System           Units                                     Best suited for

SI (MKS)         Meters, Kilograms, Seconds                Newtonian mechanics, EM forces between currents
Gaussian (CGS)   Centimeters, Grams, Seconds               EM forces between particles, plasma physics, astrophysics
Particle         Meters, Electron Volts, Seconds           Particle physics
Planck           Planck length, Planck mass, Planck time   General relativity, quantum gravity


1 gram                =  .001 kg
1 cm                  =  .01  meters
1 electron Volt (eV)  =  1.602e-19 Joules
                      =  The energy gained by an electron upon descending a potential of 1 Volt

Helium atom	Particles and forces

In this plot, the diameter of each particle proportional to CubeRoot(Mass). This is what the particles would look like if they were uniform-density spheres.

The electron is exaggerated otherwise it would be invisible.

The blue particles represent the heaviest particle that can be produced by each accelerator.

At this scale, a Big Bang particle has a diameter of 10 km.

Photons, Gluons, and Gravitons are massless.

Electron neutrino < 1     eV
Muon neutrino     < 2     eV
Red photon          1.8   eV
Green photon        2.3   eV
Blue photon         3.1   eV
Electron             .51 MeV
Up quark            1.9  MeV
Down quark          4.4  MeV
Strange quark      87    MeV
Muon              105.7  MeV
Neutral pion      135    MeV
Charged pion      140    MeV
Proton            938.27 MeV
Neutron           939.57 MeV
Charm quark         1.32 GeV  Discovered at SLAC
Tau                 1.78 GeV  Discovered at SLAC
Bottom quark        4.24 GeV  Discovered at Fermilab
SLAC limit         45    GeV  Highest-energy particle that SLAC can produce
W boson            80    GeV  Discovered at the Super Proton Synchrotron
Z boson            91    GeV  Discovered at the Super Proton Synchrotron
Fermilab limti    125    GeV  Highest-energy particle that Fermilab can produce
Higgs Boson       125    GeV  Discovered at the LHC
Top quark         173    GeV  Discovered at Fermilab
LHC limit        1000    GeV  Highest-energy particle that the LHC can produce
Cosmic rays      10^12   GeV  Highest-energy events observed
Planck energy    10^19   GeV  Quantum gravity. Planck energy = 1.22e28 eV = 1.956e9 Joules

1 electron Volt (eV) = 1.602e-19 Joules ~ kT at 11,000 Kelvin

Electricity and magnetism

Quantity                         MKS units                       CGS units

Mass                             M    kg                          gram        m  = .001 M
Wire length                      Z    meter                       cm          z  = .01 Z
Radial distance from wire        R    meter                       cm          r  = .01 R
Time                             T    second                      second      t  = T
Force                            F    Newton                      dyne        f  = 100000 F
Charge                           Q    Coulomb                     Franklin    q  = 3.336e-10 Q
Electric current                 I    Ampere = Coulomb/s          Franklin/s  i  = 3.336e-10 I
Electric potential               Volt Volt
Electric field                   E    Volt/meter                  StatVolt/cm e  = 299.79 E
Magnetic field                   B    Tesla                       Gauss       b  = 10000 B
Energy                           En   Joule                       erg         en = e-7 En
Capacitance                      Cap  Farad                       cm
Inductance                       L    Henry                       s^2/cm
Velocity of a charge             V    meter/second                cm/s        v  = .01 V
Speed of light                   C    2.999e8 meter/second        cm/s        c  = 100 C
Electric force constant          Ke   =8.988e9 N m^2/C^2                      ke = 1 dyne*cm^2/Franklin^2
Magnetic force constant          Km   = 2e-7 = Ke/C^2                         km = 1/c^2
Vacuum permittivity              Eps0 = 8.854e-12 F/m =1/4/Pi/Ke
Vacuum permeability              Mu0  = 4 Pi e-7 Vs/A/m =2 Pi Km
Proton charge                    Qpro = 1.602e-19 Coulomb                     qpro= 4.803e-10 Franklin
Electric field from a charge     E    = Ke Q / R^2                            e  = q / r^2
Electric force on a charge       F    = Q  E                                  f  = q e
Electric force between charges   F    = Ke Q Q / R^2                          f  = q q / r^2
Magnetic field of moving charge  B    = Km V Q / R^2                          b  = (v/c) q / r^2
Magnetic field around a wire     B    = Km I / R                              b  = (v/c) i / r
Magnetic force on a charge       F    = Q  V B                                f  = (v/c) q b
Magnetic force on a wire         F    = Km I B Z                              f  = i b z
Magnetic force between charges   F    = Km V^2 Q Q / R^2                      f  = (v/c)^2 q q / r^2
Magnetic force between wires     F    = Km I1 I2 Z / R                        f  = i1 i2 z / r
Energy of a capacitor            En   = .5 Capacitance*Volt^2
Field energy per volume          Z    = (8 Pi Ke)^-1 (E^2 + B^2/C^2)          z = .5 (e^2 + b^2/c^2)


