.

Electromagnetic radiation (EM radiation or EMR) is a form of energy emitted and absorbed by charged particles, which exhibits wave-like behavior as it travels through space. EMR has both electric and magnetic field components, which stand in a fixed ratio of intensity to each other, and which oscillate in phase perpendicular to each other and perpendicular to the direction of energy and wave propagation. In vacuum, electromagnetic radiation propagates at a characteristic speed, the speed of light.

Electromagnetic radiation is a particular form of the more general electromagnetic field (EM field) that is defined as the field produced by moving charges. Electromagnetic radiation is associated with only the type of EM field which is far enough away from the moving charges that produced it, that absorption of the EM radiation no longer affects the behavior of these moving charges. These two types or behaviors of EM field are sometimes referred to as the near and far field. In this language, EMR is merely another name for the far-field. Charges and currents directly produce the near-field. However, charges and currents produce EMR only indirectly- rather, in EMR, both the magnetic and electric fields are produced by changes in the other type of field, not directly by charges and currents.

EMR carries energy - sometimes called radiant energy - through space continuously away from the source (this is not true of the near-field part of the EM field). EMR also carries both momentum and angular momentum. These properties may all be imparted to matter with which it interacts. EMR is produced from other types of energy when created, and it is converted to other types of energy when it is destroyed. The photon is the quantum of the electromagnetic interaction, and is the basic "unit" or constituent of all forms of EMR. The quantum nature of light becomes more apparent at high frequencies (or high photon energy). Such photons behave more like particles than lower-frequency photons do.

In classical physics, EMR is considered to be produced when charged particles are accelerated by forces acting on them. Electrons are responsible for emission of most EMR because they have low mass, and therefore are easily accelerated by a variety of mechanisms. 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. Quantum processes can also produce EMR, such as when atomic nuclei undergo gamma decay, and processes such as neutral pion decay.

EMR is classified according to the frequency of its wave. The electromagnetic spectrum, in order of increasing frequency and decreasing wavelength, consists of radio waves, microwaves, infrared radiation, visible light, ultraviolet radiation, X-rays and gamma rays. The eyes of various organisms sense a small and somewhat variable window of frequencies of EMR called the visible spectrum.

The effects of EMR upon biological systems (and also to many other chemical systems, under standard conditions) depends both upon the radiation's power and frequency. For lower frequencies of EMR up to those of visible light (i.e., radio, microwave, infrared), the damage done to cells and also to many ordinary materials under such conditions is determined mainly by heating effects, and thus by the radiation power. By contrast, for higher frequency radiations at ultraviolet frequencies and above (i.e., X-rays and gamma rays) the damage to chemical materials and living cells by EMR is far larger than that done by simple heating, due to the ability of single photons in such high frequency EMR to damage individual molecules chemically.

Physics
Theory
Shows the relative wavelengths of the electromagnetic waves of three different colors of light (blue, green, and red) with a distance scale in micrometres along the x-axis.
Main articles: Maxwell's equations and Near and far field
Maxwell’s equations for EM fields far from sources

James Clerk Maxwell first formally postulated electromagnetic waves. These were subsequently confirmed by Heinrich Hertz. 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.

According to Maxwell's equations, a spatially varying electric field causes the magnetic field to change over time. Likewise, a spatially varying magnetic field causes changes over time in the electric field. In an electromagnetic wave, the changes induced by the electric field shift the wave in the magnetic field in one direction; the action of the magnetic field shifts the electric field in the same direction. Together, these fields form a propagating electromagnetic wave, which moves out into space and never again affects the source. The EM field formed by this mechanism "radiates," hence the term for it.
Near and far fields

Looking at the original charges and currents that are the cause of the wave brings into play terms involving charges and currents ("sources") in Maxwell’s equations that produce a local type of electromagnetic field near sources that does not have the behavior of EMR. In particular, according to Maxwell, currents directly produce a magnetic field, but it is of a magnetic dipole type which dies out rapidly with distance from the current. In a similar manner, moving charges being separated from each other in a conductor by a changing electical potential (such as in an antenna) produce an electric dipole type electrical field, but this also dies away very quickly with distance. Both of these fields make up the near-field near the EMR source. Neither of these behaviors are responsible for EM radiation. Instead, they cause electromagnetic field behavior that only efficiently transfers power to a receiver very close to the source, such as the magnetic induction inside an electrical transformer, or the feedback behavior that happens close to the coil of a metal detector. Typically, near-fields have a powerful effect on their own sources, causing an increased “load” (decreased electrical reactance) in the source or transmitter, whenever energy is withdrawn from the EM field by a receiver. Otherwise, these fields do not “propagate,” freely out into space, carrying their energy away without distance-limit, but rather oscillate back and forth, returning their energy to the transmitter if it is not received by a receiver.

