# Electrostatics: boundary conditions

This question is probably simple, but I am confused.. Assuming we have an arbitrary charge density $$\rho_e$$ inside a volume $$V$$. Studying electrostatics, Gauss's law equation would be $$\nabla\cdot E=\rho_e/\epsilon_0$$ and the Poisson equation would be $$\nabla^2\Phi=\rho_e/\epsilon_0$$. I have seen that all studies on electrostatics apply a voltage value at the boundaries and solve the Poisson equation and then differentiate the potential field to find the electric field. But why don't they apply directly electric field boundary conditions and solve the much easier Gauss's law equation??

Edit: Also the finite difference time domain (FDTD) method uses only electric and magnetic sources/boundary conditions. So I would expect the same for electrostatics.

• Oct 29, 2022 at 13:07

The main problems in solving directly the Gauss's law are:

• you're not taking into account the irrotational condition $$\nabla \times \mathbf{e} = \mathbf{0}$$, that holds in electrostatics. This comes from Maxwell's equations that are the principles of electromagnetism, so you have to satisfy this condition to build a good mathematical model for describing the physics of electrostatic systems;
• it's a scalar equation while the unknown electric field $$\mathbf{e}(\mathbf{r})$$ is a vector field. Mathematically, the problem (without the irrotational condition) is not-determined: in 3D, you have one PDE for 3 unknowns, namely the components of the electric field.

In order to have a well defined problem, that satisfies the irrotational condition, you exploit it to write the electric field as the gradient of a scalar field, the potential field $$\Phi(\mathbf{r})$$, defined as $$\nabla \Phi(\mathbf{r}) = \mathbf{e}(\mathbf{r})$$, and you substitute this latter expression in the Gauss's law for the electric field to get the Poisson's equation

$$\nabla^2 \Phi(\mathbf{r}) = \dfrac{\rho(\mathbf{r})}{\varepsilon_0}$$.

This is a PDE define in a domain, and you need boundary conditions on the boundary of the domain to get a well-defined problem. The choice of the boundary conditions really depends on the problem you're studying:

• if your domain is the whole 3D space, and you're only interested to a limited regions that includes all the electrical charges that influence your system, you will probably use the condition of vanishing field $$\Phi(\mathbf{r}) \rightarrow 0$$ at infinity; from the analytical or numerical point of view, you hardly approach this problem with finite difference method, but you could easily treat it with singularity methods, boundary element methods, or other methods relying on a integral formulation of the problem using the Green's function method;

• if your domain is bounded, you can approach the problem with classical grid based numerical methods (like finite differences, finite elements, finite volumes, discontinuous Galerkin methods, ...) but you need to model a representative boundary condition of the electric field and translate it to the potential field. As an example,

• on a surface $${S_N}$$ where you know the normal component of the electric field, you can translate this condition into a boundary condition on the directional derivative of the potential field in the direction that is normal to the boundary (Neumann or natural boundary condition), as

$$e_n(\mathbf{r}_{S_N}) = \mathbf{\hat{n}}(\mathbf{r}_{S_N}) \cdot \mathbf{e}(\mathbf{r}_{S_N}) = \mathbf{\hat{n}}(\mathbf{r}_{S_N}) \cdot \nabla \Phi(\mathbf{r}_{S_N}) = \dfrac{\partial \Phi}{\partial n}(\mathbf{r}_{S_N})$$.

• on a surface $${S_D}$$ where you know the potential w.r.t. to an irrelevant offset value, you can prescribe the value of the potential (Dirichlet or essential boundary condition)

$$\Phi(\mathbf{r}_{S_D}) = \Phi_s(\mathbf{r}_{S_D})$$,

and if the surface is equipotential, so that no current occurs on it, $$\Phi(\mathbf{r}_{S_D}) = \Phi_{S_D}$$, with $$\Phi_{S_D}$$ constant on the surface $${S_D}$$

Take a look at this reference for a comparison between finite element and boundary element method: https://hrcak.srce.hr/file/265787