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

marqushogas

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 [itex]r_{2}[/itex] and that the disk has the thickness [itex] d[/itex]. We’re interested in solving the (complex) electric field [itex]E_{z} [/itex] directed in the [itex]\hat{z}[/itex] direction in this disk. When you work Maxwell’s eq. out you’re arriving at the PDE:

[itex]\nabla^2 E_{z} - j\omega\mu\sigma E_{z}=0[/itex]

where [itex]\sigma [/itex] is the conductivity, [itex]\omega[/itex] is the frequency, [itex]\mu[/itex] is the permeability and [itex]j[/itex] is the imaginary unit. This disk has cylindrical rotation symmetry so [itex]E_{z}[/itex] does not depend on [itex]\phi[/itex]. 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. [itex]E_{z}(\rho,z=0)=0[/itex] for all [itex]\rho\in[0,r_{2}][/itex]. (because there is a current on the circular plane surface not in contact with the coaxial cable).
2. [itex]\int_0^{r_{2}}E_{z}(\rho,z’ )\rho\,d\rho=0[/itex] for all [itex]z’ \in(0,d)[/itex]. (because the total current through a cross-section of the disk is zero).
3. [itex]\int_{r_{1}}^{r_{2}}E_{z}(\rho,z=d)\rho\,d\rho=-I/2\pi\sigma[/itex], where [itex]0 \leq r_{1} \leq r_{2}[/itex] the inner radius of the shield of the coax and [itex]I[/itex] is the given complex current. (because the current through the shield of the coax must be –I)
4. [itex]E_{z}(r’ ,z=d)=0[/itex] for all [itex]r’ \in[r_{0},r_{1}][/itex], where [itex]r_{0}[/itex] is the radius of the inner leader, such that [itex]0 \leq r_{0} \leq r_{1} \leq r_{2} [/itex]. (because we have a current on this surface)
5. [itex]\int_0^{r_{0}}E_{z}(\rho,z=d) \rho d \rho=I/2\pi\sigma[/itex]. (the total current penetrating into the disk from the coax cable is of course I.
6. Obviously [itex]E_z[/itex] 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!