Studying a cylindrical charge distribution - Gauss theorem

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consider a cylinder of radius R and of infinite length, uniformly loaded by volume with a volume density r> 0.

1. What is the direction of the electrostatic field at any point M of space?
2. Show that the value of the electrostatic field depends only on the distance r between M and the cylinder axis.
3. Using the Gauss theorem and specifying the used area, calculate the field in either case:

r> R

r <R

We will give E as a function of r.
4. Calculate the electric potential inside and outside of the cylinder.
We impose the condition V = 0 for r = 0.
5. The volume density ρ of the load cylinder is no longer uniform but with cylindrical symmetry (ρ is a function of r).
  we give ρ= ρo(r/R)for r <R and ρo constant.
Determine the electrostatic field in the case where r <R.

 

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