S. Gov

On the dynamic stability of the hovering magnetic top

Abstract In this paper we analyze the dynamic stability of the hovering magnetic top from first principles without using any preliminary assumptions. We write down the equations of motion for all six degrees of freedom and solve them analytically around the equilibrium solution. Using this solution we then find conditions which the height of the hovering top above the base, its total mass, and its spinning speed have to satisfy for stable hovering. The calculations presented in this paper can be used as a guide to the analysis and synthesis of magnetic traps for neutral particles.

Magnetic trapping of neutral particles: Classical and quantum-mechanical study of a Ioffe–Pritchard type trap

Recently, we developed a method for calculating the lifetime of a particle inside a magnetic trap with respect to spin flips, as a first step in our efforts to understand the quantum mechanics of magnetic traps. The one-dimensional toy model that was used in this study was physically unrealistic because the magnetic field was not curl free. Here, we study, both classically and quantum mechanically, the problem of a neutral particle with spin S, mass m, and magnetic moment μ, moving in three dimensions in an inhomogeneous magnetic field corresponding to traps of the Ioffe–Pritchard “clover-leaf” and “baseball” type. Defining by ωp, ωz, and ωr the precessional, the axial, and the lateral vibrational frequencies, respectively, of the particle in the adiabatic potential Veff =μ|B|, we find classically the region in the (ωr/ωp)−(ωz/ωp) plane where the particle is trapped. Quantum mechanically, we study the problem of a spin-one particle in the same field. Treating ωr/ωp and ωz/ωp as small parameters for the pe...

On the spinning motion of the hovering magnetic top

Abstract In this paper we analyze the spinning motion of the hovering magnetic top. We have observed that its motion looks different from that of a classical top. A classical top rotates about its own axis which precesses around a vertical fixed external axis. The hovering magnetic top, on the other hand, has its axis slightly tilted and moves rigidly as a whole about the vertical axis. We call this motion synchronous, because in a stroboscopic experiment, we see that a point at the rim of the top moves synchronously with the top axis. We show that the synchronous motion may be attributed to a small deviation of the magnetic moment from the symmetry axis of the top. We show that as a consequence, the minimum angular velocity required for stability is given by μH/(I 3 −I 1 ) for I3>I1 and by 4μHI 1 /I 3 2 for I3 4μHI 1 /I 3 2 both for I3>I1 and for I3 We also give experimental results that were taken with a top whose moment of inertia I1 can be changed. These results show very good agreement with our calculations.

Efficient moment-method solution for the centered shielded magnetoresistive head

We have solved the field equations for a magnetoresistive shielded head by a Galerkin-type moment method (MM), where the basis functions are chosen to satisfy exactly the edge condition at the heads corners. This choice of the basis expansion functions greatly improves the accuracy and convergence rate of the solution compared to those for MM expansion functions that are not singular.

Upper bounds on the height of levitation by permanent magnets

The limitations on the levitation height obtainable using permanent magnets are studied. We find that with mechanical transverse stabilization and optimal distribution of magnetization of the (infinite) base a maximum height of 12l/sub 0/ is achievable for ideal magnets. Here, l/sub 0/=M/sub 0//sup 2///spl rho/g where g is the free fall acceleration, /spl rho/ the mass density of the levitated magnet and M/sub 0/ is the magnetization of the supporting base plate and levitated magnet. For uniform magnetization this height is only 3.5l/sub 0/. With levitation based on spin transverse stabilization mechanism an additional constraint enters into the problem and we find that the maximum achievable height is only 4.7l/sub 0/. The value of l/sub 0/ for the best magnet known to us is about 1.5 meters. Note, however, that because real permanent magnets are not ideal the above results are only upper bounds on the levitation height by permanent magnets.

Levitating a spinning magnetic top above an air coil

We study the levitation of a spinning magnetic top above an air coil carrying a DC current. On the theoretical side we carry out an optimizing process for the geometry of the coil that minimizes the power consumption for a given hovering height, h, and a given top. We find that for a rectangular cross-section coil, carrying a uniform current density the normalized dimensions of the optimum coil are: d/h=0.83, R/sub 1//h=1.85 and R/sub 2//h=4.2 where d, R/sub 1/ and R/sub 2/ are the thickness, inner radius and outer radius of the coil in [cm], respectively. For this specific coil the power itself is given in [Watts] by P=A/spl rho/g/sup 2/h/sup 3///spl sigma//sup 2/. Here, A=2720 [Amp/sup 2/ emu/sup 2//erg/sup 2/ cm/sup 2/], /spl rho/ is the coils resistivity in [/spl Omega/ cm], g is the free fall acceleration in [cm/sec/sup 2/] and /spl sigma/ is the magnetization per unit mass of the top in [emu/gr]. A similar optimizing scheme is done for a semi-infinite coil. We look for the optimal distribution of the current density so as to minimize the power consumption. In this case we show that A=491[Amp/sup 2/ emu/sup 2//erg/sup 2/ cm/sup 2/]. On the experimental side we hovered the top above two different coils with measurements in reasonable agreement with the calculation.