Prof. D. Wilton ECE Dept. Notes 22 ECE 2317 Applied Electricity and Magnetism Notes prepared by the EM group, University of Houston. (used by Dr. Jackson, spring 2006)

Uniqueness Theorem Given: is unique inside (known charge density) on boundary (known B.C.) Note: We can guess the solution, as long as we verify that Poisson’s equation and the BC’s are correctly satisfied ! on boundary S

Example Guess:  B = V 0 = constant Check: Hollow PEC shell Prove that E = 0 inside a hollow PEC shell (Faraday cage effect) Therefore: the correct solution is V S

Example (cont.) Hollow PEC shell V S Hence E = 0 everywhere inside the hollow cavity.  B = V 0 = constant

Image Theory x z h  (x,y,z) q PEC Note: the electric field is zero below the ground plane ( z < 0).  = 0

Image Theory (cont.) Image picture: x z h q h -q Note: there is no ground plane in the image picture! The original charge and the image charge together give the correct electric field in the region z > 0. They do NOT give the correct solution in the region z < 0. original charge image charge

z > 0: x z h q h -q #1#1 #2#2 R1R1 R2R2 Image Theory (cont.)

x z h  (x,y,z) q PEC To see why image theory works, we construct a closed surface S with a large hemispherical cap (the radius goes to infinity). S = S h +S 0 ShSh S0S0 Image Theory (cont.)

x z h  (x,y,z) q PEC On S h :  = 0 (the surface is at infinity). On S 0 :  = 0 (the surface is on a perfect electric conductor at zero volts). ShSh S0S0 Image Theory (cont.)

z > 0: x q -q #1#1 #2#2 ShSh V S0S0 S = S 0 + S h correct charge in V Image Theory (cont.)

since R 1 = R 2 z > 0: x q -q #1#1 #2#2 ShSh V S0S0 S = S 0 + S h since r =  (correct B. C. on S ) Image Theory (cont.)

x q -q #1#1 #2#2 S V z < 0: Wrong charge inside ! Image Theory (cont.)

Final Solution z > 0: z < 0: x z h q h -q #1#1 #2#2 R1R1 R2R2

High-Voltage Power Line High-voltage power line (radius a ) earth h V0V0 The earth is modeled as a perfect conductor

Line-charge approximation: earth h High-Voltage Power Line (cont.) The power line is modeled as a line charge at the center of the line. This is valid (from Gauss’s law) as long as the charge density on the surface of the wire is uniform.

Image theory: h h Note: the line-charge density can be found by forcing the voltage to be V 0 at the radius of the wire (with respect to the earth). High-Voltage Power Line (cont.)

Image theory: h h Set V AB = V 0 A B The point A is selected as the point on the bottom surface of the power line. Line-charge formula: Note: b is the distance to the arbitrary reference “point” for the single line-charge solution. High-Voltage Power Line (cont.)

Image theory: h h Set V AB = V 0 A B High-Voltage Power Line (cont.) Note: b is chosen so that it serves as reference point for both line potentials

Image theory: Hence: h h High-Voltage Power Line (cont.)

Hence: High-Voltage Power Line (cont.) Image theory: h h x z (x,0,z)

Point charge over a semi-infinite dielectric region: Image Theory for Dielectric Region h q

Solution for air region: Image Theory for Dielectric Region (cont.) h q h q´q´

Solution for dielectric region: h q´´ Image Theory for Dielectric Region (cont.)

Flux Plot Note: the flux lines are straight lines in the dielectric region. q Image Theory for Dielectric Region (cont.)