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ეს განტოლებები იდენტურია '''E''' და '''B''' ველების მეშვეობით ზემოთ მოყვანილი ფორმისა, იმ განსხვავებით, რომ [[ელექტრული მუდმივა]] შეიცვალა გარემოს [[დიელექტრიკული შეღწევადობა|დიელექტრიკული შეღწევადობით]], ხოლო [[მაგნიტური მუდმივა]] [[მაგნიტური შეღწევადობა|მაგნიტური შეღწევადობით]]. გარდა ამისა, ახლა [[მუხტი]]და და [[ელექტრული დენი]]ს ქვეშ იგულისხმება მხოლოდ ''თავისუფალი'' მუხტი და დენი.
====Calculation of constitutive relations====
The fields in Maxwell's equations are generated by charges and currents. Conversely, the charges and currents are affected by the fields through the [[Lorentz force]] equation:
: <math>\mathbf{F} = q (\mathbf{E} + \mathbf{v} \times \mathbf{B}),</math>
where ''q'' is the charge on the particle and '''v''' is the particle velocity. (It also should be remembered that the Lorentz force is not the ''only'' force exerted upon charged bodies, which also may be subject to gravitational, nuclear, ''etc.'' forces.) Therefore, in both [[classical physics|classical]] and [[quantum physics]], the precise dynamics of a system form a set of [[simultaneous equations|coupled]] [[differential equation]]s, which are almost always too complicated to be solved exactly, even at the level of [[statistical mechanics]].<ref group="note">These complications show there is merit in separating the [[Lorentz force]] from the main four Maxwell equations. The four Maxwell's equations express the fields' dependence upon current and charge, setting apart the calculation of these currents and charges. As noted in this subsection, these calculations may well involve the Lorentz force only implicitly. Separating these complicated considerations from the Maxwell's equations provides a useful framework.</ref> This remark applies to not only the dynamics of free charges and currents (which enter Maxwell's equations directly), but also the dynamics of bound charges and currents, which enter Maxwell's equations through the [[constitutive equation]]s, as described next.
Commonly, real materials are approximated as [[continuum mechanics|continuous media]] with bulk properties such as the [[refractive index]], [[permittivity]], [[permeability (electromagnetism)|permeability]], [[Electrical conductivity|conductivity]], and/or various susceptibilities. These lead to the ''macroscopic'' Maxwell's equations, which are written ([[#Formulation in terms of free charge and current|as given above]]) in terms of free charge/current densities and '''D''', '''H''', '''E''', and '''B''' ( rather than '''E''' and '''B''' alone ) along with the constitutive equations relating these fields. For example, although a real material consists of [[atom]]s whose [[electron]]ic charge densities can be individually polarized by an applied field, for most purposes behavior at the atomic scale is not relevant and the material is approximated by an overall [[polarization density]] related to the applied field by an [[electric susceptibility]].
Continuum approximations of atomic-scale inhomogeneities cannot be determined from Maxwell's equations alone, but require some type of [[quantum mechanics|quantum mechanical]] analysis such as [[quantum field theory]] as applied to [[condensed matter physics]]. See, for example, [[density functional theory]], [[Green-Kubo relations]] and [[Green's function (many-body theory)]]. Various approximate transport equations have evolved, for example, the [[Boltzmann equation]] or the [[Fokker-Planck equation]] or the [[Navier-Stokes equations]]. Some examples where these equations are applied are [[magnetohydrodynamics]], [[fluid dynamics]], [[electrohydrodynamics]], [[superconductivity]], [[plasma modeling]]. An entire physical apparatus for dealing with these matters has developed. A different set of ''homogenization methods'' (evolving from a tradition in treating materials such as [[Conglomerate (geology)|conglomerates]] and [[laminate]]s) are based upon approximation of an inhomogeneous material by a homogeneous ''[[Effective medium approximations|effective medium]]''<ref name=Aspnes>[[David E. Aspnes|Aspnes, D.E.]], "Local-field effects and effective-medium theory: A microscopic perspective," ''Am. J. Phys.'' '''50''', p. 704-709 (1982).</ref><ref name=Kang>
{{cite book
