Intensity (physics)

Last updated July 22, 2024

In physics and many other areas of science and engineering the intensity or flux of radiant energy is the power transferred per unit area, where the area is measured on the plane perpendicular to the direction of propagation of the energy.^{[lower-alpha 1]} In the SI system, it has units watts per square metre (W/m²), or kg⋅s ⁻³ in base units. Intensity is used most frequently with waves such as acoustic waves (sound), matter waves such as electrons in electron microscopes, and electromagnetic waves such as light or radio waves, in which case the average power transfer over one period of the wave is used. Intensity can be applied to other circumstances where energy is transferred. For example, one could calculate the intensity of the kinetic energy carried by drops of water from a garden sprinkler.

Intensity can be found by taking the energy density (energy per unit volume) at a point in space and multiplying it by the velocity at which the energy is moving. The resulting vector has the units of power divided by area (i.e., surface power density). The intensity of a wave is proportional to the square of its amplitude. For example, the intensity of an electromagnetic wave is proportional to the square of the wave's electric field amplitude.

Mathematical description

If a point source is radiating energy in all directions (producing a spherical wave), and no energy is absorbed or scattered by the medium, then the intensity decreases in proportion to the distance from the object squared. This is an example of the inverse-square law.

Applying the law of conservation of energy, if the net power emanating is constant, $P=\int \mathbf {I} \,\cdot d\mathbf {A} ,$ where

$P$ is the net power radiated;
$I$ is the intensity vector as a function of position;
the magnitude $| I |$ is the intensity as a function of position;
$d A$ is a differential element of a closed surface that contains the source.

If one integrates a uniform intensity, $| I | = const.$ , over a surface that is perpendicular to the intensity vector, for instance over a sphere centered around the point source, the equation becomes $P=|I|\cdot A_{\mathrm {surf} }=|I|\cdot 4\pi r^{2},$ where

$| I |$ is the intensity at the surface of the sphere;
$r$ is the radius of the sphere;
$A_{\mathrm {surf} }=4\pi r^{2}$ is the expression for the surface area of a sphere.

Solving for $| I |$ gives $|I|={\frac {P}{A_{\mathrm {surf} }}}={\frac {P}{4\pi r^{2}}}.$

If the medium is damped, then the intensity drops off more quickly than the above equation suggests.

Anything that can transmit energy can have an intensity associated with it. For a monochromatic propagating electromagnetic wave, such as a plane wave or a Gaussian beam, if $E$ is the complex amplitude of the electric field, then the time-averaged energy density of the wave, travelling in a non-magnetic material, is given by: $\left\langle U\right\rangle ={\frac {n^{2}\varepsilon _{0}}{2}}|E|^{2},$ and the local intensity is obtained by multiplying this expression by the wave velocity, ⁠ ${\tfrac {\mathrm {c} }{n}}\!:$ ⁠ $I={\frac {\mathrm {c} n\varepsilon _{0}}{2}}|E|^{2},$ where

$n$ is the refractive index;
$c$ is the speed of light in vacuum;
$ε 0$ is the vacuum permittivity.

For non-monochromatic waves, the intensity contributions of different spectral components can simply be added. The treatment above does not hold for arbitrary electromagnetic fields. For example, an evanescent wave may have a finite electrical amplitude while not transferring any power. The intensity should then be defined as the magnitude of the Poynting vector.^[1]

Electron beams

For electron beams, intensity is the probability of electrons reaching some particular position on a detector (e.g. a charge-coupled device ^[2]) which is used to produce images that are interpreted in terms of both microstructure of inorganic or biological materials, as well as atomic scale structure.^[3] The map of the intensity of scattered electrons or x-rays as a function of direction is also extensively used in crystallography.^[3]^[4]

Alternative definitions

In photometry and radiometry intensity has a different meaning: it is the luminous or radiant power per unit solid angle . This can cause confusion in optics, where intensity can mean any of radiant intensity, luminous intensity or irradiance, depending on the background of the person using the term. Radiance is also sometimes called intensity, especially by astronomers and astrophysicists, and in heat transfer.

Footnotes

↑ The terms intensity and flux have multiple, inconsistent, definitions in physics and related fields. This article covers the concept of power per unit area, whatever one calls it. In radiometry the terms intensity and flux have different meanings, not covered here.

