ELECTROMAGNETIC WAVE EQUATION

The 'electromagnetic wave equation' is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum. The homogeneous form of the equation, written in terms of either the electric field 'E' or the magnetic field 'H', 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{H} = 0
where ''c'' is the speed of light in the medium. In a vacuum, c = 2.998 x 10⁸ meters per second, which is the speed of light in free space.
The electromagnetic wave equation derives from Maxwell's equations.
In a linear, isotropic, non-dispersive medium, the magnetic flux density 'B' is related to the magnetic field 'H' by
:$mathbf\{B\}\; =\; mu\; mathbf\{H\}$
where μ is the magnetic permeability of the medium.
It should also be noted that in most modern literature, 'B' is called the "magnetic field," and 'H' is called either the "auxiliary magnetic field," or "the H vector."
In this article, it is most appropriate to use SI units through the motivation and derivation of the homogeneous wave equation. Once the marriage between electromagnetism and light has been made, and the relationship between the permitivity/permeability and the speed of light has been derived, it is often useful to use other units, such as cgs or Lorentz-Heaviside. At that point, we display results in all three sets of units.

Contents

Speed of propagation

In vacuum

In a material medium

The origin of the electromagnetic wave equation

Conservation of charge

Ampère's Circuital Law prior to Maxwell's correction

Inconsistency between Ampère's Circuital Law and the Law of Conservation of Charge

Maxwell's correction to Ampère's Circuital Law

Maxwell - First to propose that light is an electromagnetic wave

Covariant form of the homogeneous wave equation

Homogeneous wave equation in curved spacetime

Inhomogeneous electromagnetic wave equation

Solutions to the homogeneous electromagnetic wave equation

Monochromatic, sinusoidal steady-state

Plane wave solutions

Spectral decomposition

Other solutions

References

Electromagnetism

Journal articles

Undergraduate-level textbooks

Graduate-level textbooks

Vector calculus

See also

Theory and Experiment

Applications

Biographies

Speed of propagation

In vacuum

If the wave propagation is in vacuum, then
: meters per second
is the speed of light in vacuum. The vacuum permeability $mu\_o$ and the vacuum permittivity are important physical constants that play a key role in electromagnetic theory.
{| border="1" cellspacing="0" cellpadding="8"
|- style="background-color: #aaeecc;"
! Symbol
! Name
! Numerical Value
! SI Unit of Measure
! Type
|-
|-
| $c\_0$
| Speed of light in vacuum
|$2.998\; imes\; 10^\{8\}$
| meters per second
| defined
|-
|
| electric constant
| $8.854\; imes\; 10^\{-12\}$
| farads per meter
| derived
|-
|$mu\_0$
| magnetic constant
|$4\; pi\; imes\; 10^\{-7\}$
| henries per meter
| defined
|}

In a material medium

For the purposes of this article, we will assume that all materials are linear, isotropic, and non-dispersive. In that case, the speed of light in a material medium is
:
where
:
is the refractive index of the medium, $mu\; ,$ is the magnetic permeability of the medium, and is the electric permittivity 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 (in Amperes per square meter) flowing through the surface and ρ is the charge density (in Coulombs per cubic meter) at each point in the volume.
From the divergence theorem, we can convert this relationship 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 Law (SI units) relates the magnetic field 'H' to its source, the current density 'J':
:$oint\; limits\_C\; mathbf\{H\}\; cdot\; d\; mathbf\{l\}\; =\; int\; limits\_S\; mathbf\{J\}\; cdot\; d\; mathbf\{a\}$
Again, we can convert to differential form, this time using Stokes' theorem:
:$$
abla imes mathbf{H} = mathbf{J}

Inconsistency between Ampère's Circuital Law and the Law of Conservation of Charge

If we take the divergence of both sides of Ampère's Circuital Law, we find
:$$
abla cdot (
abla imes mathbf{H} ) =
abla cdot mathbf{J}
The divergence of the curl of any vector field – in this case, the magnetic field 'H' – is always equal to zero:
:$$
abla cdot (
abla imes mathbf{H} ) = 0
Combining these two equations implies that
:$$
abla cdot mathbf{J} = 0
From the law of conservation of charge, we know that
:$$
abla cdot mathbf{J} = - { partial
ho over partial t }
Hence, as in the case of Kirchhoff's current law, 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

To understand Maxwell's correction to Ampère's Circuital Law, we need to look at another of Maxwell's Equations, namely, Gauss's Law (SI units) in integral form:
:
ho cdot dV
Again, using the divergence theorem, we can convert this equation to differential form:
:$$
abla cdot arepsilon_o mathbf{E} =
ho
Taking the derivative with respect to time of both sides, we find:
:$\{partial\; over\; partial\; t\; \}\; ($
abla cdot arepsilon_o mathbf{E} ) = {partial
ho over partial t}
Reversing the order of differentiation on the left-hand side, we obtain
:$$
abla cdot arepsilon_o {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'' sources of the magnetic field: the current density 'J', as Ampère had already established, and the so-called 'displacement current':
:
So the corrected form of Ampère's Circuital Law, which Maxwell discovered, becomes:
:$$
abla imes mathbf{H} = mathbf{J} + arepsilon_o {partial mathbf{E} over partial t }

