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Physics

Theory

File:VisibleEmrWavelengths.svg|thumb|Shows the relative wavelengths of the electromagnetic waves of three different colours of light (blue, green, and red) with a distance scale in micrometers along the x-axis.]]

Maxwellâ€™s equations

James Clerk Maxwell derived a wave form of the electric and magnetic equations, thus uncovering the wave-like nature of electric and magnetic fields and their symmetry. Because the speed of EM waves predicted by the wave equation coincided with the measured speed of light, Maxwell concluded that light itself is an EM wave.WEB,weblink Electromagnetic Waves, The Physics Hypertextbook, Elert, Glenn, 4 June 2018, WEB,weblink The Impact of James Clerk Maxwell's Work, www.clerkmaxwellfoundation.org, 2017-09-04, live,weblink" title="web.archive.org/web/20170917213509weblink">weblink 17 September 2017, dmy-all, Maxwell's equations were confirmed by Heinrich Hertz through experiments with radio waves.According to Maxwell's equations, a spatially varying electric field is always associated with a magnetic field that changes over time.Purcell, p 438, section 9.4: An Electromagnetic Wave. Likewise, a spatially varying magnetic field is associated with specific changes over time in the electric field. In an electromagnetic wave, the changes in the electric field are always accompanied by a wave in the magnetic field in one direction, and vice versa. This relationship between the two occurs without either type of field causing the other; rather, they occur together in the same way that time and space changes occur together and are interlinked in special relativity. In fact, magnetic fields can be viewed as electric fields in another frame of reference, and electric fields can be viewed as magnetic fields in another frame of reference, but they have equal significance as physics is the same in all frames of reference, so the close relationship between space and time changes here is more than an analogy. Together, these fields form a propagating electromagnetic wave, which moves out into space and need never again interact with the source. The distant EM field formed in this way by the acceleration of a charge carries energy with it that "radiates" away through space, hence the term.

