QUDT - Quantities, Units, Dimensions and Data Types in OWL and XML

October 30, 2009

Authors:: James Masters <cmasters at topquadrant.com >; Ralph Hodgson <Ralph.Hodgson at nasa.gov>; Paul J. Keller <Paul.J.Keller at nasa.gov>

Overview

Status of This Document

This document is currently "Work in Progress" and will be improved incrementally during the fall of 2009.

Backgound

The QUDT Ontologies, and derived XML Vocabularies, are being developed by TopQuadrant and NASA as part of the NASA Exploration Initiatives Ontology Models (NExIOM) project, a Constellation Program initiative at the AMES Research Center (ARC). The goals of the QUDT ontology are twofold:

To provide a unified model of, measurable quantities, units for measuring different kinds of quantities, the numerical values of quantities in different units of measure and the data structures and data types used to store and manipulate these objects in software;

To populate the model with the instance data (quantities, units, quantity values, etc.) required to meet the life-cycle needs of the Constellation Program engineering community.

The QUDT public site is currently only publishing the foundation ontologies and some examples from engineering disciplines.

QUDT Ontologies

The QUDT Ontologies are catalogued on this page of the oeGOV blog

Ontology Class Structure

The diagram below, exported from TopBraid Composer, illustrates the main class structure of the QUDT ontology in OWL.

Quantities, Quantity Kinds, and Quantity Values

A Quantity is an observable property of an object, event or system that can be measured and quantified numerically. Quantities are differentiated by two attributes which together comprise the essential parameters needed to formalize the structure of quantities: kind and magnitude. The kind attribute of a quantity identifies the observable property quantified, e.g. length, force, frequency; the magnitude of the quantity expresses its relative size compared to other quantities of the same kind.

For example, the speed of light in a vacuum and the escape velocity of the Earth are both quantities of the kind speed but are of different magnitudes. The speed of light in a vacuum is greater than the escape velocity of the Earth. More generally, the speed of light in a vacuum is comparable to the escape velocity of the Earth. Thus, if two quantities of the same kind, their magnitudes can be compared and ordered. However, the same is not true if the quantities are of different kinds. There is no intrinsic way to compare the magnitude of a quantity of mass with the magnitude of a quantity of length.

Quantities may arise from definition or convention, or they may be the result of one or a series of experiments and measurements. In the first case, the quantity is exact; in the second case, measurement uncertainty cannot be discounted so the expression of a quantity's magnitude must account for the parameters of uncertainty.

Quantity Kinds

A Quantity Kind is any observable property that can be measured and quantified numerically. Familiar examples include physical properties such as length, mass, time, force, energy, power, electric charge, etc. Less familiar examples include currency, interest rate, price to earning ratio, and information capacity.

Unit of Measure

A Unit of Measure or Unit is a particular quantity of a given kind that has been chosen as a scale for measuring other quantities of the same kind. For example, the Meter is a quantity of length that has been empirically defined by the BIPM. Any quantity of length can be expressed as a number multiplied by the unit meter.

More formally, the value of a quantity Q with respect to a unit (U) is expressed as the scalar multiple of a real number (n) and U:

Q = nU

Quantity Value

A quantity value expresses the numerical value of a quantity with respect to a chosen unit of measure. For example, the value of Planck's constant in Joule-Seconds (J s) is approximately 6.62606896E-34, whereas the value in Erg-Seconds (erg s) is approximately 6.62606896E-27.

The OWL model for the classes qud:QuantityKind, qud:Quantity, qud:QuantityValue, qud:Unit is shown below.

Systems of Quantities and Units

The art and science of defining, standardizing, and organizing quantity kinds and units is ancient and modern. Today, scientific boards and standards bodies maintain rigorous definitions for quantity kinds and units. The definitions of and relationships between quantity kinds are derived from physical laws and mathematical transformations. Units are defined by experimental observations, by the application of physical laws, as ratios of fundamental physical constants, or by reference. One significant advance in the modern treatment of metrology has been the use of logic and mathematics to organize quantity kinds and units into systems and to analyze the relationships between them.

