Miscellaneous‎ > ‎

Dimensional Units

Reality can be described (or defined) in terms of dimension units.  There are several ways to do this.  The most common method (used on this page) is to define a set of base (fundamental) units, and then describe everything else in terms of derived units.

Farther below is a large table of units (both derived and fundamental), but lets start with the Table of Fundamental Units:

Dimension Unit Description
Symbol Name Symbol Name
T Time s second 9192631770 cycles of caesium-133 (133Ce) groundstate hyperfine splitting frequency (at 0 K)
[1/86400 = 1/(24*60*60)] of an Earth day
L Length m meter Distance traveled by light (in vacuum) in 1/299792458 of a second
M Mass kg kilogram 1 liter (1 cubic decimeter = 1000 cubic centimeters) of pure water (H2O) at sea-level-pressure and at 4°C
Q Electric
Charge
C coulomb charge of 6.24150913 • 1018 protons
negative charge of 6.24150913 • 1018 electrons
Θ Temperature   Kelvin K 1/273.16 of the triple-point of water

This page considers electric charge (Q) as a fundamental unit, in contrast to the SI (a.k.a, "metric system") which considers electric current (I) as the fundamental unit. In other words, the SI considers electric charge to be a derived unit, while this page makes it a fundamental unit.

The SI considers the candella (cd) as the fundamental unit of luminous intensity. It is really just radiant intensity at a specific frequency (540 THz). Since all known light sources, except lasers, produce multiple frequencies, some form of weighting function, specific to human vision, should be employed. The SI doesn't define one, and several methods have been published, so this unit is quite murky and not considered as fundamental on this page.

The SI also considers the mole (mol) to be a dimensional unit. However, it is a pure (dimensionless) number, like a dozen or a hundred; it is a rather large number, approximately 6.0221408×1023. You may find it listed occasionally in the table below under the column of SI Units, but if you look at the corresponding Dimension Symbol, you should see the mole has no effect on the physical dimension.

Similarly, both plane angles and solid angles have no real physical dimension, but for a different reason. The way these are defined, the dimension of length (L) gets cancelled out. Thus a plane angle is listed with a length dimension of L1-1 because it is a length (L1) divided by a length, and a solid angle is listed as L2-2 because it is an area (L2) divided by an area.

In case you didn't already know and the above didn't make it clear, negative exponents in the dimensional symbol refers to division by that dimension. For example, meters per second (or miles per hour, etc.) has dimensional symbol L1T-1 because it is length divided by time (notice length has a positive exponent, while time has a negative exponent).

Usually, dimensions which do not appear in a unit of measurement are simply omitted. For consistency, I have included all the base dimensions described above for each unit in the table below. Those dimensions which are not present have an exponent of zero. For example, length is listed as L1T0M0Q0Θ0.

 
 
3 Types of Symbols  

One thing that confuses many beginning students of algebra and physics is that the same symbol can have very different meanings in different contexts.  Even professionals sometimes misunderstand the meaning of a symbol.  The main reason for this is there are many more concepts than there are letters of the Roman and Greek alphabets.  To prevent confusion (as much as possible), I will tell you this page has three (3) different types of symbols:

  • Dimensional Symbols (1st column below) are described in the table above; briefly: L=Length, T=Time, M=Mass, Q=Charge, Θ=Temperature
  • SI Unit Symbols (3rd column below) are defined by international standard; examples include: L=liter, T=Tesla or Tera, M=Mega, m=meter or milli, s = second, and many more!
  • Math Symbols (5th column below) are commonly used in engineering and mathematical formulas (there is no official standard for all, but a few are defined by SI or ISO)

In general, all symbols should be considered case-sensitive.  In particular, the second type (SI Unit Symbols) can be confusing because the same symbol (for example m) could mean two different things in the same (SI Unit) context.  Fortunately for you, I have included the corresponding name of the SI Symbol in the 4th column; this should resolve any ambiguity. 

The final Description column (in the table below) describes a dimensional unit in English text and often with SI Unit Symbols.  For example, a Watt-Hour may be described as "3600 J" which means "3600 joules" because J is the SI symbol for joule.  However, it is often informative to have a description in terms of engineering/math symbols.  When math symbols appear in the Description column, they will be enclosed in square brackets; for example, heat capacity (entropy) may be described as energy / temperature [E/T].  Note the E and T shown in square brackets are math symbols (not dimensional symbols nor SI symbols).  In summary, the description column uses English text with SI symbols in general, but also includes math symbols in square brackets for extra information.

 
 
Colored Units  

Under "SI Units" (columns 3 and 4 in the table below), most entries are "official" (explicitly listed or implicitly endorsed) by the SI system; these have a normal (white) background color.  Some of them are "recognized" or considered "acceptable" with the SI system; these have a yellow background (the most popular is the liter).  A few are not recognized (perhaps even forbidden) by the SI system, but are shown here because they are used by people anyway (for example kilowatt-hour, kW•hr), and some of these have no official SI unit; these have a light-red background.

To keep this page relatively short, I have included only SI Units (as much as possible).  There are other metric but non-SI units in use (such as currie, dyne, and cubic centimeter), and many other non-metric units in common use (such as miles and pounds).  These would also be shown with a light-red background if included (but that would greatly multiply the size of this page).