Wave model

(File:Circular.Polarization.Circularly.Polarized.Light Right.Handed.Animation.305x190.255Colors.gif|thumb|right|Representation of the electric field vector of a wave of circularly polarized electromagnetic radiation.) In homogeneous, isotropic media, electromagnetic radiation is a transverse wave,BOOK, Electromagnetic Theory, Julius Adams, Stratton, McGraw-Hill Book Company, New York, NY, 1941,weblink Chapter V Plane waves in unbounded, isotropic media, meaning that its oscillations are perpendicular to the direction of energy transfer and travel.BOOK, {{google books, y, qutFCgAAQBAJ, 48, |title=CNPS Proceedings 2015|last=Hilster|first=David de|publisher=Lulu.com|isbn=9781329313118|language=en|url-status=live|df=dmy-all|date=2015-07-21}} The electric and magnetic parts of the field stand in a fixed ratio of strengths in order to satisfy the two Maxwell equations that specify how one is produced from the other. In dissipation less (lossless) media, these E and B fields are also in phase, with both reaching maxima and minima at the same points in space (see illustrations). A common misconception is that the E and B fields in electromagnetic radiation are out of phase because a change in one produces the other, and this would produce a phase difference between them as sinusoidal functions (as indeed happens in electromagnetic induction, and in the near-field close to antennas). However, in the far-field EM radiation which is described by the two source-free Maxwell curl operator equations, a more correct description is that a time-change in one type of field is proportional to a space-change in the other. These derivatives require that the E and B fields in EMR are in-phase (see math section below).{{citation needed|date=July 2013}}An important aspect of light's nature is its frequency. The frequency of a wave is its rate of oscillation and is measured in hertz, the SI unit of frequency, where one hertz is equal to one oscillation per second. Light usually has multiple frequencies that sum to form the resultant wave. Different frequencies undergo different angles of refraction, a phenomenon known as dispersion.A wave consists of successive troughs and crests, and the distance between two adjacent crests or troughs is called the wavelength. Waves of the electromagnetic spectrum vary in size, from very long radio waves the size of buildings to very short gamma rays smaller than atom nuclei. Frequency is inversely proportional to wavelength, according to the equation:{{citation needed|date=July 2013}}
displaystyle v=flambda
where v is the speed of the wave (c in a vacuum, or less in other media), f is the frequency and Î» is the wavelength. As waves cross boundaries between different media, their speeds change but their frequencies remain constant.Electromagnetic waves in free space must be solutions of Maxwell's electromagnetic wave equation. Two main classes of solutions are known, namely plane waves and spherical waves. The plane waves may be viewed as the limiting case of spherical waves at a very large (ideally infinite) distance from the source. Both types of waves can have a waveform which is an arbitrary time function (so long as it is sufficiently differentiable to conform to the wave equation). As with any time function, this can be decomposed by means of Fourier analysis into its frequency spectrum, or individual sinusoidal components, each of which contains a single frequency, amplitude and phase. Such a component wave is said to be monochromatic. A monochromatic electromagnetic wave can be characterized by its frequency or wavelength, its peak amplitude, its phase relative to some reference phase, its direction of propagation and its polarization.Interference is the superposition of two or more waves resulting in a new wave pattern. If the fields have components in the same direction, they constructively interfere, while opposite directions cause destructive interference. An example of interference caused by EMR is electromagnetic interference (EMI) or as it is more commonly known as, radio-frequency interference (RFI).{{citation needed|date=July 2013}} Additionally, multiple polarization signals can be combined (i.e. interfered) to form new states of polarization, which is known as parallel polarization state generation.JOURNAL, She, Alan, Capasso, Federico, Parallel Polarization State Generation, Scientific Reports, 6, 26019, 10.1038/srep26019, 27184813, 4869035, 17 May 2016, 1602.04463, 2016NatSR...626019S, The energy in electromagnetic waves is sometimes called radiant energy.NEWS,weblink What Is Electromagnetic Radiation?, Live Science, 2017-09-04, live,weblink 4 September 2017, dmy-all, BOOK, {{google books, y, sjQHyn5ZVcIC, 635, |title=The Earth Around Us: Maintaining A Livable Planet|last=Schneiderman|first=Jill|date=2000-03-27|publisher=Henry Holt and Company|isbn=9781466814431|language=en|url-status=live|df=dmy-all}}BOOK, {{google books, y, AUriAAAAMAAJ, 22, |title=The Michigan Technic|date=1960|publisher=UM Libraries|language=en|url-status=live|df=dmy-all}}{{citation needed|date=July 2013}}

