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I would like to compute the electric field $\boldsymbol{E}$ in the time-harmonic case for a (thick) wire parallel to the $z$-axis, but I can't quite get to it.

What I've got so far:

The current density $\boldsymbol{J}$ only has a component in $z$-direction, so $$ \boldsymbol{J} = j(x,y) \exp(i \omega t) e_z $$ with some scalar function $j$. The same is true for the magnetic vector potential $\boldsymbol{A}$ (where $\boldsymbol{B}=\mathrm{curl}\,\boldsymbol{A}$), $$ \boldsymbol{A} = \phi(x,y) \exp(i\omega t) \boldsymbol{e}_z. $$ The goal is be to compute $j$ and $\phi$. Maxwell's equations tells us how they are connected, namely:

  • By Ampère's law $$ \mathrm{curl}\,(\mu^{-1}\boldsymbol{B}) = \boldsymbol{J} $$ which boils down to $$ -\nabla(\mu^{-1} \nabla\phi) = j $$ where $\mu$ is the magnetic permeability;

  • by Faraday's law $$ \mathrm{curl}\, \boldsymbol{E} = -i\omega\boldsymbol{B} $$ which boils down to $$ \mathrm{curl}\,((\sigma^{-1} j + i\omega \phi)e_z) = 0 $$ where $\sigma$ is the electric conductivity. From this, we get $$ \sigma^{-1} j + i\omega \phi = C $$ with some constant $C$. The constant $C$ is still somewhat mysterious for me. (The applied voltage has to sit in there somehow I guess.)

Given $C$, one could replace $j$ in the first equation and solve a PDE for $\phi$.

Can anyone help me out on $C$? Is there maybe a better way to retrieve $\boldsymbol{E}$?

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