Clausius–Mossotti relationFrom Wikipedia, the free encyclopedia

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

Fabrizio Mossotti, whose 1850 book[1] 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[2] 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.

The Clausius–Mossotti law applies to the dielectric constant of a dielectric that is

perfect, homogeneous and isotropic. It is the second of the following three

equalities:[3][4]

where

is the dielectric constant of a substance

is the permittivity of a vacuum

is the molar mass of the substance

is its density

is Avogadro's number,

is the molecular polarizability in SI-units (C·m2/V) and

is the molecular polarizability volume (m3 in SI) or the

polarizability in the CGS system of units.[4]

Contents [hide]

1 Clausius–Mossotti factor

2 Derivation

3 Richard Feynman on the Clausius–Mossotti equation

4 Dielectric constant and polarizability

5 References

Clausius–Mossotti factor [edit]

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

permittivities:[5][6][7]

Article Talk Read Edit View historyMore Search

Edit links

Main page

Contents

Featured content

Current events

Random article

Donate to Wikipedia

Wikimedia Shop

Interaction

Help

About Wikipedia

Community portal

Recent changes

Contact page

Tools

What links here

Related changes

Upload file

Special pages

Permanent link

Page information

Wikidata item

Cite this page

Print/export

Create a book

Download as PDF

Printable version

Languages

Català

Deutsch

Español

یسرافFrançais

Italiano

Português

Русский

Slovenščina

Українська

��

Create account Log in

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 of the applied electric field

i is the imaginary unit, the square root 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,[8] can be measured by electro-rotation

measurements thanks to optical trapping methods.

Derivation [edit]

Assume a simple cubic lattice of polarisable points with polarisability .

Application of an external field will induce a dipole at each site . Due to

symmetry, the local (microscopic) field inside lattice is identical at each lattice

point: .

Away from the lattice points, the electric field is given by:

where the dipole electric field

Richard Feynman on the Clausius–Mossotti equation[edit]

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

discussion deriving the Clausius–Mossotti 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 that for a gas. When this constant is corrected for a dense

material, by taking into account the local fields of nearby atoms, one ends up

with the Clausius–Mossotti Equation.[9] Feynman states the Clausius–Mossotti

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, 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.

Dielectric constant and polarizability [edit]

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

the atom by

where

is the dipole moment,

is the local electric field at the orbital[clarification needed]

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 non-

spherical atom, will be a tensor.[10]

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–Mossotti equation (or relation) is all about,[10] one 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 .

References [edit]

1. ^ Mossotti, O. F. (1850). Mem. di mathem. e fisica in Modena. 24 11. p. 49.

2. ^ Clausius, R. (1879). Die mechanische U’grmetheorie. 2. p. 62.

Privacy policy About Wikipedia Disclaimers Contact Wikipedia Developers Mobile view

This page was last modified on 1 July 2014 at 03:25.

Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply.By using this site, you agree to the Terms of Use and Privacy Policy. Wikipedia® is a registered trademarkof the Wikimedia Foundation, Inc., a non-profit organization.

3. ^ Rysselberghe, P. V. (January 1932). "Remarks concerning the Clausius–Mossotti Law". J. Phys. Chem. 36 (4): 1152–1155. doi:10.1021/j150334a007 .

4. ^ a b Atkins, Peter; de Paula, Julio (2010). "Chapter 17". Atkins' PhysicalChemistry. Oxford University Press. p. 622-629. ISBN 978-0-19-954337-3.

5. ^ Hughes, Michael Pycraft (2000). "AC electrokinetics: applications fornanotechnology" . Nanotechnology 11 (2): 124–132.Bibcode:2000Nanot..11..124P . doi:10.1088/0957-4484/11/2/314 .

6. ^ Markov, Konstantin Z. (2000). "Elementary Micromechanics of HeterogeneousMedia". In Konstantin Z. Markov and Luigi Preziosi. 'Heterogeneous Media:Modelling and Simulation' . Boston: Birkhauser. pp. 1–162. ISBN 978-0-8176-4083-5.

7. ^ Gimsa, J. (2001). "Characterization of particles and biological cells by AC-electrokinetics". In A.V. Delgado. Interfacial Electrokinetics and Electrophoresis.New York: Marcel Dekker Inc. pp. 369–400. ISBN 0-8247-0603-X.

8. ^ T. Honegger, K. Berton, E. Picard et D. Peyrade. Determination of Clausius–Mossotti factors and surface capacitances for colloidal particles. Appl. Phys.Lett., vol. 98, no. 18, page 181906, 2011.

9. ^ Feynman, R. P., Leighton, R. B.; Sands, M (1989). Feynman Lectures onPhysics. Vol. 2, chap. 32 (Refractive Index of Dense Materials), sec. 3: AddisonWesley. ISBN 0-201-50064-7.

10. ^ a b Kittel, Charles (1995). Introduction to Solid State Physics (8th ed.). Wiley.ISBN 0-471-41526-X.

Categories: Electrodynamics Electromagnetism Equations

Concepts in physics