View source for Physical quantity

{{Short description|Measurable  property of a material or system}}
{{more citations needed|date=March 2022}}

[[File:Ampèremetre.jpg|thumb|Ampèremetre (Ammeter)]]

A '''physical quantity''' (or simply '''quantity''')<ref name="ISO 80000-1"/>{{efn|"The concept 'quantity' may be generically divided into, e.g. 'physical quantity', 'chemical quantity', and 'biological quantity', or 'base quantity' and 'derived quantity'."<ref name="ISO 80000-1"/>}} is a property of a material or system that can be [[Quantification (science)|quantified]] by [[measurement]]. A physical quantity can be expressed as a ''value'', which is the algebraic multiplication of a ''[[numerical value]]'' and a ''[[unit of measurement]]''. For example, the physical quantity [[mass]], symbol ''m'', can be quantified as ''m''{{=}}''n''{{nbsp}}kg, where ''n'' is the numerical value and kg is the unit symbol (for [[kilogram]]). [[Vector quantity|Vector quantities]] have, besides numerical value and unit, direction or orientation in space.

== Principles {{anchor|Units and dimensions}} ==

=== Dimensions ===
{{Main|Dimension (physics)}}
The notion of ''dimension'' of a physical quantity was introduced by [[Joseph Fourier]] in 1822.<ref>Fourier, Joseph. ''[[Théorie analytique de la chaleur]]'', Firmin Didot, Paris, 1822. (In this book, Fourier introduces the concept of ''physical dimensions'' for the physical quantities.)</ref> By convention, physical quantities are organized in a dimensional system built upon [[base quantities]], each of which is regarded as having its own dimension.
The dimension of a quantity ''Z'' is denoted {{math|dim{{nbsp}}''Z''}} or {{math|dim(''Z'')}}.<ref name="ISO 80000-1"/>

=== Kind ===
[[Dimensional homogeneity]] is not necessarily sufficient for quantities to be comparable;<ref name="ISO 80000-1"/>
for example, both [[kinematic viscosity]] and [[thermal diffusivity]] have dimension of square length per time (in units of [[metre squared per second|m<sup>2</sup>/s]]).
Quantities of the same '''''kind''''' share extra commonalities beyond their dimension and units allowing their comparison;
for example, not all [[dimensionless quantities]] are of the same kind<ref name="ISO 80000-1"/> ({{xref|see also: [[dimensional equivalence]]}}).

=== Unit ===
{{main|Units of measurement}} 
There is often a choice of unit, though [[SI]] [[Units of measurement|units]] are usually used in scientific contexts due to their ease of use, international familiarity and prescription. For example, a quantity of mass might be represented by the symbol ''m'', and could be expressed in the units [[kilogram]]s (kg), [[Pound (mass)|pound]]s (lb), or [[Dalton (unit)|dalton]]s (Da).
The unit of a quantity {{mvar|Z}} is denoted {{math|[''Z'']}}.<ref name="ISO 80000-1"/>

=== Numerical value ===
Following [[ISO 80000-1]],<ref name="ISO 80000-1">{{cite web |title=ISO 80000-1:2009(en) Quantities and units — Part 1: General |website=International Organization for Standardization | url=https://www.iso.org/obp/ui/#iso:std:iso:80000:-1:ed-1:v1:en | access-date=2023-05-12}}</ref> any value or [[Magnitude (mathematics)|magnitude]] of a physical quantity is expressed as a comparison to a unit of that quantity. The ''value'' of a physical quantity ''Z'' is expressed as the product of a ''numerical value'' {''Z''} (a pure number) and a unit [''Z'']:

:<math>Z = \{Z\} \times [Z]</math>

For example, let <math>Z</math> be "2 metres"; then, <math>\{Z\} = 2</math> is the numerical value and <math>[Z] = \mathrm{metre}</math> is the unit.
Conversely, the numerical value expressed in an arbitrary unit can be obtained as:

:<math>\{Z\} = Z / [Z]</math>

The multiplication sign is usually left out, just as it is left out between variables in the scientific notation of formulas. The convention used to express quantities is referred to as ''[[quantity calculus]]''. In formulas, the unit [''Z''] can be treated as if it were a specific magnitude of a kind of physical dimension: see ''[[Dimensional analysis]]'' for more on this treatment.