By contrast, the EM far-field is composed of radiation that is free of the transmitter in the sense that (unlike the case in an electrical transformer) the transmitter requires the same power to send these changes in the fields out, whether the signal is immediately picked up, or not. This distant part of the electromagnetic field is "electromagnetic radiation" (also called the far-field). The far-fields propagate without ability for the transmitter to affect them, and this causes them to be independent in the sense that their existence and their energy is completely independent of both transmitter and receiver.

The far-field (EMR) depends on a different mechanism for its production than the near-field, and upon different terms in Maxwell’s equations. Whereas the magnetic part of the near-field is due to currents in the source, the magnetic field in EMR is due only to the local change in the electric field. In a similar way, while the electric field in the near-field is due directly to the charges and charge-separation in the source, the electric field in EMR is due to a change in the local magnetic field. Both of these processes for producing electric and magnetic fields have a different dependence on distance than near-field dipoles, and that is why the EMR type of EM field becomes dominant in power “far” from sources. The term “far from sources” refers to how far from the source (moving at the speed of light) any portion of the outward-moving EM field is located, by the time when source currents are changed by the signal, and the source therefore now begins to generate and a different outwardly-moving EM field.

A compact view of EMR is that the far-field that composes EMR is generally that part of the EM field that has traveled sufficient distance from the source, that it has become completely disconnected from any feedback to the charges and currents that were originally responsible for it. It now generates itself, as a result of changing fields.
Properties of EM radiation
The electric field is in a vertical plane and the magnetic field in a horizontal plane
Electromagnetic waves can be imagined as a self-propagating transverse oscillating wave of electric and magnetic fields. This 3D diagram shows a plane linearly polarized wave propagating from left to right
This 3D diagram shows a plane linearly polarized wave propagating from left to right
Electromagnetic waves can be imagined as a self-propagating transverse oscillating wave of electric and magnetic fields. This diagram shows a plane linearly polarized wave propagating from left to right. The electric field is in a vertical plane and the magnetic field in a horizontal plane.

The physics of electromagnetic radiation is electrodynamics. Electromagnetism is the physical phenomenon associated with the theory of electrodynamics. Electric and magnetic fields obey the properties of superposition. Thus, a field due to any particular particle or time-varying electric or magnetic field contributes to the fields present in the same space due to other causes. Further, as they are vector fields, all magnetic and electric field vectors add together according to vector addition. For example, in optics two or more coherent lightwaves may interact and by constructive or destructive interference yield a resultant irradiance deviating from the sum of the component irradiances of the individual lightwaves.

Since light is an oscillation it is not affected by travelling through static electric or magnetic fields in a linear medium such as a vacuum. However in nonlinear media, such as some crystals, interactions can occur between light and static electric and magnetic fields — these interactions include the Faraday effect and the Kerr effect.

In refraction, a wave crossing from one medium to another of different density alters its speed and direction upon entering the new medium. The ratio of the refractive indices of the media determines the degree of refraction, and is summarized by Snell's law. Light of composite wavelengths (natural sunlight) disperses into a visible spectrum passing through a prism, because of the wavelength dependent refractive index of the prism material (dispersion); that is, each component wave within the composite light is bent a different amount.

EM radiation exhibits both wave properties and particle properties at the same time (see wave-particle duality). Both wave and particle characteristics have been confirmed in a large number of experiments. Wave characteristics are more apparent when EM radiation is measured over relatively large timescales and over large distances while particle characteristics are more evident when measuring small timescales and distances. For example, when electromagnetic radiation is absorbed by matter, particle-like properties will be more obvious when the average number of photons in the cube of the relevant wavelength is much smaller than 1. Upon absorption of light, it is not too difficult to experimentally observe non-uniform deposition of energy. However, this alone is not evidence of "particulate" behavior of light. Rather, it reflects the quantum nature of matter.[1]

There are experiments in which the wave and particle natures of electromagnetic waves appear in the same experiment, such as the self-interference of a single photon. True single-photon experiments (in a quantum optical sense) can be done today in undergraduate-level labs.[2] When a single photon is sent through an interferometer, it passes through both paths, interfering with itself, as waves do, yet is detected by a photomultiplier or other sensitive detector only once.

