Losses Caused by Rotational Motion of Particles
In the preceding paragraph the ferromagnetic objects are assumed to remain fixed with respect to the surrounding medium. If, however, this medium is a liquid, the alternating magnetic field may also cause oscillating or rotating motions of the par ticles. In cases where the field amplitude does not exceed a certain critical value, magnetization of the particles remains essentially unaffected and the particle reacts as permanent magnet with a certain mass of inertia giving rise to losses caused by friction in the surrounding liquid. To illustrate this, the stationary rotation (frequency v) of a sphere (mass m, density p) in a liquid (viscosity j) forced by a rotating magnetic field may be investigated. According to the theory of viscous friction (Landau and Lifshitz, 1978), the losses per cycle related to the mass of the sphere will be:
Interestingly, these specific losses depend neither on the size of the sphere nor on the field amplitude, provided that the amplitude is sufficiently strong to overcome the torque exerted by the viscous friction. The required minimum field amplitude is given by:
where MR is the remanent magnetization of the rotating sphere. Introducing the viscosity of water (j = 10~3 Pa-s) and a frequency of 70 kHz, the minimum field for magnetite (Fe3O4) assuming MR a 2.3 x 105 A m-1 evaluates as Hmin a 10 kA m-1. Then, according to Eq. (4.13), the losses related to the mass of the sphere achieve Wrot a 3 J kg-1. Rotational losses in rotating fields of 5 kA m-1 at 500 Hz were investigated by Knauft et al. (2004), using macroscopic spheres of NdFeB in silicone oil. Since the losses described by Eq. (4.13) are independent of the size of the sample, small particles in an aqueous liquid may also exhibit rotational losses, provided that they are in a suspended state and not immobilized by sticking on solid tissue elements.
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