Image Processing Reference
In-Depth Information
spherical region inside the medium, which is large compared to any one unit.
The field insid
e
a uniformly polarized sphere can be approximated by a dipole
given by 4
πα
P
3, and after some integrations and manipulations, one can cal-
culate the local field at a unit in the center of this sphere. The local field,
E
loc
,
is the macroscopic field
E
ext
, minus the contribution due to the units inside
the sphere. If the units are spherically symmetric (which might well not be
the case!), then the surrounding units act in a spherically symmetric fashion
and we find
g
= (2 + ε
r
)/3 and
3
/
P
E
=
E
+
3
(3.19)
loc
ext
ε
0
Strictly speaking, the polarization is proportional to the local electric field, so
PNE
e
=α
loc
(3.20)
where α
e
now represents an atomic or molecular polarizability. This now
allows us to write
N
N
α
αε
PE
=
e
e
(3.21)
ext
1
−
(
/
3
)
0
From the definition of ε
eff
, we can now write that
DE
=
ε
=
ε
EP
+
(3.22)
eff xt
0
ext
and
N
N
α
αε
ε
=+
−
ε
e
e
(3.23)
eff
0
1
(
/
3
)
0
Thus, the relative permittivity is given by
ε
ε
N
N
αε
αε
/
ε
=
eff
=
1
+
e
0
(3.24)
r
1
−
(
/
3
)
0
e
0
which is the well-known Clausius-Mossotti relation.
If we now solve for
N
α
e
, we have
ε
−
+
ε
ε
ε
−
+
1
2
N
αε
=
3
eff
eff
0
=
3
ε
r
r
(3.25)
e
0
ε
2
ε
0
0
Knowing
N
and α
e
we can calculate the relative permittivity when the vari-
ous approximations made are satisfied Notice that when (
N
α
e
/ε
0
) < 3, the rela-
tive permittivity is negative. For naturally occurring materials, we can replace
the number density
N
by the bulk density multiplied by Avogadro's number
and divided by the molar mass.
Search WWH ::
Custom Search