A quantum theory of the interaction between electromagnetic radiation and matter such as electrons is described by the theory of quantum electrodynamics.
Wave model

Electromagnetic radiation is a transverse wave meaning that the oscillations of the waves are perpendicular to the direction of energy transfer and travel. An important aspect of the nature of light is 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 a spectrum of frequencies that sum to form the resultant wave. Different frequencies undergo different angles of refraction.

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:

\( \displaystyle v=f\lambda \)

where v is the speed of the wave (c in a vacuum, or less in other media), f is the frequency and λ is the wavelength. As waves cross boundaries between different media, their speeds change but their frequencies remain constant.

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.

The energy in electromagnetic waves is sometimes called radiant energy.
Particle model
See also: Quantization (physics) and Quantum optics

Because energy of an electromagnetic wave is quantized (see second quantization), electromagnetic energy is emitted and absorbed as discrete packets of energy, or quanta, called photons. The energy of the photons is proportional to the frequency of the wave.[3] On the converse, in a first-quantized treatment, because a photon acts as a transporter of energy, it is associated with a probability wave with frequency proportional to the energy carried. In both treatments, the energy per photon is related to the frequency via the Planck–Einstein equation:[4]

\( \displaystyle E=hf \)

where E is the energy, h is Planck's constant, and f is frequency. The energy is commonly expressed in the unit of electronvolt (eV). This photon-energy expression is a particular case of the energy levels of the more general electromagnetic oscillator, whose average energy, which is used to obtain Planck's radiation law, can be shown to differ sharply from that predicted by the equipartition principle at low temperature, thereby establishes a failure of equipartition due to quantum effects at low temperature.[5]

As a photon is absorbed by an atom, it excites the atom, elevating an electron to a higher energy level. If the energy is great enough the electron may jump to a high enough energy level that it escapes the positive pull of the nucleus and be liberated from the atom, in a process called photoionisation. The energy required is larger than about 10 electron volts (eV) corresponding with wavelengths smaller than 124 nm. This is at the high end of the ultraviolet spectrum, sometimes called "extreme UV." Electromagnetic radiation with this much energy, or more, is therefore termed ionizing radiation. There are also many other kinds of ionizing radiation made of particles with mass. Electromagnetic ionizing radiation extends from the extreme ultraviolet to all higher frequencies and shorter wavelengths, which means that all X-rays and gamma rays are ionizing radiation. It also means that most of the ultraviolet (UV), below 10 eV to where ultraviolet becomes visible light at 3.1 eV, is not ionizing. However, these UV wavelengths can damage molecules by exciting their electrons, even if the electrons are not removed. This is why ultraviolet at all wavelengths can damage DNA, and is capable of causing cancer, and skin burns (sunburn) which is far worse than would be produced by simple heating effects.

The EM spectrum in the wavelength and frequency range of visible light is associated with photons that usually has too little energy to damage molecules. There are a few exceptions (for example, photosynthesis relies on visible light to excite molecules). Infrared, microwaves, and radiowaves are usually considered to damage molecules and biological tissue only by heating, not excitation from single photons of the radiation.

An electron in an excited molecule or atom that descends to a lower energy level emits a photon of light equal 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. When such frequencies are in the visible range, this phenomena is called visible fluorescence. An example is visible light emitted from fluorescent paints, in response to ultraviolet (blacklight). Many other fluorescent emissions in other spectral bands than visible, are known.

Together, these effects explain the emission and absorption spectra of light. The 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, absorbing certain frequencies of the light between emitter and detector/eye, then emitting them in all directions, so that 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 the emitting gas is glowing 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. An example is the emission spectrum of nebulae.

Today, scientists use these phenomena to perform 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 a star is composed of. Spectroscopy is also used in the determination of the distance of a star, using the red shift.
Causality

The standard view of propagating electromagnetic waves makes sense from a local perspective,[6] but note that some prefer instead to look into the past for the source charge(s) that were the original cause of the wave.[7]
Speed of propagation
Main article: Speed of light

Any electric charge that accelerates, or any changing magnetic field, produces electromagnetic radiation. Electromagnetic information about the charge travels at the speed of light. Accurate treatment thus incorporates a concept known as retarded time (as opposed to advanced time, which is not physically possible in light of causality), which adds to the expressions for the electrodynamic electric field and magnetic field. These extra terms are responsible for electromagnetic radiation. When any wire (or other conducting object such as an antenna) conducts alternating current, electromagnetic radiation is propagated at the same frequency as the electric current. 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 oscillation, which is responsible for the phenomenon of radiative transition between quantum states of a charged particle.

