Electric field betwen laterally displaced parallel plates

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Discussion Overview

The discussion revolves around the calculation of the nonuniform electric field between two laterally displaced parallel plates, specifically focusing on the lateral electric force acting on the displaced plate. The problem is framed within the context of electrostatics and involves considerations of fringe fields and the geometry of the plates.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant expresses difficulty in finding literature on the specific case of laterally displaced parallel plates and questions whether the asymmetry complicates finding an analytical solution.
  • Another participant suggests that an exact analytic solution is unlikely due to the finite width and edges of the plates, proposing that numerical simulations might provide useful approximations.
  • A participant wonders if the problem could be simplified to a two-dimensional case by assuming the plates extend infinitely in the direction perpendicular to the drawing plane.
  • One participant confirms they have used the two-dimensional assumption and believes it is adequate for most setups.
  • However, a later reply argues that even in the two-dimensional case, the lack of suitable boundary conditions or symmetry makes the problem analytically unsolvable, questioning the applicability of conformal mapping techniques.

Areas of Agreement / Disagreement

Participants express differing views on the feasibility of finding an analytical solution, with some suggesting numerical methods while others highlight the challenges posed by the problem's geometry. No consensus is reached regarding the potential for an analytical approach.

Contextual Notes

Limitations include the dependence on the assumptions made about the geometry of the plates and the implications of those assumptions on the solvability of the problem. The discussion highlights the complexity introduced by the asymmetry and finite dimensions of the plates.

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I am struggling with a problem which involves the calculation of the nonuniform electric field fields between two identical parallel plates in which one of the plates is slightly displaced in lateral direction with respect to the other plate (i.e. they do not overlap perfectly). The gap between the overlapping parts of the plates is much smaller than the plate dimensions.

In particular, I am interested in calculating the lateral electric force acting on the displaced plate. I presume that this force is mainly generated from the change in the fringe fields between the edges of the plates.

I have searched the literature for papers which cover this special case of a parallel plate capacitor, however, I have found nothing so far. Why is that so? Is it because the asymmetry of the plate arrangement renders an analytical solution impossible or very cumbersome? Can anyone help me by directing me to published work or by providing me with the general (analytical) approach in attacking this problem?

Thanks a bunch!
 
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I would be surprised if there is an exact analytic solution. You have to take into account that the plates have a finite width, edges and so on. I think numerical simulations can give you some formula to approximate the forces.
 
I was wondering if the problem becomes analytically solvable if it becomes two-dimensional by assuming that both plates extend into infinity in the direction perpendicular to the drawing plane.
 
I already used this assumption to look at the system ;). It should be fine with most interesting setups.
 
mfb said:
I already used this assumption to look at the system ;). It should be fine with most interesting setups.


thanks for your thoughts. Even in the two-dimensional case, the fact that the problem space cannot be confined to a region of which the boundary conditions are given either by specification or by symmetry considerations, makes this problem analytically impossible to solve. I guess, that's why the use of conformal mapping may stand no chance in solving this problem, because it requires symmetry in the plate configuration.
 

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