 
 
Extra Subscripts  

Several of the "Math Symbols" (5th column below) may include a subscript letter(s) to be consistent and precise.  Some of these will rarely be seen in other documentation, because of their limited scope; few papers span all possible dimensional units!  A good example is Einstein's famous equation, which is most often written: E = mc2.  The context of most documents which use that form implies that E is mass-energy (and not kinetic energy or something else), and c is light-speed (and not some other speed, like the speed of sound).  On this page, the same equation would be written with subscripts: EM = mc02.  The symbol EM is for mass-energy, and c0 is for light-speed.

Some subscripts are generic (often "i" and "n").  For example, the math symbol for molar concentration is ci when referring to an unknown, but might written as cH when referring to Hydrogen or cAr when referring to Argon.


Before we dive into the details, it is important to note that other dimensional units are possible.  Here are three good examples:
  • Time could be defined in terms of length (or vice versa), according to Einstein/Lorentz relativity
  • Mass could be defined in terms of length, according to Schwartzchild radius
  • Electric charge could be defined in terms of the square root of mass times length (√M•L)
Anyway, below is a list of dimensional units, sorted in the order of (first) Length, Time, Mass, Charge, and (finally) Temperature...

Dimension SI Unit Math
Symbol(s)
Description
Symbol Name(s) Symbol Name
L-3T-1M0Q0Θ0 Catalytic
Concentration
kat/m3 katals per cubic meter ? catalytic activity / unit volume [z/V]; 1 mkat/L
L-3T0M0Q0Θ0 Molarity,
[Molar]
Concentration
mol/m3 moles per cubic meter ci moles / unit volume [nm/V]; 1 mmol/L
mol/L, M moles per liter moles / liter [nm/V]; reciprocal of molar volume [1/Vm]; 1 kmol/m3
Number
Concentration
m-3 per cubic meter Ci number of particles per unit volume [n/V]
Acidity,Basicity * power of Hydrogen pH, PH approximately, negative decimal logarithm of hydrogen molar concentration [-log10 cH];
pure H2O has pH = 7.0; acids are less (lemon pH ≈ 2.2), bases are more (bleach pH ≈ 13.0)
L-3T0M0Q1Θ0 Charge Density C/m3 coulombs per cubic meter ρq electric charge / volume [Q/V]
L-3T0M1Q0Θ0 [Mass] Density kg/m3 kilograms per cubic meter ρ mass / unit volume [m/V]; 1 kg/m3 = 0.0010 kg/L = 1 g/L
L-3T1M1Q0Θ0 Toxic Exposure s kg/m3 second-kilograms
per cubic meter
E mass density * time period [ρ•t]; may be multiplied by "toxicity factor" for the type of toxin
L-2-1T1M-1Q2Θ0 Conductivity,
Admittivity,
Susceptivity
S/m siemans per meter κ, ?, ? conductance * distance / area [C•r/A] = reciprocal of resistivity [1/ρ]; 1 S/m = 1 A/(V m)
L-3T2M-1Q2Θ0 [Electric]
Permittivity
F/m ferads per meter ε capacitance / distance [C/r]
vacuum permittivity ε0 [1/c02μ0]; ≈ 8.854187817 pF/m
L-2T-2M1-1Q0Θ-1 Specific
Heat Capacity
J/(K kg) joules per kelvin
per kilogram
h heat capacity / mass [H/m] = energy / temperature / mass [E/(T•m)]
L-2T-2M1Q0Θ0 Force Density,
Specific Weight
N/m3 newtons per cubic meter f force / volume [F/V] = acceleration * density [a•ρ] = standard gravity * density [g•ρ]
L-2T-1M0Q1Θ0 Current Density A/m2 amperes
per square meter
J current / directed area [I/S] = charge density * velocity [ρqv];
electric field * conductivity [E•κ]
L-2T-1M1Q0Θ0 Momentum
Density
N s/m3 newton seconds
per cubic meter
g momentum / volume [p/V]; permittivity * electric field * magnetic field (ε•E×B
Advective Flux ? volumetric flux * density [q•ρ] = velocity * density [v•ρ]
Mass Flux kg Hz/m2 kilogram hertz
per square meter
Jm mass flow rate / directed area [ṁ/S] = momentum / volume [p/V]
L-2T0M-1Q2Θ0 Reluctance H-1 per Henry R current / magnetic flux [I/ΦB] = reciprocal of permeance [1/P]
L-2T0M0Q1Θ0 Polarization
Density
C/m2 coulombs
per square meter
Pe electric dipole / volume [δp/V];
vacuum permittivity * electric susceptibility * electric field [ε0χeE]
Electric
Displacement
D vacuum permittivity * electric field + polarization density [ε0E+Pe = ε(1+χe)E]
