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By Bossavit A.

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A function is continuously differentiable in D if all its partial derivatives are in C0(D), and one denotes by C 1(D) the set of such functions (an infinite-dimensional linear space). Similarly, C k(D) or C ∞(D) denote the spaces composed of functions which have continuous partial derivatives of all orders up to k or of all orders without restriction, inside D. In common parlance, one says that a function “is C k ”, or “is C ∞ ” in some region, implying that there is a domain D such that C k(D), or C∞(D), includes the restriction of this function to D as a set element.

It deals with polarizable materials, in which charges are too strongly bound to separate from their original sites, but loose enough to be pulled a little off their equilibrium position by Coulomb forces, when the material is subject to a macroscopic electric field. This polarization phenomenon is important for some materials, dubbed dielectric. The simple reasoning (or myth . . ) that follows shows how to account for it, by a simple relation between e and the p of (3). Despite its electrical neutrality at a macroscopic scale, matter contains positive and negative charges (+ and – for brevity) which we may imagine as being attached by pairs at certain material sites.

So let us just review typical constitutive laws about magnetization. Apart from amagnetic materials (m = 0), a simple case is that of paramagnetic or diamagnetic materials, characterized by the linear law m = χh (whence b = µh, with µ = (1 + χ)µ0), where the magnetic susceptib i l i t y χ is of positive or negative sign, respectively. It can be a tensor, in the case of anisotropic materials. For most bodies, χ is too small to matter in numerical simulations, the accuracy of which rarely exceeds 1 % (χ ~ 10−4 for Al or Cu).

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