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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). |
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No explanations, historical roots, really outdated or unusual units are given
- for the fun part use the link. |
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First, the basics: |
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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. |
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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.
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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. |
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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. |
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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. |
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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? |
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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. |
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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. |
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| 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 |
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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!) |
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From this basic units all other SI units can be derived. Below are tables
with the more important secondary units. |
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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. |
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| 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 |
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Now some more involved units - including important quantities like energy
, voltage , and magnetic things. |
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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. |
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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 |
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| 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 |
| A·s |
| 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
("centigrade") |
°C |
| K |
| Radioactivity | becquerel | Bq |
| 1/s |
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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. |
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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: |
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"Old" units which may be used together with SI units without restrictions. |
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Old units which may be used for some time in parallel
to SI units. |
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Old units which are definitely out and must not be used at all
any more. |
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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. |
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Here is the first category: Some of the non-SI units you still may use without restrictions : |
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| 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 |
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What a relief! |
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Now to the old units you may use for some more time
to come in parallel to the SI units: |
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| 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 |
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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! |
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Now to the units you must not use anymore!.
We might put them into two groups: |
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1. The forerunners of the SI units, the cgs
units; i.e. the units based on the centimeter,
the gram and the second
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2. The simple old fashioned no-no's. |
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While it may appear that the cgs system is practically the same as the
SI system, this is not so! |
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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! |
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In fact, there were several
cgs systems - the electrostatic, the electromagnetic,
and the Gauss cgs system! |
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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. |
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| 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 |
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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. |
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Finally, some still fondly remembered old units you simply
do not use anymore : |
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| 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 |
micron
(micrometer is what you use!) |
µ | 1 µ = 1 µm |
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Fundamental
constants are some numbers with units that cannot (yet) be calculated from some physical theory,
but must be measured. |
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This may have three possible reasons:
- 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).
- 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.
- 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.
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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). |
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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. |
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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. |
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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? |
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Nobody knows. We run against the deepest physical questions at this point. |
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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 |
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Speed of light in vacuum |
| c0, c | 2.997,924,58 |
108m·s–1 | Truly fundamental |
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Gravitational constant |
| G | 6.673 |
10–11 m3·kg–1·s–2 |
Truly fundamental |
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Planck's constant |
| h | 6.626,068,76 |
10–34J·s | Truly fundamental |
| 4.135,6 | 10–15 eV·s |
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Elementary charge |
| e | 1.602,176,462 | 10–19C |
Truly fundamental ?
Maybe not |
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Fine structure constant |
| a = µ0·c·e2/2h |
7.297,352,533 | 10–3 |
Unitless, maybe more
fundamental than others. |
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Mass of a electron at rest |
| me | 9.109,381,88 |
10–31 kg
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Not truly fundamental; can be
calculated in principle |
| 0,510 998 902 | MeV |
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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 |
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Avogadro constant |
| NA | 6.022,141,99(47) |
1023 mol–1 |
Not truly fundamental any more |
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Faraday constant |
| F = e·N A | 96,485.3415(39) |
C·mol–1 | Not truly fundamental any more |
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Universal gas constant | | R |
8.314,472(15) |
J·mol –1·K–1 |
Not truly fundamental any more |
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Boltzmann constant |
| k = R/NA | 1.380,650,3 |
10 –23 J·K–1 |
Truly fundamental | | 8.617,269 |
10–5 eV·K–1 |
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Magnetic permeability of vacuum |
| µ0 = 1/e0c2 |
12.566,370,614 |
10–7 V·s·A–1m–1 |
Not truly fundamental |
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Electric susceptibility of vacuum |
| e 0 = 1/µ0c2 |
8.854,187,817 |
10–12A·s·V–1m–1 |
Not truly fundamental |
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Magnetic flux quant |
| P = h/2e | 2.067,833,636 |
10–15 Wb |
Smallest possible magnetic flux
Not truly fundamental |
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© H. Föll (MaWi 1 Skript)
2.1.2 Fruehe Atommodelle: Bohr und de Broglie 
Mit Frame