=== Typography ===
{{further|Mathematical notation}}

International recommendations for the use of symbols for quantities are set out in [[ISO/IEC 80000]], the [[IUPAP red book]] and the [[Quantities, Units and Symbols in Physical Chemistry|IUPAC green book]]. For example, the recommended symbol for the physical quantity "mass" is ''m'', and the recommended symbol for the quantity "electric charge" is ''Q''.

Physical quantities are normally typeset in italics.
Purely numerical quantities, even those denoted by letters, are usually printed in roman (upright) type, though sometimes in italics. Symbols for elementary functions (circular trigonometric, hyperbolic, logarithmic etc.), changes in a quantity like Δ in Δ''y'' or operators like d in d''x'', are also recommended to be printed in roman type.

Examples:
* Real numbers, such as 1 or {{radic|2}},
* e, the base of [[natural logarithms]],
* i, the [[imaginary numbers|imaginary]] unit,
* π for the ratio of a circle's circumference to its diameter, 3.14159265...
* δ''x'', Δ''y'', d''z'', representing differences (finite or otherwise) in the quantities ''x'', ''y'' and ''z''
* sin ''&alpha;'', sinh ''&gamma;'', log ''x''

== Support ==

=== Scalars ===
{{main|Scalar (physics)}}

A [[Scalar (physics)|''scalar'']] is a physical quantity that has magnitude but no direction. Symbols for physical quantities are usually chosen to be a single letter of the [[Latin alphabet|Latin]] or [[Greek alphabet]], and are printed in italic type.

=== Vectors ===
{{main|Vector quantity}}

[[Vector (mathematics and physics)|''Vectors'']] are physical quantities that possess both magnitude and direction and whose operations obey the [[axioms]] of a [[vector space]]. Symbols for physical quantities that are vectors are in bold type, underlined or with an arrow above. For example, if ''u'' is the speed of a particle, then the straightforward notations for its velocity are '''u''', <u>u</u>, or <math>\vec{u}</math>.

=== Tensors ===
{{main category|Tensor physical quantities}}

Scalar and vector quantities are the simplest '''tensor quantities''', which are [[tensor]]s that can be used to describe more general physical properties. For example, the [[Cauchy stress tensor]] possesses magnitude, direction, and orientation qualities.

== Base and derived quantities ==

=== Base quantities ===
{{main|Base quantities}}

A system of quantities relates physical quantities, and due to this dependence, a limited number of quantities can serve as a basis in terms of which the dimensions of all the remaining quantities of the system can be defined. A set of mutually independent quantities may be chosen by convention to act as such a set, and are called base quantities. The seven base quantities of the [[International System of Quantities]] (ISQ) and their corresponding [[SI]] units and dimensions are listed in the following table.<ref name="SIBrochure9thEd">{{citation |title=The International System of Units (SI) |author=International Bureau of Weights and Measures |author-link=New SI |date=20 May 2019  |edition=9th |isbn=978-92-822-2272-0 |url=https://www.bipm.org/utils/common/pdf/si-brochure/SI-Brochure-9.pdf| archive-url = https://web.archive.org/web/20211018184555/https://www.bipm.org/documents/20126/41483022/SI-Brochure-9.pdf/fcf090b2-04e6-88cc-1149-c3e029ad8232 |archive-date=18 October 2021 |url-status=live}}</ref>{{rp|page=136}} Other conventions may have a different number of [[Base unit (measurement)|base unit]]s (e.g. the [[CGS]] and [[Mks system of units|MKS]] systems of units).

