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Electromagnetism
Electricity · Magnetism
Electrodynamics
Electric current
Lorentz force law
Electromotive force
(EM) Electromagnetic induction
Displacement current
Maxwell's equations
(EMF) Electromagnetic field
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## Contents

### Theory

Electromagnetic waves were first predicted by James Clerk Maxwell and subsequently confirmed by Heinrich Hertz. Maxwell derived a wave form of the electric and magnetic equations, revealing 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. James Clerk Maxwell (13 June 1831 â€“ 5 November 1879) was a Scottish mathematician and theoretical physicist from Edinburgh, Scotland, UK. His most significant achievement was aggregating a set of equations in electricity, magnetism and inductance â€” eponymously named Maxwells equations â€” including an important modification (extension) of the AmpÃ¨res... Heinrich Rudolf Hertz (February 22, 1857 - January 1, 1894) was the German physicist and mechanician for whom the hertz, an SI unit, is named. ... Lasers used for visual effects during a musical performance. ... 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.[1] It is the speed of all electromagnetic radiation, including visible light, in a vacuum. ... For other uses, see Light (disambiguation). ...

According to Maxwell's equations, a time-varying electric field generates a magnetic field and vice versa. Therefore, as an oscillating electric field generates an oscillating magnetic field, the magnetic field in turn generates an oscillating electric field, and so on. These oscillating fields together form an electromagnetic wave. For thermodynamic relations, see Maxwell relations. ... In physics, the space surrounding an electric charge or in the presence of a time-varying magnetic field has a property called an electric field. ... Magnetic field lines shown by iron filings Magnetostatics Electrodynamics Electrical Network Tensors in Relativity This box:      In physics, the magnetic field is a field that permeates space and which exerts a magnetic force on moving electric charges and magnetic dipoles. ...

A quantum theory of the interaction between electromagnetic radiation and matter such as electrons is described by the theory of quantum electrodynamics. Quantum electrodynamics (QED) is a relativistic quantum field theory of electrodynamics. ...

### Properties

Electromagnetic waves can be imagined as a self-propagating transverse oscillating wave of electric and magnetic fields. This diagram shows a plane linearly polarised wave propagating from left to right.

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. In nonlinear media such as some crystals, however, interactions can occur between light and static electric and magnetic fields - these interactions include the Faraday effect and the Kerr effect. For other uses, see Crystal (disambiguation). ... In physics, the Faraday effect or Faraday rotation is a magneto-optical phenomenon, or an interaction between light and a magnetic field. ... The Kerr effect or the quadratic electro-optic effect is a change in the refractive index of a material in response to the intensity of an external electric field. ...

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 disperses into a visible spectrum as light is shone through a prism because of refraction. For other uses, see Density (disambiguation). ... Refraction of light at the interface between two media of different refractive indices, with n2 > n1. ... Legend Î³ = 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...

The physics of electromagnetic radiation is electrodynamics, a subfield of electromagnetism. A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... Electromagnetism is the physics of the electromagnetic field: a field, encompassing all of space, composed of the electric field and the magnetic field. ... Electromagnetism is the physics of the electromagnetic field: a field which exerts a force on particles that possess the property of electric charge, and is in turn affected by the presence and motion of those particles. ...

EM radiation exhibits both wave properties and particle properties at the same time (see wave-particle duality). The wave characteristics are more apparent when EM radiation is measured over relatively large timescales and over large distances, and the particle characteristics are more evident when measuring small distances and timescales. Both characteristics have been confirmed in a large number of experiments. Helium atom (schematic) Showing two protons (red), two neutrons (green) and two electrons (yellow). ... In physics, wave-particle duality holds that light and matter exhibit properties of both waves and of particles. ...

There are experiments in which the wave and particle natures of electromagnetic waves appear in the same experiment, such as the diffraction of a single photon. When a single photon is sent through two slits, it passes through both of them interfering with itself, as waves do, yet is detected by a photomultiplier or other sensitive detector only once. Similar self-interference is observed when a single photon is sent into a Michelson interferometer or other interferometers. In modern physics the photon is the elementary particle responsible for electromagnetic phenomena. ... Photomultipliers, or photomultiplier tubes (PMT) are extremely sensitive detectors of light in the ultraviolet, visible and near infrared. ... A Michelson interferometer for use on an optical table. ... Interferometry is the applied science of combining two or more input points of a particular data type, such as optical measurements, to form a greater picture based on the combination of the two sources. ...