L-2T1M-1Q2Θ0 Conductance,
Admittance,
Subsceptance
S Sieman G, Y, B reciprocal ohm [1/R] = 1/Ω; current / voltage [I/V]
Magnetic
Capacitance
Ω-1 per ohms xC reciprocal of (angular frequency * magnetic capacity) [1/(ω•CM)]
L-2T2M-1Q0Θ0 Coldness nat/J natural entropy
per joule
βT thermodynamic beta = change in (entropy / energy) / Boltzman's constant [δH/(δE•kB)];
1 nat/J ≈ 1.442969504089 Sh/J
L-2T2M-1Q1Θ0 Charge Affinity C/J coulombs per joule εp reciprocal voltage [1/V] = charge / energy [Q/E] = current / power [I/P]
L-2T2M-1Q2Θ0 Capacitance F ferad C charge / electric potential [Q/V]; reciprocal of elastance [1/P]; 1 F = 1 J/V2
C2/J square coulombs
per joule
square charge / energy [Q2/E]; 1 C2/J = 1 F
L1-2T-2M1Q0Θ-1 Volumetric
Heat Capacity
J/(K m3) joules per kelvin
per cubic meter
VHC heat capacity / volume [H/V] = molar heat capacity / molar volume [Hm/Vm]
= energy / temperature / volume [E/(T•V)]
N/(m2K) newtons per
square meter per kelvin
force / area / temperature [|F|/(A•T)] = pressure / temperature [|P|/T]; 1 J/(Km3)
Entropy Density nat/m3 nats per cubic meter s entropy / volume [H/V]; 1 nat/m3 ≈ 13.80648 yJ/(Km3)
L1-2T-2M1Q0Θ0 Pressure,
Stress,
Energy Density
Pa Pascal P force / area [F/A]
N/m2 newton per square meter
J/m3 joules per cubic meter ρE energy / volume [E/V]
L1-2T-1M0Q1Θ0 Magnetic Field N/Wb newtons per weber H force / electric potential / time [F/(V•t)]; 1 A/m; 1 T•m/H
T•m/H tesla meters per second magnetic flux density / permeability [B/μ]; 1 A/m; 1 N/Wb
Magnetization A/m amperes per meter M magnetic dipole moment / volume [μ/V] = magnetic susceptibility * magnetic field [χmH]
L-1T-1M1Q0Θ0 [Dynamic|Shear]
Viscosity
kg/(m s) kilograms per meter
per second
η, μ mass / absition [m/|A|]; 1 Pa•s
L1-2T1-2M1Q0Θ0 N s/m2 newton second
per square meter
momentum / area [|p|/A]; 1 Pa•s
PI poiseuille pressure * time [|P|•t] = reciprocal of fluidity [1/φ]; 1 Pa•s
L1-2T0M-1Q2Θ0 Reluctivity m/H meters per henry ? reciprocal permeability [1/μ] = conductance * speed [G•|v|]; 1 S•m/s
L-1T0M0Q0Θ0 Wavenumber m-1 per meter ν cycles / distance [n/r] = reciprocal wavelength [1/λ] = frequency / phase velocity [f/|vφ| = ν/|vφ|]
Pixel Density PPI, DPI pixels [dots] per inch ?
Usually, a color pixel consists of 3 dots, so DPI and PPI are different for color images.
Optical Power diop, dpt diopter φ reciprocal of focal-length [1/|r| = 1/f]; 1 m-1
L1-2T0M0Q0Θ0 Angular
Wavenumber
rad/m radians per meter k 2π / wavelength [2π/λ] = angular velocity / phase velocity [ω/|vφ|]; for a photon, k = 2πE/(hc0)
L-1T2M-1Q0Θ0 Fuel Economy m/J meters per joule FE "energy efficiency in transportation"; distance / energy [r/E] = reciprocal force [1/|F|]; 1 N-1;
1 m/J = 3600 m/(W•hr) ≈ 753.847530422 mi/galUS
L-1T2M-1Q1Θ0 Electric Strain C/N coulombs per newton d charge / force [Q/F] = reciprocal electric field [1/E]
m/V meters per volt distance / electric potential [r/V]; 1 m/V = 1 C/N
L1-1T-3M0Q0Θ0 Angular Jerk rad/s3 radians per cubic second ζ angular acceleration / time period [α/t]
L2-2T-3M1Q0Θ0 Luminance W/m2 watts per square meter ? radiant exposure / time [He/t]
cd/m2 candella per
square meter
Lv luminous flux / solid angle / directed area [Φv/(Ω•S)] = luminous intensity / directed area [Iv/S]
= 1 W/m2 @ 540 THz
Illuminance lx lux Ev luminous flux / area [Φv/A]; 1 lm/m2 ≈ 1.46413 mW/m2 @ 540 THz
Irradiance W/m2 watts per square meter Ee radiant flux / area [Φe/A]
Radiance W/(sr m2) watts per steradian
per square meter
Le radiant flux / solid angle / directed area [Φ/Ω/S] = radiant intensity / directed area [Ie/S]
L1-1T-2M0Q0Θ0 Angular
Acceleration
rad/s2 radians per
square second
α angular velocity / time period [ω/t]
L2-2T-2M1-1Q0Θ0 Specific Radiant
Exposure
J/(m2kg) joules per square meter
per kilogram
? radiant exposure / mass [He/m] = force / leverage [|F|/|Λ|] = square frequency [f2 = ν2];
1 J/(m2kg) = 1 N/(m•kg) = 1 Hz2
L2-2T-2M1Q0Θ0 Radiant
Exposure
J/m2 joules per square meter He irradiance * time [Eet] = radiant energy / area [Qe/A]