Particle model and quantum theory

{{see also|Quantization (physics)|Quantum optics}}(File:Laser photons.gif|thumb|right|Representation of how the alternating electric field vector of a wave of circularly polarized electromagnetic radiation emerge.)An anomaly arose in the late 19th century involving a contradiction between the wave theory of light and measurements of the electromagnetic spectra that were being emitted by thermal radiators known as black bodies. Physicists struggled with this problem unsuccessfully for many years. It later became known as the ultraviolet catastrophe. In 1900, Max Planck developed a new theory of black-body radiation that explained the observed spectrum. Planck's theory was based on the idea that black bodies emit light (and other electromagnetic radiation) only as discrete bundles or packets of energy. These packets were called quanta. In 1905, Albert Einstein proposed that light quanta be regarded as real particles. Later the particle of light was given the name photon, to correspond with other particles being described around this time, such as the electron and proton. A photon has an energy, E, proportional to its frequency, f, by
E = hf = frac{hc}{lambda} ,!
where h is Planck's constant, lambda is the wavelength and c is the speed of light. This is sometimes known as the Planckâ€“Einstein equation.BOOK
, Physical Chemistry
, Paul M. S. Monk
, John Wiley and Sons
, 2004
, 978-0-471-49180-4
, 435
,
, In quantum theory (see first quantization) the energy of the photons is thus directly proportional to the frequency of the EMR wave.BOOK
, Weinberg
, S.
, Steven Weinberg
, The Quantum Theory of Fields
, 1
, Cambridge University Press
, 1995
, 978-0-521-55001-7
, 15â€“17
,
Likewise, the momentum p of a photon is also proportional to its frequency and inversely proportional to its wavelength:
p = { E over c } = { hf over c } = { h over lambda }.
The source of Einstein's proposal that light was composed of particles (or could act as particles in some circumstances) was an experimental anomaly not explained by the wave theory: the photoelectric effect, in which light striking a metal surface ejected electrons from the surface, causing an electric current to flow across an applied voltage. Experimental measurements demonstrated that the energy of individual ejected electrons was proportional to the frequency, rather than the intensity, of the light. Furthermore, below a certain minimum frequency, which depended on the particular metal, no current would flow regardless of the intensity. These observations appeared to contradict the wave theory, and for years physicists tried in vain to find an explanation. In 1905, Einstein explained this puzzle by resurrecting the particle theory of light to explain the observed effect. Because of the preponderance of evidence in favor of the wave theory, however, Einstein's ideas were met initially with great skepticism among established physicists. Eventually Einstein's explanation was accepted as new particle-like behavior of light was observed, such as the Compton effect.{{citation needed|date=July 2013}}As a photon is absorbed by an atom, it excites the atom, elevating an electron to a higher energy level (one that is on average farther from the nucleus). When an electron in an excited molecule or atom descends to a lower energy level, it emits a photon of light at a frequency corresponding to the energy difference. Since the energy levels of electrons in atoms are discrete, each element and each molecule emits and absorbs its own characteristic frequencies. Immediate photon emission is called fluorescence, a type of photoluminescence. An example is visible light emitted from fluorescent paints, in response to ultraviolet (blacklight). Many other fluorescent emissions are known in spectral bands other than visible light. Delayed emission is called phosphorescence.WEB,weblink 7 Differences between Fluorescence and Phosphorescence, Haneef, Deena T. Kochunni, Jazir, 2017-09-04, live,weblink" title="web.archive.org/web/20170904152324weblink">weblink 4 September 2017, dmy-all, BOOK, {{google books, y, kAn4AgAAQBAJ, 93, |title=Fundamental Physics of Radiology|last=Meredith|first=W. J.|last2=Massey|first2=J. B.|date=2013-10-22|publisher=Butterworth-Heinemann|isbn=9781483284354|language=en|url-status=live|df=dmy-all}}{{citation needed|date=July 2013}}

Waveâ€“particle duality

The modern theory that explains the nature of light includes the notion of waveâ€“particle duality. More generally, the theory states that everything has both a particle nature and a wave nature, and various experiments can be done to bring out one or the other. The particle nature is more easily discerned using an object with a large mass. A bold proposition by Louis de Broglie in 1924 led the scientific community to realize that matter (e.g. electrons) also exhibits waveâ€“particle duality.BOOK, Browne, Michael, Physics for Engineering and Science, McGraw-Hill/Schaum, 2nd, 2010, 978-0-07-161399-6, Chapter 36, page 382: de Broglie Waves. "Light exhibits both wave properties (interference, diffraction, refraction) and particle properties (photoelectric effect, scattering.)"

