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Encyclopedia > Electromagnetic wave equation
Lasers used for visual effects during a musical performance.
Lasers used for visual effects during a musical performance.
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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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Microwave oven
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X-rays can reveal the details of bones and teeth
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
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
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Optical fibers

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: Image File history File links Download high resolution version (1600x1067, 348 KB) Classical Specatular featured state of the art laser and lighting designed by Durham Marenghi Photo taken at ISO 1600 with no tripod File links The following pages link to this file: Laser Lighting ... Image File history File links Download high resolution version (1600x1067, 348 KB) Classical Specatular featured state of the art laser and lighting designed by Durham Marenghi Photo taken at ISO 1600 with no tripod File links The following pages link to this file: Laser Lighting ... Download high resolution version (1784x1188, 1089 KB)Public domain. ... Download high resolution version (1784x1188, 1089 KB)Public domain. ... A combined microwave and fan-assisted oven. ... A combined microwave and fan-assisted oven. ... Photo by dene. From http://www. ... Photo by dene. From http://www. ... In the NATO phonetic alphabet, X-ray represents the letter X. An X-ray picture (radiograph) taken by Röntgen An X-ray is a form of electromagnetic radiation with a wavelength approximately in the range of 5 pm to 10 nanometers (corresponding to frequencies in the range 30 PHz... Download high resolution version (546x697, 80 KB)The only information given by the source about this image is Radar. We can deduce a few more things from this image: The dish rotates on a circular track The facility is in the tropics (because of palm trees) The dish is approximately... Download high resolution version (546x697, 80 KB)The only information given by the source about this image is Radar. We can deduce a few more things from this image: The dish rotates on a circular track The facility is in the tropics (because of palm trees) The dish is approximately... A yagi antenna Most simply, an antenna is an electronic component designed to send or receive radio waves. ... An anti-ballistic missile (ABM) is a missile designed to counter ballistic missiles. ... Kwajalein infrastructure and RTS headquarters, click to enlarge. ... Kwajalein Atoll - NASA NLT Landsat 7 (Visible Color) Satellite Image Kwajalein Atoll (Marshallese: Kuwajleen) is part of the Republic of the Marshall Islands (RMI). ... An atoll is a type of low, coral island found in tropical oceans and consisting of a coral-algal reef surrounding a central depression. ... Image File history File linksMetadata Download high resolution version (1579x1335, 525 KB) Summary Fiber optic strands. ... Image File history File linksMetadata Download high resolution version (1579x1335, 525 KB) Summary Fiber optic strands. ... In mathematics, a partial differential equation (PDE) is a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables. ... Free space is the most simple and elementary electromagnetic medium. ... Look up Vacuum in Wiktionary, the free dictionary. ... In mathematics, homogeneous may refer to: a homogeneous polynomial, in algebra a homogeneous function a homogeneous differential equation a homogeneous system of linear equations, in linear algebra homogeneous coordinates a homogeneous number a homogeneous space for a Lie group G, or more general transformation group a homogeneous ideal in a... In physics, an electric field or E-field is an effect produced by an electric charge (or a time-varying magnetic field) that exerts a force on charged objects in the field. ... To meet Wikipedias quality standards, this article may require cleanup. ...

( nabla^2 - { 1 over c^2 } {partial^2 over partial t^2} ) mathbf{E}   =   0
( nabla^2 - { 1 over c^2 } {partial^2 over partial t^2} ) 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 speed of light in a vacuum is an important physical constant denoted by the letter c for constant or the Latin word celeritas meaning swiftness. In metric units, c is exactly 299,792,458 metres per second (1,079,252,848. ... In physics, free space is a concept of electromagnetic theory, corresponding roughly to the vacuum, the baseline state of the electromagnetic field, or the replacement for the electromagnetic aether. ...


The electromagnetic wave equation derives from Maxwell's equations. Maxwells equations (sometimes called the Maxwell equations) are the set of four equations, attributed to James Clerk Maxwell, that describe the behavior of both the electric and magnetic fields, as well as their interactions with matter. ...


