- Electromagnetic wave equation
The

**electromagnetic wave equation**is a second-order partial differential equation that describes the propagation ofelectromagnetic wave s through a medium or in avacuum . The homogeneous form of the equation, written in terms of either theelectric field **E**or themagnetic field **B**, takes the form::$left(\; abla^2\; -\; \{\; 1\; over\; \{c\}^2\; \}\; \{partial^2\; over\; partial\; t^2\}\; ight)\; mathbf\{E\}\; =\; 0$

:$left(\; abla^2\; -\; \{\; 1\; over\; \{c\}^2\; \}\; \{partial^2\; over\; partial\; t^2\}\; ight)\; mathbf\{B\}\; =\; 0$

where "c" is the

speed of light in the medium. In a vacuum, "c" = "c"_{0}= 299,792,458 meters per second, which is the speed of light infree space . [*Current practice is to use "c"*]_{0}to denote the speed of light in vacuum according toISO 31 . In the original Recommendation of 1983, the symbol "c" was used for this purpose. See [*http://physics.nist.gov/Pubs/SP330/sp330.pdf NIST "Special Publication 330", Appendix 2, p. 45*]The electromagnetic wave equation derives from

Maxwell's equations .It should also be noted that in most older literature,

**B**is called the "magnetic flux density" or "magnetic induction".**peed of propagation****In vacuum**If the wave propagation is in vacuum, then

:$c\; =\; c\_o\; =\; \{\; 1\; over\; sqrt\{\; mu\_o\; varepsilon\_o\; \}\; \}\; =\; 2.99792458\; imes\; 10^8$ meters per second

is the

speed of light in vacuum , a "defined" value that sets the standard of length, themeter (unit) . Themagnetic constant $mu\_0$ and thevacuum permittivity $varepsilon\_0$ are importantphysical constant s that play a key role in electromagnetic theory. Their values (also a matter of definition) inSI units taken from [*http://physics.nist.gov/cuu/Constants/index.html NIST*] are tabulated below:**In a material medium**The speed of light in a linear, isotropic, and non-dispersive material medium is

:$c\; =\; \{\; c\_0\; over\; n\; \}\; =\; \{\; 1\; over\; sqrt\{\; mu\; varepsilon\; \}\; \}$

where

:$n\; =\; sqrt\{\; mu\; varepsilon\; over\; mu\_0\; varepsilon\_0\; \}$

is the

refractive index of the medium, $mu$ is the magnetic permeability of the medium, and $varepsilon$ is the electricpermittivity of the medium.**The origin of the electromagnetic wave equation****Conservation of charge**Conservation of charge requires that the time rate of change of the total charge enclosed within a volume "V" must equal the net current flowing into the surface "S" enclosing the volume:

:$oint\; limits\_S\; mathbf\{j\}\; cdot\; d\; mathbf\{A\}\; =\; -\; \{d\; over\; d\; t\}\; int\; limits\_V\; ho\; cdot\; dV$

where

**j**is the current density (inAmpere s per square meter) flowing through the surface and ρ is the charge density (incoulomb s per cubic meter) at each point in the volume.From the

divergence theorem , this relationship can be converted from integral form to differential form::$abla\; cdot\; mathbf\{j\}\; =\; -\; \{\; partial\; ho\; over\; partial\; t\}$

**Ampère's circuital law prior to Maxwell's correction**In its original form,

Ampère's circuital law relates the magnetic field**B**to the current density**j**::$oint\; limits\_C\; mathbf\{B\}\; cdot\; d\; mathbf\{l\}\; =\; iint\; limits\_S\; mu\; mathbf\{j\}\; cdot\; d\; mathbf\{A\}$

where "S" is an open surface terminated in the curve "C". This integral form can be converted to differential form, using

Stokes' theorem ::$abla\; imes\; mathbf\{B\}\; =\; mu\_0\; mathbf\{j\}$

**Inconsistency between Ampère's circuital law and the law of conservation of charge**Taking the divergence of both sides of Ampère's circuital law gives:

:$abla\; cdot\; (\; abla\; imes\; mathbf\{B\}\; )\; =\; abla\; cdot\; mu\_0\; mathbf\{j\}$