### Wave model

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: For other uses, see Wavelength (disambiguation). ...

v = fλ

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. 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.[1] It is the speed of all electromagnetic radiation, including visible light, in a vacuum. ...

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. For other uses, see Interference (disambiguation). ...

The energy in electromagnetic waves is sometimes called radiant energy. Radiant energy is the energy of electromagnetic waves. ...

### Particle model

Because energy of an EM wave is quantized, in the particle model of EM radiation, a wave consists of discrete packets of energy, or quanta, called photons. The frequency of the wave is proportional to the magnitude of the particle's energy. Moreover, because photons are emitted and absorbed by charged particles, they act as transporters of energy. The energy per photon can be calculated by Planck's equation: In physics quanta is the plural of quantum. ... In modern physics the photon is the elementary particle responsible for electromagnetic phenomena. ... In modern physics the photon is the elementary particle responsible for electromagnetic phenomena. ... â€œPlanckâ€ redirects here. ...

E = hf

where E is the energy, h is Planck's constant, and f is frequency. A commemoration plaque for Max Planck on his discovery of Plancks constant, in front of Humboldt University, Berlin. ...

As a photon is absorbed by an atom, it excites an electron, elevating it to a higher energy level. If the energy is great enough, so that the electron jumps to a high enough energy level, it may escape the positive pull of the nucleus and be liberated from the atom in a process called photoionisation. Conversely, an electron that descends to a lower energy level in an atom emits a photon of light equal to the energy difference. Since the energy levels of electrons in atoms are discrete, each element emits and absorbs its own characteristic frequencies. Properties For other meanings of Atom, see Atom (disambiguation). ... For other uses, see Electron (disambiguation). ... A quantum mechanical system can only be in certain states, so that only certain energy levels are possible. ... Photoionisation is a physical process in which a photon strikes an atom, ion or molecule, resulting in the ejection of an electron. ...

Together, these effects explain the absorption spectra of light. The dark bands in the spectrum are due to the atoms in the intervening medium absorbing different frequencies of the light. The composition of the medium through which the light travels determines the nature of the absorption spectrum. For instance, dark bands in the light emitted by a distant star are due to the atoms in the star's atmosphere. These bands correspond to the allowed energy levels in the atoms. A similar phenomenon occurs for emission. As the electrons descend to lower energy levels, a spectrum is emitted that represents the jumps between the energy levels of the electrons. This is manifested in the emission spectrum of nebulae. Today, scientists use this phenomenon to observe what elements a certain star is composed of. It is also used in the determination of the distance of a star, using the so-called red shift. For other uses, see Light (disambiguation). ... In physics, emission is the process by which the energy of a photon is released by another entity, for example, by an atom whose valence electrons make a transition between two electronic energy levels. ... In physics, emission is the process by which the energy of a photon is released by another entity, for example, by an atom whose valence electrons make a transition between two electronic energy levels. ... The Triangulum Emission Nebula NGC 604 The Pillars of Creation from the Eagle Nebula For other uses, see Nebula (disambiguation). ... This article is about the light phenomenon. ...

### Speed of propagation

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.) 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.[1] It is the speed of all electromagnetic radiation, including visible light, in a vacuum. ... â€œEinsteinâ€ redirects here. ... For a less technical and generally accessible introduction to the topic, see Introduction to special relativity. ...

In a medium (other than vacuum), velocity of propagation 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. Velocity of Propagation (VoP) is a parameter that characterizes the speed at which an electrical or radio signal passes through a medium. ... The refractive index (or index of refraction) of a medium is a measure for how much the speed of light (or other waves such as sound waves) is reduced inside the medium. ...

## 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

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
ELF = Extremely low frequency

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. Electromagnetic radiation can be divided into octaves — as sound waves are — winding up with eighty-one octaves.[1] For other uses, see Octave (disambiguation). ...

