Prof. David R. Jackson ECE Dept.

1. ECE 2317 Applied Electricity and Magnetism Spring 2014 Prof. David R. Jackson ECE Dept. Notes 23

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

3. Image Theory z (x,y,z) q h x Infinite PEC ground plane  = 0 Note: The electric field is zero below the ground plane (z < 0). This can be justified by the uniqueness theorem, just as we did for the Faraday cage discussion (make a closed surface by adding a large hemisphere in the lower region).

4. Image Theory (cont.) z q Original charge h x h -q Image charge Image picture: 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. (A proof that this is a valid solution is given after the next slide.)

5. Image Theory (cont.) z R1 #1 q h R2 x h -q #2 z > 0 Here is what the image solution looks like.

6. Image Theory (cont.) z Sh S = Sh+S0 q (x,y,z) h S0 x PEC To see why image theory works, we construct a closed surface S that has a large hemispherical cap (the radius goes to infinity). Original problem On S0:  =0 (the surface is on a perfect electric conductor at zero volts). On Sh:  = 0 (the surface is at infinity where E = 0,and  = 0 on the bottom).  =0 onS

7. Image Theory (cont.) S=S0+ Sh Image picture: q #1 Sh V R1 S0 x R2 -q #2 correct charge in V (charge #1) sinceR1 = R2 (correct B.C. on S) sincer =

8. Image Theory (cont.) z < 0 #1 q x V S -q #2 Wrong charge inside! Image theory does not give the correct result in the lower region (the where the image charge is).

9. Final Solution for z > 0 z R1 #1 q h R2 x h -q #2 z > 0:

10. Final Solution (All Regions) z > 0: z < 0:

11. High-Voltage Power Line V0 High-voltage power line (radius a) h Earth R The earth is modeled as a perfect conductor (E = 0). Note: This is an exact model in statics, since there will be no current flow in the earth.

12. High-Voltage Power Line (cont.) The loss tangent of a material shows how good of a conductor it is at any frequency. (This is discussed in ECE 3317.) Table showing tan for earth • 10 [Hz] 2.25 107 100 [Hz] 2.25 106 1 [kHz] 2.25105 10 [kHz] 2.25104 100 [kHz] 2.25103 1 [MHz] 225 10 [MHz] 22.5 100 [MHz] 2.25 1.0 [GHz] 0.225 10.0 [GHz] 0.0225 Assume the following parameters for earth:

13. High-Voltage Power Line (cont.) Line-charge approximation: We need to find the correct value of l0. h Earth R 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 approximately uniform.

14. High-Voltage Power Line (cont.) Image theory: h h Line charge formula: Note: bis the distance to the arbitrary reference “point” for the single line-charge solution. or The line-charge density l0can be found by forcing the voltage to be V0 at the surface of the original wire (with respect to the earth).

15. High-Voltage Power Line (cont.) Set VAB = V0 A h B h Original wire (radius a) The point A is selected as the point on the bottom surface of the power line.

16. High-Voltage Power Line (cont.) Set VAB = V0 A h B h

17. High-Voltage Power Line (cont.) h h Hence

18. High-Voltage Power Line (cont.) y Alternative calculation of l0 A h x Line charge formula: B h Along the vertical line we have: Hence

19. High-Voltage Power Line (cont.) Performing the integration, we have: Hence

20. High-Voltage Power Line (cont.) y R1 (x, y) R2 h x h Final solution z > 0

21. Image Theory for Dielectric Region z Note: We are now interested in bothregions (z > 0 and z < 0). q h x Point charge over a semi-infinite dielectric region (Please see the textbook for a derivation.)

22. Image Theory for Dielectric Region (cont.) Observation point Solution for air region (z > 0): q h h q Note: A very large permittivity acts like a PEC (q' = - q).

23. Image Theory for Dielectric Region (cont.) Observation point Solution for dielectric region (z < 0): q'' h

24. Image Theory for Dielectric Region (cont.) q Sketch of electric field Note: The flux lines are straight lines in the dielectric region. The flux lines are bent away from the normal (as proven earlier from B.C.s).