= Uniform current in a straight homogeneous wire
{title2=$\mathbf j=I\mathbf e_z/(\pi a^2)$}
In a homogeneous <Ohmic conductor>, a steady <electric current density> obeys $\nabla\cdot\mathbf j=0$ and $\nabla\times\mathbf j=0$. Uniform axial current in a straight cylinder follows if its side is insulated and its flat end faces are equipotentials. The <electric potential> then solves <Laplace's equation> with those boundary conditions, and uniqueness makes it linear in the axial coordinate. Steadiness alone does not impose these contact conditions: nonuniform end potentials can produce nonuniform current even with an insulating side. Bending also generally redistributes current; in an annular conductor the azimuthal current density is proportional to the inverse radius.
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