 
Boundary conditionsThe 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 fluxsurface condition, requires that the normal component vanishes on the boundary which is assumed to be sufficiently smooth, i.e.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. If (12) holds on a smooth surface, then we get
if and denote two perpendicular tangential unit vectors. Thus, in view of (5), we should formulate a Neumann condition 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 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.  
