Biomedical Engineering Reference
In-Depth Information
�
1
µ
-
1
�
r
r
3
m
=
π
a H
(9.44)
2
µ
+
2
where
a
is the dipolar distance. This induced magnetization creates the induced
magnetic field
�
1
� �
� �
H
=
(3 ( .
i
i m m
)
-
)
i
3
4
π
r
�
�
at a point defined by the vector
ri
(from the dipole center). In the total field
H
, the
induced magnetic field of a microparticle of radius
a
is
�
3
�
a
1
µ
µ
-
1
r
(9.45)
H
=
H
i
3
4
+
2
r
r
Take two same microbeads—of radius
a
—referred to by indices 1 and 2. The mag-
netic field at the center of each bead is
� � �
H H H
=
+
1
0
i
,12
� � �
H
=
H H
+
2
0
i
,21
Using (9.45) with the corresponding indices, we find the coupled system
� �
3
�
1
µ
µ
-
1
a
r
r
H H
=
+
H
1
0
2
3
4
+
2
r
(9.46)
� �
3
�
a
1
µ
µ
-
1
r
r
H
=
H
+
H
2
0
1
3
4
+
2
r
1
µ
-
1
r
r
Let
α
=
, we can solve (9.46) and we find
4
µ
+
2
� �
æ
ö
1
H H
=
1
0
ç
÷
3
è
1
-
α
u
ø
with
u = a/r
. For
r =
a (when the two beads contact)
� �
æ
1
ö
H H
=
ç
÷
1
0
è
ø
1
-
α
Using the formulation of the magnetic force, the force exerted by one sphere on the
other is
�
6
2
a
α
2
F
= -
6
µ π
H
1
0
0
4
3
r
æ
3
ö
a
r
æ
ö
1
-
α
ç
ç
÷
÷
è
ø
è
ø
