1. Applied Electricity and Magnetism
ECE 2317
Applied Electricity and Magnetism
Fall 2014
Prof. David R. Jackson
ECE Dept.
Notes 24

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  = 0
Infinite PEC ground plane
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. Note: There is no ground plane in the image picture!
Image Theory (cont.) x z h q -q
Original charge
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. Here is what the image solution looks like.
Image Theory (cont.) x z h q -q #1 #2 R1 R2
z > 0
Here is what the image solution looks like.

6. Image Theory (cont.) S = Sh+S0 Original problem z Sh q (x,y,z) h S0 x
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 x z h
(x,y,z) q
PEC
S = Sh+S0 Sh S0
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 on S

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

8. Image Theory (cont.) z < 0 #1 q x V S #2 -q 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 > 0x z h q -q #1 #2 R1 R2

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

11. High-Voltage Power Line
High-voltage power line (radius a) h V0 R
Earth (, )
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] 107
100 [Hz] 106
1 [kHz] 105
10 [kHz] 104
100 [kHz] 103
1 [MHz]
10 [MHz]
100 [MHz]
1.0 [GHz]
10.0 [GHz]
Assume the following parameters for earth:

13. High-Voltage Power Line (cont.)
Line-charge approximation: h R
Earth
We need to find the correct value of l0.
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
Line charge formula:
Note:
Here b is the distance to the arbitrary reference “point” for the single line-charge solution. or
The line-charge density l0 can 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.)
Original wire (radius a) h
Set VAB = V0 A B
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 B

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

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

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

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

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

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

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

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