Maxwell's equations
                                   MKS            CGS

Speed of light                      C              c
Electric field                      E              e
Electric field, time derivative     Et             et
Magnetic field                      B              b
Magnetic field, time derivative     Bt             bt
Charge                              Q              q
Charge density                      Qd             qd
Current density                     J              j
Electric force constant             Ke=8.988e9     ke=1
Magnetic force constant             Km=2e-7        km=2/c


Divergence(E)  =  4 Pi Ke Qd                  Divergence(e)  =  4 Pi qd
Divergence(B)  =  0                           Divergence(b)  =  0
Curl(E)        = -Bt                          Curl(e)        = -bt / c
Curl(B)        =  2 Pi Km J + Et / C^2        Curl(b)        =  4 Pi j / c  +  et / c



Magnetic field
                                     Teslas

Field generated by brain             10^-12
Wire carrying 1 Amp                  .00002     1 cm from the wire
Earth magnetic field                 .0000305   at the equator
Neodymium magnet                    1.4
Magnetic resonance imaging machine  8
Large Hadron Collider magnets       8.3
Field for frog levitation          16
Strongest electromagnet            32.2         without using superconductors
Strongest electromagnet            45           using superconductors
Neutron star                       10^10
Magnetar neutron star              10^14


Electrical and thermal conductivity of a wire
L  =  Length of wire            meters
A  =  Cross section of wire     meters^2
_______________________________________________________________________________________________________
                                                  |
                  Electric quantities             |                Thermal quantities
                                                  |
Q  =  Charge                 Coulomb              |   Ene=  Thermal energy          Joule
I  =  Current                Amperes              |   It =  Thermal current         Watts
E  =  Electric field         Volts/meter          |   Et =  Thermal field           Kelvins/meter
C  =  Electric conductivity  Amperes/Volt/meter   |   Ct =  Thermal conductivity    Watts/meter/Kelvin
J  =  Current flux           Amperes/meter^2      |   Jt =  Thermal flux            Watts/meter^2
   =  I / A                                       |      =  It / A
   =  C * E                                       |      =  Ct * Et
V  =  Voltage                Volts                |   T  =  Temperature difference  Kelvin
   =  E L                                         |      =  Et L
   =  I R                                         |      =  It Rt
R  =  Resistance             Volts/Ampere = Ohms  |   Rt =  Thermal resistance      Kelvins/Watt
   =  L / (A C)                                   |      =  L / (A Ct)
H  =  Current heating        Watts/meter^3        |
   =  E J                                         |
P  =  Current heating power  Watts                |
   =  E J L A                                     |
   =  V I                                         |


Continuum
Continuum quantity       Macroscopic quantity

     E             <->      V
     C             <->      R = L / (A C)
     J = C E       <->      I = V / R
     H = E J       <->      P = V I


Electric and thermal conductivity
         Electric  Thermal  Density   Electric   C/Ct     Heat   Heat      Melt   $/kg  Young  Tensile Poisson  Brinell
         conduct   conduct            conduct/            cap    cap                                   number   hardness
        (e7 A/V/m) (W/K/m)  (g/cm^3)  Density   (AK/VW)  (J/gK)  (J/cm^3K)  (K)         (GPa)  (GPa)             (GPa)

