curvilinear “square” even though the shape may be rectangular.
20. Method of Curvilinear Squares (cont.) CV D CV W L Theorem: The shape (aspect ratio) of the “curvilinear squares” is preserved throughout the plot. Assumption: V is constant throughout plot.
21. Method of Curvilinear Squares (cont.) Proof of constant aspect ratio property CV - D + W B L A If we integrate along the flux line, E is parallel to dr. Hence, so Therefore
22. Method of Curvilinear Squares (cont.) CV D W B L A Hence, - + Also, so Hence, (proof complete)
23. Summary of Flux Plot Rules 1) Lines are in direction of D . 2) Equipotential contours are perpendicular to the flux lines. 3) We have a fixed V between equipotential contours. 4) L / Wis kept constant throughout the plot. If all of these rules are followed, then we have the following:
24. Example Line charge y Note how the flux lines get closer as we approach the line charge: there is a stronger electric field there. D W L x l0 The aspect ratio L/W has been chosen to be unity in this plot. This distance between equipotential contours (which defines W) is proportional to the radius  (since the distance between flux lines is).
25. Example Note:L / W 0.5 A parallel-plate capacitor http://www.opencollege.com
26. Example Coaxial cable with a square inner conductor Figure 6-12 in the Hayt and Buck book.
27. Flux Plot with Conductors (cont.) Conductor Some observations: Flux lines are closer together where the field is stronger. The field is strong near a sharp conducting corner. Flux lines begin on positive charges and end on negative charges. Flux lines enter a conductor perpendicular to it. http://en.wikipedia.org/wiki/Electrostatics
28. Example of Electric Flux Plot Note: In this example, the aspect ratio L/W is not held constant. Electroporation-mediated topical delivery of vitamin C for cosmetic applications Lei Zhanga, , Sheldon Lernerb, William V Rustruma, Günter A Hofmanna aGenetronics Inc., 11199 Sorrento Valley Rd., San Diego, CA 92121, USA b Research Institute for Plastic, Cosmetic and Reconstructive Surgery Inc., 3399 First Ave., San Diego, CA 92103, USA.
29. Example of Magnetic Flux Plot Solenoid near a ferrite core (cross sectional view) Flux plots are often used to display the results of a numerical simulation, for either the electric field or the magnetic field. Magnetic flux lines Solenoid windings Ferrite core
30. Appendix: Proof of Flux Property
31. Proof of Flux Property D NC: flux lines Through C L C  L NC: # flux lines C One small piece of the contour (the length is L) C D L so   L or
32. Flux Property Proof (cont.) D  L Also, (from the definition of a flux plot) Hence, substituting into the above equation, we have Therefore, (proof complete)