Complex form of poisson equation

[fnsaceleanu]

Hi,

I am trying to solve the complex form of poisson equation ∇[ε(x)∇V(x)] = -1/εo ρ(x)
with complex permittivity.

If I introduce complex permittivity ε = εr - j σ/(εow), then I must introduce a complex potential ,V = Vr + jVi.
That means the charge density, ρ, must also take a complex form, but I'm not sure what to make of that.

For my boundary conditions, I have the real part of the voltage measured from the oscilloscope, and the real charge density I compute through the conservation equations.

Any help on this topic? I can't find much on this on internet.
Thanks!

 

[eljose79]

Hi there,

It's great that you are working on solving the complex form of the Poisson equation. This type of problem can be quite challenging, but with the right approach, it can be solved successfully.

Firstly, it is important to note that the complex permittivity and potential are introduced in order to account for the imaginary part of the permittivity, which represents the loss of energy due to conductivity. This means that the charge density will also have a complex form in order to accurately represent the loss of energy in the system.

When it comes to boundary conditions, it is important to use the real part of the voltage measured from the oscilloscope, as this is the physical quantity that is being measured. However, for the charge density, it would be more accurate to use the complex form, as it takes into account the loss of energy in the system.

I understand that there may not be a lot of information available on this topic, but I would suggest consulting with other scientists or researchers who have experience in solving similar complex equations. They may be able to offer valuable insights and guidance.

I wish you all the best in your research and I hope you are able to successfully solve the complex form of the Poisson equation. Keep up the great work!