The reduced potential option is useful for models involving a uniform or known external background field, usually originating from distant sources that might be expensive or inconvenient to include in the model geometry. A typical example is when analyzing induced magnetization in ferromagnetic objects such as ships or vehicles due to the Earth’s magnetic field. The strategy is then to solve only for the induced fields represented by the reduced vector potential Ared, introducing the substitution A = Ared + Aext, where Aext represents the known background field, into Maxwell–Ampère’s law:

For time-harmonic quasistatic systems solving for an A formulation, the reduced potential formulation results in the following PDE:

Here it is possible to interpret the term ∇ × Aext as an additional remanent magnetic flux density and the term (jωσ − ω2ε)Aext as an additional external current source.

In this case it is also possible to express the external field through a known external magnetic flux density, Bext. The domain equation in reduced form then reads