A system of quantity kinds is a set of one or more quantity kinds together with a set of zero or more algebraic equations that define relationships between quantity kinds in the set. In the physical sciences, the equations relating quantity kinds are typically physical laws and definitional relations, and constants of proportionality. Examples include Newton’s First Law of Motion, Coulomb’s Law, and the definition of velocity as the instantaneous change in position.

In almost all cases, the system identifies a subset of base quantity kinds. The base set is chosen so that all other quantity kinds of interest can be derived from the base quantity kinds and the algebraic equations.

A system of units is a set of units which are chosen as the reference scales for some set of quantity kinds together with the definitions of each unit. Units may be defined by experimental observation or by proportion to another unit not included in the system. If the unit system is explicitly associated with a quantity kind system, then the unit system must define at least one unit for each quantity kind.

Base and Derived Quantity Kinds

Many systems of quantity kinds identify a special subset of the included quantity kinds called the base quantity kinds. Base quantity kinds are typically chosen so that no base quantity kind can be expressed as an algebraic relation of one or more other base quantity kinds using only the constituent equations included in the system. A quantity kind that can be expressed as an algebraic relation of one or more base quantity kind is called a derived quantity kind. Thus, in any quantity kind system, the base set and derived set are disjoint.

Similarly, unit systems may distinguish between base units and derived units. A base unit is a unit of measurement for a base quantity, and a derived unit is a unit of measurement for a derived quantity. Unit systems define at least one base unit for each base quantity and at least one derived unit for each derived quantity.

Quantity Dimensions

Quantity kind systems that define base and derived sets have certain mathematical properties that permit quantity kinds to be manipulated symbolically. The construction goes as follows: Assign a distinct dimension symbol to each base quantity kind. For each derived quantity kind, take the formula that expresses it in terms of the base quantity kinds and replace every occurrence of a base quantity with its symbol. This is the dimension symbol for the derived quantity kind. In this way, every quantity kind maps to a dimension symbol of the form:

dim Q = (B₁)^d1(B₂)^d2…(B_n)^dn

Here {B₁,…,B_n} are the dimension symbols for the base quantities and {d₁,…,d_n} are rational numbers. Typically, the values of the d_i are between -3 and 3, however magnitudes as high as 7 are required to cover the range of quantity kinds currently defined. Using the multiplication identity for exponents AⁿA^m = A^n+m one can show that the set of dimension symbols is homomorphic to an n-dimensional vector space over the rational numbers. Multiplication of quantity kinds corresponds to vector addition, division corresponds to vector subtraction, and inverting a quantity kind corresponds to computing the additive inverse of its dimension vector.

In some cases, distinct quantity kinds may have the same dimension symbol. This often occurs in cases where physical laws are discovered and formalized independently of each other, but reduce to the same base quantity kinds. Perhaps the most commonly quoted example is the dimensional equivalence of mechanical torque and energy. Both have the same dimensions (L²MT^-2) but are defined very differently. One consequence of the equivalence is that the same units of measure are applicable to both. One salient difference between the two in this example is that torque is a pseudo-vector while energy is a scalar. However, this distinction (value type) is not accounted for in the quantity kind system formalism.

The OWL model of Dimensions is illustrated below.

Dimensionless Quantities and Units

[TBD]

Allowed Units

Some unit systems identify units that are not defined within the system but are allowed to be used in combination with units that are defined within the system. Allowed units must be commensurable with some defined unit of the system, so that quantities expressed in the allowed unit may be converted to a defined unit. The SI System explicitly allows a number of non-SI units.

