Ohm's Law and its Physical Consequences

[aaaa202]

Ohms law is in my book stated in the form:

J = $\sigma$E

My question is about the validity of this statement (I know it is just an empirical law, but therefore you can still think of its consequences). If the above is true then the volume current density will depend on the strength of the field at a specific point. But how are you guarenteed that the field through our ohmic material will vary in such a way as to not violate charges being piled up. Let me make my point more clear through an example:
Suppose we have a thin rod through which the field varies like 1/r where r is the distance from the minus pole to a point on the rod. Close to the minus pole you will then have a current density going through a crosssection of the rod which is very large. But as you move away further the current density will according to Ohms law be much less. This is not physical since charge cannot disappear and why would it pile up in a conductor?

 

[Antiphon]

Charge doesn't pile up because of the forces on the charges when it tries to.

You are confused because this is only a single constuent equation of the larger set. This is a simple equation of state without a time dependence. The full set of maxwells equations in the conductor would allow charge to bunch up dynamically in time then relax; time varying magnetic fields would also be present.

The steady-state solutions where the time-varying components have died out will have field solutions where your concerns are not an issue. There will be no charge piling up in this steady state (by definition!)