Electromagnetic wave equation

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Lasers used for visual effects during a musical performance.

Primary (42*) and secondary rainbows (51*) are visible, as well as a reflected primary and a faintly visible reflection primary. The secondary rainbow is higher than the primary and has inverted colors. (from hyperphysics.phy-astr.gsu.edu).

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This long range radar antenna, known as ALTAIR, is used to detect and track space objects in conjunction with ABM testing at the Ronald Reagan Test Site on the Kwajalein atoll

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:

$\left( \nabla^2 - { 1 \over c^2 } {\partial^2 \over \partial t^2} \right) \mathbf{E} \ \ = \ \ 0$

$\left( \nabla^2 - { 1 \over c^2 } {\partial^2 \over \partial t^2} \right) \mathbf{H} \ \ = \ \ 0$

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.

1 Speed of propagation
- 1.1 In vacuum
- 1.2 In a material medium
2 The origin of the electromagnetic wave equation
3 Covariant form of the homogeneous wave equation
4 Homogeneous wave equation in curved spacetime
5 Inhomogeneous electromagnetic wave equation
6 Solutions to the homogeneous electromagnetic wave equation
7 References
- 7.1 Electromagnetism
- 7.2 Vector calculus
8 See also

If the wave propagation is in vacuum, then

$c = c_o = { 1 \over \sqrt{ \mu_o \varepsilon_o } } = 2.998 \times 10^8$ meters per second

is the speed of light in free space. The magnetic permeability $\ \mu_o$ and the electric permittivity $\ \varepsilon_o$ are important physical constants that play a key role in electromagnetic theory.

Symbol	Name	Numerical Value	SI Unit of Measure	Type
$c \$	Speed of light	$2.998 \times 10^{8}$	meters per second	defined
$\ \varepsilon_0$	Permittivity	$8.854 \times 10^{-12}$	Farads per meter	derived
$\ \mu_0 \$	Permeability	$4 \pi \times 10^{-7}$	Henries per meter	defined

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

$c = { c_o \over n } = { 1 \over \sqrt{ \mu \varepsilon } }$

where

$n = \sqrt{ \mu \varepsilon \over \mu_o \varepsilon_o }$

is the refractive index of the medium, $\mu \,$ is the magnetic permeability of the medium, and $\varepsilon \,$ is the electric permittivity of the medium.

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:

$\oint \limits_S \mathbf{J} \cdot d \mathbf{a} = - {d \over d t} \int \limits_V \rho \cdot dV$

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:

$\nabla \cdot \mathbf{J} = - { \partial \rho \over \partial t}$

**André-Marie Ampère**

Born:	January 20, 1775
Died:	June 10, 1836 Marseille,France
Occupation:	Physicist

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:

$\nabla \times \mathbf{H} = \mathbf{J}$

If we take the divergence of both sides of Ampère's Circuital Law, we find

$\nabla \cdot ( \nabla \times \mathbf{H} ) = \nabla \cdot \mathbf{J}$

The divergence of the curl of any vector field – in this case, the magnetic field H – is always equal to zero:

$\nabla \cdot ( \nabla \times \mathbf{H} ) = 0$

Combining these two equations implies that

$\nabla \cdot \mathbf{J} = 0$

From the law of conservation of charge, we know that

$\nabla \cdot \mathbf{J} = - { \partial \rho \over \partial t }$

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.

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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:

$\oint \limits_S \varepsilon_o \mathbf{E} \cdot d \mathbf{a} = \int \limits_V \rho \cdot dV$

Again, using the divergence theorem, we can convert this equation to differential form:

$\nabla \cdot \varepsilon_o \mathbf{E} = \rho$

Taking the derivative with respect to time of both sides, we find:

${\partial \over \partial t } ( \nabla \cdot \varepsilon_o \mathbf{E} ) = {\partial \rho \over \partial t}$

Reversing the order of differentiation on the left-hand side, we obtain

$\nabla \cdot \varepsilon_o {\partial \mathbf{E} \over \partial t } = { \partial \rho \over \partial t}$

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:

${\partial \mathbf{D} \over \partial t } = \varepsilon_o {\partial \mathbf{E} \over \partial t }$

So the corrected form of Ampère's Circuital Law, which Maxwell discovered, becomes:

$\nabla \times \mathbf{H} = \mathbf{J} + \varepsilon_o {\partial \mathbf{E} \over \partial t }$

Father of Electromagnetic Theory

A postcard from Maxwell to Peter Tait.