In a linear, isotropic, non-dispersive medium, the magnetic flux density B is related to the magnetic field H by Current flowing through a wire produces a magnetic field (B, labeled M here) around the wire. ...

mathbf{B} = mu mathbf{H}

where μ is the magnetic permeability of the medium. In electromagnetism, permeability is the degree of magnetisation of a material that responds linearly to an applied magnetic field. ...


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. Cover of brochure The International System of Units. ... This article or section is in need of attention from an expert on the subject. ... Lorentz-Heaviside units for Maxwells equations are often used in relativistic calculations. ...

Contents

Speed of propagation

In vacuum

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. The speed of light in a vacuum is an important physical constant denoted by the letter c for constant or the Latin word celeritas meaning swiftness. In metric units, c is exactly 299,792,458 metres per second (1,079,252,848. ... In physics, free space is a concept of electromagnetic theory, corresponding roughly to the vacuum, the baseline state of the electromagnetic field, or the replacement for the electromagnetic aether. ... In electromagnetism, permeability is the degree of magnetisation of a material that responds linearly to an applied magnetic field. ... Permittivity is a physical quantity that describes how an electric field affects and is affected by a dielectric medium and is determined by the ability of a material to polarize in response to an applied electric field, and thereby to cancel, partially, the field inside the material. ... In science, a physical constant is a physical quantity whose numerical value does not change. ... Electromagnetism is the force observed as static electricity, and causes the flow of electric charge (electric current) in electrical conductors. ...

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

The farad (symbol F) is the SI unit of capacitance (named after Michael Faraday). ... An inductor. ...

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

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. The refractive index (or index of refraction) of a material is the factor by which the phase velocity of electromagnetic radiation is slowed in that material, relative to its velocity in a vacuum. ...


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: Charge conservation is the principle that electric charge can neither be created nor destroyed. ...

oint_S mathbf{J} cdot d mathbf{a} = - {d over d t} int_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. In physics, the ampere (symbol: A, often informally abbreviated to amp) is the SI base unit used to measure electrical currents. ... The coulomb (symbol: C) is the SI unit of electric charge. ...


From the divergence theorem, we can convert this relationship from integral form to differential form: In vector calculus, the divergence theorem, also known as Gauss theorem, Ostrogradskys theorem, or Ostrogradsky–Gauss theorem is a result that links the divergence of a vector field to the value of surface integrals of the flow defined by the field. ...

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

Ampere's Law prior to Maxwell's correction

André-Marie Ampère

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

In its original form, Ampere's Law (SI units) relates the magnetic field H to its source, the current density J: Image File history File links Ampere1. ... January 20 is the 20th day of the year in the Gregorian calendar. ... 1775 was a common year starting on Sunday (see link for calendar). ... June 10 is the 161st day of the year in the Gregorian calendar (162nd in leap years), with 204 days remaining. ... October 2, Charles Darwin returns from his voyage around the world. ...   City flag Coat of arms Motto: By her great deeds, Marseille shines in the world Coordinates Time Zone CET (GMT +1) Administration Country France Région Provence-Alpes-Côte dAzur Département Bouches-du-Rhône (13) Subdivisions 16 arrondissements (in 8 secteurs) Intercommunality Urban Community of Marseille... Physicists working in a government lab A physicist is a scientist who studies or practices physics. ... In physics, Ampères law is the magnetic equivalent of Gausss law, discovered by André-Marie Ampère. ...

oint_C mathbf{H} cdot d mathbf{l} = int_S mathbf{J} cdot d mathbf{a}

Again, we can convert to differential form, this time using Stokes' theorem: Stokess theorem in differential geometry is a statement about the integration of differential forms which generalizes several theorems from vector calculus. ...

nabla times mathbf{H} = mathbf{J}

Inconsistency between Ampere's Law and Conservation of Charge

James Clerk Maxwell, who unified the laws of electricity and magnetism, discovered an important inconsistency between Ampere's Law and the Conservation of Charge. James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematical physicist, born in Edinburgh. ...


