Poisson's Equation for Charge above Infinite Ground Plane

[aaaa202]

I want to solve the Poisson equation for a thin slab of charge held above a grounded plane at z=0. The problem is somewhat reminiscent of the classical image problem of a point charge above an infinite grounded plane but differs because we are using a slab of charge instead which extends above the whole grounded plane.
I will assume that the sheet and plane extend over a large region in the x- and y- plane so that we can assume translational invariance in these directions. Poisson's equation then reads

$$\frac{d^2\phi}{dz^2} = \frac{\rho_{0}}{\epsilon_{0}} \ \ \ a \leq z \leq b \\ \frac{d^2\phi}{dz^2} = 0 \ \ \ \text{Elsewhere}$$

My question is now: What should be the boundary conditions for this equation? In the classical Image problem you assume that
$$\phi(x,y,z)=0$$
and that
$$\phi \rightarrow 0 \ \ \ \ \ \text{for} \ \ \ \ \ \sqrt{x^2+y^2+z^2}\rightarrow \infty.$$
But if you impose this condition for [tex] \phi(z) [\tex] in my problem you get a non-physical result, which predicts that there is a field infinitely far away from the plane. This is understandable since we have assumed translational invariance, which is effectively the same as saying that we have an infinite plane of charge from which the electric field is constant when you move away. In reality, of course, the plane is not infinite and one should solve the full 3D problem with the boundary condition above. This seems however like a lot of effort. Is there a way to translate the boundary condition for the full 3D problem into a boundary condition for the 1D problem with translational invariance?

 

[eljose79]

Thank you for your question. It is an interesting problem and I will try to provide some insight into the boundary conditions for the Poisson equation in this scenario.

Firstly, let me clarify that the boundary conditions for the Poisson equation in this case will depend on the specific geometry and distribution of charge in the thin slab. However, I will provide some general guidelines that can be followed to determine the boundary conditions.

As you have correctly pointed out, the boundary conditions for the classical image problem assume an infinite plane of charge. In reality, the plane is not infinite and we need to consider the finite extent of the slab in our boundary conditions.

One possible approach is to consider the finite extent of the slab as a perturbation on the infinite plane case. This means that we can start with the boundary conditions for the infinite plane and then add correction terms to account for the finite extent of the slab.

For example, in the classical image problem, the boundary condition at infinity is given by
\phi \rightarrow 0 \ \ \ \ \ \text{for} \ \ \ \ \ \sqrt{x^2+y^2+z^2}\rightarrow \infty

In the case of a finite slab, we can modify this boundary condition as follows:
\phi \rightarrow 0 \ \ \ \ \ \text{for} \ \ \ \ \ \sqrt{x^2+y^2+(z-z_0)^2}\rightarrow \infty

Here, z_0 is the position of the top surface of the slab. This boundary condition accounts for the finite extent of the slab in the z-direction.

Similarly, we can also consider the boundary condition at the bottom surface of the slab. In the classical image problem, this boundary condition is given by \phi(x,y,z)=0. In the case of a finite slab, we can modify this boundary condition as follows:
\phi(x,y,z_0)=\phi_0

Here, \phi_0 is the potential at the bottom surface of the slab. This boundary condition accounts for the finite extent of the slab in the z-direction.

In summary, the boundary conditions for the Poisson equation in this scenario can be obtained by considering the finite extent of the slab as a perturbation on the infinite plane case. However, as I mentioned earlier, the exact form of the boundary conditions will depend on the specific geometry and distribution of charge in the thin slab.

I hope this helps. Please let me