## Electric field in a coaxial cable (PDE + boundary value)

Hi! This is a quite sophisticated problem, but it’s fun and interesting!

A coaxial current transmission line is short-circuited by a cylindrical disk in one of its ends. Let’s say that the disk and the coax cable have the radius $r_{2}$ and that the disk has the thickness $d$. We’re interested in solving the (complex) electric field $E_{z}$ directed in the $\hat{z}$ direction in this disk. When you work Maxwell’s eq. out you’re arriving at the PDE:

$\nabla^2 E_{z} - j\omega\mu\sigma E_{z}=0$

where $\sigma$ is the conductivity, $\omega$ is the frequency, $\mu$ is the permeability and $j$ is the imaginary unit. This disk has cylindrical rotation symmetry so $E_{z}$ does not depend on $\phi$. If we choose our cylindrical coordinate system so that the z-axis passes through the center and that z=0 at the circular plane of the disk not in direct contact with the coax cable; then the boundary values on the disk are the following:

1. $E_{z}(\rho,z=0)=0$ for all $\rho\in[0,r_{2}]$. (because there is a current on the circular plane surface not in contact with the coaxial cable).
2. $\int_0^{r_{2}}E_{z}(\rho,z’ )\rho\,d\rho=0$ for all $z’ \in(0,d)$. (because the total current through a cross-section of the disk is zero).
3. $\int_{r_{1}}^{r_{2}}E_{z}(\rho,z=d)\rho\,d\rho=-I/2\pi\sigma$, where $0 \leq r_{1} \leq r_{2}$ the inner radius of the shield of the coax and $I$ is the given complex current. (because the current through the shield of the coax must be –I)
4. $E_{z}(r’ ,z=d)=0$ for all $r’ \in[r_{0},r_{1}]$, where $r_{0}$ is the radius of the inner leader, such that $0 \leq r_{0} \leq r_{1} \leq r_{2}$. (because we have a current on this surface)
5. $\int_0^{r_{0}}E_{z}(\rho,z=d) \rho d \rho=I/2\pi\sigma$. (the total current penetrating into the disk from the coax cable is of course I.
6. Obviously $E_z$ must also be finite for all points in the disk.

My problem is to solve this PDE with these boundary values. I would be very thankful for any insight or idea on how to solve this problem (full solutions not necessary acquired!). So if you can help me in any way I owe you a huge amount of thankfulness and respect!
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