How to find an electric potential in anisotropic, inhomogeneous medium

[Agent007]

Hello!

(I am sorry for probable mistakes. English is not my native language. I have never written anything about mathematics and physics in English.)

I have an electrostatic problem. I need to find an electric potential $\psi$ ($\vec{E}=-\nabla\psi$) in anisotropic, inhomogeneous medium.

Let's introduce a cylindrical coordinate system ($\rho$, $\varphi$, z).

The only source of the field is the linear charge on the endless thread:
$\rho=\lambda\delta(\rho).$

Here $\rho$ is the volume charge density, $\lambda$ is a constant that describes the linear charge density.

1. If $\rho<a$, medium is homogeneus and anisotropic. Permittivity $\widehat{\varepsilon}$ is the given Hermitian matrix (3 x 3). All its entries are non-nil, some of them depend on the polar angle $\varphi$ so $\widehat{\varepsilon}=\widehat{\varepsilon}( \varphi )$.

From Gauss's flux theorem we obtain ($\rho<a$):
div($\widehat{\varepsilon}(\varphi)\nabla\psi$)=-4$\pi\rho$.

This is the hyperbolic partial differential equation due to properties of $\widehat{\varepsilon}$.

2. If $\rho\geq a$, medium is homogeneus and isotropic. Permittivity $\varepsilon=1$, its a scalar.

From Gauss's flux theorem we obtain ($\rho\geq a$):
div($\nabla\psi$)=-4$\pi\rho$=0.

This is the elliptic partial differential equation.
________

So I have to solve these equations. Unfortunately, it's impossible to separate variables in the area $\rho<a$. The only thing that may help is that nothing depends on z.

I have the boundary conditions:
$\psi(a-0)=\psi(a+0)$,
$\widehat{\varepsilon}\frac{\partial\psi}{\partial \rho}(a-0)=\frac{\partial\psi}{\partial\rho}(a+0)$,
$\psi(\rho\rightarrow\infty)\rightarrow 0$.

If somebody has any ideas, it will be great!

 

[jvicens]

Hello!

Thank you for sharing your problem. From your description, it seems like you are dealing with a challenging electrostatic problem in an anisotropic and inhomogeneous medium. Your approach using Gauss's flux theorem to obtain the partial differential equations is correct, and it is not surprising that the equations are hyperbolic and elliptic in different regions of the medium.

One possible approach to solving these equations is to use numerical methods, such as finite element or finite difference methods. These methods allow for the solution of complex partial differential equations in non-separable coordinate systems. You can discretize your domain and use iterative methods to approximate the solution to your equations. There are many software packages available that can help with the implementation of these methods.

Another approach is to try to simplify the problem by making certain assumptions or approximations. For example, if the anisotropy of the medium is small, you may be able to approximate the permittivity matrix as a scalar, making the problem easier to solve. Additionally, you can try to find analytical solutions in simpler cases, such as when the medium is isotropic and homogeneous, and use those as a starting point for solving the more complex problem.

I hope these suggestions help in finding a solution to your problem. Good luck!