{| class="wikitable"
|+ style="font-size:larger;font-weight:bold;"|[[International System of Quantities]] base quantities
! colspan=2|Quantity
! colspan=2|SI unit
! rowspan=2|Dimension<br>symbol
|-
! Name(s)
! (Common) symbol(s)
! Name
! Symbol
|-
| [[Length]]
| ''l'', ''x'', ''r''
| [[metre]]
| m 
| L
|-
| [[Time]] 
| ''t'' 
| [[second]]
| s 
| T
|-
| [[Mass]] 
| ''m'' 
| [[kilogram]]
| kg 
| M
|-
| [[Thermodynamic temperature]] 
| ''T''
| [[kelvin]]
| K 
| Θ
|-
| [[Amount of substance]] 
| ''n''
| [[Mole (unit)|mole]] 
| mol 
| N
|-
| [[Electric current]] || ''i, I''
| [[ampere]]
| A
| I
|-
| [[Luminous intensity]] || ''I''<sub>v</sub>
| [[candela]] 
| cd 
| J
|}

The angular quantities, [[plane angle]] and [[solid angle]], are defined as derived dimensionless quantities in the SI. For some relations, their units [[radian]] and [[steradian]] can be written explicitly to emphasize the fact that the quantity involves plane or solid angles.<ref name="SIBrochure9thEd"/>{{rp|page=137}}

=== General derived quantities ===
{{further|SI derived unit}}
Derived quantities are those whose definitions are based on other physical quantities (base quantities).

==== Space ====
Important applied base units for space and time are below. [[Area]] and [[volume]] are thus, of course, derived from the length, but included for completeness as they occur frequently in many derived quantities, in particular densities.

{| class="wikitable"
|-
! colspan=2|Quantity
! rowspan=2| SI unit
! rowspan=2| Dimensions
|-
! Description
! Symbols
|-
| (Spatial) [[position (vector)]] 
| '''r''', '''R''', '''a''', '''d''' 
| m 
| L
|-
| Angular position, angle of rotation (can be treated as vector or scalar)
| ''θ'', '''θ'''
| rad 
| ''None''
|-
| Area, cross-section 
| ''A'', ''S'', Ω 
| m<sup>2</sup> 
| L<sup>2</sup>
|-
| [[Vector area]] (Magnitude of surface area, directed normal to [[tangent]]ial plane of surface)
| <math> \mathbf{A} \equiv A\mathbf{\hat{n}}, \quad \mathbf{S}\equiv S\mathbf{\hat{n}}</math> 
| m<sup>2</sup> 
| L<sup>2</sup>
|-
| Volume 
| ''τ'', ''V'' 
| m<sup>3</sup> 
| L<sup>3</sup>
|-
|}

==== Densities, flows, gradients, and moments ====
Important and convenient derived quantities such as densities, [[flux]]es, [[Fluid dynamics|flows]], [[Electric current|current]]s are associated with many quantities. Sometimes different terms such as ''current density'' and ''flux density'', ''rate'', ''frequency'' and ''current'', are used interchangeably in the same context; sometimes they are used uniquely.

To clarify these effective template-derived quantities, we use ''q'' to stand for ''any'' quantity within some scope of context (not necessarily base quantities) and present in the table below some of the most commonly used symbols where applicable, their definitions, usage, SI units and SI dimensions – where [''q''] denotes the dimension of ''q''.

For time derivatives, specific, molar, and flux densities of quantities, there is no one symbol; nomenclature depends on the subject, though time derivatives can be generally written using overdot notation. For generality we use ''q<sub>m</sub>'', ''q<sub>n</sub>'', and '''F''' respectively. No symbol is necessarily required for the gradient of a scalar field, since only the [[Del|nabla/del operator]] ∇ or [[Gradient|grad]] needs to be written. For spatial density, current, current density and flux, the notations are common from one context to another, differing only by a change in subscripts.

For current density, <math> \mathbf{\hat{t}}</math> is a unit vector in the direction of flow, i.e. tangent to a flowline. Notice the [[dot product]] with the unit normal for a surface, since the amount of current passing through the surface is reduced when the current is not normal to the area. Only the current passing perpendicular to the surface contributes to the current passing ''through'' the surface, no current passes ''in'' the (tangential) plane of the surface.

The calculus notations below can be used synonymously.

