1. 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)

2. 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

3. 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

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

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

6. 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

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

8. 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.)

9. 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.)

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

11. 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.)

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

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

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

15. 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.

16. 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.)

17. 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.)

18. 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

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

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

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

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

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

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