# Interdisciplinary Applied Mathematics

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Magnetic Forces

In a permeable magnetic medium the magnetic flux density B (units of T) is due to the external field and the induced magnetic moments within the material. The governing equations are

B = Mq(H + M),

VB = 0,

VxH = j;

Here j is the macroscopic current density that is generated externally and M is the magnetic dipole density (Am 1) induced in the material. The latter is written in terms of the magnetic field strength H (units of Am-1) as

M = xH,

where x is the magnetic susceptibility of the material. For a particle, the overall dipole moment m is the volume integral of the dipole density over the particle volume,

m = / M d3x,

so for a uniform sphere of radius a subject to an external uniform magnetic field with flux density Bq,

m

4

3

^a3xH

4

-тта

3

Bo

X—•

Mo

In general, the susceptibility will be a nonlinear function of the field strength, and the induced dipole moment density will saturate at some level. Saturation often occurs when the magnetic flux density in the material reaches about 1.0 to 1.5 tesla. The Langevin theory provides an estimate for the susceptibility including the effect of thermal energy. In a medium in which the dipole elements interact through their induced fields, a modified estimate for the overall dipole moment must be made that leads to a higher effective susceptibility. For the beads used in the experiments of (Promislow et al., 1995), the effective volumetric susceptibility was x = 0.9.

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