# Poisson's Equation for Charge above Infinite Ground Plane

• aaaa202
In summary, the boundary conditions for the Poisson equation in a thin slab of charge held above a grounded plane at z=0 can be obtained by considering the finite extent of the slab as a perturbation on the infinite plane case. The boundary conditions will depend on the specific geometry and distribution of charge in the slab, but a general approach is to modify the boundary conditions for the classical image problem to account for the finite extent of the slab in the z-direction.
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?

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

## 1. What is Poisson's equation for charge above an infinite ground plane?

Poisson's equation for charge above an infinite ground plane is a differential equation in electrostatics that describes the electric potential around a point charge placed above a conducting ground plane. It takes into account the influence of the charge and the grounding plane on the electric potential.

## 2. How is Poisson's equation derived?

Poisson's equation is derived from the more general Laplace's equation, which describes the electric potential in the absence of any charges. By adding the influence of the point charge and the grounded plane to Laplace's equation, we arrive at Poisson's equation.

## 3. What is the physical significance of Poisson's equation?

Poisson's equation has several important physical implications. It shows the relationship between the electric potential and the charge distribution in a system. It also helps us understand how charges interact with conducting surfaces, such as the ground plane in this case.

## 4. Can Poisson's equation be solved analytically?

In simple cases, Poisson's equation can be solved analytically using techniques such as separation of variables or Green's function. However, in more complex situations, numerical methods may be necessary to obtain a solution.

## 5. How is Poisson's equation used in practical applications?

Poisson's equation is used in a wide range of applications, including analyzing and designing electrical circuits, modeling the electric fields around antennas and other devices, and understanding the behavior of charged particles in electric fields.

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