We present an alternative formulation of the magnetostatic boundary value problem that is useful for calculating the magnetic field around a magnetic material placed in the vicinity of steady currents. The formulation differs from the standard approach in that a single-valued scalar potential plus a vector field that depends on the given currents but not on the magnetic material are used to obtain the total magnetic field instead of a magnetic vector potential. We illustrate the method with examples.
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This separation of the field into two parts follows from an application of the Poincare lemma: We know that for any given scalar field ψ, always. The Poincare lemma then assures us that if a vector field has the property that in a given region of space, there exists a scalar function ϕ defined in that region such that .
