Physical Quantities and units

August 24, 2011

A physical quantity is a physical property of a phenomenon, body, or substance, that can be quantified by measurement.^[1]

Formally, the International Vocabulary of Metrology, 3rd edition (VIM3) defines quantity as:

property of a phenomenon, body, or substance, where the property has a magnitude that can be expressed as a number and a reference^[2]

Hence the value of a physical quantity Q is expressed as the product of a numerical value {Q} and a unit of measurement [Q].

Q = {Q} x [Q]

Quantity calculus describes how to do maths with quantities.

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[edit] Examples

If the temperature T of a body is quantified (measured) as 300 kelvin this is written

T = 300 x K = 300 K

where T is the symbol for the physical quantity, and K is the symbol for the unit. (NB degrees Celsius cannot be treated in this way)

If a person weighs 120 pounds, then “120” is the numerical value and “pound” is the unit. This physical quantity mass would be written as “120 lbs”, or

m = 120 lbs

If a person traveling with a yardstick, measures the length of such yardstick, the physical quantity of length would be written as

L = 36 inches

Note that different observers may get different values of L. While L is a physical property, L is not physically invariant. L (like all physical quantities) depends on a number of parameters picked by the observer; including the coordinate system, and metric.

An example employing SI units and scientific notation for the number, might be a measurement of power written as

P = 42.3 x 10³ W,

Here, P represents the physical quantity of power, 42.3 x 10³ is the numerical value {P}, and W is the symbol for the unit of power [P], the watt

[edit] Symbols for physical quantities (philosophy)

Symbols for physical quantities are usually chosen to be a single letter of the Latin or Greek alphabet, and are printed in italic type. Symbols can be modified by subscripts and superscripts, to specify what they refer to — for instance E_k is usually used to denote kinetic energy and c_p heat capacity at constant pressure. (Note the difference in the style of the subscripts: k is the abbreviation of the word kinetic, whereas p is the symbol for the physical quantity pressure rather than an abbreviation of the word “pressure”.)

Symbols for quantities should be chosen according to the international recommendations from ISO 80000, the IUPAP red book and the IUPAC green book. For example, the recommended symbol for the physical quantity ‘mass’ is m, and the recommended symbol for the quantity ‘charge’ is Q.

Symbols for physical quantities that are vectors are bold type. If, e.g., u is the speed of a particle, then the straightforward notation for its velocity is u.

Numerical quantities, even those denoted by letters, are usually printed in Roman (upright) type, e.g.: 1, 2, e (for the base of natural logarithm), i (for the imaginary unit) or π (for 3.14…). Symbols for numerical functions such as sin α are Roman type too. Although the recommendation is not followed by Wikipedia, operators like d in dx are recommended also to be printed in Roman type.

[edit] Units of physical quantities (philosophy)

Most physical quantities Q include a unit [Q] (where [Q] means “unit of Q“). Neither the name of a physical quantity, nor the symbol used to denote it, implies a particular choice of unit. For example, a quantity of mass might be represented by the symbol m, and could be expressed in the units kilograms (kg), pounds (lb), or Daltons (Da). SI units are usually preferred today.

[edit] Base quantities, derived quantities and dimensions

The notion of physical dimension of a physical quantity was introduced by Fourier in 1822.^[3] 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 seven base quantities of the International System of Quantities (ISQ) and their corresponding SI units are listed in the following table. Other conventions may have a different number of fundamental units (e.g. the CGS and MKS systems of units).

International System of Units base quantities
Quantity Name/s	(Common) Quantity Symbol/s	SI Unit Name	SI Unit Symbol	Dimension Symbol
Length, width, height, depth	a, b, c, d, h, l, r, w, x, y, z	metre	m	[L]
Time	t	second	s	[T]
Mass	m	kilogram	kg	[M]
Temperature	T, θ	kelvin	K	[Θ]
Amount of Substance,Number of Moles	n	mole	mol	[N]
Electric Current	i, I	Ampere	A	[I]
Luminous Intensity	I_v	Candela	Cd	[J]
Plane Angle	α, β, γ, θ, φ, χ	radian	rad	dimensionless
Solid Angle	ω, Ω	steradian	sr	dimensionless

The last two angular units; plane and solid angle are subsidiary units used in the SI, but treated dimensionless. The subsidiary Units are used for convenience to differentiate between a truly dimensionless quantity (pure number) and an angle.

