HEJ, HU ISSN 1418-7108Manuscript no.: ANM-981030-A

## Boundary conditions

The physical model would lead us to a problem in an infinite domain with Sommerfeld's radiation condition, cf. [10]. However, in many practical cases we may assume that the field lines of the magnetic induction do not enter or leave a bounded domain representing the device (e.g. the machine). The latter condition, called flux-surface condition, requires that the normal component vanishes on the boundary which is assumed to be sufficiently smooth, i.e.
 (11)

Suppose for the moment that the normal direction is in the z direction of the Cartesian coordinate system. Then we have

and is an immediate consequence of

Therefore, we will demand that the tangential components of vanish on , i.e.

 (12)

If (12) holds on a smooth surface, then we get
 (13)

if and denote two perpendicular tangential unit vectors. Thus, in view of (5), we should formulate a Neumann condition
 (14)

for the normal component rather than a Dirichlet condition. Before we discuss the interface conditions, we perform the standard procedure for getting a variational formulation from (8), i.e. we multiply with a test function v and obtain

 (15)

The second integral will be zero if , i.e. . The latter will be satisfied if is taken from an appropriate test space related to , see Subsection 3.1.
 HEJ, HU ISSN 1418-7108Manuscript no.: ANM-981030-A