# Basic question about the generalized Poisson Equation

1. Jul 26, 2013

### maverick280857

Hi,

Suppose we look at two dimensional Poisson's equation in a medium with spatially varying (but real) dielectric constant:

$$\nabla(\epsilon_r\nabla \varphi) = -\frac{\rho(x,y)}{\epsilon_0}$$

Consider the problem of solving this using the Finite Difference method on a rectangular grid, subject to Dirichlet boundary conditions at the 4 edges (which are assumed to be conducting sheets, surrounding some dielectric medium). This is supposed to model a semiconducting device I am trying to work on numerically for a thesis project.

Now if $\rho(x,y) = 0$ I know from basic electromagnetic theory (c.f. Griffiths, for instance) that in a sub-region where the dielectric constant doesn't vary at all, the potential can have no local maxima or minima, and that it can only take extreme values at the boundaries. This is because the equation is then Laplace's equation, and this is a property of all Harmonic functions. I get this.

(1) Now suppose $\rho(x,y) = 0$ still, but the permittivity varies spatially -- it is a constant for one region, and another constant for another region, etc. In this case, can I make any statements about the limits on the values of the potential?

(2) If $\rho(x,y)$ is now nonzero but permittivity does not vary spatially, I just have the regular Poisson equation. If $\rho(x,y)$ gets contributions only from the doping and voltage induced local charge density but no horrible delta function like isolated charges, is it still correct to say that the potential in the region cannot exceed the values at the boundaries?

So if I were to apply 1 V to the right edge and 0.5 V to the top edge, then as long as there are no isolated point charges in the dielectric medium, can I generally say that the potential in the interior will never exceed 0.5 V or 1 V?

I tried to answer this question in one-dimension where the solution to a constant charge density is a parabolic potential. I can fit the potential to the end point Dirichlet boundary condition values, but in the intermediate region I see no reason why it shouldn't be allowed to go beyond these values.

Any insights?

2. Jul 27, 2013

### Staff: Mentor

This is analogous to a steady state heat conduction problem with a variable thermal conductivity and a distributed heat source (or sink). The temperature within the region can certainly be higher than at any of the boundaries if heat is being generated. The boundaries would just then be providing cooling to remove the heat.