ABSTRACT
Combining Equation 4.5 and Equation 4.3, the force on a small spherical weakly magnetic particle placed in the external magnetic fi eld can be written as
F m [ m
or in a simplifi ed form (assuming that k << 1),
(1/2) m 0 k V ( H 2 ) (4.7)
If a paramagnetic particle (volume magnetic susceptibility: k p ) is immersed in the fl uid (volume
magnetic susceptibility: k f ), the magnetic force per unit volume acting on a particle is given by
Equation 4.7 (using k k p – k f ). For practical calculations it is sometimes advantageous to replace
the magnetic fi eld strength by the magnetic induction B . Then, Equation 4.7 reads as follows:
F m ( k / m
Here B is considered as the external magnetic induction, and B is the gradient of the magnetic induction. Thus, in the direction of x , the magnetic force F
H and magnetization k H x I
m 0 V k ( H x · d H x / d x ) m 0 V I x · d H x / d x (4.9)
Magnetic force is proportional to the product of the external magnetic fi eld and the fi eld gradient and has the direction of the gradient. In a homogeneous magnetic fi eld, in which B 0 or d H
0, the force to change the position of a particle is zero. Along the axis of symmetry, and for values of the external magnetic fi eld H
x0 lower than the bulk saturation value H
the expression
H x H
where a is the rod radius and r is the radius from the center of the rod, as shown in Figure 4.1. For H
H x H
< H s along the x axis of the particle, when the magnetization of
the particles is small, is given as
d H x / dx –2 H
If the Equation 4.10, Equation 4.12, and Equation 4.6 are combined, the approximate expression for the magnetic force on a pointlike spherical weakly magnetic particle is given as
F m –(8/3) p m
where b is the particle radius. Equation 4.13 can be rewritten for a matched system a 3 b as
F m –(75/128) p m
TA B
LE 4
.1
