Units and Constants

General Remarks

This is the no-nonsense module with the hard facts about units, constants and transformations from one system of units into an another one (after this paragraph, that is).
No explanations, historical roots, really outdated or unusual units are given - for the fun part use the link.
First, the basics:
In physics we always have two things: a physical quantity - e.g. the speed of something, or the strain of something under load - and some units to measure the quantity in question.
The physical quantity is what it is - it does not depend on how you express it in numbers. Somebody on some other planet will for sure do it differently from you and me.
The number you will give to the physical quantity is strictly a function of the units you chose. You might use m/s, oder lightyears/s, or wersts/year - that will just change the number for the speed of the moving object a lot, but not the speed itself. Trivial, but often forgotten.
To make life easier for everybody (at least for scientists), the choice of units was taken away from you and me, and everybody is now required to strictly adhere to the international standard system, abbreviated in any language as SI units.
Well, by now you, and I, and most others scientists, do comply with the SI system (which was not always the case) - but the public at large does not give shit; especially in the USA. Tell the gas station attendant any number you like in pascal or bar for the tire pressure, and he (or she) will just look at you as if you escaped from the lunatic asylum. Its psi or bust! And on occasion, even engineers or scientists do not use SI units - with disastrous consequences if you have tough luck.
The question now is: how many basic units do we need, so we can express everything else in these units? And which ones do we take?
This is one of the deeper questions of humankind. Physicists claim that we just need one more truly basic constant of nature - and we do not need units at all anymore. Velocities, for instance, can always be given using the absolutely constant speed of light (in vacuum) as the unit; your typical car speed than would be something like 0.000,001.
But redundancy tends to make life easier (just look at your typical Sheik and his harem), and the SI system gives us 7 basic units which are independent of each other.

