The Clausius–Mossotti relation is named after the Italian physicist Ottaviano-Fabrizio Mossotti

Ottaviano-Fabrizio Mossotti

Ottaviano-Fabrizio Mossotti was an Italian physicist exiled from Italy for his liberal ideas. He later taught astronomy and physics at the University of Buenos Aires. His name is associated with a type of multiple-element lens correcting spherical aberration and coma, but not chromatic...

, whose 1850 book analyzed the relationship between the dielectric constant

Dielectric constant

The relative permittivity of a material under given conditions reflects the extent to which it concentrates electrostatic lines of flux. In technical terms, it is the ratio of the amount of electrical energy stored in a material by an applied voltage, relative to that stored in a vacuum...

s of two different media, and the German physicist Rudolf Clausius

Rudolf Clausius

Rudolf Julius Emanuel Clausius , was a German physicist and mathematician and is considered one of the central founders of the science of thermodynamics. By his restatement of Sadi Carnot's principle known as the Carnot cycle, he put the theory of heat on a truer and sounder basis...

, who gave the formula explicitly in his 1879 book in the context not of dielectric constant

Dielectric constant

The relative permittivity of a material under given conditions reflects the extent to which it concentrates electrostatic lines of flux. In technical terms, it is the ratio of the amount of electrical energy stored in a material by an applied voltage, relative to that stored in a vacuum...

s 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

Refraction

Refraction is the change in direction of a wave due to a change in its speed. It is essentially a surface phenomenon . The phenomenon is mainly in governance to the law of conservation of energy. The proper explanation would be that due to change of medium, the phase velocity of the wave is changed...

, in which it is known as the Lorentz–Lorenz equation.

The Clausius–Mossotti law applies to the dielectric constant of a dielectric that is perfect, homogeneous and isotropic:


where

• is the dielectric constant
  Dielectric constant
  The relative permittivity of a material under given conditions reflects the extent to which it concentrates electrostatic lines of flux. In technical terms, it is the ratio of the amount of electrical energy stored in a material by an applied voltage, relative to that stored in a vacuum...
  
   of a substance
• is the permittivity of a vacuum
• is the molar mass
  Molar mass
  Molar mass, symbol M, is a physical property of a given substance , namely its mass per amount of substance. The base SI unit for mass is the kilogram and that for amount of substance is the mole. Thus, the derived unit for molar mass is kg/mol...
  
   of the substance
• is its density
• is Avogadro's number, and
• is the molecular polarizability.

Clausius–Mossotti factor

The Clausius–Mossotti factor can be expressed in terms of complex permittivities:



where

• is the permittivity (the subscript p refers to a lossless dielectric sphere suspended in a medium m)
• is the conductivity
• is the angular frequency
  Angular frequency
  In physics, angular frequency ω is a scalar measure of rotation rate. Angular frequency is the magnitude of the vector quantity angular velocity...
  
   of the applied electric field
  Electric field
  In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...
  
  
• i
  Imaginary unit
  In mathematics, the imaginary unit allows the real number system ℝ to be extended to the complex number system ℂ, which in turn provides at least one root for every polynomial . The imaginary unit is denoted by , , or the Greek...
  
   is the imaginary unit
  Imaginary unit
  In mathematics, the imaginary unit allows the real number system ℝ to be extended to the complex number system ℂ, which in turn provides at least one root for every polynomial . The imaginary unit is denoted by , , or the Greek...
  
  , the square root
  Square root
  In mathematics, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square is x...
  
   of -1

In the context of electrokinetic manipulation, the real part of the Clausius-Mossotti 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. Whereas can be directly measured by application of different AC potentials directly on electrodes , can be measured by electro-rotation measurments thanks to optical trapping methods.

Richard Feynman on the Clausius–Mossotti equation

In his Lectures on Physics (Vol.2, Ch32), Richard Feynman

Richard Feynman

Richard Phillips Feynman was an American physicist known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics and the physics of the superfluidity of supercooled liquid helium, as well as in particle physics...

 has a background discussion deriving the Clausius-Mosotti 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 Clausius-Mosotti Equation. Feynman states the Clausius-Mosotti equation as follows:


Where:

• is the number of particles per unit volume of the capacitor
• is the atomic polarizability
• is the refractive index

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 that depends on the frequency. This constant is a complex number

Complex number

A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

, 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

Electric field

In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...

.

Dielectric constant and polarizability

The polarizability , of an atom is defined in terms of the local electric field at the atom:

Where:

• is the dipole moment
• is the Local Electrical Field at the atom

The polarizability is an atomic property, but the dielectric constant

Dielectric constant

The relative permittivity of a material under given conditions reflects the extent to which it concentrates electrostatic lines of flux. In technical terms, it is the ratio of the amount of electrical energy stored in a material by an applied voltage, relative to that stored in a vacuum...

 will depend on the manner in which the atoms are assembled to form a crystal. For a non-spherical atom, will be a tensor

Tensor

Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...

.

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 Clausius-Mosotti equation (or relation) is all about, you must consider that the results will depend on the relation that holds between the macroscopic electric field and the local electric field.


Where:

• is the concentration
• is the polarizability of atoms, j
• Local Electrical Field at atom sites