Applied Electricity and Magnetism ECE 3318 Applied Electricity and Magnetism Spring 2017 Prof. David R. Jackson ECE Dept. Notes 24
Uniqueness Theorem Inside region (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 Infinite PEC ground plane Point charge 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).
Image Theory (cont.) Image picture: 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.) #1 #2 z > 0 Here is what the image solution looks like.
Image Theory (cont.) Original problem PEC To see why image theory works, we construct a closed surface S that has a large hemispherical cap (the radius goes to infinity). PEC 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).
Image Theory (cont.) Image picture: #1 #2 correct charge in V (charge #1) since R1 = R2 (correct B.C. on S) since R1 = R2 =
Image Theory (cont.) z < 0 #1 #2 Wrong charge inside! Image theory does not give the correct result in the lower region (the region where the image charge is). (The original problem does not have a charge here.)
Final Solution for z > 0 #1 #2
Final Solution (All Regions) z > 0: z < 0:
High-Voltage Power Line High-voltage power line (radius a) Earth (, ) Infinite in z 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] 2.25105 10 [kHz] 2.25104 100 [kHz] 2.25103 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: Earth We need to find the correct value of l0. The power line is modeled as an effective 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: Line charge formula: Note: In the formula 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) The point A is selected as the point on the bottom surface of the power line.
High-Voltage Power Line (cont.) Original wire (radius a)
High-Voltage Power Line (cont.) Hence
High-Voltage Power Line (cont.) 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.) Final solution (x, y) 20
High-Voltage Power Line (cont.) Line charge formula: Electric Field 21
High-Voltage Power Line (cont.) Electric Field on Earth 22
High-Voltage Power Line (cont.) Hence, we have on the surface on the Earth: (x, 0) 23
High-Voltage Power Line (cont.) Final result: Earth 24
High-Voltage Power Line (cont.) Example Find the electric field at the surface of the Earth. Find the electric field at the bottom surface of the line. Parameters: In MKS units: V0 = 230 [kV] V0 = 2.30105 [V] h = 50 [m] h = 50 [m] a = 1 [cm] a = 0.01 [m] 25
High-Voltage Power Line (cont.) Part (a) At the surface of the Earth we have This gives us : 26
High-Voltage Power Line (cont.) Part (b) Original wire (radius a) Along the vertical line we have: Hence, at the bottom of the line (y = h-a) we have: 27
High-Voltage Power Line (cont.) Original wire (radius a) so that Recall: This gives us: Recall: Ec = 3.0106 [V/m] 28
Image Theory for Dielectric Region Point charge over a semi-infinite dielectric region (Please see the textbook for a derivation.) Note: We are now interested in both regions (z > 0 and z < 0). Note: A line charge could also be done the same way.
Image Theory for Dielectric Region (cont.) Solution for air region (z > 0): 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): Observation point
Image Theory for Dielectric Region (cont.) 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).