Example Quantity Kind and Unit Systems

This section contains tables of several of the quantity kind and unit systems that are currently defined in the ontology. The table columns are:

Category – Either “Base᾿, “Derived᾿, or “TBD᾿
Quantity Kind – The name of the quantity kind
Quantity QName – The QName of the quantity kind
Dimension Symbol – The dimension symbol for the quantity kind. For derived quantity kinds, the symbol is a linear combination of the base quantity symbols, as described above.
Unit – The name of a unit in the unit system that is the defined unit for the quantity kind
Unit QName – The QName of the unit
Unit Symbol – A common symbol or abbreviation for the unit

The SI System

SI Base and Derived Quantities and Units

Category	Quantity Kind	QName	Dimension Symbol	Unit	QName	Unit Symbol
Base	Dimensionless	quantity:Dimensionless	U	Unity	unit:Unity	U
	Length	quantity:Length	L	Meter	unit:Meter	m
	Mass	quantity:Mass	M	Kilogram	unit:Kilogram	kg
	Time	quantity:Time	T	Second	unit:SecondTime	s
	Electric Current	quantity:ElectricCurrent	I	Ampere	unit:Ampere	A
	Temperature	quantity:ThermodynamicTemperature	Θ	Kelvin	unit:Kelvin	K
	Amount of Substance	quantity:AmountOfSubstance	N	Mole	unit:Mole	mol
	Luminous Intensity	quantity:LuminousIntensity	J	Candela	unit:Candela	cd
Derived	Absorbed Dose	quantity:AbsorbedDose	L²T^-2	Gray	unit:Gray	Gy
	Absorbed Dose Rate	quantity:AbsorbedDoseRate	L²T^-3	Gray per second	unit:GrayPerSecond	Gy/s
	Activity	quantity:Activity	T^-1	Becquerel	unit:Becquerel	Bq
	Amount of Substance Per Unit Volume	quantity:AmountOfSubstancePerUnitVolume	L^-3N¹	Mole per cubic meter	unit:MolePerCubicMeter	mol/m^3
	Amount of Substance per Unit Mass	quantity:AmountOfSubstancePerUnitMass	M^-3N¹	Mole per kilogram	unit:MolePerKilogram	mol/kg
	Angular Acceleration	quantity:AngularAcceleration	U¹T^-2	Radian per second squared	unit:RadianPerSecondSquared	rad/s^2
	Angular Mass	quantity:AngularMass	L²M¹	Kilogram Meter Squared	unit:KilogramMeterSquared	kg-m^2
	Angular Momentum	quantity:AngularMomentum	L²M¹T^-1	Joule Second	unit:JouleSecond	J s
	Angular Velocity	quantity:AngularVelocity	U¹T^-1	Radian per second	unit:RadianPerSecond	rad/s
	Area	quantity:Area	L²	Square meter	unit:SquareMeter	m^2
	Area Angle	quantity:AreaAngle	U¹L²	Square meter steradian	unit:SquareMeterSteradian	m^2-sr
	Area Temperature	quantity:AreaTemperature	L²Θ¹	Square meter kelvin	unit:SquareMeterKelvin	m^2-K
	Area Thermal Expansion	quantity:AreaThermalExpansion	L²Θ^-1	Square meter per kelvin	unit:SquareMeterPerKelvin	m^2/K
	Capacitance	quantity:Capacitance	L^-2M^-1T⁴I²	Farad	unit:Farad	F
