In the following exercise:

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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)?


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