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Point source
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A point source is a single identifiable localized source of something. A point source has a negligible extent, distinguishing it from other source geometries. Sources are called point sources because, in mathematical modeling, these sources can usually be approximated as a mathematical point to simplify analysis.

The actual source need not be physically small if its size is negligible relative to other length scales in the problem. For example, in astronomy, stars are routinely treated as point sources, even though they are in actuality much larger than the Earth.

In three dimensions, the density of something leaving a point source decreases in proportion to the inverse square of the distance from the source, if the distribution is isotropic, and there is no absorption or other loss.

Mathematics

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In mathematics, a point source is a singularity from which flux or flow is emanating. Although singularities such as this do not exist in the observable universe, mathematical point sources are often used as approximations to reality in physics and other fields.

Visible electromagnetic radiation (light)

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Generally, a source of light can be considered a point source if the resolution of the imaging instrument is too low to resolve the source's apparent size. There are two types and sources of light: a point source and an extended source.

Mathematically an object may be considered a point source if its angular size, , is much smaller than the resolving power of the telescope:
,
where is the wavelength of light and is the telescope diameter.

Examples:

Other electromagnetic radiation

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Radio wave sources that are smaller than one radio wavelength are also generally treated as point sources. Radio emissions generated by a fixed electrical circuit are usually polarized, producing anisotropic radiation. If the propagating medium is lossless, however, the radiant power in the radio waves at a given distance will still vary as the inverse square of the distance if the angle remains constant to the source polarization.

Gamma ray and X-ray sources may be treated as a point source if sufficiently small. Radiological contamination and nuclear sources are often point sources. This has significance in health physics and radiation protection.

Examples:

  • Radio antennas are often smaller than one wavelength, even though they are many meters across
  • Pulsars are treated as point sources when observed using radio telescopes
  • In nuclear physics, a "hot spot" is a point source of radiation

Sound

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Sound is an oscillating pressure wave. As the pressure oscillates up and down, an audio point source acts in turn as a fluid point source and then a fluid point sink. (Such an object does not exist physically, but is often a good simplified model for calculations.)

Examples:

A coaxial loudspeaker is designed to work as a point source to allow a wider field for listening.

Ionizing radiation

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Geiger-Müller counter with dual counts/dose rate display measuring a "point source".

Point sources are used as a means of calibrating ionizing radiation instruments. They are usually sealed capsules and are most commonly used for gamma, x-ray and beta-measuring instruments.

Heat

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A mushroom cloud as an example of a thermal plume. A nuclear explosion can be treated as a thermal point source in large-scale atmospheric simulations.

In a vacuum, heat escapes as radiation isotropically. If the source remains stationary in a compressible fluid such as air, flow patterns can form around the source due to convection, leading to an anisotropic pattern of heat loss. The most common form of anisotropy is the formation of a thermal plume above the heat source. Examples:

  • Geological hotspots on the surface of the Earth which lie at the tops of thermal plumes rising from deep inside the Earth
  • Plumes of heat studied in thermal pollution tracking.

Fluid

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Fluid point sources are commonly used in fluid dynamics and aerodynamics. A point source of fluid is the inverse of a fluid point sink (a point where fluid is removed). Whereas fluid sinks exhibit complex rapidly changing behavior such as is seen in vortices (for example water running into a plug-hole or tornadoes generated at points where air is rising), fluid sources generally produce simple flow patterns, with stationary isotropic point sources generating an expanding sphere of new fluid. If the fluid is moving (such as wind in air or currents in water) a plume is generated from the point source.

Examples:

  • Air pollution from a power plant flue gas stack in a large-scale analysis of air pollution
  • Water pollution from an oil refinery wastewater discharge outlet in a large-scale analysis of water pollution
  • Gas escaping from a pressurized pipe in a laboratory
  • Smoke is often released from point sources in a wind tunnel in order to create a plume of smoke which highlights the flow of the wind over an object
  • Smoke from a localized chemical fire can be blown in the wind to form a plume of pollution

Pollution

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Main article: Point source pollution

Sources of various types of pollution are often considered as point sources in large-scale studies of pollution.[1]

See also

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References

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Revisions and contributorsEdit on WikipediaRead on Wikipedia

Point source

View on Grokipedia
from Grokipedia
A point source in physics is a localized emitter of waves or , such as , , or ionizing particles, whose physical dimensions are negligible relative to the to the observer or the of the emission, permitting approximation as radiation originating from a single geometric point with isotropic . This idealization simplifies modeling, as the intensity from such a source follows the , decreasing proportionally to the reciprocal of the squared from the source. In , point sources are fundamental to analyzing and resolution limits; for instance, the smallest resolvable spot from a point source through an of D exhibits an angular spread governed by the λ, approximately θ ≈ λ/D under far-field conditions, distinguishing point-like behavior from extended sources. This holds when θ ≪ λ/D, enabling precise treatments in theory without accounting for source extent. For , point sources are critical in and shielding calculations, where detectors like Geiger counters measure assuming spherical spreading from the emission point, as exemplified in controlled experiments or historical events involving concentrated releases. Such models underpin safety protocols, emphasizing distance as a primary attenuator since absorption and effects vary by medium but the geometric dilution remains universal for true point approximations.