:If ''X'' is a [[Multivariable calculus|''n''-variable]] [[Function (mathematics)|function]] <math> X \equiv X \left ( x_1, x_2 \cdots x_n \right ) </math>, then

:'''''Differential''''' The differential [[n-dimensional space|''n''-space]] [[volume element]] is <math> \mathrm{d}^n x \equiv \mathrm{d} V_n \equiv \mathrm{d} x_1 \mathrm{d} x_2 \cdots \mathrm{d} x_n  </math>,

:'''''Integral''''': The [[Multiple integral|''multiple'' integral]] of ''X'' over the ''n''-space volume is <math> \int X \mathrm{d}^n x \equiv \int X \mathrm{d} V_n \equiv \int \cdots \int \int X \mathrm{d} x_1 \mathrm{d} x_2 \cdots \mathrm{d} x_n  </math>.

{| class="wikitable"
! scope="col" width="150" | Quantity
! scope="col" width="150" | Typical symbols
! scope="col" width="250" | Definition
! scope="col" width="200" | Meaning, usage
! scope="col" width="100" | Dimensions
|-
| Quantity 
| ''q'' 
| ''q'' 
| Amount of a property
| [q]
|-
| Rate of change of quantity, [[time derivative]]
| <math> \dot{q} </math> 
| <math> \dot{q} \equiv \frac{\mathrm{d} q}{\mathrm{d} t} </math>
| Rate of change of property with respect to time
| [q]T<sup>−1</sup>
|-
| Quantity spatial density 
| ''ρ'' = volume density (''n'' = 3), ''σ'' = surface density (''n'' = 2), ''λ'' = linear density (''n'' = 1)

No common symbol for ''n''-space density, here ''ρ<sub>n</sub>'' is used. 
| <math> q = \int \rho_n  \mathrm{d} V_n </math> 
| Amount of property per unit n-space <br />
(length, area, volume or higher dimensions)
| [q]L<sup>−''n''</sup>
|-
| [[Specific quantity]]
| ''q<sub>m</sub>'' 
| <math> q_m = \frac{\mathrm{d} q}{\mathrm{d} m} </math> 
| Amount of property per unit mass
| [q]M<sup>−1</sup>
|-
| Molar quantity
| ''q<sub>n</sub>''
| <math> q_n = \frac{\mathrm{d} q}{\mathrm{d} n} </math> 
| Amount of property per mole of substance
| [q]N<sup>−1</sup>
|-
| Quantity gradient (if ''q'' is a [[scalar field]]).
|
| <math> \nabla q </math> 
| Rate of change of property with respect to position 
|| [q]L<sup>−1</sup>
|-
| Spectral quantity (for EM waves)
| ''q<sub>v</sub>, q<sub>ν</sub>, q<sub>λ</sub>''
| Two definitions are used, for frequency and wavelength:<br />
<math> q=\int q_\lambda \mathrm{d} \lambda </math><br />
<math> q=\int q_\nu \mathrm{d} \nu </math> 
| Amount of property per unit wavelength or frequency.
| [q]L<sup>−1</sup> (''q<sub>λ</sub>'')<br />
[q]T (''q<sub>ν</sub>'')
|-
| Flux, flow (synonymous) 
| ''Φ<sub>F</sub>'', ''F'' 
| Two definitions are used:<br />
[[Transport phenomena (engineering & physics)|Transport mechanics]], [[nuclear physics]]/[[particle physics]]: <br />
<math> q = \iiint F \mathrm{d} A \mathrm{d} t </math>

[[Vector field]]: <br />
<math> \Phi_F = \iint_S \mathbf{F} \cdot \mathrm{d} \mathbf{A}</math> 
| Flow of a property though a cross-section/surface boundary.
| [q]T<sup>−1</sup>L<sup>−2</sup>, [F]L<sup>2</sup>
|-
| Flux density 
| '''F''' 
| <math> \mathbf{F} \cdot \mathbf{\hat{n}} = \frac{\mathrm{d} \Phi_F}{\mathrm{d} A} </math> 
| Flow of a property though a cross-section/surface boundary per unit cross-section/surface area
| [F]
|-
| Current 
| ''i'', ''I''
| <math> I = \frac{\mathrm{d} q}{\mathrm{d} t} </math> 
| Rate of flow of property through a cross-section/surface boundary
| [q]T<sup>−1</sup>
|-
| Current density (sometimes called flux density in transport mechanics)
| '''j''', '''J'''
| <math> I = \iint \mathbf{J} \cdot \mathrm{d}\mathbf{S}</math>
| Rate of flow of property per unit cross-section/surface area
| [q]T<sup>−1</sup>L<sup>−2</sup>
|-
| [[Moment (physics)|Moment]] of quantity
| '''m''', '''M''' 
|
''k''-vector ''q'': <math> \mathbf{m} = \mathbf{r} \wedge q </math> 
* scalar ''q'': <math> \mathbf{m} = \mathbf{r} q </math> {{br}}
* 3D vector '''q''', equivalently{{efn|via [[Hodge duality]]}} <math> \mathbf{m} = \mathbf{r} \times \mathbf{q} </math>
| Quantity at position '''r''' has a moment about a point or axes, often relates to tendency of rotation or [[potential energy]].
| [q]L
|}