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. Animation of the dispersion of light as it travels through a triangular prism. ... Spiral Galaxy ESO 269-57 Astrophysics is the branch of astronomy that deals with the physics of the universe, including the physical properties (luminosity, density, temperature, and chemical composition) of celestial objects such as stars, galaxies, and the interstellar medium, as well as their interactions. ... This article is about the chemistry of hydrogen. ... Properties For other meanings of Atom, see Atom (disambiguation). ... In physics, emission is the process by which the energy of a photon is released by another entity, for example, by an atom whose valence electrons make a transition between two electronic energy levels. ... 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. ... For other uses, see Wavelength (disambiguation). ... A centimetre (American spelling centimeter, symbol cm) is a unit of length that is equal to one hundredth of a metre, the current SI base unit of length. ...

### Light

Main article: light

EM radiation with a wavelength between approximately 400 nm and 700 nm is 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 the visibility to humans is not relevant. For other uses, see Light (disambiguation). ... For other uses, see Wavelength (disambiguation). ... A nanometre (American spelling: nanometer, symbol nm) (Greek: Î½Î¬Î½Î¿Ï‚, nanos, dwarf; Î¼ÎµÏ„ÏÏŽ, metrÏŒ, count) is a unit of length in the metric system, equal to one billionth of a metre (or one millionth of a millimetre), which is the current SI base unit of length. ... This article is about modern humans. ... For other uses, see Eye (disambiguation). ... For other uses, see Light (disambiguation). ...

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. In psychology, visual perception is the ability to interpret visible light information reaching the eyes which is then made available for planning and action. ...

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. The coding used in such data is similar to that used with radio waves. Optical fibers An optical fiber (or fibre) is a glass or plastic fiber designed to guide light along its length. ...

Radio waves can be made to carry information by varying a combination of the amplitude, frequency and phase of the wave within a frequency band. It has been suggested that this article or section be merged with Radio waves. ...

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. In science and engineering, conductors, such as copper or aluminum, are materials with atoms having loosely held valence electrons. ... Radio frequency induction or RF induction is an electrical phenomenon in which an electromagnetic wave passing through a conductor causes electric current to flow through it. ... The skin effect is the tendency of an alternating electric current (AC) to distribute itself within a conductor so that the current density near the surface of the conductor is greater than that at its core. ... Microwave oven A microwave oven, or microwave, is a kitchen appliance employing microwave radiation primarily to cook or heat food. ...

## Derivation

Electromagnetic waves as a general phenomenon were predicted by the classical laws of electricity and magnetism, known as Maxwell's equations. If you inspect Maxwell's equations without sources (charges or currents) then you will find that, along with the possibility of nothing happening, the theory will also admit nontrivial solutions of changing electric and magnetic fields. Beginning with Maxwell's equations for a vacuum: Electricity (from New Latin Ä“lectricus, amberlike) is a general term for a variety of phenomena resulting from the presence and flow of electric charge. ... For thermodynamic relations, see Maxwell relations. ...

$nabla cdot mathbf{E} = 0 qquad qquad qquad (1)$
$nabla times mathbf{E} = -frac{partial}{partial t} mathbf{B} qquad qquad (2)$
$nabla cdot mathbf{B} = 0 qquad qquad qquad (3)$
$nabla times mathbf{B} = mu_0 epsilon_0 frac{partial}{partial t} mathbf{E} qquad (4)$
where
$nabla$ is a vector differential operator (see Del).

One solution, In vector calculus, del is a vector differential operator represented by the nabla symbol: âˆ‡. Del is a mathematical tool serving primarily as a convention for mathematical notation; it makes many equations easier to comprehend, write, and remember. ...

$mathbf{E}=mathbf{B}=mathbf{0}$,

is trivial.

To see the more interesting one, we utilize vector identities, which work for any vector, as follows: This article lists a few helpful mathematical identities which are useful in vector algebra. ...

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

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:

$nabla times left(nabla times mathbf{E} right) = nablaleft(nabla cdot mathbf{E} right) - nabla^2 mathbf{E} = - nabla^2 mathbf{E} qquad quad (6) ,$
where we simplified the above by using equation (1).

Evaluate the right hand side:

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

Equations (6) and (7) are equal, so this results in a vector-valued differential equation for the electric field, namely Visualization of airflow into a duct modelled using the Navier-Stokes equations, a set of partial differential equations. ...

