# American Institute of Mathematical Sciences

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## Magnetic forces in and on a magnet

 Laboratoire de Génie électrique et électronique de Paris (GeePs), Universities UPMC and UPSud, Gif-sur-Yvette, France

Thanks to Alain Léger and Frédéric Bouillault for pointed questions

Received  January 2018 Revised  April 2018 Published  November 2018

Given the shape of a magnet and its magnetization, point by point, which force does it exert on itself, also point by point? We explain what 'force' means in such a context and how to define it by using the Virtual Power Principle. Mathematically speaking, this force is a vector-valued distribution, with Dirac-like concentrations on surfaces across which the magnetization is discontinuous, i.e., material interfaces. To find these concentrations, we express the force as the divergence of a (symmetric) 2-tensor which generalizes a little the classical Maxwell tensor.

Citation: Alain Bossavit. Magnetic forces in and on a magnet. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2019108
##### References:
 [1] J. G. van Bladel, Unusual boundary conditions at an interface, IEEE A.P. Mag., 33 (1991), 57-58. [2] A. Bossavit, Forces inside a magnet, Int. Compumag Soc. Newsletter, 11 (2004), 4-12. [3] A. Bossavit, Bulk forces and interface forces in assemblies of magnetized pieces of matter, IEEE Trans. Magn., 52 (2016), Art. 7003504. [4] H. S. Choi, I. H. Park and S. H. Lee, Concept of virtual airgap and its application for forces computations, IEEE Trans. Magn., 42 (2006), 663-666. [5] H. Gouin and J.-F. Debieve, Variational principle involving the stress tensor in elastodynamics, Int. J. Engng Sc., 24 (1986), 1057-1066. [6] E. Kröner, Benefits and shortcomings of the continuous theory of dislocations, Int. J. Solids & Structures, 38 (2001), 1115-1134. [7] A. R. Lee and T. M. Kalotas, A note on unconventional Gaussian surfaces, Am. J. Phys., 54 (1986), 753-754. [8] J. E. Marsden and T. J. Hughes, Mathematical Foundations of Elasticity, Prentice Hall, Englewood Cliffs, 1983. [9] P. Penfield Jr., Hamilton's principle for fluids, Phys. Fluids, 9 (1966), 1184-1194. [10] K. Reichert, H. Freundl and W. Vogt, The calculation of forces and torques within numerical magnetic field calculation methods, Compumag, (1976), 64-73. [11] W. G. V. Rosser, Classical Electromagnetism via Relativity, An Alternative Approach to Maxwell's Equations, Butterworths, London, 1968. [12] J. M. Souriau, Physics and geometry, Found. Phys., 13 (1983), 133-151. doi: 10.1007/BF01889416. [13] A. N. Wignall, A. J. Gilbert and S. J. Yang, Calculation of forces on magnetised ferrous cores using the Maxwell stress tensor, IEEE Trans. Magn., 24 (1988), 459-462.

show all references

##### References:
 [1] J. G. van Bladel, Unusual boundary conditions at an interface, IEEE A.P. Mag., 33 (1991), 57-58. [2] A. Bossavit, Forces inside a magnet, Int. Compumag Soc. Newsletter, 11 (2004), 4-12. [3] A. Bossavit, Bulk forces and interface forces in assemblies of magnetized pieces of matter, IEEE Trans. Magn., 52 (2016), Art. 7003504. [4] H. S. Choi, I. H. Park and S. H. Lee, Concept of virtual airgap and its application for forces computations, IEEE Trans. Magn., 42 (2006), 663-666. [5] H. Gouin and J.-F. Debieve, Variational principle involving the stress tensor in elastodynamics, Int. J. Engng Sc., 24 (1986), 1057-1066. [6] E. Kröner, Benefits and shortcomings of the continuous theory of dislocations, Int. J. Solids & Structures, 38 (2001), 1115-1134. [7] A. R. Lee and T. M. Kalotas, A note on unconventional Gaussian surfaces, Am. J. Phys., 54 (1986), 753-754. [8] J. E. Marsden and T. J. Hughes, Mathematical Foundations of Elasticity, Prentice Hall, Englewood Cliffs, 1983. [9] P. Penfield Jr., Hamilton's principle for fluids, Phys. Fluids, 9 (1966), 1184-1194. [10] K. Reichert, H. Freundl and W. Vogt, The calculation of forces and torques within numerical magnetic field calculation methods, Compumag, (1976), 64-73. [11] W. G. V. Rosser, Classical Electromagnetism via Relativity, An Alternative Approach to Maxwell's Equations, Butterworths, London, 1968. [12] J. M. Souriau, Physics and geometry, Found. Phys., 13 (1983), 133-151. doi: 10.1007/BF01889416. [13] A. N. Wignall, A. J. Gilbert and S. J. Yang, Calculation of forces on magnetised ferrous cores using the Maxwell stress tensor, IEEE Trans. Magn., 24 (1988), 459-462.
Notations for the 'pillbox trick'. The pillbox $\Sigma$ is a flat volume containing a part of $S$. The normal $n$ to $S$ goes from $D$ (magnetized region, here) to $D'$ (non-magnetized, air for instance). We reserve the square brackets, as here in $[M],$ to denote the jump of some quantity. The jump $[M]$ of the magnetization $M$ across surface $S$ is its value on the "upstream" side of $S$ minus its value on the "downstream" side, as both defined by the direction of the normal field $n$. Jumps of other vector or scalar quantities are defined similarly.
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