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.
No explanations, historical roots, really outdated or unusual units are given - for the fun part use this link.
First, some basics about measurements and units.
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 do it differently from you and me for sure
The number you will give to the physical quantity is strictly a function of the units that you chose. You might use m/s, lightyears/s or wersts/year - that will just change the number you assign to the speed of the moving object 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. Everybody is now required to strictly adhere to the international standard system, abbreviated in any language as SI units.
Well, by now I and most others scientists, do comply with the SI system (which was not always the case); about you I don't know. The public at large, of course, does not give shit; especially in the USA. Tell the gas station attendant any number you like for the tire pressure in Pascal, and he (or she) will just look at you as if you escaped from the lunatic asylum. Its psi or bust! On occasion, even (american) engineers or scientists do not use SI units - with disastrous consequences like satellites lost in space.
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 fundamental 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 with the numerical value 1. Your typical car speed than would be something like 0,000001.
But redundancy tends to make life easier and more pleasant for some (just look at your typical Sheik and his harem). The SI system gives us 7 basic units which are independent of each other and plenty of derived (and thus redundant) ones.

Basic Units

Here are the seven basic units of the SI system:
Quantity 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
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 the second.
Quantity 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 let's look at 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. These symbols should not be used for something else
Quantity Name Normal Symbol Symbol SI
Secondary Basic
Plane angle Radiant rad   m / m = 1
Frequency Hertz Hz   s–1
Force Newton N   m kgs–2
Pressure, stress Pascal Pa N/m2 m–1kgs–2
Energy, work, quantity of heat Joule J Nm m2 ·kgs–2
Power, energy flux Watt W J/s m2kgs–3
Electric charge Coulomb C
Electric potential, voltage Volt V W/A mkgs-3A–1
Capacitance Farad F C/V m–2kg–1s4·A2
Electric resistance Ohm W V/A m2kgs–3·A–2
Conductance Siemens S A/V m–2kg-1s3A2
Magnetic flux Weber Wb Vs m2kgs–2A-1
Magnetic flux density Tesla T Wb/m2 kgs-2A-1
Inductance Henry H Wb/A m2kgs–2A–2
Celsius temperature Degree Celsius oC
Radioactivity Becquerel Bc
Mercifully, the members of the "Comite International des Poids and Mesures" are human (up to a point). 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. I 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:
Quantity Name Unit
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
1o = (p/180) rad
1 ' = (1/60) o
1 '' = (1/60) ' = (1/3600) o
Liter l, L 1 l = 1 dm3 = 10–3 m3
Ton t 1 t = 103 kg
Electronvolt eV 1 eV = 1,602 540 2 · 10–19 J
Atomic mass unit u 1 u = 1,660 540 2 · 10–27 kg
Now to the old units you may use for some more time to come in parallel to the SI units:
Quantity Name Unit
Ångstrom 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 fm = 10–28 m2
Curie Ci 1 Ci = 3,7 · 1010 Bq
Roentgen R 1 R 0 2,58 · 104 Ci/kg
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! I will not go deeper into this, however.
Finally, some still fondly remembered old units you simply do not use anymore:
Quantity Name Unit
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
Calory cal 1 cal = 4,186 8 J
(Micrometer is what you use!)
µ 1 = 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 to calculate fundamental constants. They have the value they have because of an act of God (or, as it should be written in our enlightened times: an act of one or more gods and/or godesses), or they are purely random. In this case we just happen to live in an universe where their value is what we measure. In some other universe, or some other corner of our universe, it might be different.
  2. There is presently no theory, but some day there will be one. Fundamental constants will then be calculated from scratch and then are no longer fundamental.
  3. There already is a theory for some of those constants, we just are not yet smart enough to see the obvious or to do the numerics. Masses of elementary particles, e.g., might fall into this category.
Hot-shot physicists have some ideas, which constants 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? Interestingly, 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 also some quantities that are actually computable from others by now, I 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 58 108 ms–1 Truly fundamental
Gravitational constant
G 6,673 10–11 m3kg–1s–2 Truly fundamental
Planck's constant
h 6,626 068 76 10–34 Js Truly fundamental
4,1356 10–15 eVs
Elementary charge
e 1,602 176 462 10–19 C Truly fundamental ?
Maybe not
Fine structure constant
a = µ0ce2/2h 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
Lohschmidt constant (also known as Avogadro constant)
NA 6,022 141 99(47) 1023 mol–1 Not truly fundamental
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) Jmol–1K–1 Not truly fundamental any more
Boltzmann constant
k = R/NA 1,380 6503 10–23 JK–1 Truly fundamental
8,617269 10–5 eVK–1
Magnetic permeability of vacuum
µ0 = 1/e0c2 12,566 370 614 10–7 VsA–1m–1 Not truly fundamental
Electric susceptibility of vacuum
e0 = 1/µ0c2 8,854 187 817 10–12 AsV–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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