Register to reply

How to find an electric potential in anisotropic, inhomogeneous medium

Share this thread:
Agent007
#1
Nov25-12, 11:52 PM
P: 1
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 [itex]\psi[/itex] ([itex]\vec{E}=-\nabla\psi[/itex]) in anisotropic, inhomogeneous medium.

Let's introduce a cylindrical coordinate system ([itex]\rho[/itex], [itex]\varphi[/itex], z).

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

Here [itex]\rho[/itex] is the volume charge density, [itex]\lambda[/itex] is a constant that describes the linear charge density.

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

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

This is the hyperbolic partial differential equation due to properties of [itex]\widehat{\varepsilon}[/itex].

2. If [itex]\rho\geq a[/itex], medium is homogeneus and isotropic. Permittivity [itex]\varepsilon=1[/itex], its a scalar.

From Gauss's flux theorem we obtain ([itex]\rho\geq a[/itex]):
div([itex]\nabla\psi[/itex])=-4[itex]\pi\rho[/itex]=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 [itex]\rho<a[/itex]. The only thing that may help is that nothing depends on z.

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

If somebody has any ideas, it will be great!
Phys.Org News Partner Science news on Phys.org
'Office life' of bacteria may be their weak spot
Lunar explorers will walk at higher speeds than thought
Philips introduces BlueTouch, PulseRelief control for pain relief

Register to reply

Related Discussions
Electric Potential energy in dielectric medium Advanced Physics Homework 0
Wave equation in inhomogeneous medium Advanced Physics Homework 4
Anisotropic dielectric medium Advanced Physics Homework 2
Inhomogeneous and anisotropic dust models Special & General Relativity 6
Cosmos: more anisotropic or inhomogeneous? Astronomy & Astrophysics 0