Depending on the circumstances, electromagnetic radiation may behave as a wave or as particles. As a wave, it is characterized by a velocity (the speed of light), wavelength, and frequency. When considered as particles, they are known as photons, and each has an energy related to the frequency of the wave given by Planck's relation E = hν, where E is the energy of the photon, h = 6.626 × 10−34 J·s is Planck's constant, and ν is the frequency of the wave.

One rule is always obeyed regardless of the circumstances: EM radiation in a vacuum always travels at the speed of light, relative to the observer, regardless of the observer's velocity. (This observation led to Albert Einstein's development of the theory of special relativity.)

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.
Thermal radiation and electromagnetic radiation as a form of heat
Main article: Thermal radiation

The basic structure of matter involves charged particles bound together in many different ways. When electromagnetic radiation is incident on matter, it causes the charged particles to oscillate and gain energy. The ultimate fate of this energy depends on the situation. It could be immediately re-radiated and appear as scattered, reflected, or transmitted radiation. It may also get dissipated into other microscopic motions within the matter, coming to thermal equilibrium and manifesting itself as thermal energy in the material. With a few exceptions related to high-energy photons (such as fluorescence, harmonic generation, photochemical reactions, the photovoltaic effect for ionizing radiations at far ultraviolet, X-ray, and gamma radiation), absorbed electromagnetic radiation simply deposits its energy by heating the material. This happens both for infrared, microwave, and radio wave radiation. Intense radio waves can thermally burn living tissue and can cook food. In addition to infrared lasers, sufficiently intense visible and ultraviolet lasers can also easily set paper afire.

Ionizing electromagnetic radiation creates high-speed electrons in a material and breaks chemical bonds, but after these electrons collide many times with other atoms in the material eventually most of the energy is downgraded to thermal energy; this whole process happens in a tiny fraction of a second. This process makes ionizing radiation far more dangerous per unit of energy than non-ionizing radiation. This caveat also applies to the ultraviolet (UV) spectrum, even though almost all of it is not ionizing, because UV can damage molecules due to electronic excitation which is far greater per unit energy than heating effects produce.

Infrared radiation in the spectral distribution of a black body is usually considered a form of heat, since it has an equivalent temperature, and is associated with an entropy change per unit of thermal energy. However, the word "heat" is a highly technical term in physics and thermodynamics, and is often confused with thermal energy. Any type of electromagnetic energy can be transformed into thermal energy in interaction with matter. Thus, any electromagnetic radiation can "heat" a material when it is absorbed.

The inverse or time-reversed process of absorption is responsible for thermal radiation. Much of the thermal energy in matter consists of random motion of charged particles, and this energy can be radiated away from the matter. The resulting radiation may subsequently be absorbed by another piece of matter, with the deposited energy heating the material. Thermal radiation is an important mechanism of heat transfer.

The electromagnetic radiation in an opaque cavity at thermal equilibrium is effectively a form of thermal energy, having maximum radiation entropy. The thermodynamic potentials of electromagnetic radiation can be well-defined as for matter. By means of Planck's law, the energy density in a cavity due to thermal radiation is

\( {U\over V} = \frac{8\pi^5(kT)^4}{15 (hc)^3}, \)

Differentiating the above with respect to temperature, we may say that the electromagnetic radiation field has an effective volumetric heat capacity given by

\( C_v = \frac{32\pi^5 k^4 T^3}{15 (hc)^3}, \)

Electromagnetic spectrum
Main article: Electromagnetic spectrum
Electromagnetic spectrum with light highlighted
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 = Ultrahigh frequency
VHF = Very high frequency
HF = High frequency
MF = Medium frequency
LF = Low frequency
VLF = Very low frequency
VF = Voice frequency
ULF = Ultra-low frequency
SLF = Super-low frequency
ELF = Extremely low frequency

In general, EM radiation (the designation 'radiation' excludes static electric and magnetic and near fields) is classified by wavelength into radio, microwave, infrared, the visible region we perceive as light, ultraviolet, X-rays, and gamma rays. Arbitrary electromagnetic waves can always be expressed by Fourier analysis in terms of sinusoidal monochromatic waves, which can be classified into these regions of the spectrum.