Related Research Articles

In physics, the cross section is a measure of the probability that a specific process will take place in a collision of two particles. For example, the Rutherford cross-section is a measure of probability that an alpha particle will be deflected by a given angle during an interaction with an atomic nucleus. Cross section is typically denoted $σ$ (sigma) and is expressed in units of area, more specifically in barns. In a way, it can be thought of as the size of the object that the excitation must hit in order for the process to occur, but more exactly, it is a parameter of a stochastic process.

$Diffraction Phenomenon of the motion of waves$

Diffraction is the interference or bending of waves around the corners of an obstacle or through an aperture into the region of geometrical shadow of the obstacle/aperture. The diffracting object or aperture effectively becomes a secondary source of the propagating wave. Italian scientist Francesco Maria Grimaldi coined the word diffraction and was the first to record accurate observations of the phenomenon in 1660.

In physics and mathematics, wavelength or spatial period of a wave or periodic function is the distance over which the wave's shape repeats. In other words, it is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, troughs, or zero crossings. Wavelength is a characteristic of both traveling waves and standing waves, as well as other spatial wave patterns. The inverse of the wavelength is called the spatial frequency. Wavelength is commonly designated by the Greek letter lambda (λ). The term "wavelength" is also sometimes applied to modulated waves, and to the sinusoidal envelopes of modulated waves or waves formed by interference of several sinusoids.

<span class="mw-page-title-main">Reflectance</span> Capacity of an object to reflect light

The reflectance of the surface of a material is its effectiveness in reflecting radiant energy. It is the fraction of incident electromagnetic power that is reflected at the boundary. Reflectance is a component of the response of the electronic structure of the material to the electromagnetic field of light, and is in general a function of the frequency, or wavelength, of the light, its polarization, and the angle of incidence. The dependence of reflectance on the wavelength is called a reflectance spectrum or spectral reflectance curve.

Flux describes any effect that appears to pass or travel through a surface or substance. Flux is a concept in applied mathematics and vector calculus which has many applications to physics. For transport phenomena, flux is a vector quantity, describing the magnitude and direction of the flow of a substance or property. In vector calculus flux is a scalar quantity, defined as the surface integral of the perpendicular component of a vector field over a surface.

<span class="mw-page-title-main">Poynting vector</span> Measure of directional electromagnetic energy flux

In physics, the Poynting vector represents the directional energy flux or power flow of an electromagnetic field. The SI unit of the Poynting vector is the watt per square metre (W/m²); kg/s³ in base SI units. It is named after its discoverer John Henry Poynting who first derived it in 1884. Nikolay Umov is also credited with formulating the concept. Oliver Heaviside also discovered it independently in the more general form that recognises the freedom of adding the curl of an arbitrary vector field to the definition. The Poynting vector is used throughout electromagnetics in conjunction with Poynting's theorem, the continuity equation expressing conservation of electromagnetic energy, to calculate the power flow in electromagnetic fields.

The Stefan–Boltzmann law, also known as Stefan's law, describes the intensity of the thermal radiation emitted by matter in terms of that matter's temperature. It is named for Josef Stefan, who empirically derived the relationship, and Ludwig Boltzmann who derived the law theoretically.

In the physical sciences, the wavenumber, also known as repetency, is the spatial frequency of a wave, measured in cycles per unit distance or radians per unit distance. It is analogous to temporal frequency, which is defined as the number of wave cycles per unit time or radians per unit time.

Electrostatics is a branch of physics that studies slow-moving or stationary electric charges.

In electromagnetism, the magnetic moment or magnetic dipole moment is the combination of strength and orientation of a magnet or other object or system that exerts a magnetic field. The magnetic dipole moment of an object determines the magnitude of torque the object experiences in a given magnetic field. When the same magnetic field is applied, objects with larger magnetic moments experience larger torques. The strength of this torque depends not only on the magnitude of the magnetic moment but also on its orientation relative to the direction of the magnetic field. Its direction points from the south pole to north pole of the magnet.

In radiometry, irradiance is the radiant flux received by a surface per unit area. The SI unit of irradiance is the watt per square metre (W⋅m⁻²). The CGS unit erg per square centimetre per second (erg⋅cm⁻²⋅s⁻¹) is often used in astronomy. Irradiance is often called intensity, but this term is avoided in radiometry where such usage leads to confusion with radiant intensity. In astrophysics, irradiance is called radiant flux.

In physics, the reciprocal lattice emerges from the Fourier transform of another lattice. The direct lattice or real lattice is a periodic function in physical space, such as a crystal system. The reciprocal lattice exists in the mathematical space of spatial frequencies, known as reciprocal space or k space, which is the dual of physical space considered as a vector space, and the reciprocal lattice is the sublattice of that space that is dual to the direct lattice.

