# Electro Static: solved equations

## Introduction

In an Electro Static application the equations used for computation are:

- the corresponding Maxwell's equations for an electrical system, and
- the constitutive equations that characterize the dielectric materials

The conditions of computation of an Electro Static application are the following:

- the computation concerns the
**D**and E fields; the**B**and**H**fields are not computed. The equations of the electric fields**D**,**E**and of the magnetic fields**B**,**H**are decoupled. - the fields are time independent (static):
**d/dt = 0** - the current density is null:
**J = 0**

## Equations and conditions

In the previously defined conditions of computation , the equations are summarized as follows:

Equations | Description | |
---|---|---|

E: electric field strength (in V/m) D: electric flux density (in C/m V: electric potential (in V) q: density of electric charges (in C/m |
||

ε_{r} : relative
permittivityε |

## Solved equation

The second order equation solved by the finite element method in Flux in case of an Electro Static application is the following:

where:

- is the tensor of relative permittivity of the medium
- ε
_{0}is the permittivity of the vacuum; ε_{0}= 1/(36π10^{9}) (in F/m) - V is the electric potential (in V)
- q is the density of electric charges (in C/m
^{3})

## State variable

The state variable of the Electro Static application is the electric potential **V**
(written **Ve** in Flux 3D).

The uniqueness condition of the scalar field of the electric potential **V** requires
that the value of this potential be assigned to at least one point of the computation
domain.