Formulations in 3D Electrostatics

Electrostatic problems are described by the Poisson's equation for scalar electric potential U (E = -gradU, where E - electric field intensity vector). The equation for 3D formulation is:

Δ(εU) = -ρ,

where electric permittivity ε, electric charge density ρ are constants within each block of the model. Anisotropy is not accounted for in 3D formulation.

The field source in electrostatic problem is the electric charge. In 3D problem it is possible to specify:

• volume charge density associated with a body;
• surface charge density associated with a face;
• line charge density associated with an edge;
• point charge in the vertex of the model.

The charge density can be specified as a function of coordinates: ρ = ρ(x,y,z).

The following boundary conditions can be specified at outward and inner boundaries of the region:

1. Dirichlet condition specifies a known value of electric potential U₀ on faces, edges or vertices of the model.
2. Neumann condition specifies a known value of normal component of electric induction D_n= σ on external faces and edges.
3. Constant potential boundary condition is used to describe surface of an isolated «floating» conductor that has constant but unknown potential value.

Non-zero Dirichlet and Neumann boundary conditions can be specified as a function of coordinates.