In magnetostatics where no electric currents are present, Maxwell’s equations (Equation 2-1) reduce to two equations: magnetic Gauss’ law and ∇ × H = 0. By introducing the magnetic scalar potential Vm and defining H = −∇Vm, the two equations are reduced to a single equation since ∇ × ∇Vm = 0 always holds. As it can be seen, this is analogous to The V Formulation. Using the constitutive relation B = μ0(H + M) between the magnetic flux density B and the magnetic field H, the magnetic Gauss’ law becomes

where μ0 is the permeability of vacuum, M is the magnetization vector. This formulation, referred to as the Vm formulation, is used in the The Magnetic Fields, No Currents Interface and The Magnetic Fields, No Currents, Boundary Elements Interface. The resulting Equation 2-7 is also a Poisson’s equation that is easy to solve. Therefore, The Magnetic Fields, No Currents Interface is the primary choice for modeling permanent magnets and domains without currents.