Electromagnetic force density in electrically and magnetically polarizable media

The force density induced by electromagnetic fields in electrically and magnetically polarizable media is studied analytically. Different formulations of the force density as a function of field-related quantities, including the spatial derivatives of the fields, gradients of the field intensity, phase gradients, electromagnetic power flow (Poynting vector field), and kinetic momentum flow, are introduced. These formulations retain certain symmetries with respect to the force expressions introduced in previous works for an isolated particle but also point out fundamental differences, such as the suppression of recoil forces, negative radiation pressure, and far-field gradient forces. It is shown how these analytical formulations also provide the necessary means to elucidate the sign of the force density in complex media and how they can assist the design of sources to manipulate clouds of particles. The theory is illustrated with numerical examples of an insulated Hertzian dipole immersed in different media, including lossy dielectrics, media with negative permittivity and permeability, and zero-index media.