Luminous
Exposure
lx s lux-second Hv illuminance * time [Evt] = luminous energy / area [Qv/A];
1 lx•s ≈ 1.46413 mJ/m2 @ 540 THz
Spectral
Irradianceν
W/(m2Hz) watts per square meter
per hertz
Ee,v irradiance / frequency [Ee/ν] = radiance * solid angle / frequency [LeΩ/ν]
Spectral
Radianceν
W/
(sr m2Hz)
watts per steradian per
square meter per hertz
Le,v radiance / frequency [Le/ν] = radiant flux / solid angle / area / frequency [Φe/(Ω•S•ν)]
L0T-1M-1Q0Θ0 Specific
Activity
Bq/kg bacquerel per kilogram a decay constant * Avagrado's number / molar mass [λ•NA/mm] = activity / mass [A/m];
natural log 2 / half-life * Avagrado's number / molar mass [(NAln 2)/(t1/2mm)]; 1 Hz/kg
(s kg)-1 per second per kilogram reciprocal of mass exposure [1/?]; 1 (s kg)-1 = 1 Hz/kg = 1 Bq/kg
L0T-1M0Q0Θ0 Frequency Hz hertz f, ν cycles per second [n/t]; 1 s-1
[Radio]Activity Bq becquerel A nuclear decays per time period [n/t]; 1 Hz
Catalytic Activity kat katal
z quantity of substance that converts another substance at 1 mole / second
L2-2T-1M1-1Q0Θ0 Fluid
Exchange Rate
Pa/PI pascals per poiseuille
? force / viscosity / area [|F|/(η•A)] = pressure / viscosity [|P|/η]; 1 Hz
ACH
air changes per hour fluid flow rate / volume [Q/V]; 277.77 μHz
L1-1T-1M0Q0Θ0 Angular Speed rad/s radians per second ω 2π * frequency [2πf = 2πν] = tangential speed / radius [|v|/r]; 1 rad/s ≈ 57.2957795 °/s
Angular Velocity ω (r × v) / r2; the omega vector is perpendicular to both r and v; 1 rad/s ≈ 159.154943 mHz
L0T-1M0Q1Θ0 [Electric] Current A ampere I electric charge / time [Q/t] = voltage / resistance [V/R]; 1 A = 1 C/s = V/Ω
L2-2T-1M1Q-1Θ0 Magnetic Flux
Density
T tesla B magnetic flux / directed area [ΦB/S]; 1 T = 1 Wb/m2
Wb/m2 webers per square meter
L0T-1M1Q0Θ0 Mass Flow Rate,
Mass Current
kg/s kilograms per second mass / time [m/t] = density * fluid flow rate [ρ•|Q|];
density * volumetric flux * directed area [ρ•qS]; 1 kg/s = 1 kg•Hz
L0T0M-1Q1Θ0 Ionizing
Exposure,
Specific Charge
C/kg coulombs per kilogram γ charge / mass [Q/m]
Gyromagnetic
Ratio
e-/me electron
charge-to-mass ratio
γe ≈ -175.88200 GC/kg
p+/mp proton
charge-to-mass ratio
γp ≈ +95.788332 MC/kg
L3-3T0M0Q0Θ-1 Coefficient of
Thermal
Expansion
m/(mK)
m2/(m2K)
m3/(m3K)
dim meters per
dim meter per kelvin
αL, αA, αV change in (length / temperature) / length [(δr/δT)/r]
change in (area / temperature) / area [(δA/δT)/A]
change in (volume / temperature) / volume [(δV/δT)/V]
Coldness 1/K reciprocal kelvin βT reciprocal temperature [1/T] = thermodynamic beta * Boltzman's constant [β•kB];
1/K ≈ 13.80648 ynat/J
L0T0M1-1Q0Θ0 Mass Fraction kg/kg parts per million|billion... wi component mass / total mass[mi/m]
Mass Ratio ζi component mass / remaining mass [mi/(m-mi)]
L0T0M1-1Q0Θ1-1 Adiabatic Index JK/(JK) joule kelvins
per joule-kelvin
γ ratio of fixed-pressure to fixed-volume heat capacities [Cp/Cv]
L1-1T0M0Q0Θ0 Plane Angle rad radian θ, φ 2π radians per revolution; 1 rad ≈ 57.2957795131°
? π pi   ratio of circumference of a circle to its diameter; π ≈ 3.14159265359
L1-1T1-1M0Q1-1Θ0 Magnetic
Susceptibility
  magnetic susceptibility χm magnetization / magnetic field [M/H]; vacuum χm = 0
relative permeability μr 1 + magnetic susceptibility [1+χm] = permeability / vacuum permeability [μ/μ0]
Magnetic
Modulus?
  Mm ?
reciprocal relative permeability [1/μr]
L2-2T0M0Q0Θ0 Solid Angle sr steradian Ω 4π steradians per sphere; 2π/3 sr per cube face measured from cube center
L2-2T0M0Q1-1Θ0 Electric
Susceptibility
  electric susceptibility χe polarization density / electric field / vacuum permittivity [Pe/(E•ε0)];
polarization density / (electric displacement - polarization density) [Pe/(D-Pe)]; vacuum χe = 0
relative permeability εr 1 + electric susceptibility [1+χe] = permittivity / vacuum permittivity [ε/ε0]
Electric Modulus   Me reciprocal relative permittivity [1/εr]
L2-2T2-2M1-1Q0Θ0 Energy
Effeciency,
Energy Intensity
J/J joules per joule ? energy efficiency = output energy / input energy [Eo/Ei];
energy intensity = input energy / output energy [Ei/Eo]
