Electrostatics Model

The electrostatics model describes a static electric field \(\mathbf{E}\) in terms of the scalar electric potential \(\phi\) and solves the following equation:

(1)\[\nabla \cdot \left(\varepsilon \nabla \phi\right) = -\rho,\]

where \(\varepsilon\) is the electric permittivity and \(\rho\) is the density of free electric charges. For a certain choice of boundary conditions, this equation allows one to find the distribution of the electric potential \(\phi\) in the region of interest, and hence the corresponding distribution of the static electric field \(\mathbf{E}\), using the following definition: \(\mathbf{E}=-\nabla\phi\). The electrostatic model is used to study electrostatic phenomena that arise due to the forces with which stationary electric charges act on each other. It can be applied to study capacitors and the energy stored in them, the forces acting on distributions of electric chargers, as well as various shielding effects (Faraday cages, etc.). The electrostatic model can be used to study the polarization in dielectrics and space charge distributions in semiconductors and insulators. Additionally this model is very useful in field analysis for visualizing field lines and equipotential surfaces.

Materials

In the most general case, the electric permittivity \(\varepsilon\) in Eq. (1) can be a spatially varying quantity, which can also depend on some additional physical properties, such as temperature. To define \(\varepsilon\) as a function of all these parameters we use a corresponding material. A material is represented by a set of functions that introduce \(\varepsilon\) as a function of various parameters in the form of analytic expressions, tabulated data, or external models. For more details, please refer to Electrostatics Material

To simplify the setup of the electric permittivity \(\varepsilon\) for the most common cases, we provide a set of predefined material functions, listed below:

Predefined electrostatics materials
Name	Description
Constant	Defines a material with a constant relative electric permittivity \(\varepsilon_\text{usr}\): \[\begin{split}\varepsilon &= \varepsilon_0 \varepsilon_\text{usr}, \\\end{split}\] where \(\varepsilon_0\) is the vacuum electric permittivity.

Conditions

To solve Eq. (1) in a given region, it is necessary to specify boundary conditions at the boundaries of this region. In addition, it may be necessary to specify a specific distribution of electric charges or to fix the electrical potential at a specific point within the volume of interest. For all these cases, we can use the following list of supported conditions:

List of supported conditions
Name	Supported Entities	Description
Charge Density	Volume, Boundary	This condition specifies the distribution of charge density \(\rho\) in a volume, on a surface or along a line (wire).
Displacement Field	Boundary	This condition sets a specific value of the electric displacement field \(\mathbf{D}\) at the selected boundary
Electric Potential	Volume, Boundary	This condition establishes a fixed electric potential \(\phi\) in the volume or at the boundary of the region of interest.
Floating Potential	Volume	This condition imposes a constraint on the electric potential \(\phi\), requiring that its value in a given region be constant.
Potential Jump	Boundary	This condition defines a discontinuous electric potential \(\phi\) with a given difference in value at the boundary of two contacting regions.

Reports

List of reports
Name	Type	Description

Coefficient Functions

The following functions are available for in the electrostatics model for visualization or querying:

List of functions
Name	Type	Description
Electric Displacement Field	Vector Field	Electric displacement field vector: \[\mathbf{D} = \varepsilon \mathbf{E}\]
Electric Energy Density	Scalar Field	Electric energy density: \[u = \frac{1}{2} \mathbf{E} \cdot \mathbf{D}\]
Electric Field	Vector Field	Electric field vector: \[\mathbf{E} = -\nabla \phi\]
Electric Permittivity	Scalar Field	Electric permittivity: \[\varepsilon\]
Electric Potential	Scalar Field	Electric potential: \[\phi\]
Element Order	Scalar Field	Order of each finite element in the mesh
Mesh Refinement Error	Scalar Field	Local error used to decide on the mesh refinement of a given finite element