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

$\nabla \cdot \mathbf{E} = 0$

$\nabla \times \mathbf{E} = -\mu_o \frac{\partial \mathbf{H}} {\partial t}$

$\nabla \cdot \mathbf{H} = 0$

$\nabla \times \mathbf{H} =\varepsilon_o \frac{ \partial \mathbf{E}} {\partial t}$

If we take the curl of the curl equations we obtain

$\nabla \times \nabla \times \mathbf{E} = -\mu_o \frac{\partial } {\partial t} \nabla \times \mathbf{H} = -\mu_o \varepsilon_o \frac{\partial^2 \mathbf{E} } {\partial t^2}$

$\nabla \times \nabla \times \mathbf{H} = \varepsilon_o \frac{\partial } {\partial t} \nabla \times \mathbf{E} = -\mu_o \varepsilon_o \frac{\partial^2 \mathbf{H} } {\partial t^2}$

If we note the vector identity

$\nabla \times \left( \nabla \times \mathbf{V} \right) = \nabla \left( \nabla \cdot \mathbf{V} \right) - \nabla^2 \mathbf{V}$

where $\mathbf{V}$ is any vector function of space, we recover the wave equations

${\partial^2 \mathbf{E} \over \partial t^2} \ - \ c^2 \cdot \nabla^2 \mathbf{E} \ \ = \ \ 0$

${\partial^2 \mathbf{H} \over \partial t^2} \ - \ c^2 \cdot \nabla^2 \mathbf{H} \ \ = \ \ 0$

where

$c = { 1 \over \sqrt{ \mu_o \varepsilon_o } } = 2.998 \times 10^8$ meters per second

is the speed of light in free space.

Time dilation in transversal motion. The requirement that the speed of light is constant in every inertial reference frame leads to the theory of Special Relativity

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

$A^{\mu}=(\varphi, \mathbf{A} c)$ $\left( SI \right)$

$A^{\mu}=(\varphi, \mathbf{A} )$ $\left( cgs \right)$

with the Lorenz gauge

$\partial_{\mu} A^{\mu} = 0\,$ .

Here

$\Box = \nabla^2 - { 1 \over c^2} \frac{ \partial^2} { \partial t^2}$ is the d'Alembertian operator. The square box is not a typographical error; it is the correct symbol for this operator.

Main article: 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.

$- {A^{\alpha ; \beta}}_{; \beta} + {R^{\alpha}}_{\beta} A^{\beta} = 0$

where

${R^{\alpha}}_{\beta}$

is the Ricci curvature tensor and the semicolon indicates covariant differentiation.

We have assumed the generalization of the Lorenz gauge in curved spacetime

${A^{\mu}}_{ ; \mu} =0$ .

Main article: 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.

Main article: 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.

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.

Main article: 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.

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

Electromagnetic spectrum illustration.

$\mathbf{E} ( \mathbf{r}, t ) = \mathbf{E}_0 \cos( \omega t - \mathbf{k} \cdot \mathbf{r} + \phi_0 )$

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.

Spherically symmetric and cylindrically symmetric analytic solutions to the electromagnetic wave equations are also possible.

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

Griffiths, David J. (1998). Introduction to Electrodynamics (3rd ed.). Prentice Hall. ISBN 0-13-805326-X.
Tipler, Paul (2004). Physics for Scientists and Engineers: Electricity, Magnetism, Light, and Elementary Modern Physics (5th ed.). W. H. Freeman. 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.

Jackson, John D. (1998). Classical Electrodynamics (3rd ed.). Wiley. 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.
Maxwell, James C. (1954). A Treatise on Electricity and Magnetism. Dover. 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.)

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.

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Sponsored Links This section contains paid listings which have been purchased by companies that want to have their sites appear for specific search terms and related content. These listings are administered, sorted and maintained by a third party and are not endorsed by Search.com.

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