L2-2T3-3M1-1Q0Θ0 Power Ratio,
Power
Effeciency,
Power Intensity
W/W watts per watt ? power efficiency = output power / input power [Po/Pi];
power intensity = input power / output power [Pi/Po]
dB decibel Lp 10 log10 (measured power / reference power) [10 log10 (P/P0) = 10 log10 (V2/V02)]
Lf 20 log10 (measured field / reference field) [20 log10 (V/V0)]
L3-3T0M0Q0Θ0 Volume Fraction m3/m3 cubic meters
per cubic meter
φi component volume / total volume [Vi/V]
L0T0M0Q0Θ1 Temperature K kelvin T 0 K is 273.16 °C below triple point of water (which is defined as 0.01 °C)
L0T0M0Q1Θ0 [Electric] Charge C coulomb Q electric current * time [I•t]; charge of approximately 6.24150913 Ee (quintillion protons)
L0T0M1Q0Θ0 Mass, Inertia kg kilogram m originally, mass of one liter of water at 4°C at sea-level pressure
eV/c2 electron-volts per
light-speed squared
? 1 eV/c2 ≈ 160.217662 zJ ≈ 1.78266191•10-36 g
 (per Einstein's mass/energy equivalence)
L0T1M0Q0Θ0 Time s second t 9192631770 cycles of caesium-133 (133Ce) groundstate hyperfine splitting period (at 0 K)
F Ω farad ohm capacitance * resistance [C•R] = inductance / resistance [L/R]; 1 H/Ω = 1 s
L0T1M0Q0Θ1 ??? K s kelvin-second kDTS Dam Thanh Son limit = viscosity / entropy density [η/s];
reduced plank constant / 4π / Boltzman's constant [ℏ/(4π•kB] ≈ 607.8306 fKs
L0T1M1Q0Θ0 Mass Exposure kg/Bq kilogram per bacquerel ? reciprocal specific activity [1/a] = mass / activity [m/A] = action / absorb dose [S/D]; 1 Js/Gy
s kg second-kilogram time * mass [t•m] = toxic exposure * volume [E•V] = payload distance / speed [Λ/v]; 1 kg/Hz
L0T2M0Q0Θ0 ? s2 square second t2 time period / frequency [t/f = t/ν]; 1 s/Hz
F H farad henry capacitance * inductance [C•L]; 1 s2
L1T-3M0Q0Θ0 Jerk m/s3 meters per cubic second j acceleration / time period [a/t]
L1T-3M1Q0Θ-1 Thermal
Conductivity
W/(m•K) watts per meter-kelvin ? spectral flux in wavelength / temperature [Φe,λ/T]
N/(s•K) newtons per
second-kelvin
yank / temperature [Y/T]; 1 W/(m•K)
L1T-3M1Q0Θ0 Yank N/s newtons per second Y force / time period [F/t] = energy / absition [E/A]; 1 J/(m•s)
Spectral Fluxλ W/m watts per meter Φe,λ radiant flux / wavelength [Φe/λ] = power / distance [P/|r|]
L1T-2M0Q0Θ0 Acceleration m/s2 meters per
square second
a, g velocity / time period [v/t]; standard (Earth surface) gravity [gc]= 9.806650 m/s2
Specific Force N/kg newtons per kilogram |a| force / mass [|F|/m]
L1T-2M1Q-1Θ0 Electric Field,
Electric Flux
Density
N/C newtons per coulomb E force / charge [F/Q]
V/m volts per meter electric potential / distance [V/r]; 1 N/C
C/(F•m) coulombs per ferad-meter (electric displacement - polarization density) / vacuum permittivity [(D-Pe)/ε0]
electric displacement / permittivity [D/ε]; 1 N/C
J/(C•m) joules per coulomb-meter energy / charge / distance [E/(Q•r)]; 1 N/C
W/(A•m) watts per ampere-meter power / current / distance [P/(I•r)] = power / charge / velocity [P/(Q•v)]; 1 N/C
L1T-2M1Q0Θ-1 Thermic Force? N/K newtons per kelvin ? force / temperature [|F|/T] = delta (entropy / distance) [δH/(δr)]
L1T-2M1Q0Θ0 Force, Weight N newton F mass * acceleration [m•a] = momentum / time period [p/t]; 1 kg•m/s2;
magnetic flux * magnetic field [ΦBH] = charge * electric field [Q•E]; 1 N = Wb•A/m = 1 C•V/m
Fuel
Consumption
J/m joules per meter FC "energy intensity in transportation"; energy / distance [E/r]; 1 N; 322 J/m ≈ 1 Lpetrol/(100 km)
kW hr/km kilowatt-hours
per kilometer
energy / distance [E/r]; 1 kW•hr/km = 3600 J/m = 3600 N
L1T-1M0Q0Θ0 Speed, Velocity m/s meters per second c, v, v distance / time period [r/t]; 1 m/s = 1 kL•Hz/m2
  light-speed c0 speed of light in vacuum: 299792458 m/s (by definition)
Volumetric Flux L Hz/m2 liter hertz
per square meter
q fluid flow rate / directed area [Q/S]; 1 L•Hz/m2 = 1 mm/s
L1T-1M1-1Q0Θ0 Specific
Momentum
N s/kg newton-seconds
per kilogram
v momentum / mass [p/m]; 1 m/s
L1T-1M0Q1Θ0 Magnetic
Pole Strength
A m ampere-meters qm magnetization * directed area [MS]
C m/s coulomb-meters
per second
charge * speed [Q•|v|]; 1 Am