	Catalytic Activity	quantity:CatalyticActivity	T^-1N¹	Katal	unit:Katal	kat
	Coefficient of Heat Transfer	quantity:CoefficientOfHeatTransfer	M¹T^-3Θ^-1	Watt per square meter kelvin	unit:WattPerSquareMeterKelvin	W/(m^2-K)
	Density	quantity:Density	L^-3M¹	Kilogram per cubic meter	unit:KilogramPerCubicMeter	kg/m^3
	Dose Equivalent	quantity:DoseEquivalent	L²T^-2	Sievert	unit:Sievert	Sv
	Dynamic Viscosity	quantity:DynamicViscosity	L^-1M¹T^-1	Pascal second	unit:PascalSecond	Pa-s
	Electric Charge	quantity:ElectricCharge	T¹I¹	Coulomb	unit:Coulomb	C
	Electric Charge Line Density	quantity:ElectricChargeLineDensity	L^-1T¹I¹	Coulomb per meter	unit:CoulombPerMeter	C/m
	Electric Charge Volume Density	quantity:ElectricChargeVolumeDensity	L^-3T¹I¹	Coulomb per cubic meter	unit:CoulombPerCubicMeter	C/m^3
	Electric Charge per Amount of Substance	quantity:ElectricChargePerAmountOfSubstance	T¹I¹N^-1	Coulomb per mole	unit:CoulombPerMole	C/mol
	Electric Current Density	quantity:ElectricCurrentDensity	L^-2I¹	Ampere per square meter	unit:AmperePerSquareMeter	A/m^2
	Electric Current per Angle	quantity:CurrentPerAngle	U^-1I¹	Ampere per radian	unit:AmperePerRadian	A/rad
	Electric Dipole Moment	quantity:ElectricDipoleMoment	L¹T¹I¹	Coulomb meter	unit:CoulombMeter	C-m
	Electric Field Strength	quantity:ElectricFieldStrength	L¹M¹T^-3I^-1	Volt per Meter	unit:VoltPerMeter	V/m
	Electric Flux Density	quantity:ElectricFluxDensity	L^-2T¹I¹	Coulomb per Square Meter	unit:CoulombPerSquareMeter	C/m^2
	Electrical Conductivity	quantity:ElectricalConductivity	L^-2M^-1T³I²	Siemens	unit:Siemens	S
	Electromotive Force	quantity:ElectromotiveForce	L²M¹T^-3I^-1	Volt	unit:Volt	V
	Energy Density	quantity:EnergyDensity	L^-1M¹T^-2	Joule per cubic meter	unit:JoulePerCubicMeter	J/m^3
	Energy and Work	quantity:EnergyAndWork	L²M¹T^-2	Joule	unit:Joule	J
	Energy per Unit Area	quantity:EnergyPerUnitArea	M¹T^-2	Joule per square meter	unit:JoulePerSquareMeter	J/m^2
	Exposure	quantity:Exposure	M^-1T¹I¹	Coulomb per kilogram	unit:CoulombPerKilogram	C/kg
	Force	quantity:Force	L¹M¹T^-2	Newton	unit:Newton	N
	Force per Electric Charge	quantity:ForcePerElectricCharge	L¹M¹T^-3I^-1	Newton per coulomb	unit:NewtonPerCoulomb	N/C
	Force per Unit Length	quantity:ForcePerUnitLength	M¹T^-2	Newton per meter	unit:NewtonPerMeter	N/m
	Frequency	quantity:Frequency	T^-1	Hertz	unit:Hertz	Hz
	Frequency	quantity:Frequency	T^-1	Inverse second time	unit:InverseSecondTime	s^-1
	Gravitational Attraction	quantity:GravitationalAttraction	L³M^-1T^-2	Cubic meter per kilogram second squared	unit:CubicMeterPerKilogramSecondSquared	m^3/(kg-s^2)
	Heat Capacity and Entropy	quantity:HeatCapacityAndEntropy	L²M¹T^-2Θ^-1	Joule per kelvin	unit:JoulePerKelvin	J/K