Wave and particle effects of electromagnetic radiation

Together, wave and particle effects fully explain the emission and absorption spectra of EM radiation. The matter-composition of the medium through which the light travels determines the nature of the absorption and emission spectrum. These bands correspond to the allowed energy levels in the atoms. Dark bands in the absorption spectrum are due to the atoms in an intervening medium between source and observer. The atoms absorb certain frequencies of the light between emitter and detector/eye, then emit them in all directions. A dark band appears to the detector, due to the radiation scattered out of the beam. For instance, dark bands in the light emitted by a distant star are due to the atoms in the star's atmosphere. A similar phenomenon occurs for emission, which is seen when an emitting gas glows due to excitation of the atoms from any mechanism, including heat. As electrons descend to lower energy levels, a spectrum is emitted that represents the jumps between the energy levels of the electrons, but lines are seen because again emission happens only at particular energies after excitation.Browne, p 376: "Radiation is emitted or absorbed only when the electron jumps from one orbit to the other, and the frequency of radiation depends only upon on the energies of the electron in the initial and final orbits. An example is the emission spectrum of nebulae.{{citation needed|date=July 2013}} Rapidly moving electrons are most sharply accelerated when they encounter a region of force, so they are responsible for producing much of the highest frequency electromagnetic radiation observed in nature.These phenomena can aid various chemical determinations for the composition of gases lit from behind (absorption spectra) and for glowing gases (emission spectra). Spectroscopy (for example) determines what chemical elements comprise a particular star. Spectroscopy is also used in the determination of the distance of a star, using the red shift.WEB, Spectroscopy,weblink National Redshift Project, National Redshift Project, 19 January 2017, live,weblink" title="web.archive.org/web/20170201234024weblink">weblink 1 February 2017, dmy-all,

Propagation speed

When any wire (or other conducting object such as an antenna) conducts alternating current, electromagnetic radiation is propagated at the same frequency as the current. In many such situations it is possible to identify an electrical dipole moment that arises from separation of charges due to the exciting electrical potential, and this dipole moment oscillates in time, as the charges move back and forth. This oscillation at a given frequency gives rise to changing electric and magnetic fields, which then set the electromagnetic radiation in motion.{{citation needed|date=July 2013}}At the quantum level, electromagnetic radiation is produced when the wavepacket of a charged particle oscillates or otherwise accelerates. Charged particles in a stationary state do not move, but a superposition of such states may result in a transition state that has an electric dipole moment that oscillates in time. This oscillating dipole moment is responsible for the phenomenon of radiative transition between quantum states of a charged particle. Such states occur (for example) in atoms when photons are radiated as the atom shifts from one stationary state to another.{{citation needed|date=July 2013}}As a wave, light is characterized by a velocity (the speed of light), wavelength, and frequency. As particles, light is a stream of photons. Each has an energy related to the frequency of the wave given by Planck's relation E = hf, where E is the energy of the photon, h = 6.626 Ã— 10âˆ’34 JÂ·s is Planck's constant, and f is the frequency of the wave.{{citation needed|date=July 2013}}One rule is obeyed regardless of circumstances: EM radiation in a vacuum travels at the speed of light, relative to the observer, regardless of the observer's velocity. (This observation led to Einstein's development of the theory of special relativity.){{citation needed|date=July 2013}}In a medium (other than vacuum), velocity factor or refractive index are considered, depending on frequency and application. Both of these are ratios of the speed in a medium to speed in a vacuum.{{citation needed|date=July 2013}}

Special theory of relativity

By the late nineteenth century, various experimental anomalies could not be explained by the simple wave theory. One of these anomalies involved a controversy over the speed of light. The speed of light and other EMR predicted by Maxwell's equations did not appear unless the equations were modified in a way first suggested by FitzGerald and Lorentz (see history of special relativity), or else otherwise that speed would depend on the speed of observer relative to the "medium" (called luminiferous aether) which supposedly "carried" the electromagnetic wave (in a manner analogous to the way air carries sound waves). Experiments failed to find any observer effect. In 1905, Einstein proposed that space and time appeared to be velocity-changeable entities for light propagation and all other processes and laws. These changes accounted for the constancy of the speed of light and all electromagnetic radiation, from the viewpoints of all observersâ€”even those in relative motion.

