# Distribution of ions in a cylindrical container

• Mat15
In summary, the students were unable to solve the problem because they did not understand why the Hall effect was suggested and they did not understand the equilibrium conditions of the system.
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.

## The Attempt at a Solution

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

Mat15 said:

## 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.

## 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.

berkeman said:
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.

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

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

Can anybody help me to solve this problem?

## 1. What is the purpose of studying the distribution of ions in a cylindrical container?

The distribution of ions in a cylindrical container is studied in order to understand the behavior of charged particles in confined spaces. This can have implications in fields such as electrochemistry, materials science, and biophysics.

## 2. How are ions distributed in a cylindrical container?

Ions in a cylindrical container are distributed based on factors such as their charge, size, and concentration. They tend to accumulate near the walls of the container due to the influence of the electric field.

## 3. What factors affect the distribution of ions in a cylindrical container?

The distribution of ions in a cylindrical container can be affected by factors such as the shape and size of the container, the type and concentration of ions present, and the presence of external electric fields.

## 4. How does the distribution of ions in a cylindrical container impact the overall system behavior?

The distribution of ions in a cylindrical container can have a significant impact on the overall behavior of the system. It can affect properties such as conductivity, diffusion, and chemical reactions, and can also influence the stability and performance of devices such as batteries and fuel cells.

## 5. Can the distribution of ions in a cylindrical container be controlled?

Yes, the distribution of ions in a cylindrical container can be controlled through various methods such as changing the shape or size of the container, adjusting the concentration of ions, or applying external electric fields. This allows for manipulation of the system behavior and can be useful in various applications.

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