Page 231 - Electromagnetics

within a dielectric matrix of permittivity 1 . If we assume that a is much smaller than
the wavelength of the electromagnetic field, and that the spheres are sparsely distributed
within the matrix, then we may ignore any mutual interaction between the spheres. Since
the expression for the permittivity of a uniform dielectric given by (4.98) describes the
effect produced by dipoles in free space, we can use the Clausius–Mosotti formula to
define an effective permittivity e for a material consisting of spheres in a background
dielectric by replacing 0 with 1 to obtain
3 + 2Nα/ 1
e = 1 . (4.99)
3 − Nα/ 1
In this expression α is the polarizability of a single dielectric sphere embedded in the
background dielectric, and N is the number density of dielectric spheres. To find α
we use the static field solution for a dielectric sphere immersed in a field (§ 3.2.10).
Remembering that p = αE and that for a uniform region of volume V we have p = V P,
we can make the replacements 0 → 1 and → 2 in (3.117) to get
2 − 1
α = 3 1 V . (4.100)
2 + 2 1
Defining f = NV as the fractional volume occupied by the spheres, we can substitute
(4.100) into (4.99) to find that
1 + 2 fy
e = 1
1 − fy
where

2 − 1
y = .
2 + 2 1
This is known as the Maxwell–Garnett mixing formula. Rearranging we obtain
e − 1 2 − 1
= f ,
e + 2 1 2 + 2 1
which is known as the Rayleigh mixing formula. As expected, e → 1 as f → 0.Even
though as f → 1 the formula also reduces to e = 2 , our initial assumption that f 1
(sparsely distributed spheres) is violated and the result is inaccurate for non-spherical
inhomogeneities [90]. For a discussion of more accurate mixing formulas, see Ishimaru
[90] or Sihvola [175].

The dispersion formula of classical physics. We may determine the frequency de-
pendence of the permittivity by modeling the time response of induced dipole moments.
This was done by H. Lorentz using the simple atomic model we introduced earlier. Con-
sider what happens when a molecule consisting of heavy particles (nuclei) surrounded by
clouds of electrons is exposed to a time-harmonic electromagnetic wave. Using the same
arguments we made when we studied the interactions of fields with a plasma in § 4.6.1,
we assume that each electron experiences a Lorentz force F e =−q e E . We neglect the

magnetic component of the force for nonrelativistic charge velocities, and ignore the mo-
tion of the much heavier nuclei in favor of studying the motion of the electron cloud.
However, several important distinctions exist between the behavior of charges within a
plasma and those within a solid or liquid material. Because of the surrounding polarized

matter, any molecule responds to the local field E instead of the applied field E. Also,
as the electron cloud is displaced by the Lorentz force, the attraction from the positive

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