Abstract
There is a great variety of formulas purporting to describe the force field inside magnetized matter. They don’t always agree, which is puzzling. Starting from the Korteweg–Helmholtz formula, obtained thanks to the highly reliable Virtual Power Principle (VPP), we show how variant expressions can result from algebraic manipulations that assume, without making this explicit, extra physical hypotheses. Those we discuss here, assuming a B = μH magnetic law, are (1) Dependence of μ on density, (2) Incompressibility of the magnetic medium in which the magnetic forces develop.
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- 1.
We shall not attempt to deal with anisotropy here. For this, one should make \(\tilde {\nu }\), no longer a scalar but a symmetric 3 × 3 matrix, a function of the symmetrized gradient ∇ S u instead of div u, subject to a condition of invariance with respect to rotations that comply with material frame indifference.
- 2.
Which derives from the tensor
, not from 
. - 3.
One made of two close magnetic charges of opposite signs, not of a small current loop.
, not from 
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Bossavit, A. (2017). Topics in Magnetic Force Theory: Some Avatars of the Helmholtz Formula. In: Quintela, P., et al. Progress in Industrial Mathematics at ECMI 2016. ECMI 2016. Mathematics in Industry(), vol 26. Springer, Cham. https://doi.org/10.1007/978-3-319-63082-3_19
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DOI: https://doi.org/10.1007/978-3-319-63082-3_19
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