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5.1.2. Dielectrophoresis in Media
For a particle with complex dielectric permittivity
e
p
in a medium
e
m
, the internal
electric displacement is the sum of the particle polarization and the external
displacement:
*
i
¼ e
p
*
i
¼
*
þ e
m
*
i
:
Therefore, the polarization scales with the internal electric field as
*
i
:
*
e
p
e
m
¼
This polarization generates an additional surface charge dQ on each surface
element dF:
dF
¼
*
d
*
dQ
¼ s
pol
:
So, applying Coulomb's Law and combining this result with the unperturbed field,
we find that
ZZ
ZZ
h
i
*
12
4
pe
m
r
12
*
12
4
pe
m
r
12
*
i
ð
*
1
Þ¼
*
e
ð
*
1
Þ
dQ
2
¼
*
e
ð
*
1
Þ
*
ð
*
2
Þ
d
*
2
ZZ
h
i
*
12
4
pe
m
r
12
¼
*
e
ð
*
1
Þ e
p
e
m
*
i
ð
*
2
Þ
d
*
2
:
Since we assumed that the particle is linearly polarizable and, for approximation
purposes, considering only the polarization parallel to the electric field, it follows
that:
*
i
*
¼ a
*
e
¼
e
p
e
m
(
)
ZZ
h
i
*
12
4
pe
m
r
12
a
e
p
e
m
*
e
ð
*
1
Þ
*
e
ð
*
1
Þa
Proj
*
e
ð
*
1
Þ
*
e
ð
*
2
Þ
d
*
2
!
"
#
ZZ
*
12
*
e
ð
*
1
Þ
*
2
*
e
ð
*
2
Þ
d
*
2
r
12
a
e
p
e
m
a
*
e
ð
*
1
Þ
*
e
ð
*
1
Þ
4
pe
m
*
e
ð
*
1
Þ
!
e
ð
*
1
Þ
e
p
e
m
! a
e
m
"
#
ZZ
*
12
*
e
ð
*
1
Þ
*
2
*
e
ð
*
2
Þ
d
*
2
r
12
e
p
e
m
4
p
e
m
þ
e
ð
*
1
Þ
e
m
is termed the Clausius-Mossotti factor and is
denoted by the frequency-dependent function K(
o
).
The above expression for
a
=
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