Your question sounds to be about something I did some work on - inverted fireballs FB, so I will try to answer that.

First of all I need to know if you are asking about the plasma potential inside or outside the electrode.

As far as I know there is not much difference that depends of the order of switching on the plasma and bias the electrode as soon as the inverted FB is formed (however, I have not measured the potential development in the plasma-switching phase (i.e. the moment when you turn on the bias). - That would be actually something interesting to do...

There might be some short living difference as in the case of background plasma first then bias, the background is often quite homogenious (except for the edge regions), while in the second case (so, bias first and no background plasma) you will create also plasma outside the grid because you have to have a bias above the ionisation potential of your working gas and thus will have some impact ionisation processes (dependent of the mean free path length, of course).

As soon as the inverted FB is established the potential inside the grid will be flat, except for the edges at the grid where it forms small (positive) humps, which trap the low energetic ions.

Outside you will usually observe a monotonically rising potential whith a rise in magnitude. This is due to the partial extraction of slow electrons from the bulk und due to the injection of high energetic ions that can leave the grid.

Thank you, Dr. Johannes Gruenwald. I have roughly read your article about the creation of inverted FB ( I failed to download it) and it is interesting. Did the potential difference between the electrode and the plasma inside it vary with some parameter?

The situation in my question is a little different from your inverted FB experiment. The plasma is produced inside a metal box made of metal plates, which is somewhat gas-tight except for a slot used to pass through electron beam. The accelerating voltage power-supply of electron beam is based on the metal box, insulated from earth.  What confuses me is about the plasma potential inside the metal box. If the metal box was biased on different voltages, like 10V, 100V, 1000V, 10 kV, would the plasma potential remain unchanged, or change a little?

Thx for your interest :)

you mean the plasma potential within the box? - this should vary with the bias as the current continuity requires that the product of electron density and velocity has to be constant and the density is coupled via Poisson's equation with the potential (at least with the curvature of it). Roughly spoken the density should go down when the velocity increases (if you assume that all electrons created are absorbed by your box).

Do you have any means to measure Upl in your box directly?

Well, I have not measured the Upl yet. Direct measuring needs to conquer difficulty of the highly biased voltage, usually in the magnitude of 10 kV.

This problem remains me of another situation. A biased metal ball immersed in a plasma will be surrounded by plasma sheath.Outside the sheath, the plasma potential almost remains unchanged. All the potential difference is contained in the plasma sheath. However, in my question, the plasma is surrounded by the metal box. The sheath would be produced. Why not the potential difference between the metal box and the plasma inside it also be contained in the sheath? I haven't figured it out yet.

I am not sure, that there will be a sheath on the inside as the metal box forms a Faraday cage and is field free on the inside.

@Tsanko Vaskov Tsankov 

Thank you for your enlightening answer. I will search for more detailed knowledge in the articles you have recommended. 

As you say,  the potential in the plasma interior relative to ground is raised by the same amount when raising the wall potential, but it does not like this in other situations. Usually, when a double Langmuir probes immerse in a plasma, the negatively biased one has a negative potential relative to the plasma. But the plasma potential outside the sheath surrounding the probe is considered unchanged, even at different probe potentials.

Does that mean an essential difference between them?

To add a little more complexity to the system, let me introduce also the role of secondary emission into the charge balance...Usually, the sheath potential, roughly 3kT for H plasmas, is quite low in gas discharges, certainly well below the external potentials you are using. This sheath potential is very much dependent on the Secondary Electron Emission (SEE)  of the surface,which will quickly increase as the energy of the impinging electrons  is increased by the external biasing. At SEE=1, no sheath is formed, and the flux of electrons to the wall dominates. As you are injecting electrons into the plasma through the SEE process, the actual potential near the surface is decreased w.r.t. its nominal, external value.I don´t think there is any simple expression to date which gives you an exact solution, unfortuntely...

@Francisco L Tabares

Thank you very much! Your answer makes me more clear to the question.

As you say, the sheath potential quickly increases when the biasing voltage increases. At SEE=1, no sheath is formed. Is that means an unstable process will occur to increase the plasma potential after the sheath disappears? The sheath will be re-established after the plasma potential increases to a proper value, is that right?

Maybe this explains that the plasma potential increases with the biasing. But it seems that the increasing curve of the plasma potential v.s. biasing potential is somewhat jumping at points where "SEE=1" occurs. Can I say that?

It hard to say...people working on thrusters for space propulsion  have investigated these effects in detail, the so call multipactor coefficient. See for example M D Campanell et al, PRL 108, 255001 (2012)....Is your box made out of plain steel?

@Francisco L Tabares 

Thank you for the article you recommend. In fact, the box has two kinds. One kind is made of stainless steel, another kind graphite. Now, only the graphite one is still used.

@Tsanko Vaskov Tsankov

Yeah, I agree with your viewpoints basically.

However, the plasma inside the box couldn't sees a grounded surface, but its initial potential could be as low as ground. For example, the box is not biased when the plasma produces (by ionization of high energy electron beam), then bias the box to a high potential relative to ground. In this situation, will not high-voltage sheaths appear?

Tsanko is right, as I mentioned before, there should not be any sheaths in steady state operation within your box (except, maybe in the vicinity of your hole where the electron beam enters).

There can be, however, local small variations in your potential as the (local) plasma potential is connected with differences in your charge densities. The plasma potential within your box should be constant and in the order of magnitude of your box bias.

As there is no electric field the curvature of your potential has to be zero (i.e. a straight, constant line) - this is due to the structure of Poisson's equation.

@Tsanko Vaskov Tsankov; @Johannes Gruenwald

Your points are right. In application of the biased box, the plasma potential is considered equal to of the box, which ignores the sheath potential.

I'm much more clear to my question now, thanks a lot! @Tsanko Vaskov Tsankov; @Johannes Gruenwald;@Francisco L Tabares.