	Heat Flow Rate	quantity:HeatFlowRate	L²M¹T^-3	Watt	unit:Watt	W
	Heat Flow Rate per Unit Area	quantity:HeatFlowRatePerUnitArea	M¹T^-3	Watt per square meter	unit:WattPerSquareMeter	W/m^2
	Illuminance	quantity:Illuminance	U¹L^-2J¹	Lux	unit:Lux	lx
	Inductance	quantity:Inductance	L²M¹T^-2I^-2	Henry	unit:Henry	H
	Inverse Amount of Substance	quantity:InverseAmountOfSubstance	N^-1	Per mole	unit:PerMole	mol^(-1)
	Inverse Permittivity	quantity:InversePermittivity	L³M¹T^-4I^-2	Meter per farad	unit:MeterPerFarad	m/F
	Kinematic Viscosity	quantity:KinematicViscosity	L²T^-1	Square meter per second	unit:SquareMeterPerSecond	m^2/sec
	Length Mass	quantity:LengthMass	L¹M¹	Meter kilogram	unit:MeterKilogram	m-kg
	Length Temperature	quantity:LengthTemperature	L¹Θ¹	Meter kelvin	unit:MeterKelvin	m-K
	Linear Acceleration	quantity:LinearAcceleration	L¹T^-2	Meter per second squared	unit:MeterPerSecondSquared	m/s^2
	Linear Momentum	quantity:LinearMomentum	L¹M¹T^-1	Kilogram Meter Per Second	unit:KilogramMeterPerSecond	kg-m/s
	Linear Thermal Expansion	quantity:LinearThermalExpansion	L¹Θ^-1	Meter per kelvin	unit:MeterPerKelvin	m/K
	Linear Velocity	quantity:LinearVelocity	L¹T^-1	Meter per second	unit:MeterPerSecond	m/s
	Luminance	quantity:Luminance	L^-2J¹	Candela per square meter	unit:CandelaPerSquareMeter	cd/m^2
	Luminous Flux	quantity:LuminousFlux	U¹J¹	Lumen	unit:Lumen	lm
	Magnetic Dipole Moment	quantity:MagneticDipoleMoment	L²I¹	Joule per Tesla	unit:JoulePerTesla	J/T
	Magnetic Field Strength	quantity:MagneticFieldStrength	L^-1I¹	Ampere Turn per Meter	unit:AmpereTurnPerMeter	At/m
	Magnetic Field Strength	quantity:MagneticFieldStrength	L^-1I¹	Ampere per meter	unit:AmperePerMeter	A/m
	Magnetic Flux	quantity:MagneticFlux	L²M¹T^-2I^-1	Weber	unit:Weber	Wb
	Magnetic Flux Density	quantity:MagneticFluxDensity	M¹T^-2I^-1	Tesla	unit:Tesla	T
	Magnetomotive Force	quantity:MagnetomotiveForce	U¹I¹	Ampere Turn	unit:AmpereTurn	At
	Mass Temperature	quantity:MassTemperature	M¹Θ¹	Kilogram kelvin	unit:KilogramKelvin	kg-K
	Mass per Time	quantity:MassPerUnitTime	M¹T^-1	Kilogram per second	unit:KilogramPerSecond	kg/s
Mass per Unit Area	quantity:MassPerUnitArea	L^-2M¹	Kilogram per square meter	unit:KilogramPerSquareMeter	kg/m^2
Mass per Unit Length	quantity:MassPerUnitLength	L^-1M¹	Kilogram per meter	unit:KilogramPerMeter	kg/m
Molar Energy	quantity:MolarEnergy	L²M¹T^-2N^-1	Joule per mole	unit:JoulePerMole	J/mol
Molar Heat Capacity	quantity:MolarHeatCapacity	L²M¹T^-2Θ^-1N^-1	Joule per mole kelvin	unit:JoulePerMoleKelvin	J/(mol-K)
Permeability	quantity:Permeability	L¹M¹T^-2I^-2	Henry per meter	unit:HenryPerMeter	H/m