If we take the divergence of both sides of Ampere's 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
A multiband rotary directional antenna for Amateur Radio use
A multiband rotary directional antenna for Amateur Radio use

Combining these two equations implies that Image File history File linksMetadata Antenna_d44ac. ... Image File history File linksMetadata Antenna_d44ac. ...

nabla cdot mathbf{J} = 0

From the conservation of charge, we know that

nabla cdot mathbf{J} = - { partial rho over partial t }
{ partial rho over partial t } = 0

This last result suggests that the net charge density at any point in space is a fixed constant that cannot ever change, which is of course absurd. Not only is this outcome contrary to all physical intuition, it also directly contradicts the empirical results of thousands of laboratory experiments. It requires not only that electrical charge is conserved, but that it cannot be re-distributed from one place to another. But we know that electrical currents can and do re-distribute electrical charge. As long as the total amount of charge remains constant, conservation of charge allows for the movement of charge from one place to another. So this last result is incorrect.


Something was clearly missing from Ampere's Law, and Maxwell figured out what it was.


Maxwell's correction to Ampere's Law

To understand Maxwell's correction to Ampere's Law, we need to look at another of Maxwell's Equations, namely, Gauss's Law (SI units) in integral form: In physics and mathematical analysis, Gausss law gives the relation between the electric or gravitational flux flowing out a closed surface and, respectively, the electric charge or mass enclosed in the surface. ...

oint_S varepsilon_o mathbf{E} cdot d mathbf{a} = int_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 Ampere's Law and the conservation of charge equation, suggests that there are actually two sources of the magnetic field: the current density J, as Ampere had already established, and the so-called displacement current: Displacement current is a quantity related to a changing electric field. ...

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

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

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

Maxwell - First to propose that light is an electromagnetic wave

Father of Electromagnetic Theory
Father of Electromagnetic Theory
A postcard from Maxwell to Peter Tait.
Enlarge
A postcard from Maxwell to Peter Tait.

Maxwell's correction of Ampere's Law set the stage for an even more important and, at the time, startling discovery made by Heinrich Rudolph Hertz. Maxwell realized that the equations of electromagnetism suggest that electric and magnetic fields can propagate through free space – in other words, in the absence of matter – as electromagnetic waves, and further, that the speed of propagation of these waves is exactly the same as the speed of light. Reflecting on his discovery in 1865, Maxwell wrote: Image File history File links File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Image File history File links File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Postcard from James Clerk Maxwell to Peter Guthrie Tait. ... Postcard from James Clerk Maxwell to Peter Guthrie Tait. ... Peter Tait Peter Guthrie Tait (April 28, 1831 - July 4, 1901) was a Scottish mathematical physicist. ... Heinrich Hertz Heinrich Rudolf Hertz (February 22, 1857 - January 1, 1894), was the German physicist for whom the hertz, the SI unit of frequency, is named. ... The speed of light in a vacuum is an important physical constant denoted by the letter c for constant or the Latin word celeritas meaning swiftness. In metric units, c is exactly 299,792,458 metres per second (1,079,252,848. ... 1865 (MDCCCLXV) is a common year starting on Sunday. ...

This velocity is so nearly that of light, that it seems we have strong reason to conclude that light itself . . . is an electromagnetic disturbance in the form of waves propagated through the electromagnetic field according to electromagnetic laws.

To obtain electromagnetic waves in a vacuum note that Maxwell's equations (SI units) in a vacuum are Electromagnetic radiation or EM radiation is a combination (cross product) of oscillating electric and magnetic fields perpendicular to each other, moving through space as a wave, effectively transporting energy and momentum. ...

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.


Covariant form of the homogeneous wave equation

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

These relativistic equations can be written in covariant form as Image File history File links This animated GIF is meant to be used as an illustration for the time dilation article. ... The special theory of relativity was proposed in 1905 by Albert Einstein in his article On the Electrodynamics of Moving Bodies. Some three centuries earlier, Galileos principle of relativity had stated that all uniform motion was relative, and that there was no absolute and well-defined state of rest... In special relativity, in order to more clearly express the fact that Maxwells equations (in vacuum) take the same form in any inertial coordinate system, the vacuum Maxwells equations are written in terms of four-vectors and tensors in the manifestly covariant form (cgs units): , and where is... It has been suggested that this article or section be merged into Covariant transformation. ...

