# American Institute of Mathematical Sciences

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October  2019, 12(6): 1589-1600. doi: 10.3934/dcdss.2019108

## 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, 2019, 12 (6) : 1589-1600. doi: 10.3934/dcdss.2019108
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##### References:
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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