# Solution of Maxwell's equation for simple, time-harmonic wire

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 AmpĂ¨re'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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