Spatial variation of currents and fields due to localized scatterers in metallic conduction

TLDR: In this article, the electric field is associated only with space charge but not with a current, and approximate space charge neutrality is restored, by adding a particular solution of the transport equation in which the electric fields are associated with a specific dipole field about each scatterer.

Abstract: Localized scatterers can be expected to give rise to spatial variations in the electric field and in the current distribution. The transport equation allowing for spatial variations is solved by first considering the homogeneous transport equation which omits electric fields. The homogeneous solution gives the purely diffusive motion of current carriers and involves large space charges. The electric field is then found, and approximate space charge neutrality is restored, by adding a particular solution of the transport equation in which the electric field is associated only with space charge but not with a current. The presence of point scatterers leads to a dipole field about each scatterer. The spatial average of a number of these dipole fields is the same as that obtained by the usual approach which does not explicitly consider the spatial variation. Infinite plane obstacles with a reflection coefficient r are also considered. These produce a resistance proportional to r/(1-r).