Prof. D. Wilton ECE Dept. Notes 9 ECE 2317 Applied Electricity and Magnetism Notes prepared by the EM group, University of Houston.

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

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

Current Flux Through Surface J S I

Electric Flux Through Surface q D S

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

Spherical Surface (cont.)

3D Flux Plot for a Point Charge

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

Flux Plot (2D) Rules: 1) Flux lines are in direction of D 2) l0l0 D  L  = small length perpendicular to the flux vector N L = # 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.

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

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

Flux Property N S : flux lines Through S S 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)

Flux Property (Proof)  N S : # flux lines  SS D N S : flux lines Through S S D SS   S S D SS

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

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

Equipotential Surfaces (Contours) D  C V Proof: On C V : C V : (V = constant ) dr CVCV D

Equipotential Surfaces (cont.) CVCV D 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” 2D flux plot

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

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

Example Line charge  l0 D x y

Example Flux plot for two line charges h x y h R1R1 R2R2 r = (x, y) l0l0 -l0-l0

flux lines equipotential lines line charges of opposite sign

line charges of same sign

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

Example + -

Example Software for calculating cross-sectional view of 3D flux plot for two point charges: