I have simplified the model, fixed the mesh and I am still getting the same results. I am wondering can it be a numerical thing?

I am adding two figures for reference

calcs.png
mesh.PNG

P.S. ran an additional test just to see whether the same issue occurs if I only have a cylinder and a wire connected. Under this case it seems that the current is uniform, as expected.
calcs2.png

Thank you very much.
Tadas

Another finding, whenever the conductivity of the material is set as 5e-9, then the current density in the wire is not uniform, changing the conductivity of the middle part to 1 (added to figures to explain which part is the middle), makes the calculations reasonable.

Your electric conductivity is ranging from 2e-11 to 6e7 so that is over 18 orders of magnitude.

This causes several problems:

* You should be very careful how you solve the equation so that the linear system tolerances don't overshadow the small variations of the potential. You should rather solve at least in the same accuracy as your orders of magnitude but that is impossible with the double precision used.

* No matter what you do there will be a challenge with flux conservation. Default FEM formulation does not guarantee conservation and the problem gets worse when there are large variations of conductivity. The possible remedy for this is to use a mixed formulation with Hdiv basis for the fluxes. Even in Elmer we have a proto equation for this but it has not been applied to static current conduction and I doubt that this is not the correct solution for this problem.

It is not an accident that Maxwell's equations have often different approximations for conductors and insulators. I don't know what the question is you're trying to solve but maybe you could use hierarchical solution of the different regions. It is often a good idea to write down the problem you're trying to solve. I didn't realize the huge range of your coefficients before you brought the problem up. It is the mother of all your other problems.