Possible to create global potential from local distribution?

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Discussion Overview

The discussion revolves around the possibility of creating a global potential from a local distribution of potential values, specifically in the context of a grid-based representation. Participants explore the implications of known potential distributions and their relationship to the Poisson and Laplace equations.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • One participant presents a scenario with specific potential values distributed across a grid and questions whether it is possible to create contour lines that span the entire space based on these local potentials.
  • Another participant references a previous discussion about solving potential distributions using the Poisson equation and suggests that the current problem may relate to that discussion.
  • There is a suggestion that the Poisson equation is applicable when the charge distribution is known, but the participant expresses uncertainty about how this relates to their specific case of known potential distributions.
  • A clarification is made that the discrete version of the Poisson equation can be used if the charge distribution is zero in the regions of interest.
  • One participant proposes that the Laplace equation might be more appropriate since the potential is already known in certain areas, and questions whether averaging neighboring points would suffice to determine the potential elsewhere.
  • Another participant confirms that if the charge density is zero, the Laplace and Poisson equations are effectively the same, and that iterating averages can approximate the potential in other regions.

Areas of Agreement / Disagreement

Participants express varying degrees of understanding regarding the application of the Poisson and Laplace equations, with some agreeing on the use of the Laplace equation under certain conditions, while others remain uncertain about the implications of their specific potential distributions.

Contextual Notes

There are limitations regarding the assumptions about charge distributions and the specific conditions under which the equations apply, which remain unresolved in the discussion.

KFC
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Hi all,
I saw some figures about potential (contour) plot in some articles that has some beautiful gradient lines. The shape is quite weird but you can see clearly there are some strong concentration of potential at some places. For example, if you have 300x300 grids, there shows strong potential of 3V in region of radius 10 grids at somewhere close to bottom-left corner, 5V in region of radius 15 girds at top-right corner, -1V in region of radius 15 grids in middle. With this distribution is it possible to create all contour potential lines based on those 3 strong regions such that the potential line are gradually change to span whole space (grids)?
 
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Thanks. After reading that post, base on my understanding, it is suggested to solve the spatial distribution of potential with Poisson equation. The excel file found there use the average of neighboring points to estimate the value of the current point (that's the idea of solving Poisson equation?) But after reading something about Poisson equation, it is about potential distribution if the charge distribution is known. I wonder if it is the same case since here I know some spatial distribution of potential, I am looking for how the other part of potential in space look like.
 
KFC said:
(that's the idea of solving Poisson equation?)
It is the discrete version of Poisson's equation in cartesian coordinates.
The charge distribution is known in every cell where you want to calculate the potential - it is zero (if it is not, the formulas need an additional term to account for that). You don't know the charge distribution at the electrodes, but there you know the potential.
 
Oh, so you mean we actually use the Laplace equation instead of Poisson equation to solve the problem since we know the potential in space already? So just average each point with its neighbor will do the work, is that correct?
 
If you have zero charge density where you want to calculate the potential (the usual case), both equations are identical.
You can fix the potential at some points, iterating the averages over neighbors will give an approximation for the potential elsewhere.
 
got it. Thanks mfb.
 

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