# Deriving magnitudes for $\mathbf J$ and $\mathbf E$ from the shape of a conductor

In the following exercise:

I have no idea how to infer the magnitude of $$\mathbf J$$ nor $$\mathbf E$$ given the shape of the wire. The only clear thing to me here is that A, B and C all have different volumes, but I don't know how to relate this with the current density, nor $$\mathbf E$$.

My only guess is that since there is a steady current this implies in all regions $$dQ/dt$$ is constant, so in regions of smaller volume this may imply that $$|\mathbf J |$$ must be largest there, but it's just a guess. As far as $$\mathbf E$$ is concerned, I know that the relation $$\mathbf J = \sigma \mathbf E$$ is an apparenty common approximation, but I don't known when this approximation can be willfully applied.

If I assume it is here, then $$|\mathbf E|$$ is largest where $$|\mathbf J|$$ is largest, so the magnitude of $$\mathbf E$$ is largest in region B as well. However, I find this odd because since this is a conducting wire and therefore a conductor, I expect it to have an electric field of $$0$$, as this is what I am told is one of the key characteristics of a conductor.

Where am I going wrong in my interpretations for (a)?