Ponderomotive forces in «axially magnetized torus - soft magnetic orb» system

Аuthors

Andreev A. K.

e-mail: alexande_andreev@yahoo.com

Abstract

The field H(r,z) of axially magnetized torus in the vicinity of its axis of symmetry Z has been numerically studied. It is shown that the profile of the field allows construct a simple mathematical model to determine ponderomotive forces in the torus — orb system. Unloaded (in the absence of external forces f = 0) soft magnetic orb in the field of the torus has three equilibrium positions near H_max: in the center of the torus, over the torus and under the torus (areas C, A and B, respectively) — Fig. 1

Stable position of the orb under the load f is realized under condition of dH(r,z)/dz < 0. Integral equation for orb magnetizing can not be solved by such model, since it assumes a simplified calculation of the orb polarization in the field of the torus with surface charges averaging over two hemispheres of the orb, and solution of the forces affecting magnetic charges of these hemispheres.

The model allows further simplification by introducing into consideration of effective magnetic charges located in the center of gravity of the hemispheres charges on the Z-axis. The forces acting on the point charges are calculated. Effective charges provide the charge neutrality of the orb. In this manner, the model is reduced to one-dimensional model. The model simplifies the calculations without loss of accuracy. Radial r-component of the field of the torus is much smaller than Z-component. —component of the field H(r,z) is relatively linear in the computation domain. It serves as a basis for the introduction of the one-dimensional model.

In this paper, the calculations were made for the torus of the size of 9,7 × 4 × 1,25 cm. The magnetization of the torus is set to M_z = 915 emu/cm3. The saturation magnetization of the orb is set to M_z = 1620 emu/cm3.

The diameter of the orb is 2R = 1,15 cm, the gravity weight is equal to 6,121·103 dyne. The displacements of the orb under its own weight relative to the unloaded orb: in the area A — ∆A = 0,353 cm, in the area B — ∆B = 0,741 cm, where ∆A < ∆B .

The maximum values of ponderomotive forces, at which the position the of the orb is steady, in area A — F_Amax = 2,072·104 dyne, in area B — F_Bmax = 6,846· 103 dyne. The ratio of the maximum forces in the areas A and B is equal to: F_Amax^/F_Bmax = 3,027. When |f | > F_max, an abrupt reposition of the orb occurs: fr om area A to area C, and from areas C and B to (- ∞).

Levitation ranges in the area A — ∆A = 1.164 cm, in the area B — ∆B = 1.217 cm, wh ere ∆A < ∆B Displacement of the unloaded orb relative to the maximum value of the field is equal to: +10-3 cm in the area A and −103 cm in the area B. The area C is not considered in this paper.

Numerical calculations explain the experimentally observed significant difference in the displacement of the orb in the field of gravity and maximal forces F_Amax and F_Bmax above and under the torus. When the diameter of the orb is greater than it reaches the area of the sign reversal of the torus field, and the maximum value of the force is not defined in the model.

The experiment showed that the ratio of maximum calculated force F_max to the force measured in the area A is equal to 1,37. It is obvious that the accuracy of the model increases for the samples of smaller diameter.

Taking into account the given above data it can be concluded that the model can be implemented for semi- quantitative evaluation of the power characteristics of the system torus-orb.

Keywords:

magnetic levitation, magnetization, axially magnetized torus, ponderomotive forces, soft magnetic orb, magnetostatic potential, magnetic charges, demagnetization factor

References

1. Urman Yu.M., Bugrova N.A., Lapin N.I. Zhurnal tekhnicheskoi fiziki, 2010, vol. 80, no. 9, pp. 25-33.

2. Kuznetsov S.I., Urman Yu.M. Zhurnal tekhnicheskoi fiziki, 2006, vol. 76, no. 3, pp. 7-14.

3. Tamm I.E. Osnovy teorii elektrichestva (Basics of the theory of electricity), Moscow, Nauka, 1976, 660 p.

4. Geim A. Everyones magnetism. Physics today, 1998, vol. 51, pp. 36-39.

5. Simon M.D., Geim A.K. Diamagnetic levitation; flying frogs and floating magnets. Journal of applied physics, 2000, vol. 87, no. 9, pp. 6200-6204.

6. Simon M.D., Heflinder L.O., Geim A.K. Diamagnetically stabilized magnet levitation. American journal of physics, 2001, vol. 69, no. 6, pp. 702-713.

7. Martynenko Yu.G. Dvizhenie tverdogo tela v elektricheskikh i magnitnykh polyakh (Movement of a solid body in electric and magnetic fields), Moscow, Nauka, 1988, 368 p.

8. Martynenko Yu.G. Sorosovskii obrazovatelnyi zhurnal, 1996, no. 3, pp. 82-86.

9. Andreev A.K. Magnitostatika ferromagnetikov (Magnetistatics of ferromagnetics), Moscow, MAI, 2011, 162 p.

10. Magnitnye polya v gerkonakh (Magnetic fields in reed switches), http://tube.sfu-kras.ru/video/271, 1983.

11. Chernyshov N.N., Lupikov V.S., Baida E.I., Kryukova N.V., Gelyarovskii O.A., Mvudzho E. Voprosy atomnoi nauki i tekhniki, 2008, no. 1, pp. 115-122.

12. Andreev A.K. Svidetelstvo o Gosregistracii 2012614673, 25.05.2012.

13. Kogen-Dalin V.V., Komarov E.V. Raschet i ispytanie sistem s postoyannymi magnitami (Calculation and test of systems with permanent magnets), Moscow, Energiya, 1977, 247 p.

14. Vladimirov V.S. Uravneniya matematicheskoi fiziki (Equations of mathematical physics), Moscow, Nauka, 1971, 512 p.

15. Parsell E. Elektrichestvo i magnetism (Electricity and magnetism), Moscow, Nauka, 1975, 439 p.