Electromagnetic spectrum

Radio waves have the least amount of energy and the lowest frequency. When radio waves impinge upon a conductor, they couple to the conductor, travel along it and induce an electric current on the conductor surface by moving the electrons of the conducting material in correlated bunches of charge. Such effects can cover macroscopic distances in conductors (such as radio antennas), since the wavelength of radiowaves is long.Electromagnetic radiation phenomena with wavelengths ranging from as long as one meter to as short as one millimeter are called microwaves; with frequencies between 300 MHz (0.3 GHz) and 300 GHz.At radio and microwave frequencies, EMR interacts with matter largely as a bulk collection of charges which are spread out over large numbers of affected atoms. In electrical conductors, such induced bulk movement of charges (electric currents) results in absorption of the EMR, or else separations of charges that cause generation of new EMR (effective reflection of the EMR). An example is absorption or emission of radio waves by antennas, or absorption of microwaves by water or other molecules with an electric dipole moment, as for example inside a microwave oven. These interactions produce either electric currents or heat, or both.

Infrared

Like radio and microwave, infrared also is reflected by metals (as is most EMR into the ultraviolet). However, unlike lower-frequency radio and microwave radiation, Infrared EMR commonly interacts with dipoles present in single molecules, which change as atoms vibrate at the ends of a single chemical bond. It is consequently absorbed by a wide range of substances, causing them to increase in temperature as the vibrations dissipate as heat. The same process, run in reverse, causes bulk substances to radiate in the infrared spontaneously (see thermal radiation section below).

Visible light

Natural sources produce EM radiation across the spectrum. EM radiation with a wavelength between approximately 400 nm and 700 nm is directly detected by the human eye and perceived as visible light. Other wavelengths, especially nearby infrared (longer than 700 nm) and ultraviolet (shorter than 400 nm) are also sometimes referred to as light.As frequency increases into the visible range, photons have enough energy to change the bond structure of some individual molecules. It is not a coincidence that this happens in the "visible range," as the mechanism of vision involves the change in bonding of a single molecule (retinal) which absorbs light in the rhodopsin in the retina of the human eye. Photosynthesis becomes possible in this range as well, for similar reasons, as a single molecule of chlorophyll is excited by a single photon. Animals that detect infrared make use of small packets of water that change temperature, in an essentially thermal process that involves many photons (see infrared sensing in snakes). For this reason, infrared, microwaves and radio waves are thought to damage molecules and biological tissue only by bulk heating, not excitation from single photons of the radiation.Visible light is able to affect a few molecules with single photons, but usually not in a permanent or damaging way, in the absence of power high enough to increase temperature to damaging levels. However, in plant tissues that conduct photosynthesis, carotenoids act to quench electronically excited chlorophyll produced by visible light in a process called non-photochemical quenching, in order to prevent reactions that would otherwise interfere with photosynthesis at high light levels. Limited evidence indicate that some reactive oxygen species are created by visible light in skin, and that these may have some role in photoaging, in the same manner as ultraviolet A.JOURNAL, Liebel, F., Kaur, S., Ruvolo, E., Kollias, N., Southall, M. D., Irradiation of Skin with Visible Light Induces Reactive Oxygen Species and Matrix-Degrading Enzymes, 10.1038/jid.2011.476, Journal of Investigative Dermatology, 132, 7, 1901â€“1907, 2012, 22318388,

Ultraviolet

As frequency increases into the ultraviolet, photons now carry enough energy (about three electron volts or more) to excite certain doubly bonded molecules into permanent chemical rearrangement. In DNA, this causes lasting damage. DNA is also indirectly damaged by reactive oxygen species produced by ultraviolet A (UVA), which has energy too low to damage DNA directly. This is why ultraviolet at all wavelengths can damage DNA, and is capable of causing cancer, and (for UVB) skin burns (sunburn) that are far worse than would be produced by simple heating (temperature increase) effects. This property of causing molecular damage that is out of proportion to heating effects, is characteristic of all EMR with frequencies at the visible light range and above. These properties of high-frequency EMR are due to quantum effects that permanently damage materials and tissues at the molecular level.{{citation needed|date=July 2013}}At the higher end of the ultraviolet range, the energy of photons becomes large enough to impart enough energy to electrons to cause them to be liberated from the atom, in a process called photoionisation. The energy required for this is always larger than about 10 electron volt (eV) corresponding with wavelengths smaller than 124 nm (some sources suggest a more realistic cutoff of 33 eV, which is the energy required to ionize water). This high end of the ultraviolet spectrum with energies in the approximate ionization range, is sometimes called "extreme UV." Ionizing UV is strongly filtered by the Earth's atmosphere).{{citation needed|date=July 2013}}

