In electrostatics where no magnetic field is present, Maxwell’s equations (Equation 2-1) reduce to two equations: Gauss’ law and ∇ × E = 0. By introducing the electric potential V and defining E = −∇V, the two equations are reduced to a single equation since ∇ × ∇V = 0 always holds. Giving the constitutive relationship D = ε0E + P between the electric displacement field D and the electric field E, Gauss’ law is rewritten as

where ε0 is the permittivity of vacuum, P is the electric polarization vector, and ρ is the space charge density. Equation 2-6 is the Poisson’s equation, which is easy to solve from the numerical point of view. Compared to using direct representation of the field components or a vector potential, the scalar potential reduces the number of unknowns by a factor 3 and the computational load (for Direct solvers) up to an order of magnitude. It is also attractive from a theoretical point of view, as it is generally much easier to find analytical solutions to problems formulated using a scalar potential.

For general cases of static electric field and currents, Equation 2-6 has to be solved together with the equation of continuity (Equation 2-2). The V formulation is used by all electric field and current interfaces in the AC/DC Module. Depending on specific situations, different physics interfaces together with different study types are available. Deciding what specific physics interface and study type to use requires a basic understanding of the Charge Relaxation Theory. See Theory for the Electrostatics Interface and Theory for the Electric Currents Interface for more details.