By R. Belmans
By analogy to the electric field, the magnetic field strength is calculated as the gradient of a scalar potential. It must be distinguished between currentcarrying and currentfree regions.
Currentfree regions
The magnetostatic problem without conducting currents can be formulated in terms of the magnetic scalar potential
With the vector identityand Ampere’s law, a magnetic scalar potentialcan easily be defined by evaluating:
This potential formulation is not suitable for problems inside regions with conducting currents. A typical application for this type of potential is the calculation of a magnetic shielding.
With the zero divergence condition of the magnetic flux density the of the scalar magnetic potential is introduced:
This formulation is a Laplace equation.is a scalar and A in the vector potential formulation of the magnetic field is a vector quantity. By using the same numerical discretisation, the scalar potential problem consists of a third of unknown when compared to the formulation using the vector potential. This example makes clear that an appropriate choice of the potential formulation has significant influence on the size of the problem.
This definition of the scalar magnetic potential,causes problems in multiple connected domains.
Fig. 4.2. Multiple connected domain.
The domain is free of conducting current. There is a current / carrying conductor leading through the opening in the domain (Fig. 4.2). Applying Ampere’s law on the contour X inside the domain yields:
The potential differencecan only be nonzero ifis discontinuous inside the domain. Therefore, a discontinuity, a cut, is defined in the way that by considering the cut as an outside boundary, the domain is not further multiple connected.
Therefore, the boundary condition applied to the cut is
This is a periodic boundary condition.
Currentcarrying regions
In the case of currentcarrying regions it is not possible to define a magnetic scalar potential. However, by again using an arbitrary vector field T, it is possible to define a similar potential (Silvester & Ferrari’03).
The electric vector potential T and Ampere’s law eq.(ii), (3.27) yield:
With the vector identity
the gradient of the scalar magnetic potential d> is now defined by:
The zero divergence condition of the magnetic fieldand the material equation combining B and H yields an equation of Poisson type:
A disadvantage in solving this problem is that the solution of the magnetic field problem has to be obtained in three steps:
1. Determine the auxiliary potential function T .
2. Solve the Poisson equationto find the magnetic scalar potential.
3. Evaluateand T to obtain the required overall solution of H .
is called aThe magnetic scalar potential has the dimension [A]. This potential formulation is in common use in magnetostatic and diffusion problems. To determine T, BiotSavart’s law can be evaluated (Hafner47).
By using the finite element method, T can be determined in the following way:
1. Create a tree of mesh edges.
2. T= 0 for all treeedges.
3. Apply Ampere’s law to each element and determine T for all cotree edges.
A tree in the topology of the finite element mesh is defined as a set of edges reaching all nodes of the mesh but forming no loops in this mesh. The associated cotree is the set of the remaining edges. T is constructed as a field built of edge elements
The coefficientsof the elements associated with the tree edges are 0. The coefficients of the elements associated with the cotree edges are calculated from Ampere’s law (Fig. 4.3):
Fig. 4.3. Tree definition and finite element with imposed current.
With Table 4.2 a comparison of the properties of the magnetic vector and scalar potential is possible.
Table 4.2. Comparison between A and
Aformulation 

potential formulation 

implicitly fulfilled equation 

explicitly fulfilled equations 

source field 
T has to be determined 

additional condition element type 
gauge edge 
cut node 