Permittivity	quantity:Permittivity	L^-3M^-1T⁴I²	Farad per meter	unit:FaradPerMeter	F/m
Plane Angle	quantity:PlaneAngle	U¹	Radian	unit:Radian	rad
Power	quantity:Power	L²M¹T^-3	Watt	unit:Watt	W
Power per Angle	quantity:PowerPerAngle	U^-1L²M¹T^-3	Watt per steradian	unit:WattPerSteradian	W/sr
Power per Area Angle	quantity:PowerPerAreaAngle	U^-1M¹T^-3	Watt per square meter steradian	unit:WattPerSquareMeterSteradian	W/(m^2-sr)
Power per Unit Area	quantity:PowerPerUnitArea	M¹T^-3	Watt per square meter	unit:WattPerSquareMeter	W/m^2
Pressure or Stress	quantity:PressureOrStress	L^-1M¹T^-2	Pascal	unit:Pascal	Pa
Resistance	quantity:Resistance	L²M¹T^-3I^-2	Ohm	unit:Ohm	Ohm
Solid Angle	quantity:SolidAngle	U¹	Steradian	unit:Steradian	sr
Specific Energy	quantity:SpecificEnergy	L²T^-2	Joule per kilogram	unit:JoulePerKilogram	J/kg
Specific Heat Capacity	quantity:SpecificHeatCapacity	L²T^-2Θ^-1	Joule per kilogram kelvin	unit:JoulePerKilogramKelvin	J/(kg-K)
Specific Heat Pressure	quantity:SpecificHeatPressure	L³M^-1Θ^-1	Joule per kilogram kelvin per pascal	unit:JoulePerKilogramKelvinPerPascal	J/(km-K-Pa)
Specific Heat Volume	quantity:SpecificHeatVolume	L^-1T^-2Θ^-1	Joule per kilogram kelvin per cubic meter	unit:JoulePerKilogramKelvinPerCubicMeter	J/(kg-K-m^3)
Temperature Amount of Substance	quantity:TemperatureAmountOfSubstance	Θ¹N¹	Mole kelvin	unit:MoleKelvin	mol-K
Thermal Conductivity	quantity:ThermalConductivity	L¹M¹T^-3Θ^-1	Watt per meter kelvin	unit:WattPerMeterKelvin	W/(m*K)
Thermal Diffusivity	quantity:ThermalDiffusivity	L²T^-1	Square meter per second	unit:SquareMeterPerSecond	m^2/sec
Thermal Insulance	quantity:ThermalInsulance	M^-1T³Θ¹	Square meter Kelvin per watt	unit:SquareMeterKelvinPerWatt	(K^2)m/W
Thermal Resistance	quantity:ThermalResistance	L^-2M^-1T³Θ¹	Kelvin per watt	unit:KelvinPerWatt	K/W
Thermal Resistivity	quantity:ThermalResistivity	L^-1M^-1T³Θ¹	Meter Kelvin per watt	unit:MeterKelvinPerWatt	K-m/W
Thrust to Mass Ratio	quantity:ThrustToMassRatio	L¹T^-2	Newton per kilogram	unit:NewtonPerKilogram	N/kg
Time Squared	quantity:TimeSquared	T²	Second time squared	unit:SecondTimeSquared	s^2
Torque	quantity:BendingMomentOrTorque	L²M¹T^-2	Newton meter	unit:NewtonMeter	N-m
Volume	quantity:Volume	L³	Cubic Meter	unit:CubicMeter	m^3
Volume Thermal Expansion	quantity:VolumeThermalExpansion	L³Θ^-1	Cubic meter per kelvin	unit:CubicMeterPerKelvin	m^3/K
Volume per Unit Time	quantity:VolumePerUnitTime	L³T^-1	Cubic meter per second	unit:CubicMeterPerSecond	m^3/s
Volumetric heat capacity	quantity:VolumetricHeatCapacity	L^-1M¹T^-2Θ^-1	Joule per cubic meter kelvin	unit:JoulePerCubicMeterKelvin	J/(m^3 K)

