Biomedical Engineering Reference
In-Depth Information
�
external field,
H
is a Neumann boundary condition for (9.20). After solving for
f
,
0
�
®
�
H
is obtained by
H
= -
grad
φ
.
( )
�
�
�
Note that
H
is different from
H
in the vicinity of the rod:
H
is the sum of the
0
�
�
H
. The magnetic force field on the paramag-
netic microparticles is then obtained from
H
external field,
H
and the induced field
i
0
�
by calculating
�
� �
µ
0
2
F
=
µ χ χ
v
(
-
) (
H H
. )
Ñ
=
v
(
χ χ
-
)
Ñ
H
(9.21)
m
0
p
p
f
p
p
f
2
and the value of the magnetic force field is then plugged into the trajectory equation
�
� �
dV
1
æ
ö
p
2
m
=
v
D Ñ
χ
H
-
6
πη
r V V
(
-
)
+
g v
D
ρ
y
ˆ
ç
÷
p
h
p
f
p
è
ø
d t
2
This is the general case. However, in the special case of the ferromagnetic wire, the
calculation of the particles trajectories may be further detailed because the magnetic
field may be calculated analytically [13].
9.5.2 Analytical Solution for the Magnetic Field
The total magnetic field in the vicinity of the rod is the sum of the external magnetic
field plus the induced field due to the presence of the rod. It can be shown [9] that,
at a distance
r
from the rod center and at angle
q
from the external magnetic field
H
0
, the magnetic potential is
1
æ
ö
2
-
1
φ
= -
H r
+
Ma r
cos
θ
(9.22)
ç
÷
ext
0
è
ø
2
At this stage, two different cases may be distinguished:
1. At very high external field, the rod is magnetically saturated and the total
field,
H
�
is given by
A
æ
ö
ext
r
H
=
cos
θ
H
+
ç
÷
0
è
2
ø
r
(9.23)
A
æ
ö
ext
H
=
sin
θ
-
H
+
ç
÷
0
θ
è
2
ø
r
1
2
2
where
A
=
M a
. The magnetic force is then
s
A
æ
ö
+
H
cos2
θ
A
0
ç
÷
2
F
= -
2
µ χ χ
(
-
)
v
(9.24)
r
m
0
p
f
P
3
ç
÷
r
è
H
sin2
θ
ø
0
2. At sufficiently low magnetic field, the magnetization is linear homogeneous
and (9.22) leads to
