Page 230 - Electromagnetics

or for a randomly-structured gas, the contribution from dipoles within the sphere is zero.
Indeed, it is convenient and reasonable to assume that for most dielectrics the effects of
the dipoles immediately surrounding the observation point cancel so that E 3 = 0. This
was first suggested by O.F. Mosotti in 1850 [52].
With E 2 approximated as (4.95) and E 3 assumed to be zero, we have the value of the
resulting local field:

P(r)

E (r) = E(r) + . (4.96)
3 0
This is called the Mosotti field. Substituting the Mosotti field into (4.94) and using
P = Np, we obtain

P(r)

P(r) = NαE (r) = Nα E(r) + .
3 0
Solving for P we obtain
3 0 Nα

P(r) = E(r) = χ e 0 E(r).
3 0 − Nα
So the electric susceptibility of a dielectric may be expressed as
3Nα
χ e = . (4.97)
3 0 − Nα
Using χ e = r − 1 we can rewrite (4.97) as

3 + 2Nα/ 0
= 0 r = 0 , (4.98)
3 − Nα/ 0
which we can arrange to obtain

3 0 r − 1
α = α e + α a + α d = .
N r + 2
This has been named the Clausius–Mosotti formula, after O.F. Mosotti who proposed it
in 1850 and R. Clausius who proposed it independently in 1879. When written in terms of
2
the index of refraction n (where n = r ), it is also known as the Lorentz–Lorenz formula,
after H. Lorentz and L. Lorenz who proposed it independently for optical materials in
1880. The Clausius–Mosotti formula allows us to determine the dielectric constant from
the polarizability and number density of a material. It is reasonably accurate for certain
simple gases (with pressures up to 1000 atmospheres) but becomes less reliable for liquids
and solids, especially for those with large dielectric constants.
The response of the microscopic structure of matter to an applied field is not instanta-
neous. When exposed to a rapidly oscillating sinusoidal field, the induced dipole moments
may lag in time. This results in a loss mechanism that can be described macroscopically
by a complex permittivity. We can modify the Clausius–Mosotti formula by assuming
that both the relative permittivity and polarizability are complex numbers, but this will
not model the dependence of these parameters on frequency. Instead we shall (in later
paragraphs) model the time response of the dipole moments to the applied field.
An interesting application of the Clausius–Mosotti formula is to determine the permit-
tivity of a mixture of dielectrics with different permittivities. Consider the simple case
in which many small spheres of permittivity 2 , radius a, and volume V are embedded

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