Editing Maxwell's equations (section)

== Solutions ==
Maxwell's equations are [[partial differential equations]] that relate the electric and magnetic fields to each other and to the electric charges and currents. Often, the charges and currents are themselves dependent on the electric and magnetic fields via the [[Lorentz force|Lorentz force equation]] and the [[#Constitutive relations|constitutive relations]]. These all form a set of coupled partial differential equations which are often very difficult to solve: the solutions encompass all the diverse phenomena of [[classical electromagnetism]]. Some general remarks follow.

As for any differential equation, [[boundary condition]]s<ref name=Monk>
{{cite book
 |author=Peter Monk
 |title=Finite Element Methods for Maxwell's Equations
 |page =1 ff
 |publisher=Oxford University Press
 |location=Oxford UK
 |isbn=978-0-19-850888-5
 |url=https://books.google.com/books?id=zI7Y1jT9pCwC&q=electromagnetism+%22boundary+conditions%22&pg=PA1
 |year=2003
}}</ref><ref name=Volakis>
{{cite book
 |author=Thomas B. A. Senior & John Leonidas Volakis
 |title=Approximate Boundary Conditions in Electromagnetics
 |page =261 ff
 |publisher=Institution of Electrical Engineers
 |location=London UK
 |isbn=978-0-85296-849-9
 |url=https://books.google.com/books?id=eOofBpuyuOkC&q=electromagnetism+%22boundary+conditions%22&pg=PA261
 |date=1995-03-01
}}</ref><ref name=Hagstrom>
{{cite book
 |author=T Hagstrom (Björn Engquist & Gregory A. Kriegsmann, Eds.)
 |title=Computational Wave Propagation
 |page =1 ff
 |publisher=Springer
 |location=Berlin
 |isbn=978-0-387-94874-4
 |url=https://books.google.com/books?id=EdZefkIOR5cC&q=electromagnetism+%22boundary+conditions%22&pg=PA1
 |year=1997
}}</ref> and [[initial condition]]s<ref name=Hussain>
{{cite book
 |author=Henning F. Harmuth & Malek G. M. Hussain
 |title=Propagation of Electromagnetic Signals
 |page =17
 |publisher=World Scientific
 |location=Singapore
 |isbn=978-981-02-1689-4
 |url=https://books.google.com/books?id=6_CZBHzfhpMC&q=electromagnetism+%22initial+conditions%22&pg=PA45
 |year=1994
}}</ref> are necessary for a [[Electromagnetism uniqueness theorem|unique solution]]. For example, even with no charges and no currents anywhere in spacetime, there are the obvious solutions for which '''E''' and '''B''' are zero or constant, but there are also non-trivial solutions corresponding to electromagnetic waves. In some cases, Maxwell's equations are solved over the whole of space, and boundary conditions are given as asymptotic limits at infinity.<ref name=Cook>
{{cite book
 |author=David M Cook
 |title=The Theory of the Electromagnetic Field
 |year=2002
 |page =335 ff
 |publisher=Courier Dover Publications
 |location=Mineola NY
 |isbn=978-0-486-42567-2
 |url=https://books.google.com/books?id=bI-ZmZWeyhkC&q=electromagnetism+infinity+boundary+conditions&pg=RA1-PA335
}}</ref> In other cases, Maxwell's equations are solved in a finite region of space, with appropriate conditions on the boundary of that region, for example an [[Perfectly matched layer|artificial absorbing boundary]] representing the rest of the universe,<ref name=Lourtioz>
{{cite book
 |author=Jean-Michel Lourtioz
 |title=Photonic Crystals: Towards Nanoscale Photonic Devices
 |page =84
 |publisher=Springer
 |location=Berlin
 |isbn=978-3-540-24431-8
 |url=https://books.google.com/books?id=vSszZ2WuG_IC&q=electromagnetism+boundary++-element&pg=PA84
 |date=2005-05-23
}}</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> or [[periodic boundary conditions]], or walls that isolate a small region from the outside world (as with a [[waveguide]] or cavity [[resonator]]).<ref>
{{cite book
 |author=S. F. Mahmoud
 |title=Electromagnetic Waveguides: Theory and Applications
 |page =Chapter 2
 |publisher=Institution of Electrical Engineers
 |location=London UK
 |isbn=978-0-86341-232-5
 |url=https://books.google.com/books?id=toehQ7vLwAMC&q=Maxwell%27s+equations+waveguides&pg=PA2
 |no-pp=true
 |year=1991
}}</ref>

[[Jefimenko's equations]] (or the closely related [[Liénard–Wiechert potential]]s) are the explicit solution to Maxwell's equations for the electric and magnetic fields created by any given distribution of charges and currents. It assumes specific initial conditions to obtain the so-called "retarded solution", where the only fields present are the ones created by the charges. However, Jefimenko's equations are unhelpful in situations when the charges and currents are themselves affected by the fields they create.

[[Numerical partial differential equations|Numerical methods for differential equations]] can be used to compute approximate solutions of Maxwell's equations when exact solutions are impossible. These include the [[finite element method]] and [[finite-difference time-domain method]].<ref name=Monk/><ref name=Hagstrom/><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=1998
 |page =79 ff
 |publisher=Wiley IEEE
 |location=New York
 |isbn=978-0-7803-3425-0
 |url=https://books.google.com/books?id=55q7HqnMZCsC&q=electromagnetism+%22boundary+conditions%22&pg=PA79
}}</ref><ref name= Friedman>
{{cite book
 |author=Bernard Friedman
 |title=Principles and Techniques of Applied Mathematics
 |year= 1990
 |publisher=Dover Publications
 |location=Mineola NY
 |isbn=978-0-486-66444-6
}}</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=978-1-58053-832-9
 |no-pp=true
}}</ref> For more details, see [[Computational electromagnetics]].