The Clausius–Mossotti equation is named after the Italian physicist OttavianoFabrizio Mossotti, whose 1850 book analyzed the relationship between the dielectric constants of two different media, and the German physicist Rudolf Clausius, who gave the formula explicitly in his 1879 book in the context not of dielectric constants but of indices of refraction. The same formula also arises in the context of conductivity, in which it is known as Maxwell's formula. It arises yet again in the context of refractivity, in which it is known as the Lorentz–Lorenz equation.
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The Clausius–Mossotti factor can be expressed in terms of complex permittivities:
where
In the context of electrokinetic manipulation, the real part of the ClausiusMossotti factor is a determining factor for the dielectrophoretic force on a particle, whereas the imaginary part is a determining factor for the electrorotational torque on the particle. Other factors are, of course, the geometries of the particle to be manipulated and the electric field.
In his Lectures on Physics, (Vol.2, Ch32), Richard Feynman has a background discussion deriving the ClausiusMosotti Equation, in reference to the Index of Refraction for dense materials. He starts with the derivation of an equation for the index of refraction for gases, and then shows how this must be modified for dense materials, modifying it, because in dense materials, there are also electric fields produced by other nearby atoms, creating local fields. In essence, Feynman is saying that for dense materials the polarization of a material is proportional to its electric field, but that it has a different constant of proportionality than for that of a gas. When this constant is corrected for a dense material, by taking into account the local fields of nearby atoms, you end up with the ClausiusMosotti Equation.^{[1]} Feynman states the ClausiusMosotti equation as follows:
Where:
Feynman discusses "atomic polarizability" and explains it in these terms: When there is a sinusoidal electric field acting on a material, there is an induced dipole moment per unit volume which is proportional to the electric field  with a proportionality constant α that depends on the frequency. This constant is a complex number, meaning that the polarization does not exactly follow the electric field, but may be shifted in phase to some extent. At any rate, there is a polarization per unit volume whose magnitude is proportional to the strength of the electric field.
The polarizability α, of an atom is defined in terms of the local electric field at the atom:
Where:
The polarizability is an atomic property, but the dielectric constant will depend on the manner in which the atoms are assembled to form a crystal. For a nonspherical atom, α will be a tensor.^{[2]}
The polarization of a crystal may be expressed approximately as the product of the polarizabilities of the atoms times the local electric field:
Now, to relate the dielectric constant to the polarizability, which is what the ClausiusMosotti equation (or relation) is all about^{[3]}, you must consider that the results will depend on the relation that holds between the macroscopic electric field and the local electric field.
Where:
