  In this paper, we study the theory of second gradient electromagnetostatics
as the static version of second gradient electrodynamics. The theory of second
gradient electrodynamics is a linear generalization of higher order of
classical Maxwell electrodynamics whose Lagrangian is both Lorentz and
U(1)-gauge invariant. Second gradient electromagnetostatics is a gradient field
theory with up to second-order derivatives of the electromagnetic field
strengths in the Lagrangian. Moreover, it possesses a weak nonlocality in space
and gives a regularization based on higher-order partial differential
equations. From the group theoretical point of view, in second gradient
electromagnetostatics the (isotropic) constitutive relations involve an
invariant scalar differential operator of fourth order in addition to scalar
constitutive parameters. We investigate the classical static problems of an
electric point charge, and electric and magnetic dipoles in the framework of
second gradient electromagnetostatics, and we show that all the electromagnetic
fields (potential, field strength, interaction energy, interaction force) are
singularity-free unlike the corresponding solutions in the classical Maxwell
electromagnetism as well as in the Bopp-Podolsky theory. The theory of second
gradient electromagnetostatics delivers a singularity-free electromagnetic
field theory with weak spatial nonlocality.