The divergence of the curl of any vector field, including the magnetic field

**B**, is always equal to zero::$abla\; cdot\; (\; abla\; imes\; mathbf\{B\})\; =\; 0$

Combining these two equations implies that

:$abla\; cdot\; mu\_0\; mathbf\{j\}\; =\; 0$

Because $mu\_0$ is nonzero constant, it follows that

:$abla\; cdot\; mathbf\{j\}\; =\; 0$

However, the law of conservation of charge tells that

:$abla\; cdot\; mathbf\{j\}\; =\; -\; \{\; partial\; ho\; over\; partial\; t\; \}$

Hence, as in the case of

Kirchhoff's circuit laws , Ampère's circuital law would appear only to hold in situations involving constant charge density. This would rule out the situation that occurs in the plates of a charging or a discharging capacitor.**Maxwell's correction to Ampère's circuital law**Gauss's law in integral form states::$oint\; limits\_S\; mathbf\{E\}\; cdot\; d\; mathbf\{A\}\; =\; frac\{1\}\{varepsilon\_0\}\; int\; limits\_V\; ho\; cdot\; dV\; ,$

where "S" is a closed surface enclosing the volume "V". This integral form can be converted to differential form using the divergence theorem:

:$abla\; cdot\; varepsilon\_0\; mathbf\{E\}\; =\; ho$

Taking the time derivative of both sides and reversing the order of differentiation on the left-hand side gives:

:$abla\; cdot\; varepsilon\_0\; \{partial\; mathbf\{E\}\; over\; partial\; t\; \}\; =\; \{\; partial\; ho\; over\; partial\; t\}$

This last result, along with Ampère's circuital law and the conservation of charge equation, suggests that there are actually "two" origins of the magnetic field: the current density

**j**, as Ampère had already established, and the so-called:displacement current :$\{partial\; mathbf\{D\}\; over\; partial\; t\; \}\; =\; varepsilon\_0\; \{partial\; mathbf\{E\}\; over\; partial\; t\; \}$

So the corrected form of Ampère's circuital law becomes:

:$abla\; imes\; mathbf\{B\}\; =\; mu\_0\; mathbf\{j\}\; +\; mu\_0\; varepsilon\_0\; \{partial\; mathbf\{E\}\; over\; partial\; t\; \}$

**Maxwell's hypothesis that light is an electromagnetic wave**In his 1864 paper entitled

A Dynamical Theory of the Electromagnetic Field , Maxwell utilized the correction to Ampère's circuital law that he had made in part III of his 1861 paper [*http://vacuum-physics.com/Maxwell/maxwell_oplf.pdf On Physical Lines of Force*] . In PART VI of his 1864 paper which is entitled 'ELECTROMAGNETIC THEORY OF LIGHT' [*[*] , Maxwell combined displacement current with some of the other equations of electromagnetism and he obtained a wave equation with a speed equal to the speed of light. He commented:*http://www.zpenergy.com/downloads/Maxwell_1864_4.pdf Maxwell 1864 4*] (page 497 of the article and page 9 of the pdf link):"The agreement of the results seems to show that light and magnetism are affections of the same substance, and that light is an electromagnetic disturbance propagated through the field according to electromagnetic laws." [

*See [*]*http://www.zpenergy.com/downloads/Maxwell_1864_5.pdf Maxwell 1864 5*] , page 499 of the article and page 1 of the pdf linkMaxwell's derivation of the electromagnetic wave equation has been replaced in modern physics by a much less cumbersome method involving combining the corrected version of Ampère's circuital law with

Faraday's law of induction .To obtain the electromagnetic wave equation in a vacuum using the modern method, we begin with the modern 'Heaviside' form of Maxwell's equations. In a vacuum, these equations are:

:$abla\; cdot\; mathbf\{E\}\; =\; frac\; \{\; ho\}\; \{epsilon\_0\}$

:$abla\; imes\; mathbf\{E\}\; =\; -frac\{partial\; mathbf\{B\; \{partial\; t\}$

:$abla\; cdot\; mathbf\{B\}\; =\; 0$

:$abla\; imes\; mathbf\{B\}\; =mu\_0\; varepsilon\_0\; frac\{\; partial\; mathbf\{E\; \{partial\; t\}$