Basic Units

Quantity Unit name Symbol
Length meter m
Mass kilogram kg
Time second s
Electrical current ampere A
Thermodynamic temperature kelvin K
Amount of substance mol mol
Luminous intensity candela cd
Note that in English only the names of persons (as well as of animals and fictitious characters) are written with the first letter capitalized. Therefore, all units must be written with small letters only. (The same holds for the chemicel elements, by the way: small letters only!)
From this basic units all other SI units can be derived. Below are tables with the more important secondary units.
First, we look at some secondary units just invoking basic units and a length. While we often do use special symbols for these quantities (e.g. r for density), these symbols are not really necessary and thus were not pronounced immutable and sacred as, e.g., the "m" for meter or the "s" for second.
Quantity Unit name Symbol
Area square meter m2
Volume cubic meter m3
Velocity meter per second m/s; ms–1
Acceleration meter per square second m/s2; ms–2
Wave number reciprocal meter m–1
Density kilogram per cubic meter kg/m3
Specific volume cubic meter per kilogram m3/kg
Electrical current density ampere per square meter A/m2
Magnetic field strength ampere per meter A/m
Substance concentration mol per cubic meter mol/m3
Luminance candela per sqare meter cd/m2
Now some more involved units - including important quantities like energy, voltage, and magnetic things.
They are more involved, because we usually do not express them in SI basic units - which is perfectly possible - but in secondary units. We will also find one case where there is no unit - it just cancels out.
These units often have their own symbols for reasons that become clear if you look at the SI units, and these symbols should not be used for something else
Quantity Unit name Symbol Conversion
in secondary units in basic units
Plane angle radian rad   m / m = 1
Frequency hertz Hz   s–1
Force newton N   m · kg · s–2
Pressure, stress pascal Pa N/m2 m–1 · kg · s–2
Energy, work, quantity of heat joule J N·m m2 · kg · s–2
Power, energy flux watt W J/s m2 · kg · s–3
Quantity of electricity
Electric charge
coulomb C
Electric potential, voltage volt V W/A m2·kg·s-3·A–1
Capacitance farad F C/V m–2·kg–1·s4·A2
Electric resistance ohm W V/A m2·kg·s–3·A–2
Conductance siemens S A/V m–2·kg–1·s3·A2
Magnetic flux weber Wb V·s m2·kg·s–2·A–1
Magnetic flux density tesla T Wb/m2 kg·s-2·A–1
Inductance henry H Wb/A m2·kg·s–2·A–2
Celsius temperature degree celsius
Radioactivity becquerel Bq
Again: By using small letters it is clear that here it's all about the unit names; capitalizing the first letter would refer to the person after which this unit was named.
Mercifully, the members of the "Comité international des poids et mesures" are human (up to a point, at least). In consequence they did not outlaw all older units in one fell stroke, but sorted them into three groups:
"Old" units which may be used together with SI units without restrictions.
Old units which may be used for some time in parallel to SI units.
Old units which are definitely out and must not be used at all any more.
Some of the units in the second category are regional and you probably have never heard of them. We will not include them here. The number of outlawed units is legion, we just include the still tempting ones.
Here is the first category: Some of the non-SI units you still may use without restrictions:
Unit name Symbol Conversion
minute min 1 min = 60 s
hour h 1h = 60 min = 3600 s
day d 1 d = 24 hr = 86400 s
angle degree
angle minute
angle second
1° = (p/180) rad
1 ' = (1/60) °
1 '' = (1/60) ' = (1/3600) °
liter l, L 1 l = 1 dm3 = 10–3 m3
ton t 1 t = 103 kg
electron volt eV 1 eV = 1.602,176,6 · 10–19 J
atomic mass unit (amu) u 1 u = 1.660,539,1 · 10–27 kg
What a relief!
Now to the old units you may use for some more time to come in parallel to the SI units:
Unit name Symbol Conversion
angstrom / ångström Å 1 Å = 0.1 nm
ar a 1 a = 100 m2
hectar ha 1 ha = 100 a
bar bar 1 bar = 0.1 MPa
barn b 1 b = 100 fm2 = 10–28 m2
curie Ci 1 Ci = 3.7 · 1010 Bq
roentgen R 1 R = 2.58 · 10–4 C/kg = 2.58 · 10–4 As/kg
Note that the letter Å is not pronounced as the a in "far", instead, it sounds like the o in "of" (cf. en.wiktionary.org/wiki/%C3%85ngstr%C3%B6m). Germans seem to think that it has to be pronounced as a mixture of German o and a (i.e., as an a-ish variant of o), but that's wrong!
Now to the units you must not use anymore!. We might put them into two groups:
1. The forerunners of the SI units, the cgs units; i.e. the units based on the centimeter, the gram and the second.
2. The simple old fashioned no-no's.
While it may appear that the cgs system is practically the same as the SI system, this is not so!
Of course, the cm, g, and s are essentially the same basic units as in the SI system, the abbreviation "cgs", however, does not tell you anything about the other necessary basic units in this system - and that is where the problems come in!
In fact, there were several cgs systems - the electrostatic, the electromagnetic, and the Gauss cgs system!
We will not unravel all the intricacies for cgs systems and the conversion to SI units here - this is done in its own module - but just give some of the more common units and their conversion.
Unit name Symbol Conversion
erg erg 1 erg = 10–7 J
dyne dyn 1 dyn = 10–5 N
poise P 1 P = 1 dyn·s/cm–2 = 0.1 Pa·s
gauss Gs, G 1 G corresponds to 10–4 T
maxwell Mx 1 Mx (= 1 G·cm2) corresponds to 10–8 Wb
oersted Oe 1 Oe (= 1 dyn/Mx) corresponds to (1000/4p) A/m
The "corresponds to" instead of simply "=" is an indication that while the three quantities in question do have SI units that correspond to magnetic flux density, magnetic field strength, and magnetic flux, they are not exactly the same thing.
Finally, some still fondly remembered old units you simply do not use anymore:
Unit name Symbol Conversion
torr Torr 1 Torr = (101,325/760) Pa
» 133.32 Pa
physical atmosphere atm 1 atm = 101,325 Pa
kilopond kp 1 kp = 9.806,65 N
calorie cal 1 cal = 4.184 J
(micrometer is what you use!)
µ 1 µ = 1 µm