The CGS System

CGS Base and Derived Quantity Kinds and Units

Category	Quantity Kind	QName	Dimension Symbol	Unit	QName	Unit Symbol
Base	Dimensionless	quantity:Dimensionless	U	Unity	unit:Unity	U
	Length	quantity:Length	L	Centimeter	unit:Centimeter	cm
	Mass	quantity:Mass	M	Gram	unit:Gram	g
	Time	quantity:Time	T	Second	unit:SecondTime	s
Derived	Angular Momentum	quantity:AngularMomentum	L²M¹T^-1	Erg second	unit:ErgSecond	erg s
	Area	quantity:Area	L²	Square centimeter	unit:SquareCentimeter	cm^2
	Dynamic Viscosity	quantity:DynamicViscosity	L^-1M¹T^-1	Poise	unit:Poise	P
	Energy Density	quantity:EnergyDensity	L^-1M¹T^-2	Erg per cubic centimeter	unit:ErgPerCubicCentimeter	erg/cm^3
	Energy and Work	quantity:EnergyAndWork	L²M¹T^-2	Erg	unit:Erg	erg
	Force	quantity:Force	L¹M¹T^-2	Dyne	unit:Dyne	dyn
	Frequency	quantity:Frequency	T^-1	Inverse second time	unit:InverseSecondTime	s^-1
	Linear Acceleration	quantity:LinearAcceleration	L¹T^-2	Centimeter per second squared	unit:CentimeterPerSecondSquared	cm/s^2
	Linear Velocity	quantity:LinearVelocity	L¹T^-1	Centimeter per second	unit:CentimeterPerSecond	cm/s
	Power	quantity:Power	L²M¹T^-3	Erg per second	unit:ErgPerSecond	erg/s
	Power per Unit Area	quantity:PowerPerUnitArea	M¹T^-3	Erg per square centimeter second	unit:ErgPerSquareCentimeterSecond	erg/(cm^2-s)
	Pressure or Stress	quantity:PressureOrStress	L^-1M¹T^-2	Dyne per square centimeter	unit:DynePerSquareCentimeter	dyn/cm^2
	Time Area	quantity:TimeArea	L²T¹	Square centimeter second	unit:SquareCentimeterSecond	cm^2-s
	Torque	quantity:BendingMomentOrTorque	L²M¹T^-2	Dyne centimeter	unit:DyneCentimeter	dyn-cm
	Volume	quantity:Volume	L³	Cubic Centimeter	unit:CubicCentimeter	cm^3

CGS Units for Electricity and Magnetism

There are two different approaches to defining electric and magnetic quantities using the base CGS mechanical quantities of length, mass and time. The Electromagnetic Unit (EMU) approach derives electric charge from Coulomb’s Law, while the Electrostatic Unit (ESU) approach derives electric charge from Ampere’s Law.

EMU Derived Units

Coulomb’s Law states that the force exerted between two charged particles, q₁ and q₂, is inversely proportional to the square of their distance, r.

F=k(q₁q₂)/r²

Retaining only the terms of the quantity kinds involved (force, electric charge, distance), this equation can be rearranged to express electric charge as length multiplied by the square root of force. The CGS Electromagnetic Unit is called the Abcoulomb. The table below contains the dimension symbols and corresponding units of other electricity and magnetism quantity kinds in terms of the base CGS quantity kinds and the definition of electric charge above.

CGS EMU Derived Units for Electricity and Magnetism

Quantity Kind	QName	Dimension Symbol	Unit	QName	Unit Symbol
Capacitance	quantity:Capacitance	L^-1T²	Abfarad	unit:Abfarad	abF
Electric Charge	quantity:ElectricCharge	L^0.5M^0.5	Abcoulomb	unit:Abcoulomb	abC
Electric Current	quantity:ElectricCurrent	L^0.5M^0.5T^-1	Abampere	unit:Abampere	abA
Electric Field Strength	quantity:ElectricFieldStrength	L^0.5M^0.5T^-2	Abvolt per Centimeter	unit:AbvoltPerCentimeter	abV/cm
Electric Flux Density	quantity:ElectricFluxDensity	L^-1.5M^0.5	Abcoulomb per square centimeter	unit:AbcoulombPerSquareCentimeter	abC/cm^2
Electrical Conductivity	quantity:ElectricalConductivity	L^-1T¹	Absiemen	unit:Absiemen	aS
Electromotive Force	quantity:ElectromotiveForce	L^1.5M^0.5T^-2	Abvolt	unit:Abvolt	abV
Inductance	quantity:Inductance	L¹	Abhenry	unit:Abhenry	abH
Magnetic Field Strength	quantity:MagneticFieldStrength	L^-0.5M^0.5T^-1	Abtesla	unit:Abtesla	abT
Magnetic Flux	quantity:MagneticFlux	L^1.5M^0.5T^-1	Abvolt Second	unit:AbvoltSecond	abV-s
Magnetic Flux Density	quantity:MagneticFluxDensity	L^-0.5M^0.5T^-1	Abtesla	unit:Abtesla	abT
Magnetomotive Force	quantity:MagnetomotiveForce	L^0.5M^0.5T^-1	Gilbert	unit:Gilbert	Gi
Permeability	quantity:Permeability	U¹	Relative permeability	unit:RelativePermeability	μ _r
Permittivity	quantity:Permittivity	L^-2T²	Abfarad per centimeter	unit:AbfaradPerCentimeter	abF/cm
Resistance	quantity:Resistance	L¹T^-1	Abohm	unit:Abohm	abOhm

