Carnegie Mellon
Department of Mathematical 

Anja Schloemerkemper, Mathematics Institute, University of Oxford.

"Discrete-to-continuum limit of the magnetic force between two parts of a rigid magnetized body"


In this talk we will discuss mathematically rigorous derivations of formulas for the magnetic force between two parts of a magnetic continuous body.

Starting with a discrete, periodic setting of magnetic dipoles we consider the force between the dipoles in a part of a bounded subset of $\mathbb{R}^3$ and in its complement. For the passage to the continuum an appropriate force formula is formulated on a scaled lattice and the continuum limit is performed by letting the scaling parameter tend to zero. This involves a regularization of the occurring hypersingular kernel. It turns out that the limiting force formula is different from a corresponding formula which has been known for long in the physics literature (cf. e.g. W.F. Brown). If time allows, we will briefly look at a mathematical derivation of this formula. While the classical force formula contains a surface density which depends nonlinearly on the normal, the discrete-to-continuum limit yields a surface term which involves a certain lattice sum and which is linear in the normal. This agrees with Cauchys theorem in continuum mechanics.

TUESDAY, November 19, 2002
Time: 1:30 P.M.
Location: PPB 300