X-rays and gamma rays

Electromagnetic radiation composed of photons that carry minimum-ionization energy, or more, (which includes the entire spectrum with shorter wavelengths), is therefore termed ionizing radiation. (Many other kinds of ionizing radiation are made of non-EM particles). Electromagnetic-type ionizing radiation extends from the extreme ultraviolet to all higher frequencies and shorter wavelengths, which means that all X-rays and gamma rays qualify. These are capable of the most severe types of molecular damage, which can happen in biology to any type of biomolecule, including mutation and cancer, and often at great depths below the skin, since the higher end of the X-ray spectrum, and all of the gamma ray spectrum, penetrate matter.

Atmosphere and magnetosphere

File:Atmospheric electromagnetic opacity.svg|thumb|upright=2.25|Rough plot of Earth's atmospheric absorption and scattering (or opacity) of various wavelengthwavelengthMost UV and X-rays are blocked by absorption first from molecular nitrogen, and then (for wavelengths in the upper UV) from the electronic excitation of dioxygen and finally ozone at the mid-range of UV. Only 30% of the Sun's ultraviolet light reaches the ground, and almost all of this is well transmitted.Visible light is well transmitted in air, as it is not energetic enough to excite nitrogen, oxygen, or ozone, but too energetic to excite molecular vibrational frequencies of water vapor.{{citation needed|date=July 2013}}Absorption bands in the infrared are due to modes of vibrational excitation in water vapor. However, at energies too low to excite water vapor, the atmosphere becomes transparent again, allowing free transmission of most microwave and radio waves.{{citation needed|date=July 2013}}Finally, at radio wavelengths longer than 10 meters or so (about 30 MHz), the air in the lower atmosphere remains transparent to radio, but plasma in certain layers of the ionosphere begins to interact with radio waves (see skywave). This property allows some longer wavelengths (100 meters or 3 MHz) to be reflected and results in shortwave radio beyond line-of-sight. However, certain ionospheric effects begin to block incoming radiowaves from space, when their frequency is less than about 10 MHz (wavelength longer than about 30 meters).JOURNAL, Dabas, R S, Ionosphere and its influence on radio communications, Resonance, en, 5, 7, 28â€“43, 10.1007/bf02867245, 0971-8044, July 2000,

Biological effects

Use as weapon

{{see also|Directed energy weapons#Microwave weapons}}The heat ray is an application of EMR that makes use of microwave frequencies to create an unpleasant heating effect in the upper layer of the skin. A publicly known heat ray weapon called the Active Denial System was developed by the US military as an experimental weapon to deny the enemy access to an area.{{citation needed|date=June 2018}} A death ray is a weapon that delivers heat ray electromagnetic energy at levels that injure human tissue. The inventor of the death ray, Harry Grindell Matthews, claims to have lost sight in his left eye while developing his death ray weapon based on a primitive microwave magnetron from the 1920s (a typical microwave oven induces a tissue damaging cooking effect inside the oven at about 2 kV/m).