Taking the curl of the curl equations gives::$abla\; imes\; abla\; imes\; mathbf\{E\}\; =\; -frac\{partial\; \}\; \{partial\; t\}\; abla\; imes\; mathbf\{B\}\; =\; -mu\_0\; varepsilon\_0\; frac\{partial^2\; mathbf\{E\}\; \}\; \{partial\; t^2\}$

:$abla\; imes\; abla\; imes\; mathbf\{B\}\; =\; mu\_0\; varepsilon\_0\; frac\{partial\; \}\; \{partial\; t\}\; abla\; imes\; mathbf\{E\}\; =\; -mu\_o\; varepsilon\_o\; frac\{partial^2\; mathbf\{B\{partial\; t^2\}$

By using the vector identity

:$abla\; imes\; left(\; abla\; imes\; mathbf\{V\}\; ight)\; =\; abla\; left(\; abla\; cdot\; mathbf\{V\}\; ight)\; -\; abla^2\; mathbf\{V\}$

where $mathbf\{V\}$ is any vector function of space, it turns into the wave equations:

:$\{partial^2\; mathbf\{E\}\; over\; partial\; t^2\}\; -\; \{c\_0\}^2\; cdot\; abla^2\; mathbf\{E\}\; =\; 0$

:$\{partial^2\; mathbf\{B\}\; over\; partial\; t^2\}\; -\; \{c\_0\}^2\; cdot\; abla^2\; mathbf\{B\}\; =\; 0$

where

:$c\_0\; =\; \{\; 1\; over\; sqrt\{\; mu\_0\; varepsilon\_0\; \}\; \}\; =\; 2.99792458\; imes\; 10^8$ meters per second

is the speed of light in free space.

**Covariant form of the homogeneous wave equation** These relativistic equations can be written in covariant form as

inertial reference frame leads to the theory of Special Relativity:$Box\; A^\{mu\}\; =\; 0$

where the

electromagnetic four-potential is:$A^\{mu\}=(varphi,\; mathbf\{A\}\; c)$

with the

Lorenz gauge condition ::$partial\_\{mu\}\; A^\{mu\}\; =\; 0,$.

Here

:$Box\; =\; abla^2\; -\; \{\; 1\; over\; c^2\}\; frac\{\; partial^2\}\; \{\; partial\; t^2\}$ is the

d'Alembertian operator. The square box is not a typographical error; it is the correct symbol for this operator.**Homogeneous wave equation in curved spacetime**The electromagnetic wave equation is modified in two ways, the derivative is replaced with the

covariant derivative and a new term that depends on the curvature appears.:$-\; \{A^\{alpha\; ;\; eta\_\{;\; eta\}\; +\; \{R^\{alpha\_\{eta\}\; A^\{eta\}\; =\; 0$

where

:$\{R^\{alpha\_\{eta\}$

is the

Ricci curvature tensor and the semicolon indicates covariant differentiation.The generalization of the

Lorenz gauge condition in curved spacetime is assumed::$\{A^\{mu\_\{\; ;\; mu\}\; =0$.

**Inhomogeneous electromagnetic wave equation**Localized time-varying charge and current densities can act as sources of electromagnetic waves in a vacuum. Maxwell's equations can be written in the form of a wave equation with sources. The addition of sources to the wave equations makes the

partial differential equations inhomogeneous.**Solutions to the homogeneous electromagnetic wave equation**The general solution to the electromagnetic wave equation is a linear superposition of waves of the form

:$mathbf\{E\}(\; mathbf\{r\},\; t\; )\; =\; g(phi(\; mathbf\{r\},\; t\; ))\; =\; g(\; omega\; t\; -\; mathbf\{k\}\; cdot\; mathbf\{r\}\; )$

and

:$mathbf\{B\}(\; mathbf\{r\},\; t\; )\; =\; g(phi(\; mathbf\{r\},\; t\; ))\; =\; g(\; omega\; t\; -\; mathbf\{k\}\; cdot\; mathbf\{r\}\; )$

for virtually "any" well-behaved function "g" of dimensionless argument φ, where:$omega$ is the

angular frequency (in radians per second), and :$mathbf\{k\}\; =\; (\; k\_x,\; k\_y,\; k\_z)$ is thewave vector (in radians per meter).Although the function "g" can be and often is a monochromatic