Maxwell - First to propose 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 On Physical Lines of Force. In PART VI of his 1864 paper which is entitled 'ELECTROMAGNETIC THEORY OF LIGHT' [1] (page 497 of the article and page 9 of the pdf link), 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,
''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 [2], page 499 of the article and page 1 of the pdf link)
Maxwell'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 electromagnetic 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. Using (SI units) in a vacuum, these equations are
:$$
abla cdot mathbf{E} = 0
:$$
abla imes mathbf{E} = -mu_o rac{partial mathbf{H}} {partial t}
:$$
abla cdot mathbf{H} = 0
:$$
abla imes mathbf{H} =arepsilon_o rac{ partial mathbf{E}} {partial t}
If we take the curl of the curl equations we obtain
:$$
abla imes
abla imes mathbf{E} = -mu_o rac{partial } {partial t}
abla imes mathbf{H} = -mu_o arepsilon_o rac{partial^2 mathbf{E} } {partial t^2}
:$$
abla imes
abla imes mathbf{H} = arepsilon_o rac{partial } {partial t}
abla imes mathbf{E} = -mu_o arepsilon_o rac{partial^2 mathbf{H} } {partial t^2}
If we note 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, we recover the wave equations
:$\{partial^2\; mathbf\{E\}\; over\; partial\; t^2\}\; -\; c^2\; cdot$
abla^2 mathbf{E} = 0
:$\{partial^2\; mathbf\{H\}\; over\; partial\; t^2\}\; -\; c^2\; cdot$
abla^2 mathbf{H} = 0
where
: 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
:$Box\; A^\{mu\}\; =\; 0\; quad\; mbox\{(SI\; units)\}$
:$Box\; A^\{mu\}\; =\; 0\; quad\; mbox\{(cgs\; units)\}$
where the electromagnetic four-potential is
: $left(\; SI$
ight)
: $left(\; cgs$
ight)
with the Lorenz gauge
:$partial\_\{mu\}\; A^\{mu\}\; =\; 0,$.
Here
:$Box\; =$
abla^2 - { 1 over c^2} rac{ 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

Main articles: Maxwell's equations 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.
:
where
:
is the Ricci curvature tensor and the semicolon indicates covariant differentiation.
We have assumed the generalization of the Lorenz gauge in curved spacetime
:$\{A^\{mu\}\}\_\{\; ;\; mu\}\; =0$.

Inhomogeneous electromagnetic wave equation

Main articles: 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

Main articles: 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
:$mathbf\{H\}(\; mathbf\{r\},\; t\; )\; =\; g(phi(\; mathbf\{r\},\; t\; ))\; =\; g(\; omega\; t\; -\; mathbf\{k\}\; cdot\; mathbf\{r\}\; )$
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 the wave 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 the wavelength.

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

★ $j\; =\; sqrt\{-1\}\; ,$ is the imaginary unit,

★ $omega\; =\; 2\; pi\; f\; ,$' is the angular frequency in radians per second,

★ $f\; ,$ is the' frequency in hertz, and

★ $e^\{j\; omega\; t\}\; =\; cos(omega\; t)\; +\; j\; sin(omega\; t)\; ,$ is Euler's formula.

Plane wave solutions

Main articles: Sinusoidal plane-wave solutions of the electromagnetic wave equation

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
:$mathbf\{H\}(mathbf\{r\})\; =\; H\_0\; e^\{-j\; mathbf\{k\}\; cdot\; mathbf\{r\}\; \}$
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
:$c\; mu\_o\; \{partial\; H\; over\; partial\; z\}\; =\; \{partial\; E\; over\; partial\; t\}$.
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.

Spectral 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
:$mathbf\{H\}\; (\; mathbf\{r\},\; t\; )\; =\; mathbf\{H\}\_0\; cos(\; omega\; t\; -\; mathbf\{k\}\; cdot\; mathbf\{r\}\; +\; phi\_0\; )$
where
:$t$ is time (in seconds),
:$omega$ is the angular frequency (in radians per second),
:$mathbf\{k\}\; =\; (\; k\_x,\; k\_y,\; k\_z)$ is the wave vector (in radians per meter), and
:$phi\_0\; ,$ is the phase 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 the wavelength.
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.

References

Electromagnetism

Journal articles

★ James Clerk Maxwell, "A Dynamical Theory of the Electromagnetic Field", ''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.)
http://www.zpenergy.com/downloads/Maxwell_1864_1.pdf
http://www.zpenergy.com/downloads/Maxwell_1864_2.pdf
http://www.zpenergy.com/downloads/Maxwell_1864_3.pdf
http://www.zpenergy.com/downloads/Maxwell_1864_4.pdf
http://www.zpenergy.com/downloads/Maxwell_1864_5.pdf
http://www.zpenergy.com/downloads/Maxwell_1864_6.pdf

Undergraduate-level textbooks

★ Introduction to Electrodynamics (3rd ed.), Griffiths, David J., , , Prentice Hall, 1998, ISBN 0-13-805326-X

★ Physics for Scientists and Engineers: Electricity, Magnetism, Light, and Elementary Modern Physics (5th ed.), Tipler, Paul, , , W. H. Freeman, 2004, 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.

Graduate-level textbooks

★ Classical Electrodynamics (3rd ed.), Jackson, John D., , , Wiley, 1998, 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.

★ A Treatise on Electricity and Magnetism, Maxwell, James C., , , Dover, 1954, 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

★ 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.

See 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

Biographies

★ Andre Marie Ampere

★ Albert Einstein

★ Michael Faraday

★ Heinrich Hertz

★ Oliver Heaviside

★ James Clerk Maxwell