ESU Derived Units

Ampere’s Law of Magnetic Induction states that the force per unit length exerted between two infinite parallel wires at a distince, d, and carrying electric currents I₁ and I₂ is proportional to their product divided by the distance between them. I.e.

dF/dl = k(I₁I₂/d)

Retaining only the terms of the quantity kinds involved (force, electric current, distance), this equation can be rearranged to express electric current as the square root of force. The CGS Electrostatic Unit of electric current is called the Statampere. The table below contains the dimension symbols and corresponding units of other electricity and magnetism quantity kinds in terms of the base CGS quantity kinds and the definition of electric current above.

CGS ESU Derived Units for Electricity and Magnetism

Quantity Kind	QName	Dimension Symbol	Unit	QName	Unit Symbol
Capacitance	quantity:Capacitance	L¹	Statfarad	unit:Statfarad	statF
Electric Charge	quantity:ElectricCharge	L^1.5M^0.5T^-1	Statcoulomb	unit:Statcoulomb	statC
Electric Current	quantity:ElectricCurrent	L^1.5M^0.5T^-2	Statampere	unit:Statampere	statA
Electric Field Strength	quantity:ElectricFieldStrength	L^-0.5M^0.5T^-1	Statvolt per centimeter	unit:StatvoltPerCentimeter	statV/cm
Electric Flux Density	quantity:ElectricFluxDensity	L^-0.5M^0.5T^-1	Statcoulomb per square centimeter	unit:StatcoulombPerSquareCentimeter	statC/cm^2
Electromotive Force	quantity:ElectromotiveForce	L^0.5M^0.5T^-1	Statvolt	unit:Statvolt	statV
Inductance	quantity:Inductance	L^-1T²	Stathenry	unit:Stathenry	statH
Magnetic Field Strength	quantity:MagneticFieldStrength	L^0.5M^0.5T^-2	Oersted	unit:Oersted	Oe
Magnetic Flux	quantity:MagneticFlux	L^0.5M^0.5	Maxwell	unit:Maxwell	Mx
Magnetic Flux Density	quantity:MagneticFluxDensity	L^-1.5M^0.5	Gauss	unit:Gauss	G
Magnetomotive Force	quantity:MagnetomotiveForce	L^1.5M^0.5T^-2	Oersted centimeter	unit:OerstedCentimeter	Oe-cm
Permeability	quantity:Permeability	L^-2T²	Stathenry per centimeter	unit:StathenryPerCentimeter	statH/cm
Permittivity	quantity:Permittivity	U¹	Relative permittivity	unit:RelativePermittivity	ε _r
Resistance	quantity:Resistance	L^-1T¹	Statohm	unit:Statohm	statOhm

Glossary

NExIOM: NASA Exploration Intiatives Ontology Models
TBD: To Be Done
TBR: To Be Revised

Acknowledgements

[TBR]

NASA AMES Research Center for sponsoring and content for different engineering disciplines
TopQuadrant, Inc., for Ontology Architecture, foundation ontologies and tooling support
European Space Agency (ESA), for constructive dialog and input to the ontology models

References

The NIST Guide for the use of the International System of Units
International Vocabulary of Metrology – Basic and General Concepts and Associated Terms
SI Brochure, 8th Edition
Dimensional Analysis, Percy Williams Bridgman, Yale University Press (1922)

Last Updated October 30, 2009

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