Prof. David R. Jackson ECE Dept. Fall 2014 Notes 9 ECE 2317 Applied Electricity and Magnetism 1

Flux Density q E Define: “flux density vector” 2 From Coulomb’s law: We then have

Flux Through Surface q D S Define flux through a surface: 3 The electric flux through a surface is analogous to the current flowing through a surface. The total current (amps) through a surface is the “flux of the current density.” In this picture, flux is the flux crossing the surface in the upward sense. Cross-sectional view

Flux Through Surface (cont.) 4 Analogy with electric current A small electrode in a conducting medium spews out current equally in all directions. I S Cross-sectional view

Flux Through Surface (cont.) 5 Analogy

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

Example (cont.) 7 We will see later that his has to be true from Gauss’s law.

Water Analogy x y z w 8 Each electric flux line is like a stream of water. w = flow rate of water [ liters/s ] Each stream of water carries w / N f [ liters/s ]. Water nozzle N f streamlines Water streams Note that the total flow rate through the surface is w.

Water Analogy (cont.) 9 Here is a real “flux fountain” (Wortham fountain, on Allen Parkway).

Flux in 2D Problems ll D C The flux is now the flux per meter in the z direction. 10 We can also think of the flux through a surface S that is the contour C extruded h in the z direction. h [m] S ll C

Flux Plot (2D) 1) Lines are in direction of D 2)  L  = small length perpendicular to the flux vector N L = # flux lines through  L  l0l0 D x y Rules: 11 Rule #2 tells us that a region with a stronger electric field will have flux lines that are closer together. Flux lines

Example Draw flux plot for a line charge Hence  l0 [C/m] N f lines x y  12 Rule 2:

 l0 [C/m] 13 Example (cont.) 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.) N f lines x y 

Example (cont.) Note: If N f = 16, then each flux line represents  l0 / 16 [C/m] Choose N f = 16  l0 [C/m] x y Note: Flux lines come out of positive charges and end on negative charges. Flux line can also terminate at infinity. 14

Flux Property N C is the number of flux lines through C. C The flux (per meter)  l through a contour is proportional to the number of flux lines that cross the contour. 15 Note: In 3D, we would have that the total flux through a surface is proportional to the number of flux lines crossing the surface. Please see the Appendix for a proof.

Example N f = 16  l0 = 1 [C/m] x y C Graphically evaluate 16 Note: The answer will be more accurate if we use a plot with more flux lines!

Line charge example l0l0 D x y 17 Equipotential Contours Equipotential contours C V  = 1 [V]  = -1 [V]  = 0 [V] Flux lines The equipotential contour C V is a contour on which the potential is constant.

Equipotential Contours (cont.) D  C V 18 Property: The flux line are always perpendicular to the equipotential contours. C V : (  = constant ) CVCV D (proof on next slide)

Equipotential Contours (cont.) Proof: On C V : 19 The  r vector is tangent (parallel) to the contour C V. CVCV D A B rr Two nearby points on an equipotential contour are considered. Proof of perpendicular property:

Method of Curvilinear Squares Assume a constant voltage difference  V between adjacent equipotential lines in a 2D flux plot. 2D flux plot Note: Along a flux line, the voltage always decreases as we go in the direction of the flux line. 20 Note: It is called a curvilinear “square” even though the shape may be rectangular. (If we integrate along the flux line, E is in the same direction as to dr.) “Curvilinear square” CVCV D A B + -

Method of Curvilinear Squares (cont.) Theorem: The shape (aspect ratio) of the “curvilinear squares” is preserved throughout the plot. CVCV D CVCV W L 21 Assumption:  V is constant throughout plot.

Proof of constant aspect ratio property If we integrate along the flux line, E is in the same direction as dr. so Therefore Method of Curvilinear Squares (cont.) Hence, 22 CVCV D W L A B + -

Also, Hence, so Method of Curvilinear Squares (cont.) 23 Hence, (proof complete) CVCV D W L A B + -

Summary of Flux Plot Rules 24 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 / W is kept constant throughout the plot. If all of these rules are followed, then we have the following:

Example Line charge 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 l0l0 D x y L W Note how the flux lines get closer as we approach the line charge: there is a stronger electric field there.

Example 26 A parallel-plate capacitor Note: L / W  0.5

Example Figure 6-12 in the Hayt and Buck book. Coaxial cable with a square inner conductor 27

28  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. Some observations: Flux Plot with Conductors (cont.) Conductor

Example of Electric Flux Plot 29 Electroporation-mediated topical delivery of vitamin C for cosmetic applications Lei Zhang a,, Sheldon Lerner b, William V Rustrum a, Günter A Hofmann a a Genetronics Inc., 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. Note: In this example, the aspect ratio L/W is not held constant.

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. 30 Magnetic flux lines Ferrite core Solenoid windings

Appendix: Proof of Flux Property 31

Proof of Flux Property N C : flux lines Through C so or 32  N C : # flux lines  CC D LL One small piece of the contour (the length is  L ) C LL CC  LL D LL 

Flux Property Proof (cont.) Also, (from the definition of a flux plot) Hence, substituting into the above equation, we have Therefore, 33  LL D (proof complete)