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მაქსველის განტოლებები იძლევა ფიზიკურ ახსნას, თუ რატომ ვრცელდება ასეთი ტიპის ტალღები სივრცეში: ცვლადი [[მაგნიტური ველი]] აგენერირებს ცვლად [[ელექტრული ველი|ელექტრულ ველს]] [[ფარადეის ინდუქციის კანონი]]ს შესაბამისად, ხოლო ელექტრული ველი, თავის მხრივ, აგენერირებს ცვლად მაგნიტურ ველს (მაქსველის მიერ შესწორებული [[ამპერის კანონი]]ს შესაბამისად). ეს ურთიერთკავშირი განაპირობებს იმას, რომ ასეთ ტალღებს ([[ელექტრომაგნიტური ტალღა]]) შეუძლიათ გავრცელება სივრცეში ე.წ. [[სინათლის სიჩქარე|სინათლის]] ''c'' სიჩქარით.
==Boundary conditions: using Maxwell's equations==
Although Maxwell's equations apply throughout space and time, practical problems are finite and solutions to Maxwell's equations inside the solution region are joined to the remainder of the universe through [[boundary conditions]]<ref name=Monk>
{{cite book
|author=Peter Monk
|title=Finite Element Methods for Maxwell's Equations
|year= 2003
|page =1 ff
|publisher=Oxford University Press
|location=Oxford UK
|isbn=0198508883
|url=http://books.google.com/books?id=zI7Y1jT9pCwC&pg=PA1&dq=electromagnetism+%22boundary+conditions%22&lr=&as_brr=0&sig=cOnigjBvtJ4Hbjxgfm8wTjozqxE
}}</ref><ref name=Volakis>
{{cite book
|author=Thomas B. A. Senior & John Leonidas Volakis
|title=Approximate Boundary Conditions in Electromagnetics
|year= 1995
|page =261 ff
|publisher=Institution of Electrical Engineers
|location=London UK
|isbn=0852968493
|url=http://books.google.com/books?id=eOofBpuyuOkC&pg=PA261&dq=electromagnetism+%22boundary+conditions%22&lr=&as_brr=0&sig=9ZDw3dIu82v4uo0QMnDK9U9Bcdo#PPA261,M1
}}</ref><ref name=Hagstrom>
{{cite book
|author=T Hagstrom (Björn Engquist & Gregory A. Kriegsmann, Eds.)
|title=Computational Wave Propagation
|year= 1997
|page =1 ff
|publisher=Springer
|location=Berlin
|isbn=0387948740
|url=http://books.google.com/books?id=EdZefkIOR5cC&pg=PA1&dq=electromagnetism+%22boundary+conditions%22&lr=&as_brr=0&sig=XzLe2I7PkXdMGJRL4xHPDwldGpY
}}</ref> and started in time using [[initial conditions]].<ref name=Hussain>
{{cite book
|author=Henning F. Harmuth & Malek G. M. Hussain
|title=Propagation of Electromagnetic Signals
|year= 1994
|page =17
|publisher=World Scientific
|location=Singapore
|isbn=9810216890
|url=http://books.google.com/books?id=6_CZBHzfhpMC&pg=PA45&dq=electromagnetism+%22initial+conditions%22&lr=&as_brr=0&sig=FoOvL5l7XjUdY0LWumWteOdfKqY#PPA17,M1
}}</ref>
In particular, in a region without any free currents or free charges, the electromagnetic fields in the region originate elsewhere, and are introduced via boundary and/or initial conditions. An example of this type is a an electromagnetic scattering problem, where an electromagnetic wave originating outside the scattering region is scattered by a target, and the scattered electromagnetic wave is analyzed for the information it contains about the target by virtue of the interaction with the target during scattering.<ref name=inverse>
{{cite book
|title=Qualitative methods in inverse scattering theory
|author=Fioralba Cakoni
|isbn=3540288449
|year=2006
|publisher=Springer Science & Business
|url=http://books.google.com/books?id=7DqqOjPJetYC&pg=PR6#PPA61,M1
|chapter=The inverse scattering problem for an imperfect conductor
|page=61
}}, {{cite book
|title=An introduction to inverse scattering and inverse spectral problems
|author=Khosrow Chadan ''et al.''