Derivation from electromagnetic theory

Electromagnetic waves are predicted by the classical laws of electricity and magnetism, known as Maxwell's equations. There are nontrivial solutions of the homogeneous Maxwell's equations (without charges or currents), describing waves of changing electric and magnetic fields. Beginning with Maxwell's equations in free space:{{NumBlk|:|nabla cdot mathbf{E} = 0|{{EquationRef|1}}}}{{NumBlk|:|nabla times mathbf{E} = -frac{partial mathbf{B}}{partial t}|{{EquationRef|2}}}}{{NumBlk|:|nabla cdot mathbf{B} = 0|{{EquationRef|3}}}}{{NumBlk|:|nabla times mathbf{B} = mu_0 varepsilon_0 frac{partial mathbf{E}}{partial t}|{{EquationRef|4}}}}
where
mathbf{E} and mathbf{B} are the vector fields of the Electric Field (measured in V/m or N/C) and the Magnetic Field (measured in T or Wb/m2), respectively; nabla cdot X yields the divergence and nabla times X the curl of a vector field X; frac{partial mathbf{B}}{partial t} and frac{partial mathbf{E}}{partial t} are partial derivatives (rate of change in time, with location fixed) of the magnetic and electric field; mu_0 is the permeability of a vacuum (4pi x 10âˆ’7 (H/m)), and varepsilon_0 is the permittivity of a vacuum (8.85Ã—10âˆ’12 (F/m));
Besides the trivial solution
mathbf{E}=mathbf{B}=mathbf{0},
useful solutions can be derived with the following vector identity, valid for all vectors mathbf{A} in some vector field:
nabla times left( nabla times mathbf{A} right) = nabla left( nabla cdot mathbf{A} right) - nabla^2 mathbf{A}.
Taking the curl of the second Maxwell equation ({{EquationNote|2}}) yields:{{NumBlk|:|nabla times left(nabla times mathbf{E} right) = nabla times left(-frac{partial mathbf{B}}{partial t} right)|{{EquationRef|5}}}}Evaluating the left hand side of ({{EquationNote|5}}) with the above identity and simplifying using ({{EquationNote|1}}), yields:{{NumBlk|:| nabla times left(nabla times mathbf{E} right) = nablaleft(nabla cdot mathbf{E} right) - nabla^2 mathbf{E} = - nabla^2 mathbf{E}.|{{EquationRef|6}}}}Evaluating the right hand side of ({{EquationNote|5}}) by exchanging the sequence of derivations and inserting the fourth {{nowrap|Maxwell equation ({{EquationNote|4}}),}} yields:{{NumBlk|:|nabla times left(-frac{partial mathbf{B}}{partial t} right) = -frac{partial}{partial t} left( nabla times mathbf{B} right) = -mu_0 varepsilon_0 frac{partial^2 mathbf{E}}{partial t^2}|{{EquationRef|7}}}}Combining ({{EquationNote|6}}) and ({{EquationNote|7}}) again, gives a vector-valued differential equation for the electric field, solving the homogeneous Maxwell equations:{{Equation box 1|border=2|border colour=#ccccff|equation=nabla^2 mathbf{E} = mu_0 varepsilon_0 frac{partial^2 mathbf{E}}{partial t^2}}}Taking the curl of the fourth Maxwell equation ({{EquationNote|4}}) results in a similar differential equation for a magnetic field solving the homogeneous Maxwell equations:{{Equation box 1|border=2|border colour=#ccccff|equation=nabla^2 mathbf{B} = mu_0 varepsilon_0 frac{partial^2 mathbf{B}}{partial t^2}.}}Both differential equations have the form of the general wave equation for waves propagating with speed c_0, where f is a function of time and location, which gives the amplitude of the wave at some time at a certain location:
nabla^2 f = frac{1}{hide}c_0}^2} frac{partial^2 f}{partial t^2}
This is also written as:
Box f = 0
where Box denotes the so-called d'Alembert operator, which in Cartesian coordinates is given as:
Box = nabla^2 - frac{1}{{c_0}^2} frac{partial^2}{partial t^2} = frac{partial^2}{partial x^2} + frac{partial^2}{partial y^2} + frac{partial^2}{partial z^2} - frac{1}{{c_0}^2} frac{partial^2}{partial t^2}
Comparing the terms for the speed of propagation, yields in the case of the electric and magnetic fields:
c_0 = frac{1}{sqrt{mu_0 varepsilon_0{edih}.
