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ECE 3318 Applied Electricity and Magnetism Spring 2016 Prof. David R. Jackson ECE Dept. Notes 9 1 Flux Density From Coulomb’s law: E q E 4 0 r 2 rˆ 0 8.854187818 1012 F/m Define: q D 0 E We then have D q 4 r 2 “flux density vector” rˆ [C/m 2 ] 2 Flux Through Surface D Define flux through a surface: n̂ D nˆ dS C S q S 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. Cross-sectional view I J nˆ dS A S The total current (amps) through a surface is the “flux of the current density.” 3 Current Analogy Conducting medium Analogy with electric current n̂ I J nˆ dS A S I S J A small electrode in a conducting medium spews out current equally in all directions. J I 4 r 2 rˆ Cross-sectional view 4 Current Analogy (cont.) D I q 4 r 2 rˆ D nˆ dS S Conducting medium Free-space q D J [C/m 2 ] C J I 4 r 2 rˆ I J nˆ dS [A/m 2 ] A S 5 Water Analogy z N streamlines Each electric flux line is like a stream of water. Water streams Each stream of water carries F / N [liters/s]. F y x Water nozzle F = flow rate of water [liters/s] Note that the total flow rate through the surface is F. 6 Water Analogy (cont.) Here is a real “flux fountain” (Wortham fountain, on Allen Parkway). 7 Example z Find the flux from a point charge going out through a spherical surface. D r D nˆ dS S q y S x D rˆ dS S S nˆ rˆ (We want the flux going out.) S q rˆ rˆ dS 2 4 r q dS 2 4 r 8 Example (cont.) 2 0 q 2 0 4 r 2 r sin d d q 4 2 sin d d 0 0 q 2 sin d 4 0 q 2 2 4 q [ C] We will see later that his has to be true for any surface that surrounds the charge, from Gauss’s law. 9 D Flux in 2D Problems The flux is now the flux per meter in the z direction. n̂ l D nˆ dl C/m C l C We can also think of the flux through a surface S that is the contour C extruded h in the z direction. l D nˆ dl S C S h [m] C l h m 10 Flux Plot (2D) Rules: 1) Lines are in direction of D 2) D # flux lines length NL D L L = small length perpendicular to the flux vector NL = # flux lines through L y D L x s Flux lines l0 Rule #2 tells us that a region with a stronger electric field will have flux lines that are closer together. L s spacing 1/ D NL 11 Example Draw flux plot for a line charge l 0 E ˆ 2 0 V/m D ˆ l 0 2 C/m 2 y Rule 2: Nf lines D C1 L x D # lines L # lines # lines C1 L C1 # lines Hence l0 [C/m] l 0 C1 # lines l 0 # lines constant 2 2 C1 12 Example (cont.) # lines constant 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. y Nf lines This is actually a consequence of Gauss’s law. L (This is discussed later.) x l0 [C/m] 13 Example (cont.) y Choose Nf = 16 Note: Flux lines come out of positive charges and end on negative charges. l0 [C/m] Flux lines can also terminate at infinity. x # lines constant Note: If Nf = 16, then each flux line represents l0 / 16 [C/m] 14 Flux Property The flux (per meter) l through a contour is proportional to the number of flux lines that cross the contour. NC is the number of flux lines through C. C l D nˆ dl NC 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 l0 = 1 [C/m] y Choose: Nf = 16 Graphically evaluate l D nˆ dl C C x 1 C/m /line 16 l 4 lines Note: The answer will be more accurate if we use a plot with more flux lines! 1 l 4 C/m 16 Equipotential Contours The equipotential contour CV is a contour on which the potential is constant. y Line charge example D Flux lines l0 x = -1 [V] Equipotential contours CV = 1 [V] = 0 [V] 17 Equipotential Contours (cont.) Property: D CV CV D The flux line are always perpendicular to the equipotential contours. (proof on next slide) CV: ( = constant ) 18 Equipotential Contours (cont.) Proof of perpendicular property: Proof: Two nearby points on an equipotential contour are considered. B VAB E dr On CV : 0 A B E dr CV E r 0 A D E r B D r 0 r A D r The r vector is tangent (parallel) to the contour CV. 19 Method of Curvilinear Squares 2D Flux Plot V + CV - B D Assume a constant voltage difference V between adjacent equipotential lines in a 2D flux plot. A Note: Along a flux line, the voltage always decreases as we go in the direction of the flux line. B B B A A A V VAB E d r E d r E dl 0 “Curvilinear square” (If we integrate along the flux line, E is in the same direction as to dr .) Note: It is called a curvilinear “square” even though the shape may be rectangular. 20 Method of Curvilinear Squares (cont.) Theorem: The shape (aspect ratio) of the “curvilinear squares” is preserved throughout the plot. Assumption: V is constant throughout plot. CV V D CV W L V L constant W 21 Method of Curvilinear Squares (cont.) Proof of constant aspect ratio property V - CV B VAB E d r V D + A W A L B If we integrate along the flux line, E is in the same direction as dr . B Hence, E d r V A B so E dl V Therefore E L V A 22 Method of Curvilinear Squares (cont.) V - CV Hence, D + L V L E Also, W L NL NL 1 D C1 C1 L L W B A so C1 W D Hence, L V D W E C1 V D V 0 constant C1 C1 E (proof complete) 23 Summary of Flux Plot Rules Rule 1: Lines are in direction of D. Rule: 2 Equipotential contours are perpendicular to the flux lines. Rule 3: L / W is kept constant throughout the plot. If all of these rules are followed, then we have the following properties: Property 1: D # flux lines length Property 2: We have a fixed V between equipotential contours. 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 x L W 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 A parallel-plate capacitor Note: L / W 0.5 http://www.opencollege.com 26 Example Coaxial cable with a square inner conductor L 1 W 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 a Genetronics 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 Ferrite core Solenoid windings 30 Appendix: Proof of Flux Property 31 Proof of Flux Property NC : flux lines Through C C C L D n̂ L One small piece of the contour (the length is L) NC : # flux lines C l D nˆ L D cos L so l D or D L cos l D L L L 32 Flux Property Proof (cont.) l D L D Also, N C D L L (from the definition of a flux plot) Hence, substituting into the above equation, we have N C l D L L N C L Therefore, l NC (proof complete) 33