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.