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It is known that the following integral equation describes the electrostatic field produced by a capacitor consisting of two parallel circular plates, derived in this paper (download for free)

$$f(x)=1+\frac{1}{\pi}\int_{-1}^1 \frac{\kappa}{\kappa^2+(x-y)^2}f(y)dy,$$

in which $\kappa$ is the distance between the plates and when dimensionless variables are taken so that the plates have a unit radius. This is the relevant equation when the potentials of the plates are equal in magnitude but opposite in sign. Its numerical solution can be used to determine the field line and thus the edge effect could be determined.

Now, I need an analog for 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.

enter image description here

Actually, I try to use such an equation to plot the potential lines and then estimate the length of a significant influence of the electric field on the lower plate, that is $l+2\delta$ in the figure. This problem is also related to the previous one. Please see the comments there. So, is there any paper or lecture note about such a configuration? If you know, please share it. Thank you for any suggestions!

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  • $\begingroup$ You drew thick plates, but did not indicate the thickness. It is not clear what is $\delta $? $\endgroup$ – Alex Trounev Oct 26 '19 at 11:12
  • $\begingroup$ @AlexTrounev can we assume the plate to be a mathematic plane with zero thickness? The $\delta$ means the edge-effect length of the electric field on the lower plate, for example, the density of the field lines decreases by 10% as compared with the homogeneous region. Sorry for the confusion. $\endgroup$ – Enter Oct 26 '19 at 15:19
  • $\begingroup$ That is similar to a well studied structure in engineering electromagnetics called microstrip transmission line. The dielectric in a typical microstrip is not homogeneous and so your problem is simpler. My suggestion is to search first the literature for the solution of the microstrip line and then adapt to your problem the solutions found. $\endgroup$ – Massimo Ortolano Oct 26 '19 at 18:51
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Perhaps this will be useful: the electric field of a thin cylinder (coin) above a grounded plane. Magnitude (left), field lines (center), field on a grounded surface (right) Figure 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. Figure 2

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

Figure 3

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  • $\begingroup$ very useful. For coin+plane setting: 1. is the grounded plate placed at $y=0$? 2. Besides the separation (0.5 for upper and 0.25 for lower), in the 1st example, are there any other differences between upper and lower cases? 3. What is the aspect ratio between coin and plane? $\endgroup$ – Enter Oct 27 '19 at 1:19
  • $\begingroup$ May I know what equation has been used to plot the 2nd example? Could you please share the code that produces the plots? It seems like you used Mathematica. Thank you for your help! @Alex Trounev $\endgroup$ – Enter Oct 27 '19 at 7:54
  • $\begingroup$ @Enter 1. Yes. 2. No, there is the same potential on the coin $U = 1$. 3. Theoretically 1 to infinity, but in the numerical model it is 1: 3. 4. This is a solution of the Laplace equation in cylindrical coordinates using FEM and Mathematica 12. Start the topic at mathematica.stackexchange.com give me a link and I'll post the code there. $\endgroup$ – Alex Trounev Oct 27 '19 at 10:20
  • $\begingroup$ Thank you for the reply. I have started a question at MMA SE. $\endgroup$ – Enter Oct 27 '19 at 16:01
  • $\begingroup$ @Enter These are not round plates but rectangular plates of limited width but unlimited length? $\endgroup$ – Alex Trounev Oct 27 '19 at 18:02

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