  In this work the fundamentals of gradient field theories are presented and
reviewed. In particular, the theories of gradient magnetostatics and gradient
elasticity are investigated and compared. For gradient magnetostatics,
non-singular expressions for the magnetic vector gauge potential, the
Biot-Savart law, the Lorentz force and the mutual interaction energy of two
electric current loops are derived and discussed. For gradient elasticity,
non-singular forms of all dislocation key-formulas (Burgers equation, Mura
equation, Peach-Koehler stress equation, Peach-Koehler force equation, and
mutual interaction energy of two dislocation loops) are presented. In addition,
similarities between an electric current loop and a dislocation loop are
pointed out. The obtained fields for both gradient theories are non-singular
due to a straightforward and self-consistent regularization.