All other quantities are derived quantities since their dimensions are derived from those of base quantities by multiplication and division. For example, the physical quantity velocity is derived from base quantities length and time and has dimension L/T. Some derived physical quantities have dimension 1 and are said to be dimensionless quantities. A physical quantity is not necessarily physically invariant, that is an incorrect use of the term.

Further information: dimensional analysis

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

(Common) Quantity Name/s	(Common) Quantity Symbol/s	SI Unit	Dimension
(Spatial) Position Vector	r, R, a, d	m	[L]
Angular Position, Angle of Rotation (can be treated as vector or scalar)	θ, θ	rad	dimensionless
Area, Cross-Section	A, S, Ω	m²	[L]²
Area Vector(Magnitude of surfacearea, directed normal to tangential plane of surface)	$\mathbf{A} = A\mathbf{\hat{n}},\mathbf{S}= S\mathbf{\hat{n}} \,\!$	m²	[L]²
Volume	D, V	m³	[L]³

[edit] General Derived Quantities

Important/convenient derived quantities such as densities, fluxes, flows, currents are associated with many quantities. Sometimes they are used interchangeably in the same context, sometimes they are used uniqueley.

To clarify these terms, we let q be any quantity (not necessarily fundamental) and present in the table below commonly used symbols (which differ usually by changing subscripts for different situations), their definitions and their forms of usage.

For specific, molar, and flux densities of quantities there is no one symbol, nomenclature depends on subject, for generality we use q_m, q_n, and F respectively. No symbol is necessarily required for gradeint, since only the nabla/del operator ∇ or grad needs to be written.

The notations below can be used synonymously.

If F is a n-variable function $F \equiv F \left ( x_1, x_2 \cdots x_n \right ) \,\!$ , then:

the differential n-space volume element is $\mathrm{d}^n x \equiv \mathrm{d} V_n \equiv \mathrm{d} x_1 \mathrm{d} x_2 \cdots \mathrm{d} x_n \,\!$ ,
the multiple integral of F over the n-space volume is $\int F \mathrm{d}^n x \equiv \int F \mathrm{d} V_n \equiv \int \cdots \int \int F \mathrm{d} x_1 \mathrm{d} x_2 \cdots \mathrm{d} x_n \,\!$ .

For current density, $\mathbf{\hat{e}} \,\!$ is a unit vector in the direction of flow.

