Interdisciplinary Applied Mathematics

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In vacuum, B = mqH, where mq = 4п x 10-7 Hm-1 is the permeability of free space, so a magnetic field strength H = 1000 Am-1 would correspond to a magnetic flux density B = 1.257 x 10-3 T. In (Doyle et al., 2002), an external magnetic flux density of about 10 mT was used, corresponding to


a magnetic    field    strength    H =    10,000    Am-1.    In    (Hayes    et    al.,    2001),    the


magnetic flux density imposed in the liquid phase was about 50 mT, or 500 gauss in terms of cgs units. Also, the flux densities used in (Furst et al., 1998), were around 3 mT, corresponding to 30 gauss or H = 2300 Am-1.


The magnetic flux density due to an isolated dipole moment is


B



fv(


4n



m • x


гз



)•



In the presence of an external uniform magnetic field with flux density Bo, we obtain



B



Po (3(m • r)r 4-7Г I r5



m


з



+ Bo.


The corresponding force acting on the dipole in the presence of an ambient magnetic field is


F = Vx (B x m)= m • VB.


This force may be written in terms of a scalar potential F = —VU with U = —m • B. The torque acting on a dipole that has a fixed moment is


G = m x B,


and the dipole will tend to turn to align with the magnetic field.


Paramagnetic beads placed in a uniform magnetic field will acquire a dipole moment    aligned    with    the field.    The    dipoles    will    then    attract    each

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