This is the speed of light in vacuum. Thus Maxwell's equations connect the vacuum permittivity varepsilon_0, the vacuum permeability mu_0, and the speed of light, c0, via the above equation. This relationship had been discovered by Wilhelm Eduard Weber and Rudolf Kohlrausch prior to the development of Maxwell's electrodynamics, however Maxwell was the first to produce a field theory consistent with waves traveling at the speed of light.These are only two equations versus the original four, so more information pertains to these waves hidden within Maxwell's equations. A generic vector wave for the electric field.
mathbf{E} = mathbf{E}_0 fleft( hat{mathbf{k}} cdot mathbf{x} - c_0 t right)
Here, mathbf{E}_0 is the constant amplitude, f is any second differentiable function, hat{mathbf{k}} is a unit vector in the direction of propagation, and {mathbf{x}} is a position vector. fleft( hat{mathbf{k}} cdot mathbf{x} - c_0 t right) is a generic solution to the wave equation. In other words,
nabla^2 fleft( hat{mathbf{k}} cdot mathbf{x} - c_0 t right) = frac{1}{{c_0}^2} frac{partial^2}{partial t^2} fleft( hat{mathbf{k}} cdot mathbf{x} - c_0 t right),
for a generic wave traveling in the hat{mathbf{k}} direction.This form will satisfy the wave equation.
nabla cdot mathbf{E} = hat{mathbf{k}} cdot mathbf{E}_0 f'left( hat{mathbf{k}} cdot mathbf{x} - c_0 t right) = 0 mathbf{E} cdot hat{mathbf{k}} = 0
The first of Maxwell's equations implies that the electric field is orthogonal to the direction the wave propagates.
nabla times mathbf{E} = hat{mathbf{k}} times mathbf{E}_0 f'left( hat{mathbf{k}} cdot mathbf{x} - c_0 t right) = -frac{partial mathbf{B}}{partial t} mathbf{B} = frac{1}{c_0} hat{mathbf{k}} times mathbf{E}
The second of Maxwell's equations yields the magnetic field. The remaining equations will be satisfied by this choice of mathbf{E},mathbf{B}.The electric and magnetic field waves in the far-field travel at the speed of light. They have a special restricted orientation and proportional magnitudes, E_0 = c_0 B_0, which can be seen immediately from the Poynting vector. The electric field, magnetic field, and direction of wave propagation are all orthogonal, and the wave propagates in the same direction as mathbf{E} times mathbf{B}. Also, E and B far-fields in free space, which as wave solutions depend primarily on these two Maxwell equations, are in-phase with each other. This is guaranteed since the generic wave solution is first order in both space and time, and the curl operator on one side of these equations results in first-order spatial derivatives of the wave solution, while the time-derivative on the other side of the equations, which gives the other field, is first-order in time, resulting in the same phase shift for both fields in each mathematical operation.From the viewpoint of an electromagnetic wave traveling forward, the electric field might be oscillating up and down, while the magnetic field oscillates right and left. This picture can be rotated with the electric field oscillating right and left and the magnetic field oscillating down and up. This is a different solution that is traveling in the same direction. This arbitrariness in the orientation with respect to propagation direction is known as polarization. On a quantum level, it is described as photon polarization. The direction of the polarization is defined as the direction of the electric field.More general forms of the second-order wave equations given above are available, allowing for both non-vacuum propagation media and sources. Many competing derivations exist, all with varying levels of approximation and intended applications. One very general example is a form of the electric field equation,JOURNAL, Kinsler, P., 2010, Optical pulse propagation with minimal approximations, Phys. Rev. A, 81, 1, 013819, 10.1103/PhysRevA.81.013819, 0810.5689, 2010PhRvA..81a3819K,
which was factorized into a pair of explicitly directional wave equations, and then efficiently reduced into a single uni-directional wave equation by means of a simple slow-evolution approximation.

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