Mansuripur’s Paradox

An electrically neutral current-loop,1 with magnetic dipole m0, that is at rest in a static, uniform electric field E0 experiences no force or torque. However, if that system is observed in the lab frame where the loop has velocity v parallel to E0, and v c, where c is the speed of light, then there appears to be an electric dipole moment,2 p = v/c×m0 associated with the loop (in Gaussian units). Näıvely, the torque on this moment due to the electric field (which has strength E = E0 +O(v2/c2) in the lab frame) is τ = p × E ≈ E0vm0/c. Can/should the torque be different in different frames of reference? This paradox was first posed by Spavieri [3]3 and more recently by Mansuripur [7].4,5 It is a conceptual variant of a famous problem by Shockley [16] that introduced the concept of “hidden mechanical momentum.”6 The paradox is compounded by supposing the static, “uniform ” electric field is due to a single electric charge q at large, fixed distance from the magnetic moment in the rest frame of the latter, and the lab-frame velocity v is along the line of centers of the charge and moment. Discuss the force on charge q in the lab frame.7 1This problem avoids the delicate issue of whether a current loop actually is electrically neutral, näıvel