|author=Habib Ammari & Hyeonbae Kang
|title=Inverse problems, multi-scale analysis and effective medium theory : workshop in Seoul, Inverse problems, multi-scale analysis, and homogenization, June 22–24, 2005, Seoul National University, Seoul, Korea
|year= 2006
|url=http://books.google.com/books?id=dK7JwVPbUkMC&printsec=frontcover&dq=%22effective+medium%22&lr=&as_brr=0&sig=7mfnQhzzEABN5mDB9KxX9ivBL5k#PPP11,M1
|publisher=American Mathematical Society
|location=Providence RI
|isbn=0821839683
}}</ref> (valid for excitations with [[wavelength]]s much larger than the scale of the inhomogeneity).<ref name= Zienkiewicz>
{{cite book
|author=O. C. Zienkiewicz, Robert Leroy Taylor, J. Z. Zhu, Perumal Nithiarasu
|title=The Finite Element Method
|year= 2006
|edition=Sixth Edition
|page=550 ff
|url=http://books.google.com/books?id=rvbSmooh8Y4C&printsec=frontcover&dq=finite+element+inauthor:Zienkiewicz&lr=&as_brr=0&sig=Mj1sHnBhQ_zdwxRZ0Wtb33Zg63Y#PPA550,M1
|publisher=Butterworth-Heinemann
|location=Oxford UK
|isbn=0750663219
}}</ref><ref>N. Bakhvalov and G. Panasenko, ''Homogenization: Averaging Processes
in Periodic Media'' (Kluwer: Dordrecht, 1989); V. V. Jikov, S. M. Kozlov and O. A. Oleinik, ''Homogenization of Differential Operators and Integral Functionals'' (Springer: Berlin, 1994).</ref><ref name=Felsen>
{{cite journal
|title=Multiresolution Homogenization of Field and Network Formulations for Multiscale Laminate Dielectric Slabs
|author=Vitaliy Lomakin, Steinberg BZ, Heyman E, & Felsen LB
|volume=51
|issue=10
|year= 2003
|pages=2761 ff
|url=http://www.ece.ucsd.edu/~vitaliy/A8.pdf
|journal=IEEE Transactions on Antennas and Propagation
|doi=10.1109/TAP.2003.816356
}}</ref><ref name=Coifman>
{{cite book
|title=Topics in Analysis and Its Applications: Selected Theses
|author=AC Gilbert (Ronald R Coifman, Editor)
|year= 2000
|page=155
|url=http://books.google.com/books?id=d4MOYN5DjNUC&printsec=frontcover&dq=homogenization+date:2000-2009&lr=&as_brr=0#PPA156,M1
|publisher=World Scientific Publishing Company
|location=Singapore
|isbn=9810240945
}}</ref>
Theoretical results have their place, but often require fitting to experiment. Continuum-approximation properties of many real materials rely upon measurement,<ref name=Palik>
{{cite book
|author=Edward D. Palik & Ghosh G
|title=Handbook of Optical Constants of Solids
|year= 1998
|publisher=Academic Press
|location=London UK
|isbn=0125444222
|url=http://books.google.com/books?id=AkakoCPhDFUC&dq=optical+constants+inauthor:Palik&lr=&as_brr=0&source=gbs_summary_s&cad=0
}}</ref> for example, [[ellipsometry]] measurements.
In practice, some materials properties have a negligible impact in particular circumstances, permitting neglect of small effects. For example: [[nonlinear optics|optical nonlinearities]] can be neglected for low field strengths; [[dispersion (optics)|material dispersion]] is unimportant where frequency is limited to a narrow [[bandwidth (signal processing)|bandwidth]]; [[Absorption (electromagnetic radiation)|material absorption]] can be neglected for wavelengths where a material is transparent; and [[metal]]s with finite conductivity often are approximated at [[microwave]] or longer wavelengths as [[perfect conductor|perfect metals]] with infinite conductivity (forming hard barriers with zero [[skin depth]] of field penetration).
And, of course, some situations demand that Maxwell's equations and the Lorentz force be combined with other forces that are not electromagnetic. An obvious example is [[gravity]]. A more subtle example, which applies where electrical forces are weakened due to charge balance in a solid or a molecule, is the [[Casimir force]] from [[quantum electrodynamics]].<ref>[https://www.editorial.seas.harvard.edu/capasso/publications/Capasso_STJQE_13_400_2007.pdf F Capasso, JN Munday, D. Iannuzzi & HB Chen ''Casimir forces and quantum electrodynamical torques: physics and nanomechanics'']</ref>
The connection of Maxwell's equations to the rest of the physical world is via the fundamental charges and currents. These charges and currents are a response of their sources to electric and magnetic fields and to other forces. The determination of these responses involves the properties of physical materials.
===In vacuum===
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