In physics, a wave vector is a vector used in describing a wave, with a typical unit being cycle per metre. It has a magnitude and direction. Its magnitude is the wavenumber of the wave, and its direction is perpendicular to the wavefront. In isotropic media, this is also the direction of wave propagation.

The Ewald sphere is a geometric construction used in electron, neutron, and x-ray diffraction which shows the relationship between:

$Low-energy electron diffraction Technique for determining surface structures$

Low-energy electron diffraction (LEED) is a technique for the determination of the surface structure of single-crystalline materials by bombardment with a collimated beam of low-energy electrons (30–200 eV) and observation of diffracted electrons as spots on a fluorescent screen.

In condensed matter physics and crystallography, the static structure factor is a mathematical description of how a material scatters incident radiation. The structure factor is a critical tool in the interpretation of scattering patterns obtained in X-ray, electron and neutron diffraction experiments.

The transmission coefficient is used in physics and electrical engineering when wave propagation in a medium containing discontinuities is considered. A transmission coefficient describes the amplitude, intensity, or total power of a transmitted wave relative to an incident wave.

$X-ray diffraction Elastic interaction of x-rays with electrons$

X-ray diffraction is a generic term for phenomena associated with changes in the direction of X-ray beams due to interactions with the electrons around atoms. It occurs due to elastic scattering, when there is no change in the energy of the waves. The resulting map of the directions of the X-rays far from the sample is called a diffraction pattern. It is different from X-ray crystallography which exploits X-ray diffraction to determine the arrangement of atoms in materials, and also has other components such as ways to map from experimental diffraction measurements to the positions of atoms.

Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics that calculates the amount of force between two electrically charged particles at rest. This electric force is conventionally called the electrostatic force or Coulomb force. Although the law was known earlier, it was first published in 1785 by French physicist Charles-Augustin de Coulomb. Coulomb's law was essential to the development of the theory of electromagnetism and maybe even its starting point, as it allowed meaningful discussions of the amount of electric charge in a particle.

Kirchhoff's diffraction formula approximates light intensity and phase in optical diffraction: light fields in the boundary regions of shadows. The approximation can be used to model light propagation in a wide range of configurations, either analytically or using numerical modelling. It gives an expression for the wave disturbance when a monochromatic spherical wave is the incoming wave of a situation under consideration. This formula is derived by applying the Kirchhoff integral theorem, which uses the Green's second identity to derive the solution to the homogeneous scalar wave equation, to a spherical wave with some approximations.

References

↑ Paschotta, Rüdiger. "Optical Intensity". Encyclopedia of Laser Physics and Technology. RP Photonics.
↑ Spence, J. C. H.; Zuo, J. M. (1988-09-01). "Large dynamic range, parallel detection system for electron diffraction and imaging". Review of Scientific Instruments. 59 (9): 2102–2105. doi:10.1063/1.1140039. ISSN 0034-6748.
1 2 Cowley, J. M. (1995). Diffraction physics. North Holland personal library (3rd ed.). Amsterdam: Elsevier. ISBN 978-0-444-82218-5.
↑ Cullity, B. D.; Stock, Stuart R. (2001). Elements of X-ray diffraction (3rd ed.). Upper Saddle River, NJ: Prentice Hall. ISBN 978-0-201-61091-8.

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[1] The terms intensity and flux have multiple, inconsistent, definitions in physics and related fields. This article covers the concept of power per unit area, whatever one calls it. In radiometry the terms intensity and flux have different meanings, not covered here.

[2] Paschotta, Rüdiger. "Optical Intensity". Encyclopedia of Laser Physics and Technology. RP Photonics.

[3] Spence, J. C. H.; Zuo, J. M. (1988-09-01). "Large dynamic range, parallel detection system for electron diffraction and imaging". Review of Scientific Instruments. 59 (9): 2102–2105. doi:10.1063/1.1140039. ISSN 0034-6748.

[:0-4] 1 2 Cowley, J. M. (1995). Diffraction physics. North Holland personal library (3rd ed.). Amsterdam: Elsevier. ISBN 978-0-444-82218-5.

[5] Cullity, B. D.; Stock, Stuart R. (2001). Elements of X-ray diffraction (3rd ed.). Upper Saddle River, NJ: Prentice Hall. ISBN 978-0-201-61091-8.

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