Distribution of ions in a cylindrical container

[Mat15]

Homework Statement

In a cylindrical container of radius $R$ and height $h \ll R$, with long axis $\hat{z}$, in the presence of gravity acceleration $\vec{g} = - g \vec{z}$ is contained a neutral ionized gas, whose ions can be described as material points with electric charge $q> 0$ and mass $M$ (heavy positive ions) and material points with electric charge $-q$ and mass $m <M$ ( light negative ions). The system is maintained at a constant and uniform temperature, and it is assumed for the sake of simplicity that the ions can not bind to each other by neutralizing themselves. Discuss qualitatively the distribution of ions in equilibrium conditions and calculate, in particular, the value of the electric field inside the cylinder.

Homework Equations

The Attempt at a Solution

My idea was to use Hall potential. But I have no idea how to do.

 

[berkeman]

Homework Statement

In a cylindrical container of radius $R$ and height $h \ll R$, with long axis $\hat{z}$, in the presence of gravity acceleration $\vec{g} = - g \vec{z}$ is contained a neutral ionized gas, whose ions can be described as material points with electric charge $q> 0$ and mass $M$ (heavy positive ions) and material points with electric charge $-q$ and mass $m <M$ ( light negative ions). The system is maintained at a constant and uniform temperature, and it is assumed for the sake of simplicity that the ions can not bind to each other by neutralizing themselves. Discuss qualitatively the distribution of ions in equilibrium conditions and calculate, in particular, the value of the electric field inside the cylinder.

Homework Equations

The Attempt at a Solution

My idea was to use Hall potential. But I have no idea how to do.

By the Hall Potential, do you mean the Hall Effect?

https://en.wikipedia.org/wiki/Hall_effect

Why was that your first thought? What is generating the continuous currents that would cause any Hall Effects and Potentials?

It sounds like they are asking about when the system is basically in equilibrium. What forces are in play when this system is in equilibrium? How will those forces affect the distribution of the point charges? How will that distribution lead to the Electric Field distribution in the volume of the cylinder?

Also, you didn't explicitly state it, but does the problem statement mention whether the cylinder is conducting or not? It would be a lot simpler if the container is non-conducting, I think.

 

[Mat15]

By the Hall Potential, do you mean the Hall Effect?

https://en.wikipedia.org/wiki/Hall_effect

Why was that your first thought? What is generating the continuous currents that would cause any Hall Effects and Potentials?

It sounds like they are asking about when the system is basically in equilibrium. What forces are in play when this system is in equilibrium? How will those forces affect the distribution of the point charges? How will that distribution lead to the Electric Field distribution in the volume of the cylinder?

Also, you didn't explicitly state it, but does the problem statement mention whether the cylinder is conducting or not? It would be a lot simpler if the container is non-conducting, I think.

This is an exercise of the 2016 admission test of a school of excellence in Pisa (Italy).

None of the students was able to solve it and during the oral session they had the impression that the examiner suggested using the Hall effect.

I read the text in Italian again and it seems to me that the cylinder isn’t conducting.

 

[berkeman]

So other than the Hall Effect, what forces would contribute to some distribution of the ions?

 

[Mat15]

I think that forces would contribute to some distribution of the ions are:
electrostatic force, gravitational force and buoyant force.

 

[Mat15]

Can anybody help me to solve this problem?