Quantity	Typical Symbols	Definition	Meaning, Usage	Dimension
Quantity	q	q	Amount of a property	[q]
Quantity Rate of Change	$\dot{q} \,\!$	$\dot{q} \equiv \frac{\mathrm{d} q}{\mathrm{d} t} \,\!$	Rate of change of property with respectto time	[q] [T]⁻¹
Quantity Density	ρ = volume density, σ = surface density, λ = linear density No common symbol for n-space density, here ρ_n is used.	$\rho_n = \frac{\mathrm{d} q}{\mathrm{d} V_n} = \frac{\mathrm{d}^n q}{\mathrm{d} x_n \cdots \mathrm{d} x_2 \mathrm{d} x_1} \,\!$	Amount of property per unit n-space (length, area, volume or higher dimensions)	[q] [L]^–n
Specific Quantity	q_m	$q_m = \frac{\mathrm{d} q}{\mathrm{d} m} \,\!$	Amount of property per unit mass	[q] [L]^–n
Molar Quantity	q_n	$q_n = \frac{\mathrm{d} q}{\mathrm{d} n} \,\!$	Amount of property per mole of substance	[q] [L]^–n
Quantity Gradient (only if q is a scalar field.		$\nabla q \,\!$	Rate of change of property with respect to position	[q] [L]⁻¹
Spectral Quantity (for EM waves)	q_v, q_ν, q_λ	Two defninitions are used, for frequency and wavelengh: $q_\lambda = \frac{\mathrm{d} q}{\mathrm{d} \lambda} \,\!$ $q_\lambda = \frac{\mathrm{d} q}{\mathrm{d} \nu} \,\!$	Amount of property per unit wavelength or frequency.	q_λ: [q] [L]^-1 q_ν : [q] [T]
Flux, Flow (synonymous)	Φ_F, F	Two definitions are used;Transport Dynamics, Nuclear/Particle Physics: $F = \frac{\mathrm{d}^2 q}{\mathrm{d} A \mathrm{d} t} \,\!$ Vector Fields: $\Phi_F = \int_S \mathbf{F} \cdot \mathrm{d} \mathbf{A} \,\!$	Flow of a property though across-section/surface boundary.	[q] [T]⁻¹ [L]⁻², [F] [L]²
Flux Density	F	$\mathbf{F} = \frac{\mathrm{d} \Phi_F}{\mathrm{d} A} \,\!$	Flow of a property though a cross- section/surface boundary per unit cross- section/surface area	[F]
Current	i, I	$I = \frac{\mathrm{d} q}{\mathrm{d} t} \,\!$	Rate of flow of property through a crosssection/ surface boundary	[q] [T]⁻¹
Current Density(sometimes called Flux in Transport Dynamics)	j, J	$\mathbf{J} = \mathbf{\hat{e}} \frac{\mathrm{d} I}{\mathrm{d} t} = \mathbf{\hat{e}} \frac{\mathrm{d}^2 q}{\mathrm{d} A \mathrm{d} t} \,\!$	Rate of flow of property per unit cross-section/surface area	[q] [T]⁻¹ [L]⁻²
Moment of Quantity	m, M	Two definitions can be used; q is a scalar: $\mathbf{m} = \mathbf{r} q \,\!$ q is a vector: $\mathbf{m} = \mathbf{r} \times \mathbf{q} \,\!$	Quantity at position r has a moment about a point or axes, often relates to potential energy.	[q] [L]

[edit] Extensive and intensive quantities (philosophy)

A quantity (philosophy) is called:

extensive when its magnitude is additive for subsystems (volume, mass, etc.)
intensive when the magnitude is independent of the extent of the system (temperature, pressure, etc.)

There are also physical quantities that can be classified as neither extensive nor intensive, for example angular momentum, area, force, length, and time.

[edit] Physical quantities as coordinates over spaces of physical qualities (philosophy)

The meaning of the term physical quantity is generally well understood (everyone understands what is meant by the frequency of a periodic phenomenon, or the resistance of an electric wire). The term physical quantity does not imply a physically invariant quantity. Length for example is a physical quantity, yet it is variant under coordinate change in special and general relativity. The notion of physical quantities is so basic and intuitive in the realm of science, that it does not need to be explicitly spelled out or even mentioned. It is universally understood that scientists will (more often then not) deal with quantitative data, as opposed to qualitative data. Explicit mention and discussion of physical quantities is not part of any standard science program, and is more suited for a philosophy of science or philosophy program.

It should be noted that the notion of physical quantities is seldom used in physics, nor is it part of the standard physics vernacular. The idea is often misleading, as its name implies “a quantity that can be physically measured”, yet is often incorrectly used to mean a physical invariant. Due to the rich complexity of physics, many different fields posse different physical invariants. There is no known physical invariant sacred in all possible fields of physics. Energy, Space, momentum, torque, position, and length (just to name a few) are all found to be experimentally variant in some particular scale and system. Additionally, the notion that it is possible to measure “physical quantities” comes into question, particular in Quantum field theory and Normalization techniques. As infinites are produces by the theory, the actual “measurements” we make are not really those of the physical universe (as we cannot measure infinities), they are those of the renormalization scheme which is expressly depended on our measurement scheme, coordinate system and metric system.

It is not always possible to define the distance between two points of any quality space, and this distance is —inside a given theoretical context— not uniquely defined. The notion of a distance, even in the context of quality space, relies on a concept of a metric space. Without a metric space, any notion of distance, physical or otherwise is undefined.

[edit] See also

[edit] Notes

^ http://www.bipm.org/utils/common/documents/jcgm/JCGM_200_2008.pdf
^ Joint Committee for Guides in Metrology (JCGM), International Vocabulary of Metrology, Basic and General Concepts and Associated Terms (VIM), III ed., Pavillon de Breteuil : JCGM 200:2008, 1.1 (on-line)
^ 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.)