The dynamics of a particle attracted by a magnetized wire

Abstract

The dynamics of a particle attracted by a magnetized wire is studied for nonvanishing gravitational forces and a broad range of Stokes number K. The Newtonian equation of motion for the particle is integrated for 10⁻²⩽K⩽10², a range which includes conditions where the particle inertia cannot be ignored. Families of trajectories, typical of low and high K, reveal the dominance of viscous forces at low K, as expected, and show oscillatory approach to capture for high K, where inertia is significant. Capture distances in the interval 1⩽X_c⩽8 are given as a function of three independent dimensionless parameters which measure the strengths of the magnetic, viscous, and gravitational forces. The range of conditions is established for which it is permissible to neglect, for the purpose of computing capture distances, both the inertia and the radially attractive short‐range part of the magnetic force. The equation of motion in which the inertia and the short‐range term are neglected is studied. An integral of this equation is found which extends the trajectory equations of Zebel and Luborsky to include the gravitational force. A general approach to the construction of the integral of motion shows how to find the trajectory equation for a particle moving in a more complicated incompressible viscous flow with higher multipole contributions to the magnetic field of force.