Radiation force and balance of electromagnetic momentum

Some force densities can be expressed as a divergence of a stress tensor, as is the case with the electromagnetic force density. We have shown elsewhere that from the Maxwell equations several balance equations of electromagnetic momentum can be derived, depending on the form these equations are expressed in terms of fields E, D, B, H, and polarisations P and M. These balance equations imply different force densities and different stress tensors, providing a great flexibility to solve particular problems. Among these force densities we have found some proposed in the past with plausibility arguments, like the Einstein–Laub force density, while other proposed force densities appear as particular or limit cases of these general force densities, like the Helmholtz force density. We calculate the radiation force of an electromagnetic wave incident on a semi-infinite negligibly absorbing material using these balance equations, corroborating in this way that the surface integration of the stress tensor gives the same result that the calculation made through a volume integration of the force density, as done by Bohren. As is usual in applications of Gauss's theorem, the surface on which the surface integral is to be performed must be chosen judiciously, and due care of discontinuities on the boundary conditions must be taken. Advanced undergraduates and graduate students will find a different approach to new aspects of the interaction of radiation with matter.