Electric field generated by ions in a box

[Bestfrog]

Homework Statement

A ionized gas (overall neutral) is in a cylindrical box with radius $R$ and $h<<R$, with is main axis along $\hat{z}$ (there is acceleration of gravity $\vec{g}=-g \hat{z}$). The ions are points: some of mass $M$ and charge $q>0$ (heavy positive ions) and some of mass $m<M$ and charge $q<0$ (light negative ions).
The system is at uniform and constant temperature and we can neglect the fact that ions don't tie themselves. (Sorry for bad English)
What is the distribution of ions in a condition of equilibrium and what is the electric field in the cylinder?

Homework Equations

The Attempt at a Solution

I figured to put firstly the positive ions. They go on the perimeter of the down face of the cylinder (down because of gravity, on the perimeter because of the Coulomb's force). If I put now the negative ions they are attracted by the positive ones and by gravity (also them would stay on the perimeter) but in this situation positive and negative ions are attached one another. Is it correct?

 

[BvU]

In other words: if you would do this exercise with neutral molecules, they would all go on the down face of the cylinder ?
Do collect a few relevant equations and list them under 2. !

 

[Bestfrog]

In other words: if you would do this exercise with neutral molecules, they would all go on the down face of the cylinder ?

No, they don't. If they have a velocity $\vec{v}$, the components $v_x ,v_y$ change after bumps (on average they remain the same). Only the component $v_z$ is accelerated. But if I search an equilibrium condition, the ions can't be stationary, they have to move with a steady speed, I don't how.

 

[BvU]

Still don't see no equations. In what section of your textbook is this exercise ?

 

[haruspex]

My first thought was to try modelling it as the positive ions having a linearly reducing density with height, and the negative ions a correspondingly increasing density. You could then calculate the overall electrical and gravitational potential energies. The actual density gradient should be whatever minimises this PE.
However, I don't think this works. The electrical field would not be uniform, I suspect.
Second thought is to try alternating infinitesimal layers of positive and negative, starting with positive at the bottom. Find the layer thickness which minimises the PE.

 

[Bestfrog]

Still don't see no equations. In what section of your textbook is this exercise ?

I didn't put any equation because I think it is useless until I understand the situation (because the use of equations can change from one situation to another). This is a problem taken from an admission test :)

 

[BvU]

How can I help you to get under way with this without giving it away ? Admission tests sometimes test ingenuity and creativity instead of sheer knowledge.

In this case I think post #2 contains enough of a hint to point you at barometric height formulas (yes, equations!) and encourage you to somehow combine with electrostatics.

To be fair, I don't have the full answer for you, just try to help.
@haruspex : any interest in this one ?

 

[Bestfrog]

How can I help you to get under way with this without giving it away ? Admission tests sometimes test ingenuity and creativity instead of sheer knowledge.

You took the point

In this case I think post #2 contains enough of a hint to point you at barometric height formulas (yes, equations!) and encourage you to somehow combine with electrostatics.

Thank you, now I try to write an answer

@haruspex : any interest in this one ?

He already answered me in the section "Introductory Homeworks" (It is a my mistake to put this exercise in that section, but I don't know how to delete a post)

 

[Bestfrog]

Here I am again: I found that one guy solved it by starting from this formula: $\rho (z)$ density of gas, $\phi (z)$ density of charge. So the equation is $$\rho(z) gz + \phi(z) Ez + \int_z^h \rho(z')g dz' = constant$$ I would like to know where this equation comes from (it is the Bernoulli equation for fluids that don't flow) because on internet I don't find anything.

 

[haruspex]

Here I am again: I found that one guy solved it by starting from this formula: $\rho (z)$ density of gas, $\phi (z)$ density of charge. So the equation is $$\rho(z) gz + \phi(z) Ez + \int_z^h \rho(z')g dz' = constant$$ I would like to know where this equation comes from (it is the Bernoulli equation for fluids that don't flow) because on internet I don't find anything.

Shouldn't there be a γ/(γ-1) factor on the integral? See https://en.m.wikipedia.org/wiki/Bernoulli's_principle#Compressible_flow_equation

 

[Bestfrog]

I
Second thought is to try alternating infinitesimal layers of positive and negative, starting with positive at the bottom. Find the layer thickness which minimises the PE.

I think that the key idea is that if the gravitational and electric forces act on a ion, the vectorial sum must be 0. So in the stationary state I must impose that the PE is minimum. But how to translate in numbers this idea?

 

[haruspex]

I think that the key idea is that if the gravitational and electric forces act on a ion, the vectorial sum must be 0. So in the stationary state I must impose that the PE is minimum. But how to translate in numbers this idea?

That would be a Calculus of Variations problem. Are you familiar with that subject?

 

[Bestfrog]

That would be a Calculus of Variations problem. Are you familiar with that subject?

No, I will study it. Can you show me only the set up?