Fundamental Constants

Fundamental constants are some numbers with units that cannot (yet) be calculated from some physical theory, but must be measured.
This may have three possible reasons:
  1. There is presently no theory, and there never will be a theory, that allows us to calculate fundamental constants. They have the value they have because an act of one or more gods and/or godesses, or they are purely random (i.e we just happen to live in an universe, where the value is what we measure. In some other universe, or some other corner of our universe, it will be arbitrarily different).
  2. There is presently no theory, but some day there will be one. Some fundamental constants will then be calculated and then are no longer fundamental.
  3. There already is a theory, or at least a general theoretical framework; we just are not yet smart enough to see the obvious or to do the numerics. Masses of elementary particles, e.g., might be "fundamental constants" that fall into this category.
Hot-shot physicists have some ideas, which constant might fall into which category. Speculations along this line are a lot of fun - but of no consequence so far. So I will not dwell on this. (Of course, you may check for yourself which one of the three possibilities you are going to embrace and thus get some idea of what kind of person you are).
Fundamental physical theories usually introduce one new fundamental constant. Mechanics (including gravitation) needs the gravity constant G, quantum theory has Plancks constant h, statistical thermodynamics introduces Boltzmanns constant k, the special theory of relativity (or Maxwells theory of electromagnetism which is really part of the relativity theory) needs the speed of light c.
New theories sometimes "explain" old constants of nature because they can calculate them, or replace them by something more fundamental. Boltzmann's constant k, for example, is more fundamental than the "fundamental" gas constant R, because it relates its number to a fundamental unit of matter (1 particle) and not to an arbitrary one like 1 mol.
How many truly fundamental constants are there? Why do they have the values they have? (Just slight deviations in the values of some constants would make carbon based life impossible; this is where the so-called "anthropic principle" comes in). Will we eventually be able, with a "Theory of Everything" (TOE) to calculate all natural constants?
Nobody knows. We run against the deepest physical questions at this point.
So let's just look at what we have. Since it is customary to list as natural constants some quantities that are actually computable from others, we include some of these "constants" here, too (together with the conversion formula).
Symbol and formula Numerical value Magnitude and unit Remarks
Speed of light in vacuum
c0, c 2.997,924,58 108m·s–1 Truly fundamental
Gravitational constant
G 6.673 10–11m3·kg–1·s–2 Truly fundamental
Planck's constant
h 6.626,068,76 10–34J·s Truly fundamental
4.135,6 10–15 eV·s
Elementary charge
e 1.602,176,462 10–19C Truly fundamental ?
Maybe not
Fine structure constant
a = µ0·c·e2/2h 7.297,352,533 10–3 Unitless, maybe more
fundamental than others.
Mass of a electron at rest
me 9.109,381,88 10–31 kg
Not truly fundamental; can be
calculated in principle
0,510 998 902 MeV
Mass of a proton at rest
mp 1.672,621,58 10–27 kg Not truly fundamental, can be
calculated in principle
1.007,276,466 u
938.271,998(38) MeV
Avogadro constant
NA 6.022,141,99(47) 1023 mol–1 Not truly fundamental any more
Faraday constant
F = e·NA 96,485.3415(39) C·mol–1 Not truly fundamental any more
Universal gas constant
R 8.314,472(15) J·mol–1·K–1 Not truly fundamental any more
Boltzmann constant
k = R/NA 1.380,650,3 10–23 J·K–1 Truly fundamental
8.617,269 10–5 eV·K–1
Magnetic permeability of vacuum
µ0 = 1/e0c2 12.566,370,614 10–7 V·s·A–1m–1 Not truly fundamental
Electric susceptibility of vacuum
e0 = 1/µ0c2 8.854,187,817 10–12A·s·V–1m–1 Not truly fundamental
Magnetic flux quant
P = h/2e 2.067,833,636 10–15 Wb Smallest possible magnetic flux
Not truly fundamental

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