How Does Electrical Conductivity Behave in Non-Uniform Conductors?

harshant
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Electrical conductivity ??

I have 3 doubts regarding the situation depicted in the attachment (sorry for my bad drawing). The diagram below shows an electrical conductor with the ends mantained at constant potentials (red higher than blue), and current density being shown by the black arrows.
1. Is the electric field inside the conductor uniform, and what direction is it pointing?
2. If the electric field is uniform, can the conductivity be considered as a scalar (because then the current density would not always be along the wire) or it has to be considered as a tensor which transforms the uniform electric field to a current density along the wire?
3. Is the conductivity tensor constant along the conductor?
 

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Somebody please help!
 
The wire can still be modeled as one-dimensional, even though it is not straight.

- Warren
 

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