3.1 Introduction

In the previous chapter it was shown how one may introduce the concept of excess current and charge densities and excess fields as a way to describe properties of a thin boundary layer without considering the detailed behavior of these quantities as a function of position in the direction normal to this layer. In particular the discontinuity of the extrapolated bulk field at the dividing surface could be expressed in term of these excesses. In this chapter it will be shown how these excess quantities may be described as singular contributions to the corresponding field [16]. The first to introduce singularities in the polarization and magnetization along the surface in order to obtain discontinuities in the extrapolated fields, were Stahl and Wolters [19]. They did not consider the effects of singularities in the normal direction, however. In section 3.2 explicit expressions for the current and charge densities and the electromagnetic fields, containing such singular contributions, are given and in section 3.3 the validity of the Maxwell equations is then discussed, for these singular fields. It is found in this context that at the dividing surface the Maxwell equations reduce to the boundary conditions, given in section 2.4, for the fields. It is precisely this fact, which makes the extension of the validity of the Maxwell equations to the singular fields possible. In section 3.4 charge conservation is treated. The boundary conditions for the fields are used to obtain a charge conservation relation at the surface. In a general...