# How to find an electric potential in anisotropic, inhomogeneous medium

1. Nov 25, 2012

### 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!