sine wave , it does not have to be sinusoidal, or even periodic. In practice, "g" cannot have infinite periodicity because any real electromagnetic wave must always have a finite extent in time and space. As a result, and based on the theory of Fourier decomposition, a real wave must consist of the superposition of an infinite set of sinusoidal frequencies.In addition, for a valid solution, the wave vector and the angular frequency are not independent; they must adhere to the

dispersion relation ::$k\; =\; |\; mathbf\{k\}\; |\; =\; \{\; omega\; over\; c\; \}\; =\; \{\; 2\; pi\; over\; lambda\; \}$

where "k" is the

wavenumber and λ is thewavelength .**Monochromatic, sinusoidal steady-state**The simplest set of solutions to the wave equation result from assuming sinusoidal waveforms of a single frequency in separable form:

:$mathbf\{E\}\; (\; mathbf\{r\},\; t\; )\; =\; mathrm\; \{Re\}\; \{\; mathbf\{E\}\; (mathbf\{r\}\; )\; e^\{\; j\; omega\; t\; \}\; \}$

where

* $j\; ,$ is theimaginary unit ,

* $omega\; =\; 2\; pi\; f\; ,$"' is theangular frequency inradians per second ,

* $f\; ,$ is the"'frequency inhertz , and

* $e^\{j\; omega\; t\}\; =\; cos(omega\; t)\; +\; j\; sin(omega\; t)\; ,$ isEuler's formula .**Plane wave solutions**Consider a plane defined by a unit normal vector :$mathbf\{n\}\; =\; \{\; mathbf\{k\}\; over\; k\; \}$.

Then planar traveling wave solutions of the wave equations are:$mathbf\{E\}(mathbf\{r\})\; =\; E\_0\; e^\{-j\; mathbf\{k\}\; cdot\; mathbf\{r\}\; \}$and:$mathbf\{B\}(mathbf\{r\})\; =\; B\_0\; e^\{-j\; mathbf\{k\}\; cdot\; mathbf\{r\}\; \}$

where:$mathbf\{r\}\; =\; (x,\; y,\; z)$ is the position vector (in meters).

These solutions represent planar waves traveling in the direction of the normal vector $mathbf\{n\}$. If we define the z direction as the direction of $mathbf\{n\}$ and the x direction as the direction of $mathbf\{E\}$, then by Faraday's Law the magnetic field lies in the y direction and is related to the electric field by the relation:$c\; \{partial\; B\; over\; partial\; z\}\; =\; \{partial\; E\; over\; partial\; t\}$.Because the divergence of the electric and magnetic fields are zero, there are no fields in the direction of propagation.

This solution is the linearly polarized solution of the wave equations. There are also circularly polarized solutions in which the fields rotate about the normal vector.

**pectral decomposition**Because of the linearity of Maxwell's equations in a vacuum, solutions can be decomposed into a superposition of sinusoids. This is the basis for the

Fourier transform method for the solution of differential equations.The sinusoidal solution to the electromagnetic wave equation takes the form:$mathbf\{E\}\; (\; mathbf\{r\},\; t\; )\; =\; mathbf\{E\}\_0\; cos(\; omega\; t\; -\; mathbf\{k\}\; cdot\; mathbf\{r\}\; +\; phi\_0\; )$and:$mathbf\{B\}\; (\; mathbf\{r\},\; t\; )\; =\; mathbf\{B\}\_0\; cos(\; omega\; t\; -\; mathbf\{k\}\; cdot\; mathbf\{r\}\; +\; phi\_0\; )$

where:$t$ is time (in seconds),:$omega$ is the

angular frequency (in radians per second),:$mathbf\{k\}\; =\; (\; k\_x,\; k\_y,\; k\_z)$ is thewave vector (in radians per meter), and:$phi\_0\; ,$ is thephase angle (in radians).The wave vector is related to the angular frequency by:$k\; =\; |\; mathbf\{k\}\; |\; =\; \{\; omega\; over\; c\; \}\; =\; \{\; 2\; pi\; over\; lambda\; \}$

where "k" is the

wavenumber and λ is thewavelength .The

electromagnetic spectrum is a plot of the field magnitudes (or energies) as a function of wavelength.**Other solutions**Spherically symmetric and cylindrically symmetric analytic solutions to the electromagnetic wave equations are also possible.