Box A^{mu} = 0 quad mbox{(SI units)}
Box A^{mu} = 0 quad mbox{(cgs units)}

where the electromagnetic four-potential is Electromagnetic 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.

In special relativity, electromagnetism and wave theory, the dAlembert operator, also called dAlembertian, is the Laplace operator of Minkowski space. ...

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. To meet Wikipedias quality standards, this article or section may require cleanup. ... In mathematics, the covariant derivative is a way of specifying a derivative along vector fields on a manifold. ...

- {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. In differential geometry, the Ricci curvature tensor is (0,2)-valent tensor, obtained as a trace of the full curvature tensor. ...


We have assumed the generalization of the Lorenz gauge in curved spacetime The Lorenz gauge (or Lorenz gauge condition) was published by the Danish physicist Ludwig Lorenz. ...

{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. Localized time-varying charge and current densities can act as sources of electromagnetic waves in a vacuum. ... In mathematics, and in particular analysis, a partial differential equation (PDE) is an equation involving partial derivatives of an unknown function. ...


Solutions to the homogeneous electromagnetic wave equation

Main article: Wave equation

The general solution to the electromagnetic wave equation is a linear superposition of waves of the form The wave equation is an important partial differential equation that describes a variety of waves, such as sound waves, light waves and water waves. ... In linear algebra, the principle of superposition states that, for a linear system, a linear combination of solutions to the system is also a solution to the same linear system. ...

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. It has been suggested that this article or section be merged into Angular velocity. ... A wave vector is a vector that represents two properties of a wave: the magnitude of the vector represents wavenumber (inversely related to wavelength), and the vector points in the direction of wave propagation. ... In trigonometry, an ideal sine wave is a waveform whose graph is identical to the generalized sine function y = Asin[ω(x − α)] + C, where A is the amplitude, ω is the angular frequency (2π/P where P is the wavelength), α is the phase shift, and C... In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. ...


In addition, for a valid solution, the wave vector and the angular frequency are not independent; they must adhere to the dispersion relation: The relation between the energy of a system and its corresponding momentum is known as its dispersion relation. ...

k = | mathbf{k} | = { omega over c } = { 2 pi over lambda }

where k is the wavenumber and λ is the wavelength. Wavenumber in most physical sciences is a wave property inversely related to wavelength, having units of inverse length. ... The wavelength is the distance between repeating units of a wave pattern. ...


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

In mathematics, the imaginary unit (sometimes also represented by the Latin or the Greek iota) allows the real number system to be extended to the complex number system . ... It has been suggested that this article or section be merged into Angular velocity. ... Angular frequency is a measure of how fast an object is rotating In physics (specifically mechanics and electrical engineering), angular frequency ω (also called angular speed) is a scalar measure of rotation rate. ... Sine waves of various frequencies; the lower waves have higher frequencies than those above. ... The hertz (symbol: Hz) is the SI unit of frequency. ... Eulers formula, named after Leonhard Euler, is a mathematical formula in complex analysis that shows a deep relationship between the trigonometric functions and the complex exponential function. ...

Plane wave solutions

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

Consider a plane defined by a unit normal vector Perhaps the most useful solutions to the electromagnetic wave equation are sinusoidal plane-wave solutions. ...

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. In electrodynamics, polarization (also spelled polarisation) is the property of electromagnetic waves, such as light, that describes the direction of their transverse electric field. ...


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 In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. ...

Electromagnetic spectrum illustration.
Electromagnetic spectrum illustration.
mathbf{E} ( mathbf{r}, t ) = cos( omega t - mathbf{k} cdot mathbf{r} + phi_0 )

and Image File history File links Spectre. ... Image File history File links Spectre. ...

mathbf{H} ( mathbf{r}, t ) = 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 It has been suggested that this article or section be merged into Angular velocity. ... A wave vector is a vector that represents two properties of a wave: the magnitude of the vector represents wavenumber (inversely related to wavelength), and the vector points in the direction of wave propagation. ... The phase angle of a point on a periodic wave is the distance between the point and a specified reference point, expressed using an angular measure. ...

k = | mathbf{k} | = { omega over c } = { 2 pi over lambda }

where k is the wavenumber and λ is the wavelength. Wavenumber in most physical sciences is a wave property inversely related to wavelength, having units of inverse length. ... The wavelength is the distance between repeating units of a wave pattern. ...


