Stability of an electromagnetically levitated spherical sample in a set   of coaxial circular loops
  This paper presents a theoretical study of oscillatory and rotational instabilities of a solid spherical body, levitated electromagnetically in axisymmetric coils made of coaxial circular loops. We apply our previous theory to analyze the static and dynamic stability of the sample depending on the ac frequency and the position of the sample in the coils for several simple configurations. We introduce an original analytical approach employing a gauge transformation for the vector potential. First, we calculate the spring constants that define the frequency of small-amplitude oscillations. For static stability, the spring constants must be positive. Dynamic instabilities are characterized by critical ac frequencies that, when exceeded, may result either in a spin-up or oscillations with increasing amplitude. We found that the critical frequencies increase with the nonuniformity of the field. We show that for a spherically harmonic field, the critical frequency for the spin-up instability in a field of degree $l$ coincides with the critical frequency for the oscillatory instability in a field of degree $l+1$. 
J. Priede and G. Gerbeth