Dynamical interactions between two uniformly magnetized spheres

Abstract

Studies of the two-dimensional motion of a magnet sphere in the presence of a second, fixed sphere provide a convenient venue for exploring magnet-magnet interactions, inertia, friction, and rich nonlinear dynamical behavior. These studies exploit the equivalence of these magnetic interactions to the interactions between two equivalent point dipoles. We show that magnet-magnet friction plays a role when magnet spheres are in contact, table friction plays a role at large sphere separations, and eddy currents are always negligible. Webbased simulation and visualization software, called MagPhyx, is provided for education, exploration, and discovery.

Periodic bouncing modes for two uniformly magnetized spheres. II. Scaling

“…3 In this study, we take advantage of our recent proof that simple dipolar interactions exactly describe the magnetic interactions between uniformly magnetized spheres 4 and build upon our subsequent studies of dynamical interactions between spheres that remain in contact, both with and without friction. 5,6 Our 1497 modes exhibit a wide variety of behaviors, including simple modes with small m and n and complicated modes with large m and n. Mode (157,580,2) is the most complicated that we discovered. With E mnp = −0.003 999 and T mnp = 11 298, this mode requires about 16 000 Runge-Kutta time steps to integrate.…”

“…The former scaling relationship applies only to threshold energies E mnp , that is, the minimum energy for mode (m, n, p), and the associated period T mnp . As the energy E increases beyond this minimum for a particular mode, the period T increases for some modes [such as (2,1,2), (2,3,2), and (3,1,2)] and decreases for others [such as (1,1,2), (1,2,2), (2,5,2)].…”

Periodic bouncing modes for two uniformly magnetized spheres. I. Trajectories

“…Two conditions must be met during such collisions: (a) the angle of incidence equals the angle of reflection and (b) the kinetic energy (both rotational and translational) is conserved (Ref. 19, Sec. 7.2) (Sec.…”

“…As described in detail in Ref. 19, we simulate a free magnet's interaction with a fixed magnet using fourth-order Runge-Kutta integration of Eqs. (11).…”

Periodic nonlinear sliding modes for two uniformly magnetized spheres

“…We showed recently that this field exerts forces and torques on a second uniformly magnetized sphere that are identical to the forces and torques on a point dipole. 19,20 Namely, for a second sphere of total dipole moment m 2 centered at position r, the energy of interaction, force, and torque are, respectively, U ¼…”