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

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!

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).

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.)

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.

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

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 =

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).

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

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

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.

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:

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.

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).

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.

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

High-Voltage Power Line (cont.) Hence

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

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

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

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).

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).

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

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).