L1T-1M1Q-1Θ0 Magnetic
Potential
T•m tesla-meters A curl of magnetic flux density [∇×B]
N/A newtons per ampere force / current [F/I] = momentum / charge [q/Q]; 1 T•m
Wb/m webers per meter magnetic flux / distance [ΦB/r]; 1 T•m
L1T-1M1Q0Θ0 Momentum kg m/s kilogram-meters
per second
p mass * velocity [m•v] = √2 * mass * kinetic energy [√2m•Ek]
= leverage / time [d/t] = density * volume * phase velocity [ρ•V•v]; 1 N•s
Impulse N s newton-second delta momentum [δp] = force * time period [f•t] = charge * magnetic potential [Q•A];
1 kg•m/s = 1 N•s = C•N/A
L1T0M0Q0Θ0 Length m meter r, d,
x, y, z
1/299792458 of a light-second; approximately 1/40000000 of Earth's polar circumference
Wavelength λ distance of 1 cycle in a wave; phase velocity / frequency [|vφ|/f = |vφ|/ν]; for a photon, λ = c0
ly light-year ? distance traveled in 1 Julian year (365.25 Earth days) at light-speed; 9.46073047258080 Pm
L1T0M0Q1Θ0 Electric
Dipole Moment
C m coulomb-meters Δp (charge+ - charge-) / 2 / (position+ - position-) [δQ/(2r)]
L1T0M1Q-2Θ0 [Magnetic]
Permeability
H/m henries per meter μ inductance / distance [L/r] = reciprocal of reluctivity [1/?]
N/A2 newtons per
square ampere
force / square current [|F|/Q2]; 1 H/m;
relative permeability * vacuum permeability [μrμ0]
  vacuum permeability μ0 400 π nH/m ≈ 1.256637061 μH/m
L1T0M1Q0Θ0 Payload
Distance
m kg meter-kilogram Λ ? distance * mass [r•m] = momentum * time [p•t]
Leverage arm-length * mass [r•m] = rotational inertia / radius [I/r]
N/Hz2 newtons per square hertz torque / acceleration [τ/a = (a × τ)/a2]; 1 m•kg
L1T1M-1Q0Θ0 Fluidity m•s/kg meter-seconds
per kilogram
φ distance * time / mass [r•t/m] = absition / mass [|A|/m]
PI-1 reciprocal poiseuille reciprocal viscosity [1/η]; 1 PI-1 = 1 Hz/Pa
L1T1M0Q0Θ0 Absition,
Absement
m s meter second A distance * time period [r•t]
L2-1T2M-1Q0Θ0 Compressability Pa-1 reciprocal pascal β reciprocal pressure [1/|P|] = area / force [A/|F|]
m2/N square meters
per newton
area / force [A/|F|] = 1 / density / square phase velocity [1/(ρv2)]
m3/J cubic meters per joule volume / energy [V/E]; 1 m3/J = 1 kL/J
L2T-3M1-1Q0Θ0 Specific Power W/kg watts per kilogram ? power / mass [P/m] = energy / mass exposure [E/(m•t)]; 1 J/(kg•s); 1 Gy/s
Absorbtion Rate Gy/s grays per second absorbed dose / time period [D/t] = absorbed dose * activity [D•A]; 1 Gy•Bq; 1 W/kg
Dose Rate rem/dy rems per day 1/24 rem/hr = 1/86400 rem/s = 11.5740 μrem/s = 115.740 nSv/s
  MET metabolic equivalent
of task
energy / mass / time [E/(m•t)]; 1 kcal/(kg•hr) = 1.162 W/kg; 1 MET ≈ human sitting
L2T-3M1Q0Θ0 Rotatum N m/s newton-meters
per second
R delta torque / time period [δτ/t] = rotational inertia * angular jerk [Iζ] = force * velocity [F×v]
Power W watt P energy / time period [E/t]; 1 J/s; torque * angular velocity [τω]; 1 JHz
N m/s newton-meters
per second
force * velocity [Fv]; 1 W
V A volt amperes electric potential * electric current [V•I]; 1 W
A2Ω square ampere ohm square electric current * resistance [I2•R]; 1 W
V2 square volts per ohm square electric potential / resistance [V2/R]; 1 W
Radiant Flux W watt Φe radiant intensity * solid angle [IeΩ]
Radiant Intensity W/sr watts per steradian Ie radiant flux / solid angle [Φe/Ω]
Luminous Flux lm lumen Φv luminous intensity * solid angle [IvΩ]; 1 cd•sr; 1/683 W @ 540 THz
cd sr candella-steradian 1 lm; 1 cd over full sphere = 4π cd•sr ≈ 12.56643706144 lm ≈ 18.3987856726 mW @ 540 THz
Luminous
Intensity
cd candella Iv luminous flux / solid angle [Φv/Ω]; 1/683 W/sr (≈ 1.46412884334 mW/sr) @ 540 THz
L2T-2M1-1Q0Θ0 Specific Energy J/kg joules per kilogram ? energy / mass [E/m] = work / mass [W/m] = torque / mass [|τ|/m]; 1 N•m/kg
Absorbed Dose Gy gray D energy / mass [E/m]; 1 Gy = 1 J/kg = 100 RAD (radiation absorbed dose)
m2/s2 square meters
per square second
area / square time [A/t2] = velocity squared [vv]; 1 Gy
Equivalent Dose
Sv sievert H radiation weight * absorbed dose [WD]; 1 Gy of "X-Rays" has 5.5% risk of 'eventual' cancer
rem Roentgen equivalent man 10 mSv = 10 mGy of 'X-Rays' = 1 RAD of 'X-Rays'; has 0.055% risk of 'eventual' cancer