 $nabla^2 mathbf{E} = mu_0 epsilon_0 frac{partial^2}{partial t^2} mathbf{E}$

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

 $nabla^2 mathbf{B} = mu_0 epsilon_0 frac{partial^2}{partial t^2} mathbf{B}$.

These differential equations are equivalent to the wave equation: The wave equation is an important partial differential equation that describes the propagation of a variety of waves, such as sound waves, light waves and water waves. ...

$nabla^2 f = frac{1}{c^2} frac{partial^2 f}{partial t^2} ,$
where
c is the speed of the wave and
f describes a displacement

Or more simply:

$Box^2 f = 0$
where $Box^2$ is d'Alembertian:
$Box^2 = nabla^2 - frac{1}{c^2} frac{partial^2}{partial t^2} = frac{partial^2}{partial x^2} + frac{partial^2}{partial y^2} + frac{partial^2}{partial z^2} - frac{1}{c^2} frac{partial^2}{partial t^2}$

Notice that in the case of the electric and magnetic fields, the speed is: In special relativity, electromagnetism and wave theory, the dAlembert operator , also called the dAlembertian or the Wave operator, is the Laplace operator of Minkowski space and other solutions of the Einstein equation. ...

$c = frac{1}{sqrt{mu_0 epsilon_0}}$

Which, as it turns out, is the speed of light. Maxwell's equations have unified the permittivity of free space ε0, the permeability of free space μ0, and the speed of light itself, c. Before this derivation it was not known that there was such a strong relationship between light and electricity and magnetism. 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.[1] It is the speed of all electromagnetic radiation, including visible light, in a vacuum. ... Lasers used for visual effects during a musical performance. ...

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.

$mathbf{E} = mathbf{E}_0 fleft( hat{mathbf{k}} cdot mathbf{x} - c t right)$

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 $fleft( hat{mathbf{k}} cdot mathbf{x} - c t right)$ is a generic solution to the wave equation. In other words

$nabla^2 fleft( hat{mathbf{k}} cdot mathbf{x} - c t right) = frac{1}{c^2} frac{partial^2}{partial^2 t} fleft( hat{mathbf{k}} cdot mathbf{x} - c t right)$,

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?

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

The first of Maxwell's equations implies that electric field is orthogonal to the direction the wave propagates.

$nabla times mathbf{E} = hat{mathbf{k}} times mathbf{E}_0 f'left( hat{mathbf{k}} cdot mathbf{x} - c t right) = -frac{partial}{partial t} mathbf{B}$
$mathbf{B} = frac{1}{c} hat{mathbf{k}} times mathbf{E}$

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, E0 = cB0, 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 $mathbf{E} times mathbf{B}$. The Poynting vector describes the energy flux (JÂ·mâˆ’2Â·sâˆ’1) of an electromagnetic field. ...

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

## References

1. ^ Isaac Asimov, Isaac Asimov's Book of Facts. Hastingshouse/Daytrips Publ., 1992. Page 389.
• Hecht, Eugene (2001). Optics, 4th ed., Pearson Education. ISBN 0-8053-8566-5.
• Serway, Raymond A.; Jewett, John W. (2004). Physics for Scientists and Engineers, 6th ed., Brooks/Cole. ISBN 0-534-40842-7.
• 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.
• Reitz, John; Milford, Frederick; Christy, Robert (1992). Foundations of Electromagnetic Theory, 4th ed., Addison Wesley. ISBN 0-201-52624-7.
• Jackson, John David (1975). Classical Electrodynamics, 2nd ed, John Wiley & Sons. ISBN 0-471-43132-X.
• Allen Taflove and Susan C. Hagness (2005). Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed.. Artech House Publishers. ISBN 1-58053-832-0.

Results from FactBites:

 Electromagnetic radiation - Wikipedia, the free encyclopedia (2104 words) The physics of electromagnetic radiation is electrodynamics, a subfield of electromagnetism. EM radiation with a wavelength between approximately 400 nm and 700 nm is detected by the human eye and perceived as visible light. Electromagnetic waves as a general phenomenon were predicted by the classical laws of electricity and magnetism, known as Maxwell's equations.
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