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