L2T-2M1Q-2Θ0 Elastance F-1 per farad, daraf P reciprocal capacitance [1/C]
V/C volts per coulomb electric potential / electric charge [V/Q]; 1 F-1
J/C2 joules per
square coulomb
energy / square charge [E/Q2]; 1 F-1
Ω/s ohms per second resistance / time period [R/t]; 1 F-1
L2T-2M1Q-1Θ0 Electric
Potential,
Voltage
V volt V reciprocal charge affinity [1/εp]; electric field * distance [Er]
J/C joules per coulomb energy / electric charge [E/Q]; 1 V
W/A watts per ampere power / electric current [P/I]; 1 V
Wb/s webers per second magnetic flux / time period [ΦB/t] = magnetic potential * velocity [Av]; 1 V
C/F coulombs per farad electric charge / capacitance [Q/C]; 1 V
A Ω ampere-ohm electric current * resistance [I•R]; 1 V
L2T-2M1Q0Θ-1 Molar
Heat Capacity
J/(K mol) joules per kelvin-mole Hm energy / temperature / #moles [E/(T•n)] = entropy / #moles [H/n]
ideal gas constant R Boltzman's constant * Avogadro's number [kB•NA] ≈ 8.31446 J/(K•mol)
Entropy,
Heat Capacity
J/K joules per kelvin H, Cp, Cv energy / temperature [E/T]
W/(K Hz) watts per kelvin-hertz power / temperature / frequency [P/(T•f) = P/(T•ν)]; 1 J/K
  Boltzman's constant kB ≈ 13.80648 yJ/K ≈ 8.61733 μeV/K
nat natural unit of information H, S (1/loge 2) Shannon ≈ 1.442695041 Sh ≈ 13.80648 yJ/K
Information
Capacity
bit, b binary digit ? binary (yes/no, on/off, etc.) information unit;
1 Sh (if both states equally likely) ≈ 693.14718056 mnat ≈ 9.569923 yJ/K
L2T-2M1Q0Θ0 Energy J joule E power * time period [P•t]; 1 J = 1 Ws (watt-second)
kW•hr kilowatt-hour power * time period [P•t]; 3600 kJ = 3.60 MJ
cal (little) calorie energy to raise 1 g (1 mL) of liquid water at 1 atm by 1°C ≈ 4.184 J; 1 kcal is a 'food Calorie'
K nat kelvin-nat temperature * delta entropy [T•δH]; ≈ 13.80648 yJ
Gy kg gray-kilogram absorbed-dose * mass [D•m]; 1 J
Pa m3 pascal cubic meters pressure * volume [|P|•V]; 1 Pa•m3 = 1 J = 4053/40 L•atm (liter-atmosphere)
eV electron-volt charge * electric potential [Q•V]; 1 e•J/C ≈ 160.217662 zJ ≈ 1.78266191•10-36 g
C2/F square coulombs
per farad
square charge / capacitance [Q2/C]; 1 J
H A2 henry square amperes inductance * square current [L•I2]; 1 J
kg m2Hz2 kilogram square meters
square hertz
rotational inertia * half of square angular frequency [I•ω2/2]; 1 J
kg m2/s2 kilogram square meters
per square second
square momentum / mass [p2/m]; momentum * velocity [pv]; leverage * acceleration [da];
1 J
potential energy Ep mass * height * gravity [m•rgc] = payload distance * gravity [dgc]
kinetic energy EK mass * half of square velocity [m•v2/2]
(rest)mass energy EM mass * square light-speed [m•c02]
electromagnetic
field energy
Eem electric field energy + magnetic field energy [V•(E2ε+B2/μ)/2]
F V2 farad square volts Ee electric field energy = volume * half of square electric field * permittivity [V•E2/2•ε]
Wb2/H square webers per henry Em magnetic field energy = volume * half of square magnetic field / permeability [V•B2/2/μ]
lm s lumen second Qv luminous energy = luminous flux * time period [Φvt] ≈ 1.46412884334 mJ of 540 THz light
W s watt second Qe radiant energy = radiant flux * time period [Φet]; 1 J
Spectral Fluxν W/Hz watts per hertz Φe,ν spectral flux in frequency = radiant flux / frequency [Φe/ν = Φe/f]
Work N m newton meter W force * distance [Fr]; 1 J
C V coulomb volt charge * electric potential [Q•V] = charge * electric field * distance [Q•Er]; 1 N•m
Torque J/rad joules per radian |τ| energy / plane angle [E/θ]
kg m2Hz/s kilogram square meter
hertz per second
τ delta (angular momentum / time) [δL/δt]; 1 J/rad
kg m2/s2 kilogram square meters
per square second
leverage * acceleration [Λ×a]; 1 J/rad;
rotational inertia * angular acceleration [Iα]
C V coulomb volt electric dipole moment * electric field [Δp×E]; 1 J/rad
L2T-1M1-1Q0Θ0 Kinematic
Viscosity
m2/s square meters
per second
v viscosity / density [μ/ρ] = area / time period [A/t]
Specific Angular
Momentum
m N s/kg meter-newton-seconds
per kilogram
h angular momentum / mass [|L|/m]; 1 m2/s