The Electromagnetic spectrum is a plot of the field magnitudes (or energies) as a function of wavelength. Legend: γ = Gamma rays HX = Hard X-rays SX = Soft X-Rays EUV = Extreme ultraviolet NUV = Near ultraviolet Visible light NIR = Near infrared MIR = Moderate infrared FIR = Far infrared Radio waves: EHF = Extremely high frequency (Microwaves) SHF = Super high frequency (Microwaves) UHF = Ultra high frequency VHF = Very high frequency HF = High...


Other solutions

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


References

Electromagnetics

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.)

A Dynamical Theory of the Electromagnetic Field was the third of James Clerk Maxwells papers concerned with electromagnetism. ...

Undergraduate-level textbooks

  • 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.

Graduate-level textbooks

  • 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.)

Lev Davidovich Landau (Ле́в Дави́дович Ланда́у) (January 22, 1908 – April 1, 1968) was a prominent Soviet physicist and winner of the Nobel Prize for Physics whose broad field of work included the theory of superconductivity and superfluidity, quantum electrodynamics, nuclear physics and particle physics. ... Kip S. Thorne Professor Kip Stephen Thorne, Ph. ... John Archibald Wheeler (born July 9, 1911) is an eminent American theoretical physicist. ...

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

Maxwells equations (sometimes called the Maxwell equations) are the set of four equations, attributed to James Clerk Maxwell, that describe the behavior of both the electric and magnetic fields, as well as their interactions with matter. ... The wave equation is an important partial differential equation that describes a variety of waves, such as sound waves, light waves and water waves. ... Computational electromagnetics, computational electrodynamics or electromagnetic modeling refers to the process of modeling the interaction of electromagnetic fields with physical objects and the environment. ... Electromagnetic radiation can be imagined as a self-propagating transverse oscillating wave of electric and magnetic fields. ... Charge conservation is the principle that electric charge can neither be created nor destroyed. ... Prism splitting light Light is electromagnetic radiation with a wavelength that is visible to the eye (visible light) or, in a technical or scientific context, electromagnetic radiation of any wavelength [citation needed]. The elementary particle that defines light is the photon. ... Legend: γ = Gamma rays HX = Hard X-rays SX = Soft X-Rays EUV = Extreme ultraviolet NUV = Near ultraviolet Visible light NIR = Near infrared MIR = Moderate infrared FIR = Far infrared Radio waves: EHF = Extremely high frequency (Microwaves) SHF = Super high frequency (Microwaves) UHF = Ultra high frequency VHF = Very high frequency HF = High... Table of Opticks, 1728 Cyclopaedia Optics ( appearance or look in ancient Greek) is a branch of physics that describes the behavior and properties of light and the interaction of light with matter. ... The special theory of relativity was proposed in 1905 by Albert Einstein in his article On the Electrodynamics of Moving Bodies. Some three centuries earlier, Galileos principle of relativity had stated that all uniform motion was relative, and that there was no absolute and well-defined state of rest... General relativity (GR) is the geometrical theory of gravitation published by Albert Einstein in 1915. ... The Dynamics of photons in the double-slit experiment describes the relationship between classical electromagnetic waves and photons, the quantum counterpart of classical electromagnetic waves, in the context of the double-slit experiment. ... Photon polarization is the quantum mechanical description of the classical polarized sinusoidal plane electromagnetic wave. ... The Larmor formula is used to calculate the power radiated by a nonrelativistic electron as it accelerates. ... The theoretical and experimental justification for the Schrödinger equation motivates the discovery of the Schrödinger equation, the equation that describes the dynamics of nonrelativistic particles. ...