The behavior of EM radiation depends on its wavelength. Higher frequencies have shorter wavelengths, and lower frequencies have longer wavelengths. When EM radiation interacts with single atoms and molecules, its behavior depends on the amount of energy per quantum it carries. Spectroscopy can detect a much wider region of the EM spectrum than the visible range of 400 nm to 700 nm. A common laboratory spectroscope can detect wavelengths from 2 nm to 2500 nm. Detailed information about the physical properties of objects, gases, or even stars can be obtained from this type of device. It is widely used in astrophysics. For example, hydrogen atoms emit radio waves of wavelength 21.12 cm.

Soundwaves are not electromagnetic radiation. At the lower end of the electromagnetic spectrum, about 20 Hz to about 20 kHz, are frequencies that might be considered in the audio range. However, electromagnetic waves cannot be directly perceived by human ears. Sound waves are the oscillating compression of molecules. To be heard, electromagnetic radiation must be converted to pressure waves of the fluid in which the ear is located (whether the fluid is air, water or something else).
Light
Main article: Light

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, especially when visibility to humans is not relevant.

If radiation having a frequency in the visible region of the EM spectrum reflects off of an object, say, a bowl of fruit, and then strikes our eyes, this results in our visual perception of the scene. Our brain's visual system processes the multitude of reflected frequencies into different shades and hues, and through this not-entirely-understood psychophysical phenomenon, most people perceive a bowl of fruit.

At most wavelengths, however, the information carried by electromagnetic radiation is not directly detected by human senses. Natural sources produce EM radiation across the spectrum, and our technology can also manipulate a broad range of wavelengths. Optical fiber transmits light, which, although not suitable for direct viewing, can carry data that can be translated into sound or an image. To be meaningful both transmitter and receiver must use some agreed-upon encoding system - especially so if the transmission is digital as opposed to the analog nature of the waves.
Radio waves
Main article: Radio waves

Radio waves can be made to carry information by varying the amplitude, frequency or phase.

When EM radiation impinges upon a conductor, it couples to the conductor, travels along it, and induces an electric current on the surface of that conductor by exciting the electrons of the conducting material. This effect (the skin effect) is used in antennas. EM radiation may also cause certain molecules to absorb energy and thus to heat up; this is exploited in microwave ovens.
Biological effects

The effects of electromagnetic radiation upon living cells, including those in humans, depends upon the power and the frequency of the radiation. For low-frequency radiation (radio waves to visible light) most if not all effects are thought to be due to radiation power alone, acting through the effect of simple heating when the radiation is absorbed by the cell. In this range, the frequency is important only as it affects radiation penetration into the organism (for example microwaves penetrate better than infrared). The effects of radio waves, microwaves, infrared radiation, and visible light, however, should be distinguished from ultraviolet (UV) and ionizing radiation, as discussed below.

At higher frequencies yet (starting with ultraviolet radiation) the effects of individual photons of the radiation begin to become important, as these now have enough energy individually to damage biological molecules. From UV frequencies and higher, electromagnetic radiation does far more damage to biological systems than heating predicts. The far (or "extreme") ultraviolet, and also X-ray and gamma radiation, are referred to as ionizing radiation due to the ability of photons of this radiation to produce ions and free radicals in materials (including living tissue). Such radiation can produce severe damage to life at powers that produce very little heating, and such radiation is considered far more dangerous (in terms of damage-produced per unit of energy or power) than the rest of the electromagnetic spectrum.
Derivation

Electromagnetic waves as a general phenomenon were predicted by the classical laws of electricity and magnetism, known as Maxwell's equations. Inspection of Maxwell's equations without sources (charges or currents) results in, along with the possibility of nothing happening, nontrivial solutions of changing electric and magnetic fields. Beginning with Maxwell's equations in free space:

\( \nabla \cdot \mathbf{E} = 0 \qquad \qquad \qquad \ \ (1) \)
\( \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \qquad \qquad \ (2) \)
\( \nabla \cdot \mathbf{B} = 0 \qquad \qquad \qquad \ \ (3) \)
\( \nabla \times \mathbf{B} = \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t} \qquad \quad \ (4) \)

where

\( \nabla \) is a vector differential operator (see Del).

One solution,

\( \mathbf{E}=\mathbf{B}=\mathbf{0}, \)

is trivial.

For a more useful solution, we utilize vector identities, which work for any vector, as follows:

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

To see how we can use this, take the curl of equation (2):

\( \nabla \times \left(\nabla \times \mathbf{E} \right) = \nabla \times \left(-\frac{\partial \mathbf{B}}{\partial t} \right) \qquad \qquad \qquad \quad \ \ \ (5) \, \)

Evaluating the left hand side:

\( \nabla \times \left(\nabla \times \mathbf{E} \right) = \nabla\left(\nabla \cdot \mathbf{E} \right) - \nabla^2 \mathbf{E} = - \nabla^2 \mathbf{E} \qquad \ \ (6) \, \)

where we simplified the above by using equation (1).