Specific Action J s/kg joule-seconds
per kilogram
? action / mass [S/m]; 1 m2/s
L2T-1M0Q1Θ0 Magnetic
Dipole Moment
A m2 ampere square meter μ electric current * directed area [I•S] = pole strength * distance [qmv]
J/T joules per tesla torque / magnetic flux density [τ/B = (B×τ)/B2]; 1 Am2
  magnetic moment of
electron, proton, neutron
μe, μp, μn ≈ -9.284764 yJ/T, 0.014106067 yJ/T, -0.00966236 yJ/T
  Bohr magneton μB [ eh/(4π me) ] ≈ 9.274009 yJ/T
  nuclear magneton μN [ eh/(4π mp) ] ≈ 0.005050783 yJ/T
L2T-1M1Q-2Θ0 Resistance,
Impedance,
Reactance
Ω ohm R, Z, X reciprocal of conductance|admittance|susceptance [1/(G|Y|B)]; 1 S-1
V/A volts per ampere electric potential / electric current [V/I]; 1 Ω
V2/W square volts per watt square voltage / power [V2/P]; 1 Ω
W/A2 watts per square ampere power / square current [P/I2]; 1 Ω
s/F seconds per farad time period / capacitance [t/C]; 1 Ω
H/s henries per second inductance / time period [L/t]; 1 Ω
  vacuum impedance Z0 [ μ0c0 = 1/(ε0c0) = √μ₀/ε₀ = |E|/|H| ]; 119.91698320 π Ω ≈ 376.730313462 Ω
L2T-1M1Q-1Θ0 Magnetic Flux Wb weber ΦB magnetic flux density * directed area [B•S]; 1 T•m2 (tesla square meter)
J/A joules per ampere energy / electric current [E/I] = action / electric charge [S/Q]; 1 Wb
H A henry-ampere inductance * electric current [L•I]; 1 Wb
V s volt-second electric potential * time period [V•t]; 1 Wb
C Ω coulomb-ohm electric charge * resistance [Q•R]; 1 Wb
L2T-1M1Q0Θ0 Angular Impulse N m s newton-meter-second J torque * time period [τ•t]
Angular
Momentum
m N s meter-newton-second L momentum arm * tangential momentum [r×p]
kg m2Hz kilogram square meter
hertz
rotational inertia * angular frequency [Iω]
Action J s joule-second S energy * time period [E•t] = energy / frequency [E/f = E/ν]; 1 J/Hz
N s m newton-second-meter momentum * distance [pr]; 1 Js
kg m2/s kilogram square meters
per second
leverage * tangential velocity [Λv]; 1 Js
  Planck constant h charge * electric potential * time period [Q•V•t]; ≈ 4.1356676 feV•s ≈ 6.6260700•10-34 Js
  reduced
Planck constant
charge * electric potential / angular speed [Q•V/ω = h/(2π)];
 ≈ 658.211951 aeV/Hz ≈ 1.0545718•10-34 Js/rad
L2T0M0Q0Θ0 Area m2 square meter A, S an area equivalent to a square with each side 1 meter in length; [S = rx × ry]
L4-2T0M0Q0Θ0 Étendue m2sr square meter steradian G area * solid angle [A•Ω]
L2T0M1Q0Θ0 Rotational
Inertia
kg m2 kilogram-square-meters I angular momentum / angular velocity [L/α] = mass * area [m•A]
N m/Hz2 newton-meters
per square hertz
torque / angular acceleration [|τ/α]; 1 kg•m2
L3T-2M1-2Q0T0 Gravitational
Constant
N m2/kg2 newton-square meters
per square kilogram
Gk force * square distance / square mass [F•r2/m2]; Gk ≈ 66.74 pN•m2/kg2
L3T-1M0Q0Θ0 Fluid Flow Rate m3/s cubic meters per second Q volumetric flux * directed area [qS] = kinematic viscosity * distance [v•r] = volume / time [V/t]
L3T-1M1-1Q0Θ0 N/PI newtons per poiseuille force * fluidity [|F|•φ] = force / viscosity [|F|/η]; 1 N/PI = 1 m3/s = 1 kL/s = 1 kHzL
L3T0M-1Q0T0 Specific Volume m3/kg cubic meters per
kilogram
? volume / mass [V/m]
L3T0M0Q0Θ0 Volume m3 cubic meter V volume equivalent to a cube with each edge 1 meter in length; 1000 L
L liter volume equivalent to a cube with each edge 10 centimeter in length; 1/1000 m3
Molar Volume m3/mol cubic meter per mole Vm volume / #moles [V/nm] = molar mass / density [M/ρ] = reciprocal molar density [1/ci]


 
 
Notes  

The unit for acidity/basicity (math symbol pH) is listed with units of "per cubic length" (L-3). However, pH is actually the logarithm of that unit.  Because the logarithm of a unit(s) is undefined, pH is truly dimensionless! Other "respectable" units which involve logarithms are unit-less; for example, the decibel (dB) is relative to the logarithm of power/power, or amplitude/amplitude. In either case, the decibel is based on a unit-less quantity (because the units cancel out). In other words, pH is a bastard unit (dimensionally incoherent).


© 2016, 2017, H2Obsession
Comments