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# International System derived units

Derived units are formed from multiplication and division of the seven base units. For ease of understanding and convenience, twenty-two derived units have been given special names and symbols in the International System of Units (SI) 1, 2.

Names, symbols, and dimensions for SI derived units
derived quantity unit name unit symbol unit expansion
in practice, the symbols $$\scriptstyle \mathrm{rad}$$ and $$\scriptstyle \mathrm{sr}$$ may be used when appropriate and may be omitted –since they are of dimension $$\scriptstyle \mathrm{1}$$– if clarity is not lost
†† the Celsius temperature $$\scriptstyle \mathrm{\theta}$$ is defined by $$\scriptstyle \theta/\mathrm{ºC} \equiv (T/\mathrm{K}) - 273.15$$. Note that $$\scriptstyle \mathrm{ºC}$$ is treated as a single symbol, with no space between $$\scriptstyle \mathrm{º}$$ and $$\scriptstyle \mathrm{C}$$
††† although the becquerel is the preferred unit to be used in nuclear and radioscience; the units gray, sievert, and katal are admitted for reasons of safeguarding human health
plane angle radian $$\mathrm{rad}$$ $$\mathrm{m \; m^{-1} = 1}$$
solid angle steradian $$\mathrm{sr}$$ $$\mathrm{m^2 m^{-2} = 1}$$
frequency hertz $$\mathrm{Hz}$$ $$\mathrm{s^{-1}}$$
force newton $$\mathrm{N}$$ $$\mathrm{m \; kg \; s^{-2}}$$
pressure, stress pascal $$\mathrm{Pa}$$ $$\mathrm{N \; m^{-2} = m^{-1} kg \; s^{-2}}$$
energy, work, heat joule $$\mathrm{J}$$ $$\mathrm{N m = m^2 kg \; s^{-2}}$$
power, radiant flux watt $$\mathrm{W}$$ $$\mathrm{J \; s^{-1} = m^2 kg \; s^{-3}}$$
electric charge coulomb $$\mathrm{C}$$ $$\mathrm{A \; s}$$
electric potential, electromotive force, electric tension volt $$\mathrm{V}$$ $$\mathrm{J \; C^{-1} = m^2 kg \; s^{-3} A^{-1}}$$
electric resistance ohm $$\mathrm{\Omega}$$ $$\mathrm{V A^{-1} = m^2 kg \; s^{-3} A^{-2}}$$
electric conductance siemens $$\mathrm{S}$$ $$\mathrm{\Omega^{-1} = m^{-2} kg^{-1} s^3 A^2}$$
electric capacitance farad $$\mathrm{F}$$ $$\mathrm{C \; V^{-1} = m^{-2} kg^{-1} s^4 A^2}$$
magnetic flux weber $$\mathrm{Wb}$$ $$\mathrm{V \; s = m^2 kg \; s^{-2} A^{-1}}$$
magnetic flux density tesla $$\mathrm{T}$$ $$\mathrm{Wb \; m^{-2} = kg \; s^{-2} A^{-1}}$$
inductance henry $$\mathrm{H}$$ $$\mathrm{V \; A^{-1} s = m^2 kg \; s^{-2} A^{-2}}$$
Celsius temperature†† degree celsius $$\mathrm{°C}$$
luminous flux lumen $$\mathrm{lm}$$ $$\mathrm{cd \; sr = cd}$$
illuminance lux $$\mathrm{lx}$$ $$\mathrm{lm \; m^{-2} = cd \; m^{-2}}$$
radioactivity, activity of a radionuclide††† becquerel $$\mathrm{Bq}$$ $$\mathrm{s^{-1}}$$
absorbed dose, kerma††† gray $$\mathrm{Gy}$$ $$\mathrm{J kg^{-1} = m^2 s^{-2}}$$
dose equivalent††† sievert $$\mathrm{Sv}$$ $$\mathrm{J kg^{-1} = m^2 s^{-2}}$$
catalytic activity††† katal $$\mathrm{kat}$$ $$\mathrm{mol \; s^{-1}}$$

The values of several different quantities may be expressed using the same name and symbol for the SI unit. Thus for the quantity heat capacity as well as the quantity entropy, the SI unit is the joule per kelvin. Similarly for the base quantity electric current as well as the derived quantity magnetomotive force, the SI unit is the ampere. It is therefore important not to use the unit alone to specify the quantity. This applies not only to scientific and technical texts, but also, for example, to measuring instruments –i.e. an instrument read-out should indicate both the unit and the quantity measured– 3.

A derived unit can often be expressed in different ways by combining base units with derived units having special names. The joule, for example, may formally be written newton metre, or kilogram metre squared per second squared. This, however, is an algebraic freedom to be governed by common sense physical considerations; in a given situation some forms may be more helpful than others.

In practice, with certain quantities, preference is given to the use of certain special unit names, or combinations of unit names, to facilitate the distinction between different quantities having the same dimension. When using this freedom, one may recall the process by which the quantity is defined. For example, the quantity torque may be thought of as the cross product of force and distance, suggesting the unit newton metre, or it may be thought of as energy per angle, suggesting the unit joule per radian. The SI unit of frequency is given as the hertz, implying the unit cycles per second; the SI unit of angular velocity is given as the radian per second; and the SI unit of activity is designated the becquerel, implying the unit counts per second. Although it would be formally correct to write all three of these units as the reciprocal second, the use of the different names emphasizes the different nature of the quantities concerned. Using the unit radian per second for angular velocity, and hertz for frequency, also emphasizes that the numerical value of the angular velocity in radian per second is $$2\pi$$ times the numerical value of the corresponding frequency in hertz.

In the field of ionizing radiation, the SI unit of activity is designated the becquerel rather than the reciprocal second, and the SI units of absorbed dose and dose equivalent are designated the gray and the sievert, respectively, rather than the joule per kilogram. The special names becquerel, gray, and sievert were specifically introduced because of the dangers to human health that might arise from mistakes involving the units reciprocal second and joule per kilogram, in case the latter units were incorrectly taken to identify the different quantities involved 3.

The physical quantity heat is renamed to quantity of heat by the NIST 4 without any reason. Notice that the names of units in honor of scientists are written in small caps –newton, pascal, joule, etcetera–, but their symbols do not –$$\mathrm{N}$$, $$\mathrm{Pa}$$, $$\mathrm{J}$$, etcetera–.

Some authors equal the degree celsius $$\mathrm{°C}$$ to the kelvin $$\mathrm{K}$$ when give tables of derived units. This is misleading because $$21 \;\mathrm{ºC} \neq 21 \;\mathrm{K}$$, whereas the rest of entries on the above table represent true equalities; for instance, for energy $$0.53 \;\mathrm{J} = 0.53 \;\mathrm{m^2 kg \; s^{-2}}$$ and for catalytic activity $$7587 \;\mathrm{kat} = 7587 \;\mathrm{mol \; s^{-1}}$$.

## References and notes

1. Quantities, Units and Symbols in Physical Chemistry, IUPAC Green Book, 3rd Ed., 2nd Printing 2008: IUPAC & RSC Publishing; Cambridge.
2. International Vocabulary of Metrology, Basic and General Concepts and Associated Terms (VIM), 3rd ed. 2008: Joint Committee for Guides in Metrology (JCGM). .
3. BIPM - special names 2012 November 19 (access): http://www.bipm.org/en/si/derived_units/2-2-2.html. .
4. Essentials of the SI: Base & derived units 2012 November 16 (access): http://physics.nist.gov/cuu/Units/units.html. .

Date: 2013 January 03, 20:23:44+01:00
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