|year=1997
|isbn=0898713870
|url=http://books.google.com/books?id=y2rywYxsDEAC&pg=PA45
|page=45
|publisher=Society for Industrial and Applied Mathematics
}}</ref>
In some cases, like [[waveguides]] or cavity [[resonator]]s, the solution region is largely isolated from the universe, for example, by metallic walls, and boundary conditions at the walls define the fields with influence of the outside world confined to the input/output ends of the structure.<ref name=" Mahmoud">
{{cite book
|author=S. F. Mahmoud
|title=Electromagnetic Waveguides: Theory and Applications applications
|year= 1991
|page =Chapter 2
|publisher=Institution of Electrical Engineers
|location=London UK
|isbn=0863412327
|url=http://books.google.com/books?id=toehQ7vLwAMC&pg=PA2&dq=Maxwell%27s+equations+waveguides&lr=&as_brr=0&sig=CnpXp8Bo-iEvVyNywwGyclyIvM8#PPA30,M1
|nopp=true
}}</ref> In other cases, the universe at large sometimes is approximated by an [[Perfectly matched layer|artificial absorbing boundary]],<ref name=Lourtioz>
{{cite book
|author=Jean-Michel Lourtioz
|title=Photonic Crystals: Towards Nanoscale Photonic Devices
|year= 2005
|page =84
|publisher=Springer
|location=Berlin
|isbn=354024431X
|url=http://books.google.com/books?id=vSszZ2WuG_IC&pg=PA84&dq=electromagnetism+boundary++-element&lr=&as_brr=0&sig=9fu7mVJyhO31ybZMnJo_WcqklJ8
}}</ref><ref>S. G. Johnson, [http://math.mit.edu/~stevenj/18.369/pml.pdf Notes on Perfectly Matched Layers], online MIT course notes (Aug. 2007).</ref><ref name=Taflove>
{{cite book
|author=Taflove A & Hagness S C
|title=Computational Electrodynamics: The Finite-difference Time-domain Method
|year= 2005
|page =Chapters 6 & 7
|publisher=Artech House
|location=Boston MA
|isbn=1580538320
|url=http://www.amazon.com/gp/reader/1580538320/ref=sib_dp_pop_toc?ie=UTF8&p=S008#reader-link
|nopp=true
}}</ref> or, for example for radiating [[antennas]] or [[communication satellite]]s, these boundary conditions can take the form of ''asymptotic limits'' imposed upon the solution.<ref name=Cook>
{{cite book
|author=David M Cook
|title=The Theory of the Electromagnetic Field
|year= 2003
|page =335 ff
|publisher=Courier Dover Publications
|location=Mineola NY
|isbn=0486425673
|url=http://books.google.com/books?id=bI-ZmZWeyhkC&pg=RA1-PA335&dq=electromagnetism+infinity+boundary+conditions&lr=&as_brr=0&sig=HnH7D4EEhcJIZ18IuWAQCCeRp38
}}</ref> In addition, for example in an [[optical fiber]] or [[thin-film optics]], the solution region often is broken up into subregions with their own simplified properties, and the solutions in each subregion must be joined to each other across the subregion interfaces using boundary conditions.<ref name=Umashankar>
{{cite book
|author=Korada Umashankar
|title=Introduction to Engineering Electromagnetic Fields
|year= 1989
|page =§10.7; pp. 359ff
|publisher=World Scientific
|location=Singapore
|isbn=9971509210
|url=http://books.google.com/books?id=qp5qHvB_mhcC&pg=PA359&dq=electromagnetism+%22boundary+conditions%22&lr=&as_brr=0&sig=94v3JyP6RSRoRknmrUPNp0pZhh8
}}</ref><ref name= Stewart>
{{cite book
|author=Joseph V. Stewart
|title=Intermediate Electromagnetic Theory
|year= 2001
|page =Chapter III, pp. 111 ff Chapter V, Chapter VI
|publisher=World Scientific
|location=Singapore
|isbn=9810244703
|url=http://books.google.com/books?id=mwLI4nQ0thQC&printsec=frontcover&dq=intitle:Intermediate+intitle:electromagnetic+intitle:theory&lr=&as_brr=0&sig=YSFxsif_vgTCRFeDLKuTFULxNjE#PPA112,M1
|nopp=true
}}</ref><ref name=Chow>
{{cite book
|author=Tai L. Chow
|title=Electromagnetic theory
|year= 2006
|page =333ff and Chapter 3: pp. 89ff
|publisher=Jones and Bartlett
|location=Sudbury MA
|isbn=0-7637-3827-1
|url=http://books.google.com/books?id=dpnpMhw1zo8C&pg=PA153&dq=isbn=0763738271&sig=PgEEBA6TQEZ5fD_AhJQ8dd7MGHo#PPR333,M1
}}</ref> A particular example of this use of boundary conditions is the replacement of a material with a volume polarization by a charged surface layer, or of a material with a volume magnetization by a surface current, as described in the section [[#Bound_charge_and_current|Bound charge and current]].
Following are some links of a general nature concerning boundary value problems: [[Examples of boundary value problems]], [[Sturm-Liouville theory]], [[Dirichlet boundary condition]], [[Neumann boundary condition]], [[mixed boundary condition]], [[Cauchy boundary condition]], [[Sommerfeld radiation condition]]. Needless to say, one must choose the boundary conditions appropriate to the problem being solved. See also Kempel<ref name= Kempel>
{{cite book
|author=John Leonidas Volakis, Arindam Chatterjee & Leo C. Kempel
|title=Finite element method for electromagnetics : antennas, microwave circuits, and scattering applications
|year= 1989
|page =79 ff
|publisher=Wiley IEEE
|location=New York
|isbn=0780334256
|url=http://books.google.com/books?id=55q7HqnMZCsC&pg=PA79&dq=electromagnetism+%22boundary+conditions%22&lr=&as_brr=0&sig=MqJqkMSkMgOTyKcNF9e9BlquCcc#PPA80,M1
}}</ref> and the extraordinary book by Friedman.<ref name= Friedman>
{{cite book
|author=Bernard Friedman
|title=Principles and Techniques of Applied Mathematics
|year= 1990
|publisher=Dover Publications
|location=Mineola NY
|isbn=0486664449
|url=http://www.amazon.com/Principles-Techniques-Applied-Mathematics-Friedman/dp/0486664449/ref=sr_1_1?ie=UTF8&s=books&qisbn=1207010487&sr=1-1
}}</ref>
==გაუსის ერთეულებში==
| 