== See also ==
* [[List of physical quantities]]
* [[Photometry (optics)#Photometric quantities|List of photometric quantities]]
* [[Radiometry#Radiometric quantities|List of radiometric quantities]]
* [[Philosophy of science]]
* [[Quantity]]
** [[Observable|Observable quantity]]
** [[Specific quantity]]

== Notes ==
{{Notelist}}

== References ==
{{Reflist}}

== Further reading ==
* Cook, Alan H. ''The observational foundations of physics'', Cambridge, 1994. {{ISBN|0-521-45597-9}}
* Essential Principles of Physics, P.M. Whelan, M.J. Hodgson, 2nd Edition, 1978, John Murray, {{ISBN|0-7195-3382-1}}
* Encyclopedia of Physics, [[Rita G. Lerner|R.G. Lerner]], G.L. Trigg, 2nd Edition, VHC Publishers, Hans Warlimont, Springer, 2005, pp 12–13
* Physics for Scientists and Engineers: With Modern Physics (6th Edition), P.A. Tipler, G. Mosca, W.H. Freeman and Co, 2008, 9-781429-202657

== External links ==
; Computer implementations
* [http://sergey-l-gladkiy.narod.ru/ DEVLIB] project in [[C Sharp (programming language)|C#]] [[Programming language|Language]] and [[Delphi (programming language)|Delphi]] [[Programming language|Language]]
* [http://physicalquantities.codeplex.com/ Physical Quantities] {{Webarchive|url=https://web.archive.org/web/20140101075054/http://physicalquantities.codeplex.com/ |date=2014-01-01 }} project in [[C Sharp (programming language)|C#]] [[Programming language|Language]] at [[CodePlex|Code Plex]]
* [http://physicalmeasure.codeplex.com/ Physical Measure C# library] {{Webarchive|url=https://web.archive.org/web/20140101075148/http://physicalmeasure.codeplex.com/ |date=2014-01-01 }} project in [[C Sharp (programming language)|C#]] [[Programming language|Language]] at [[CodePlex|Code Plex]]
* [http://measures.codeplex.com/ Ethical Measures] {{Webarchive|url=https://web.archive.org/web/20140101074955/http://measures.codeplex.com/ |date=2014-01-01 }} project in [[C Sharp (programming language)|C#]] [[Programming language|Language]] at [[CodePlex|Code Plex]]
* [http://engineerjs.com Engineer JS] online calculation and scripting tool supporting physical quantities.
* [https://www.npmjs.com/package/physical-quantity physical-quantity] a web component (custom HTML element) for expressing physical quantities on the web/Internet, featuring self-contained unit conversion, a compact and clean UI, no redundant dual units, and seamless integration across all websites and platforms. [https://camo.githubusercontent.com/a34cf2a0f90ae4ad948059d183594cf21b258670ab920240a1d635f31d47b505/68747470733a2f2f666972656261736573746f726167652e676f6f676c65617069732e636f6d2f76302f622f6175746f2d63616c632d38303233372e61707073706f742e636f6d2f6f2f5051452532466578616d706c657325324646742d496e25323044656d6f2e6769663f616c743d6d6564696126746f6b656e3d65386565316266642d653330632d346438312d393365392d373033336564303932376262 Demo] 
{{Authority control}}

{{DEFAULTSORT:Physical Quantity}}
[[Category:Physical quantities| ]]