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Inspired by this question, I ask another. Theoretically, Coulomb potential in 2D is proportional to the logarithm of distance; In experiments, though electrons are constrained in a pseudo-2D environment, they are actually existing in 3-dimentional space. Why is the Coulomb potential in pseudo-2D experiments still proportional to the logarithm of distance?

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3 Answers3

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Hand-wavingly, I would say that boundary conditions are important here. If you confine charges with anything, I am not sure your charges will interact exactly in 2D. However, if the field lines cannot escape in the direction perpendicular to the trapping plates, you "force" them to exist in 2d so to speak. Most likely the slab is neutral in width but charge inhomogeneities are strong laterally which yields an effective 2d potential.

EDIT: Discussing about this problem with a colleague of mine, I realized I was wrong in the type of situation where this happens. You can prevent the field lines to leave the slab by having a strong dielectric mismatch between the bulk where the charge is and the plates confining the charge, so that $\epsilon_{bulk}/\epsilon_{plate} \rightarrow \infty$. In that case, you force the field lines to be parallel to the plates at the interfaces and therefore they do not "leave" the system and you can concentrate them more and more in the radial direction. In the case of metallic plates, this is actually the contrary that happens where the field lines are "sucked in" by the plates and that is why my previous attempts at explaining it failed.

Now, if we have interfaces where the normal field at the plate has to be parallel to it, it is equivalent to asking the interfaces to be a plane of symmetry of an equivalent system.

A system where this would be true is a chain of charged beads with periodicity $\delta z$ where $\delta z$ is the size of the slab confining the charge.

As we all know, in the limit $\delta z \rightarrow 0$, the field created by such a system behaves logarithmically with the radial distance $r$.

Now, a real system where this can happen is ions in solution trapped between two lipid membranes for instance.

I would not expect this quasi-2d field to work in low dielectric media for instance where you would trap the particles with a force field (e.g. with optical tweezers).

I will delete all nonsensical comments on metallic plates that can't do anything but confuse newcomers to this thread.

gatsu
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  • That was normal not to succeed in showing that two metallic plates would do the job as they actually do not. I have changed my answer accordingly. – gatsu May 15 '13 at 11:20
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You can simply measure the potential of a long charged rod, it acts as an effective two dimensional system.

Roughly, you have a 2d effective system if the lenght scale associated with the third dimension is either very big or very short compared to the others in play.

Ikiperu
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    You can take for example two very large conductor plates connected to ground, place a charge between them, if you reduce their distance you will see that the potential at the center has the logarithm behaviour, you can also try to solve analitically the problem if you have time, in this case it's possible. – Ikiperu May 13 '13 at 14:52
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    I mean solving the real case, with large but not infinite plate and small but not zero distance. – Ikiperu May 13 '13 at 16:41
  • yes, it is true in this limit, as it was shown by gatsu – Ikiperu May 13 '13 at 18:03
  • the constant is inessential, it's a good exercise to solve it via image charge method if you have time and will – Ikiperu May 13 '13 at 18:15
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Two scenarios let this happen in 3D space:

  1. The field is confined to die out in one direction with e.g. conducting slabs. The field can then be separated into xy and z components, i.e. f(x,y)g(z) for the right bump function g. If you're dealing with appropriately confined electrons, the z-component of the potential doesn't have any effect on dynamics and the system can be reduced to a two dimensional problem.

  2. The sources of the field have an axial symmetry, e.g. a long straight charged wire, or charged cylinder. By symmetry, there is no component of the field parallel to the axis, and for purely electrostatic interactions motion in that direction is given by conservation of momentum. Motion normal to the axis is again a two dimensional problem.

In both cases, the field has to fall off like 1/r from Gauss' law, so the potential is $V \propto \int \frac{1}{r}dr = log(r)$.

Chay Paterson
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