Electric field produced by a capacitor consisting of two parallel plates of different lengths

Question

I need to visualize the electrostatic field produced by a capacitor consisting of two parallel 1D plates of different lengths, as shown in the following figure (sorry for the crude drawing), in which the lower plate is grounded and the upper shorter one is charged at a high voltage. The two plates are assumed to be the mathematic plane with zero thickness, and the length ration of $l/L=1/5$.

Actually, I need to plot the field lines, field strength contour, and field strength distribution on the lower plate. Then the plots will be used to estimate the length of a significant electric-field influence domain on the lower plate. For example, $20$% in strength decay in the horizontal direction, that is, the field-line density decreases by $20$% as compared with the homogeneous central region.

The length is denoted as $l+2\delta$ in the figure, where $\delta$ means the edge-effect length of the electric field on the lower plate. This problem is also related to this one. I'd like to thank @Alex Trounev's answer there.

But I have further questions about Alex Trounev's answer: 1. is it reasonable to use two circular plates to represent two 1D plates?

2. If the upper surface of the small electrode is coated with an insulator, what are the plots?

Update

To plot full streamlines without segmentation, I added StreamScale -> {Full, All, 0.02} in StreamDensityPlot

StreamDensityPlot[Evaluate[ef], {x, y} \[Element] reg, 
 MaxRecursion -> 2, StreamPoints -> 40, ColorFunction -> "Rainbow", 
 PlotLegends -> Automatic, FrameLabel -> {"x", "y"}, 
 StreamStyle -> LightGray, FrameStyle -> LightGray, PlotRange -> All, 
 ImageSize -> 400, StreamScale -> {Full, All, 0.02}, 
 PerformanceGoal -> "Quality"]

We can see many streamlines don't touch the plates exactly and have different distances from the plates, see the following enlarged figure.

This issue can be seen in the mid-subfigure of @Alex's answer. How to get continuous streamlines, among which those lines end at the plate should touch the plates exactly? I have tried to use WorkingPrecision -> 20, which turns out to be useless. Is this related to the mesh? Thank you for any suggestions.

Answer 1

Two discs with an aspect ratio of 1: 2. The lower disk is grounded, the potential is on the upper disk $U=1$. On the left is the distribution of potential, in the center is the distribution of the electric field, on the right is the distribution of the electric field on the grounded plate.

Needs["NDSolve`FEM`"];
par = {H -> 1./4, h -> 1./10, l1 -> 1., l2 -> 2.}; reg1 = 
 RegionUnion[Rectangle[{-l1/2, H/2}, {l1/2, H/2 + h}], 
   Rectangle[{-l2/2, -H/2 - h}, {l2/2, -H/2}]] /. par; reg2 = 
 Rectangle[{0, -3}, {6, 3}];
reg = RegionDifference[reg2, reg1];
mesh = ToElementMesh[reg /. par, 
  MeshRefinementFunction -> 
   Function[{vertices, area}, 
    area > 0.0001 (0.1 + 10 Norm[Mean[vertices]])]]
mesh["Wireframe"]
eq = D[u[r, z], r, r] + D[u[r, z], r]/r + D[u[r, z], z, z];
bc = {DirichletCondition[u[r, z] == 0, 
     r == l2/2 && -H/2 - h <= z <= -H/2 || 
      z == -H/2 && 0 <= r <= l2/2 || z == -H/2 - h && 0 <= r <= l2/2],
     DirichletCondition[u[r, z] == 1., 
     r == l1/2 && H/2 <= z <= H/2 + h || z == H/2 && 0 <= r <= l1/2 ||
       z == H/2 + h && 0 <= r<= l1/2]} /. par;
U = NDSolveValue[{eq == 0, bc}, u, {r, z} \[Element] mesh];

ef = -Grad[U[x, y], {x, y}];

{DensityPlot[U[r, z], {r, z} \[Element] reg, 
  ColorFunction -> "Rainbow", PlotLegends -> Automatic, 
  FrameLabel -> Automatic, PlotRange -> All, PlotPoints -> 50], 
 StreamDensityPlot[Evaluate[ef], {x, y} \[Element] reg, 
  ColorFunction -> "Rainbow", PlotLegends -> Automatic, 
  FrameLabel -> {r, z}, StreamStyle -> LightGray, 
  VectorPoints -> Fine, PlotRange -> {{0, 2}, {-1, 1}}], 
 Plot[Evaluate[Norm[ef] /. y -> -H/2 /. par], {x, 0, 1}, 
  PlotRange -> All, AxesLabel -> {"r", "E"}, AxesOrigin -> {0, 0}]}

Two parallel very long plates with a width ratio of 1:2.The lower plate is grounded, the potential on the upper pate is $U=1$. On the left is the distribution of potential, in the center is the distribution of the electric field, on the right is the distribution of the electric field on the grounded plate. Changing the thickness 2 times from 1/10 to 1/20 has almost no effect on the field

Needs["NDSolve`FEM`"];
par = {H -> 1./4, h -> 1./20, l1 -> 1., l2 -> 2.}; reg1 = 
 RegionUnion[Rectangle[{-l1/2, H/2}, {l1/2, H/2 + h}], 
   Rectangle[{-l2/2, -H/2 - h}, {l2/2, -H/2}]] /. par; reg2 = 
 Rectangle[{0, -3}, {6, 3}];
reg = RegionDifference[reg2, reg1];
mesh = ToElementMesh[reg /. par, 
  MeshRefinementFunction -> 
   Function[{vertices, area}, 
    area > 0.0001 (0.1 + 10 Norm[Mean[vertices]])]]
mesh["Wireframe"]

eq = D[u[x, z], x, x] + D[u[x, z], z, z];
bc = {DirichletCondition[u[x, z] == 0, 
     x == l2/2 && -H/2 - h <= z <= -H/2 || 
      z == -H/2 && 0 <= x <= l2/2 || z == -H/2 - h && 0 <= x <= l2/2],
     DirichletCondition[u[x, z] == 1., 
     x == l1/2 && H/2 <= z <= H/2 + h || z == H/2 && 0 <= x <= l1/2 ||
       z == H/2 + h && 0 <= x <= l1/2]} /. par;
U = NDSolveValue[{eq == 0, bc}, u, {x, z} \[Element] mesh];

ef = -Grad[U[x, y], {x, y}];

{DensityPlot[U[x, y], {x, y} \[Element] reg, 
  ColorFunction -> "Rainbow", PlotLegends -> Automatic, 
  FrameLabel -> Automatic, PlotRange -> All, PlotPoints -> 50], 
 StreamDensityPlot[Evaluate[ef], {x, y} \[Element] reg, 
  ColorFunction -> "Rainbow", PlotLegends -> Automatic, 
  FrameLabel -> {"x", "y"}, StreamStyle -> LightGray, 
  VectorPoints -> Fine, PlotRange -> {{0, 2}, {-1, 1}}], 
 Plot[Evaluate[Norm[ef] /. y -> -H/2 /. par], {x, 0, 1}, 
  PlotRange -> All, AxesLabel -> {"x", "E"}, AxesOrigin -> {0, 0}]}