In cylindrical coordinates the wave equation can be written as follows:

:$mathbf\{E\}\; (\; mathbf\{r\},\; t\; )\; =\; \{mathbf\{E\}\_0\; cos(\; omega\; t\; -\; mathbf\{k\}\; cdot\; mathbf\{r\}\; +\; phi\_0\; )over\; s\}$and:$mathbf\{B\}\; (\; mathbf\{r\},\; t\; )\; =\; \{mathbf\{B\}\_0\; cos(\; omega\; t\; -\; mathbf\{k\}\; cdot\; mathbf\{r\}\; +\; phi\_0\; )over\; s\}$

**ee also****Theory and Experiment***

Maxwell's equations

*Wave equation

*Electromagnetic modeling

*Electromagnetic radiation

*Charge conservation

*Light

*Electromagnetic spectrum

*Optics

*Special relativity

*General relativity

*Photon dynamics in the double-slit experiment

*Photon polarization

* Larmor power formula

*Theoretical and experimental justification for the Schrödinger equation **Applications***

Rainbow

*Cosmic microwave background radiation

*Laser

*Laser fusion

*Photography

*X-ray

*X-ray crystallography

*RADAR

*Radio waves

*Optical computing

*Microwave

*Holography

*Microscope

*Telescope

*Gravitational lens

*Black body radiation **Notes****References****Further reading****Electromagnetism****Journal articles*** Maxwell, James Clerk, "", Philosophical Transactions of the Royal Society of London 155, 459-512 (1865). (This article accompanied a December 8, 1864 presentation by Maxwell to the Royal Society.)

**Undergraduate-level textbooks***cite book | author=Griffiths, David J.|title=Introduction to Electrodynamics (3rd ed.)| publisher=Prentice Hall |year=1998 |id=ISBN 0-13-805326-X

*cite book | author=Tipler, Paul | title=Physics for Scientists and Engineers: Electricity, Magnetism, Light, and Elementary Modern Physics (5th ed.) | publisher=W. H. Freeman | year=2004 | id=ISBN 0-7167-0810-8

* Edward M. Purcell, "Electricity and Magnetism" (McGraw-Hill, New York, 1985). ISBN 0-07-004908-4.

* Hermann A. Haus and James R. Melcher, "Electromagnetic Fields and Energy" (Prentice-Hall, 1989) ISBN 0-13-249020-X.

* Banesh Hoffmann, "Relativity and Its Roots" (Freeman, New York, 1983). ISBN 0-7167-1478-7.

* David H. Staelin, Ann W. Morgenthaler, and Jin Au Kong, "Electromagnetic Waves" (Prentice-Hall, 1994) ISBN 0-13-225871-4.

* Charles F. Stevens, "The Six Core Theories of Modern Physics", (MIT Press, 1995) ISBN 0-262-69188-4.

* Markus Zahn, "Electromagnetic Field Theory: a problem solving approach", (John Wiley & Sons, 1979) ISBN 0-471-02198-9**Graduate-level textbooks***cite book |author=Jackson, John D.|title=Classical Electrodynamics (3rd ed.)|publisher=Wiley|year=1998|id=ISBN 0-471-30932-X

* Landau, L. D., "The Classical Theory of Fields" (Course of Theoretical Physics: Volume 2), (Butterworth-Heinemann: Oxford, 1987). ISBN 0-08-018176-7.

*cite book | author=Maxwell, James C. | title=A Treatise on Electricity and Magnetism | publisher=Dover | year=1954 | id=ISBN 0-486-60637-6

* Charles W. Misner, Kip S. Thorne,John Archibald Wheeler , "Gravitation", (1970) W.H. Freeman, New York; ISBN 0-7167-0344-0. "(Provides a treatment of Maxwell's equations in terms of differential forms.)"**Vector calculus***P. C. Matthews "Vector Calculus", Springer 1998, ISBN 3-540-76180-2

*H. M. Schey, "Div Grad Curl and all that: An informal text on vector calculus", 4th edition (W. W. Norton & Company, 2005) ISBN 0-393-92516-1.**Biographies***

Andre Marie Ampere

*Albert Einstein

*Michael Faraday

*Heinrich Hertz

*Oliver Heaviside

*James Clerk Maxwell **External links**

*Wikimedia Foundation.
2010.*

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