Einstein-Laub and Lorentz optical force densities with a planar interface

A definitive theory for the optical or electromagnetic force density in condensed matter has remained elusive. Integral in such a theory is how material interfaces are treated. To build collective understanding, a general planar interface situation between vacuum and a semi-infinite, nonmagnetizable material medium is studied analytically using two electromagnetic force density formulations and an obliquely incident, $p$-polarized plane wave impinging on the material surface. A form of the Lorentz electromagnetic force density formulation and the Einstein-Laub electromagnetic force density are shown to both predict a total pressure that agrees with that predicted by the so-called ``Maxwell-Bartoli'' expression modified for oblique incidence. However, the two formulations present different distributions of the total force between the surface and bulk of the material. The quantification of this difference and of the implied outward surface pressure for this broadly applicable field geometry offers opportunities for the experimental determination of a force density theory that accurately predicts experiment.