Conceptual Foundations

Definition and Mathematical Formulation

A point source is an idealized physical model representing a source of emission—such as electromagnetic waves, , particles, or —with negligible spatial dimensions relative to the observation distance, allowing it to be treated as originating from a single location in space. This approximation holds when the source size is much smaller than the relevant or separation distance, enabling simplified analytical treatment via singularity functions like the Dirac delta distribution in mathematical models. For an isotropic point source emitting total power PP uniformly across a spherical surface, the radial intensity or I(r)I(r) at rr obeys the , I(r)=P4πr2I(r) = \frac{P}{4\pi r^2}, derived from over the expanding sphere's area 4πr24\pi r^2. This formulation applies to steady-state fields like or , where the potential scales as 1/r1/r and the field strength as 1/r21/r^2. In time-harmonic wave contexts, such as solutions to the (∇2+k2)ψ=−δ(r)(\nabla^2 + k^2) \psi = -\delta(\mathbf{r}) for k=2π/λk = 2\pi/\lambda, the outgoing from a point source at the origin is ψ(r)∝eikrr\psi(r) \propto \frac{e^{ikr}}{r}, with the far-field amplitude decaying as 1/r1/r and intensity ∣ψ∣2|\psi|^2 as 1/r21/r^2. This spherical wave representation underpins models in , acoustics, and for localized excitations.

Approximation Conditions and Limitations

The point source approximation assumes that the emitting object can be treated as having negligible spatial extent relative to the scale of the phenomenon, leading to isotropic emission and spherical wavefronts or inverse-square decay. This holds when the source diameter DD satisfies D≪λD \ll \lambda, where λ\lambda is the , as the finite size then contributes minimally to phase variations across the source. In acoustic , for instance, sealed sources like loudspeakers approximate point sources effectively at low frequencies, where λ\lambda greatly exceeds DD, enabling the for intensity (I∝1/r2I \propto 1/r^2) and amplitude (p∝1/rp \propto 1/r) in the radiated field. For electromagnetic and optical waves, validity extends to the far-field regime, where the observation distance zz must exceed z≳D2/λz \gtrsim D^2 / \lambda (Fraunhofer condition), ensuring quadratic phase terms in the become negligible and the field resembles a of the source distribution, as if from a point. This criterion, for example, requires z>2z > 2 m for a 1 mm at λ=550\lambda = 550 nm. In antenna , similar far-field distances (r>2D2/λr > 2D^2 / \lambda) stabilize patterns, treating the antenna as point-like beyond reactive near-field effects. Limitations emerge near the source or when D≈λD \approx \lambda, where extended-source effects like , , or dominate, distorting and requiring full wave solutions or distributed models. In ionizing radiation , such as , point approximations suffice for isolated pellets at radial distances beyond minimal thresholds but overestimate factors for clustered or elongated sources, demanding line or volume corrections for accuracy within 1-5 cm. For non-wave contexts like thermal point sources, validity similarly requires observer distances much larger than DD, failing in near-field dominated by conduction or evanescent modes.

Applications in Wave Phenomena

Electromagnetic Radiation

A point source in is an idealized model where the radiating element's dimensions are negligible compared to the λ, such that Δl ≪ λ, allowing the emission to be treated as originating from an volume. This approximation simplifies the analysis of fields from small antennas or oscillating dipoles, producing nearly spherical wavefronts in the far field (r ≫ λ/2π). The model assumes a localized current or charge , with fields derived from the in the Lorenz gauge. The Hertzian dipole serves as the fundamental point source, consisting of an electrically short, infinitesimally thin filament with uniform current density Ĩ e^{jωt} along length Δl. In the far field, the phasor electric field is approximately E~≈θ^jηkI~Δlsin⁡θ4πe−jkrr\widetilde{\mathbf{E}} \approx \hat{\theta} j\eta \frac{k \widetilde{I} \Delta l \sin\theta}{4\pi} \frac{e^{-jkr}}{r}
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