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Yu. M. Urman, "Method of irreducible tensors in problems of rotation of a conductive body in nonuniform magnetic fields," Mech. Solids. 36 (2), 8-15 (2001)
Year 2001 Volume 36 Number 2 Pages 8-15
Title Method of irreducible tensors in problems of rotation of a conductive body in nonuniform magnetic fields
Author(s) Yu. M. Urman (Nizhni Novgorod)
Abstract The ponderomotive interaction of a moving conductive body with a nonuniform magnetic field has a significant feature that mechanical degrees of freedom are coupled due to eddy currents. The coupling results in a "pumping" of energy from one degree of freedom to another. In addition, a number of destabilizing factors appear which are unusual for the traditional interaction of a conductor with a magnetic field. As a result, the deceleration of the body changes to its acceleration, a repulsive force changes to an attractive one, a "negative friction" and a nonconservative "circulatory force" appear.

To get full understanding of the interaction between a moving conductive body and magnetic field one should solve simultaneously the equations of electrodynamics and mechanics. However, this problem is extremely complicated and cannot be solved analytically in its general formulation even for simple geometric bodies. There are a number of papers [1-3] devoted to the solution of this problem by asymptotic methods based on the assumption that the depth of penetration of the field into the conductor is small or large. For spherical bodies (a ball, a spherical shell) and some specific types of their motion, one can successfully solve the boundary value problem in a fairly general formulation and find the ponderomotive interaction of the body and the field. Although it is a particular problem, the analysis of the ponderomotive interaction allows one to reveal a number of effects which are of interest as physical phenomena and as factors affecting the stability of the body supported by the field. Moreover, this problem has an independent value since the eddy currents due the motion of a conductive body in a magnetic field (together with viscous friction) is a widespread source of dissipative forces.

As shown in [4], the torque in general can be decomposed into the conservative and nonconservative components. The conservative component of the torque is generated by one scalar function, while the nonconservative component is generated by two scalar functions depending on the magnitude and direction of the angular momentum.

In the present paper, these functions are calculated for the case of rotation of a conductive ball in a nonuniform magnetic field. Also, it is shown that, as was the case for the net torque, the net force acting on the body can be decomposed into the conservative and nonconservative components.
References
1.  A. N. Kobrin and Yu. G. Martynenko, "Dynamics of a conductive rigid body in a high-frequency magnetic field," Doklady AN SSSR, Vol. 255, No. 5, pp. 1063-1065, 1980.
2.  A. N. Kobrin and Yu. G. Martynenko, "A motion of a conductive rigid body about the mass center in a slowly varying magnetic field," Doklady AN SSSR, Vol. 261, No. 5, pp. 1070-1073, 1981.
3.  Yu. G. Martynenko, Motion of a Rigid Body in Electrical and Magnetic Fields [in Russian], Nauka, Moscow, 1988.
4.  Yu. M. Urman, "Method of irreducible tensors in problems of evolution motions of a rigid body with a fixed point," Izv. AN. MTT [Mechanics of Solids], No. 4, pp. 10-20, 1997.
5.  J. Jackson, Classical Electrodynamics [Russian translation], Mir, Moscow, 1965.
6.  Yu. M. Urman, "Irreducible tensors and their application to the problems of rigid body motion in force fields," in Mechanics of a Rigid Body [in Russian], Vol. 15, pp. 75-87, Naukova Dumka, Kiev, 1983.
7.  D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of the Angular Momentum [in Russian], Nauka, Leningrad, 1975.
8.  L. D. Landau and E. M. Lifshits, Theoretical Physics. Volume 8. Electrodynamics of Continuous Media [in Russian], Nauka, Moscow, 1982.
9.  R. V. Lin'kov and Yu. M. Urman, "A rotating ball in a magnetic field," Zh. Tekhn. Fiziki, Vol. 43, No. 12, pp. 2472-2480, 1973.
Received 22 October 1998
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