Applications

Full featured rainbow in Wrangell-St. ... In cosmology, the cosmic microwave background radiation (most often abbreviated CMB but occasionally CMBR, CBR or MBR, also referred as relic radiation) is a form of electromagnetic radiation discovered in 1965 that fills the entire universe. ... // Experiment using a (likely argon) laser. ... In inertial confinement fusion (ICF), nuclear fusion reactions are initiated by heating and compressing a target – a pellet that most often contains deuterium and tritium – by the use of intense laser or ion beams. ... Photography is the process of making pictures by means of the action of light. ... In the NATO phonetic alphabet, X-ray represents the letter X. An X-ray picture (radiograph) taken by Röntgen An X-ray is a form of electromagnetic radiation with a wavelength approximately in the range of 5 pm to 10 nanometers (corresponding to frequencies in the range 30 PHz... X-ray crystallography is a technique in crystallography in which the pattern produced by the diffraction of X-rays through the closely spaced lattice of atoms in a crystal is recorded and then analyzed to reveal the nature of that lattice. ... 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[1]. RADAR is a system that uses radio waves to detect, determine the direction and distance and/or speed... Radio frequency, or RF, refers to that portion of the electromagnetic spectrum in which electromagnetic waves can be generated by alternating current fed to an antenna. ... An Optical Computer is a computer that performs its computation with photons as opposed to the more traditional electron-based computation. ... {{otheruses4|the radiation|the appliance|microwaves equency = 1 GHz) to 1 mm (300 GHz). ... Holography (from the Greek, Όλος-holos whole + γραφή-graphe writing) is the science of producing holograms; it is an advanced form of photography that allows an image to be recorded in three dimensions. ... A microscope (Greek: (micron) = small + (skopein) = to look at) is an instrument for viewing objects that are too small to be seen by the naked or unaided eye. ... 50 cm refracting telescope at Nice Observatory. ... This article is in need of attention from an expert on the subject. ... As the temperature decreases, the peak of the black body radiation curve moves to lower intensities and longer wavelengths. ...

Biographies


Einstein redirects here. ... James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematical physicist, born in Edinburgh. ... Michael Faraday, FRS (September 22, 1791 – August 25, 1867) was an English chemist and physicist (or natural philosopher, in the terminology of that time) who contributed significantly to the fields of electromagnetism and electrochemistry. ... Heinrich Hertz Heinrich Rudolf Hertz (February 22, 1857 - January 1, 1894), was the German physicist for whom the hertz, the SI unit of frequency, is named. ...

  General subfields within physics  v  d  e 

Classical mechanics | Electromagnetism | Thermodynamics | General relativity | Quantum mechanics  Physics (from the Greek, (phúsis), nature and (phusiké), knowledge of nature) is the science concerned with the discovery and understanding of the fundamental laws which govern matter, energy, space and time. ... Classical mechanics is a branch of physics which studies the deterministic motion of objects. ... Electromagnetism is the force observed as static electricity, and causes the flow of electric charge (electric current) in electrical conductors. ... Thermodynamics (from the Greek thermos meaning heat and dynamics meaning power) is a branch of physics that studies the effects of changes in temperature, pressure, and volume on physical systems at the macroscopic scale by analyzing the collective motion of their particles using statistics. ... General relativity (GR) is the geometrical theory of gravitation published by Albert Einstein in 1915. ... Fig. ...

Particle physics | Condensed matter physics | Atomic, molecular, and optical physics  Particles explode from the collision point of two relativistic (100 GeV per nucleon) gold ions in the STAR detector of the Relativistic Heavy Ion Collider. ... Condensed matter physics is the field of physics that deals with the macroscopic physical properties of matter. ... Atomic, molecular, and optical physics is the study of matter-matter and light-matter interactions on the scale of single atoms or structures containing a few atoms. ...


  Results from FactBites:
 
Electromagnetic wave equation - Wikipedia, the free encyclopedia (1836 words)
The electromagnetic wave equation derives from Maxwell's equations.
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.
Theoretical and experimental justification for the Schrödinger equation
wave: Definition, Synonyms and Much More from Answers.com (4581 words)
In seismology, waves moving though the earth are caused by the propagation of a disturbance generated by an earthquake or explosion.
Transverse waves are those with vibrations perpendicular to the direction of the propagation of the wave; examples include waves on a string and electromagnetic waves.
The units of the amplitude depend on the type of wavewaves on a string have an amplitude expressed as a distance (meters), sound waves as pressure (pascals) and electromagnetic waves as the amplitude of the electric field (volts/meter).
  More results at FactBites »

 
 

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