Evaluate the right hand side:

\( \nabla \times \left(-\frac{\partial \mathbf{B}}{\partial t} \right) = -\frac{\partial}{\partial t} \left( \nabla \times \mathbf{B} \right) = -\mu_0 \epsilon_0 \frac{\partial^2 \mathbf{E}}{\partial t^2} \quad \ \ \ \ (7) \)

Equations (6) and (7) are equal, so this results in a vector-valued differential equation for the electric field, namely

\( \nabla^2 \mathbf{E} = \mu_0 \epsilon_0 \frac{\partial^2 \mathbf{E}}{\partial t^2} \)

Applying a similar pattern results in similar differential equation for the magnetic field:

\( \nabla^2 \mathbf{B} = \mu_0 \epsilon_0 \frac{\partial^2 \mathbf{B}}{\partial t^2}. \)

These differential equations are equivalent to the wave equation:

\( \nabla^2 f = \frac{1}{{c_0}^2} \frac{\partial^2 f}{\partial t^2} \, \)

where

c₀ is the speed of the wave in free space and
f describes a displacement

Or more simply:

\( \Box f = 0 \)

where \Box is d'Alembertian:

\( \Box = \nabla^2 - \frac{1}{{c_0}^2} \frac{\partial^2}{\partial t^2} = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} - \frac{1}{{c_0}^2} \frac{\partial^2}{\partial t^2} \ \)

Notice that, in the case of the electric and magnetic fields, the speed is:

\( c_0 = \frac{1}{\sqrt{\mu_0 \epsilon_0}} \)

This is the speed of light in vacuum. Maxwell's equations have unified the vacuum permittivity \( \epsilon_0, \) the vacuum permeability \( \mu_0 \), and the speed of light itself, c₀. Before this derivation it was not known that there was such a strong relationship between light and electricity and magnetism.

But these are only two equations and we started with four, so there is still more information pertaining to these waves hidden within Maxwell's equations. Let's consider a generic vector wave for the electric field.

\( \mathbf{E} = \mathbf{E}_0 f\left( \hat{\mathbf{k}} \cdot \mathbf{x} - c_0 t \right) \)

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. We observe that \( f\left( \hat{\mathbf{k}} \cdot \mathbf{x} - c_0 t \right) \) is a generic solution to the wave equation. In other words

\( \nabla^2 f\left( \hat{\mathbf{k}} \cdot \mathbf{x} - c_0 t \right) = \frac{1}{{c_0}^2} \frac{\partial^2}{\partial t^2} f\left( \hat{\mathbf{k}} \cdot \mathbf{x} - c_0 t \right), \)

for a generic wave traveling in the \( \hat{\mathbf{k}} \) direction.

This form will satisfy the wave equation, but will it satisfy all of Maxwell's equations, and with what corresponding magnetic field?

\( \nabla \cdot \mathbf{E} = \hat{\mathbf{k}} \cdot \mathbf{E}_0 f'\left( \hat{\mathbf{k}} \cdot \mathbf{x} - c_0 t \right) = 0 \)
\( \mathbf{E} \cdot \hat{\mathbf{k}} = 0 \)

The first of Maxwell's equations implies that 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 \( \mathbf{E},\mathbf{B}. \)

Not only are the electric and magnetic field waves traveling at the speed of light but 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 \( \mathbf{E} \times \mathbf{B} \).

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; but 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. A great 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,[8] 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.
See also

Antenna (radio)
Antenna measurement
Bioelectromagnetism
Bolometer
Control of electromagnetic radiation
Electromagnetic field
Electromagnetic pulse
Electromagnetic radiation and health
Electromagnetic spectrum
Electromagnetic wave equation
Evanescent wave coupling
Finite-difference time-domain method
Helicon
Impedance of free space
Light
Maxwell's equations
Near and far field
Radiant energy
Radiation reaction
Risks and benefits of sun exposure
Sinusoidal plane-wave solutions of the electromagnetic wave equation

References

^ [1]
^ http://people.whitman.edu/~beckmk/QM/grangier/Thorn_ajp.pdf
^ Weinberg, S. (1995). The Quantum Theory of Fields. 1. Cambridge University Press. pp. 15–17. ISBN 0-521-55001-7.
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