Gauss' Law: Charge Density in Cylinders

[somasimple]

Hi all,

A closed cylinder of Length L, with a radius r and a thickness d is filled with m₁q₁+m₂q₂ charges. Their respective volume charge densities are $$\rho$$₁ and $$\rho$$₂.
The volume is surrounded with a neutral solute (n₁q₁+n₂q₂=0) with a volume charge density of $$\rho$$₃.
(see picture).

Case 1/
m₂>m₁ => m₁q₁+m₂q₂<0
Is there any attraction from the "core" to the outer charges?
Does it depend of the charges densities and volume thickness?

 

[AJ Bentley]

It isn't at all clear what you mean.
'Filled with charges' could mean the annular region or the central core.
m1q1 +m2q2... what does that signify? two sets of charges? do you mean to imply they are of different sign?

Whatever these things mean, you appear to be describing some sort of initial condition for a collection of charges and asking about their interaction from that position.

Put simply, each charge experiences a net force due to the summation of all the fields of each other charge in the configuration. You can work that out in terms of an integration of the charge densities but that would be an instantaneous solution for an unstable configuration, which would tell you nothing about the subsequent behaviour of the system.

 

[gabbagabbahey]

Is there any attraction from the "core" to the outer charges?
Does it depend of the charges densities and volume thickness?

The interaction between the cylinder and its surroundings depends both on the total charge density inside the material $\rho_1(\textbf{r})+\rho_2(\textbf{r})$ and the charge density outside the material $\rho_3(\textbf{r})$. The reason for this should be fairly clear; the field of a point charge falls off with distance from the charge. Hence, the force a test charge feels from a source charge depnds on the distance between the two charges. So, if you want to calculate the force on each one of the +/- charges in your cylinder due to the surrounding solution, you need to account for each infinitesimal piece of charge, since each will be at a different position relative to your test charge and hence exert a different force on it.

As a simple example, consider a single point charge $q$ at the origin and a physical dipole consisting of a charge $q$ at $x=a$ and a charge $-q$ at $x=a+d$ (assume $a\gg d>0$ if you like) connected by a rigid neutral rod. The charge at the origin will experience a net repulsive force since it is closer to the positive end of the dipole than it is to the negative end, even though the net charge on the dipole is zero.

 

[somasimple]

m1q1 +m2q2... what does that signify? two sets of charges? do you mean to imply they are of different sign?

The picture tells us that q₁ is positive when q₂ is negative.
It is initial conditions of a pure hypothesis. And these charges are homogeneously mixed within the core. They are moving charges (a fluid).

The interaction between the cylinder and its surroundings depends both on the total charge density inside the material $\rho_1(\textbf{r})+\rho_2(\textbf{r})$ and the charge density outside the material $\rho_3(\textbf{r})$.

That's what I meant but failed to explain it clearly.
If we know the charge density of a fluid (concentration of charges per volume?), we know the distance that separate these charges.
I just want to know when we add another distance (the cylinder thickness) if the action of the core may be unable to attract any moving outside charge?
Is there a way to say that inner cylinder action (on outside charges) is dependent of cylinder thickness?

 

[sardar]

Gauss' Law is a fundamental principle in electromagnetism that relates the electric flux passing through a closed surface to the charge enclosed by that surface. In the case of a closed cylinder with varying charge densities and a neutral solute, Gauss' Law can be applied to determine the electric field and potential within the cylinder.

In this scenario, the charge density within the cylinder is not constant, as there are different types of charges present. The electric field within the cylinder will depend on the distribution of these charges and their respective volume charge densities. This means that the attraction between the "core" charges and the outer charges will also depend on these factors.

Furthermore, the thickness of the cylinder will also play a role in the electric field and potential within it. A thicker cylinder will have a larger volume and thus a larger amount of charges, leading to a stronger electric field. Additionally, the distribution of charges within the cylinder may also be affected by its thickness.

In conclusion, the attraction between the "core" charges and the outer charges will depend on both the charges densities and the volume thickness. Gauss' Law can be used to analyze this scenario and determine the electric field and potential within the cylinder.