Conducting Cylinder vs Cylinder of Charge - Guass's Law

[Marcin H]

Homework Statement

I just have a general question about Gauss's Law and the cylinders above. I don't really understand what the difference is between the 2 cylinders? They are both charged, but one of them does not have an electric field inside the cylinder because it a conducting cylinder? I don't understand the difference between the 2 above.

Homework Equations

Gauss's Law

The Attempt at a Solution

This is just something I wanted clarification on. Hopefully I am posting in the correct place.

 

[Orodruin]

In the case of a conductor, there is no charge density inside the cylinder (it would result in an electric field that would drive the charges away - all the charge is on the cylinder surface) and in the other (cylinder of fixed volume charge density) there is a charge density inside the cylinder. Outside the cylinder this results in the same electric field but inside it it does not (as the enclosed charge will be different in the different cases).

 

[Marcin H]

In the case of a conductor, there is no charge density inside the cylinder (it would result in an electric field that would drive the charges away - all the charge is on the cylinder surface) and in the other (cylinder of fixed volume charge density) there is a charge density inside the cylinder. Outside the cylinder this results in the same electric field but inside it it does not (as the enclosed charge will be different in the different cases).

Ok. How will we know which one to deal with when doing problems? Will it have to explicitly say if it's a conductor or not? Or if we are given a charge density rho then we need to use the "cylinder of charge" formulas to solve the rest?

Also, I was a bit confused by the charge densities rho and lamba and when to use which. Do we use lambda for the charge density if and only if we are given a infinetely thin wire or line of charge? And rho when the cylinder or wire has a finite radius? This confuses me because the 2 pictures above show them using lambda which is Q/L, but in a homework problem like this:

"An infinitely long, cylindrical wire of radius R = 1 cm is centered along the ^z axis and carries a uniform volumetric charge density ρ_0." <-- Edit* This is supposed to be the symbol "rho"...

There is more to the problem, but basically it says to use Gauss's law to find the charge density of the wire. But the solutions uses ρ = Q/V instead of λ=Q/L like like we have in the equations above. Why is this? I hope I am making sense in where I am confused...
Here is the whole problem btw:

I got the correct answer, but I don't understand when we are supposed to use lambda vs rho for the charge density...

 

[Orodruin]

How will we know which one to deal with when doing problems?

Obviously it depends on the problem statement about the setup.

Also, I was a bit confused by the charge densities rho and lamba and when to use which. Do we use lambda for the charge density if and only if we are given a infinetely thin wire or line of charge?

They are different physical quantities. The line density $\lambda$ is the charge per length of the cylinder and $\rho$ is the volume density, which is the charge per volume.

 

[Marcin H]

Obviously it depends on the problem statement about the setup.They are different physical quantities. The line density $\lambda$ is the charge per length of the cylinder and $\rho$ is the volume density, which is the charge per volume.

ok thanks