 |
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really outdated or unusual units are given - for the fun part use the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. |
| | | |
| |
| 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 |
| 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 |
|
| | 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 | micron
(micrometer is what you use!) | µ | 1 µ = 1 µm |
|
| 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: - 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.
|
| | 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 |
|
© H. Föll (MaWi 1 Skript)
2.1.2 Fruehe Atommodelle: Bohr und de Broglie 
Mit Frame