On the electromagnetic force on a moving dipole

Abstract

If a compact bunch of charged particles or current rings moves in an electromagnetic field, the force on the bunch gets an additional contribution which has a form of interaction between the electric dipole moment of a bunch and a magnetic field. This comes besides the relativistic transformation of the electric and magnetic dipole moments of a moving bunch, and possible combined effects are illustrated with simple examples.

Relativistic Transformation of Magnetic Dipole Moment

“…We observe that I 1 = I 3 = I 2 = I 4 , which simply indicates that the contribution of convective current of positive ions into the total current (measured by means of crosssectional area resting in a laboratory) is different in the different segments of the circuit (on this subject see also Refs. [4,10,16]). As we have mentioned above, such a convective current of immovable charges (in the rest frame of dipole) does not contribute to the magnetization and magnetic dipole moment of moving medium and thus, in physically meaningful "configurational" definition of magnetization/magnetic dipole moment, we have to replace in Equations (8), (5) the total current density j by the proper charge density j pr , arriving at the new definitions…”

“…Hereinafter the subscript "//" stands for the component collinear to v, while the subject "⊥" denotes the component to be orthogonal to v. Equations (3a)-(3b) imply that the magnetic dipole moment obeys the transformation (see, e.g., Ref. [4])…”

Torque on a Moving Electric/Magnetic Dipole

“…[3,5]), and f h is the density of force due to hidden momentum contribution, whose physical meaning will be discussed below. Here ρ total , j total are the total charge density and current density, correspondingly.…”

“…(10) for small dipole, we can take the fields E(r), B(r) to be constant within the volume of such a dipole, because any terms, which include partial spatial derivatives of the electric and magnetic field become negligible, when the inequalities (11) are adopted † . Thus, in the subsequent integrals over the volume of dipole, † We notice that the approximation of constant fields cannot be adopted in calculation of force acting on a small dipole, because some of the force components are vanishing at E(r), B(r)= constant, and we have to involve the terms, containing their spatial derivatives (see, e.g., [5,15]). …”

On the electrodynamics of moving permanent dipoles in external electromagnetic fields

“…Introduction. The electrodynamics of moving media is a complex subject that has been discussed in several papers, textbooks and monographs [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18], yet continues to attract attention for its practical applications as well as its relevance to fundamental issues involving field-matter interactions. In the early days of the 20 th century, Minkowski used ideas from the newly developed theory of relativity to analyze the dynamics of arbitrarily moving bodies in the presence of electromagnetic (EM) fields.…”