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Say, I have a gas and then I ionize it (provide enough energy for the electrons to escape the nucleus' potential).And there exists, a SLIGHTLY positively charged surface beside the gas. Now, will the electrons just hit the surface with no additional energy requirement, or will the gas act like a plasma and electrons need additional energy to part themselves from the plasma and hit the surface? And how charged is the surface supposed to be to do this.

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Is there something keeping the positive plasma ions far away from the surface? Let's say there is.

Electrons go from the plasma to the surface until the surface and the plasma are at the same potential.

No other energy besides the potential difference between the surface and the plasma was needed for those electrons to go from the plasma to the surface, and that potential difference can be extremely small.

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Scenario 1
The mechanical forces holding the charged surface ruin the plasma approximation (i.e., quasi-neutrality). Since the charged surface cannot physically move, the ions and electrons cannot modify it or the charge distribution on the surface using long-range (i.e., Coulomb potentials) interactions. In the absence of pre-existing magnetic field the Lorentz force reduces to just $\mathbf{F} = q \ \mathbf{E}$. Thus, initially the electrons would move toward the charge sheet and the ions away. If the electrons were numerous enough and slowly impacted the surface (i.e., slow enough to not eject more charged particles), the electrons could eventually reduce the charge on the sheet. I do not think the sheet would ever reach zero net charge though because as the ions move away, they will start to generate a charge imbalance of their own. This will lead to another electric field that will attempt to drag electrons away from the charged surface. The system would eventually reach an equilibrium state.

Scenario 2
If you inserted a sheet of charge that was not constrained by any mechanical forces (e.g., a sheet of protons), then the electrons and pre-existing ions would respond and oscillate/readjust until a quasi-neutral state could be reached. This latter scenario would require that the initial plasma a lot of particles compared to the number inserted in the charge sheet. There may be some recombination but the extent to which this occurs would depend upon the velocity distribution of the electrons and ions (i.e., how much kinetic energy they have relative to the recombination energies).

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