Silver      6.30   429      10.49       .60      147       .235   2.47     1235    590    83   .17      .37      .024
Copper      5.96   401       8.96       .67      147       .385   3.21     1358      6   130   .21      .34      .87
Gold        4.52   318      19.30       .234     142       .129   2.49     1337  24000    78   .124     .44      .24
Aluminum    3.50   237       2.70      1.30      148       .897   2.42      933      2    70   .05      .35      .245
Beryllium   2.5    200       1.85      1.35      125      1.825   3.38     1560    850   287   .448     .032     .6
Magnesium   2.3    156       1.74      1.32      147      1.023   1.78      923      3    45   .22      .29      .26
Iridium     2.12                                           .131            2917  13000   528  1.32      .26     1.67
Rhodium     2.0    150      12.41       .161     133       .243            2237  13000   275   .95      .26     1.1
Tungsten    1.89                                           .132            3695     50   441  1.51      .28     2.57
Molybdenum  1.87   138      10.28       .182     136       .251            2896     24   330   .55      .31     1.5
Cobalt      1.7    100       8.90       .170               .421            1768     30   209   .76      .31      .7
Zinc        1.69   116                                     .388             693      2   108   .2       .25      .41
Nickel      1.4     90.9                                   .444            1728     15
Ruthenium   1.25   117                                                     2607   5600
Cadmium     1.25    96.6                                                    594      2    50   .078     .30      .20
Osmium      1.23                                           .130            3306  12000
Indium      1.19                                                            430    750    11   .004     .45      .009
Iron        1.0     80.4                                   .449            1811          211   .35      .29      .49
Palladium    .95    71.8                                                   1828
Tin          .83    66.8                                                    505     22    47   .20      .36      .005
Chromium     .79    93.9                                   .449            2180
Platinum     .95                                           .133            2041
Tantalum     .76                                           .140            3290
Gallium      .74                                                            303
Niobium      .55    53.7                                                   2750
Rhenium      .52                                           .137            3459
Vanadium     .5     30.7                                                   2183
Titanium     .25    21.9                                   .523            1941
Scandium     .18    15.8                                                   1814
Neodymium    .156                                                          1297
Mercury      .10     8.30                                  .140             234
Manganese    .062    7.81                                                  1519
Germanium    .00019                                                        1211

Dimond iso 10    40000
Diamond     e-16  2320                                     .509
Tube       10     3500         Carbon nanotube.      Electric conductivity = e-16 laterally
Tube bulk          200         Carbon nanotubes in bulk
Graphene   10     5000
Graphite    2      400                                     .709         Natural graphite
Al Nitride  e-11   180
Brass       1.5    120
Steel               45         Carbon steel
Bronze       .65    40
Steel Cr     .15    20         Stainless steel
Quartz (C)          12         Crystalline quartz.  Thermal conductivity is anisotropic
Quartz (F)  e-16     2         Fused quartz
Granite              2.5
Marble               2.2
Ice                  2
Concrete             1.5
Limestone            1.3
Soil                 1
Glass       e-12      .85
Water       e-4       .6
Seawater    1         .6
Brick                 .5
Plastic               .5
Wood                  .2
Wood (dry)            .1
Plexiglass  e-14      .18
Rubber      e-13      .16
Snow                  .15
Paper                 .05
Plastic foam          .03
Air        5e-15      .025
Nitrogen              .025                                1.04
Oxygen                .025                                 .92
Silica aerogel        .01

Siemens:    Amperes^2 Seconds^3 / kg / meters^2     =   1 Ohm^-1



Effect of temperature on conductivity

Resistivity in 10^-9 Ohm Meters

              293 K   300 K   500 K

Beryllium     35.6    37.6     99
Magnesium     43.9    45.1     78.6
Aluminum      26.5    27.33    49.9
Copper        16.78   17.25    30.9
Silver        15.87   16.29    28.7


Dielectric strength
The critical electric field for electric breakdown for the following materials is:

              MVolt/meter
Air                3
Glass             12
Polystyrene       20
Rubber            20
Distilled water   68
Vacuum            30        Depends on electrode shape
Diamond         2000


Relative permittivity

Relative permittivity is the factor by which the electric field between
charges is decreased relative to vacuum.  Relative permittivity is dimensionless.
Large permittivity is desirable for capacitors.
             Relative permittivity
Vacuum            1                   (Exact)
Air               1.00059
Polyethylene      2.5
Sapphire         10
Concrete         4.5
Glass          ~ 6
Rubber           7
Diamond        ~ 8
Graphite       ~12
Silicon         11.7
Water (0 C)     88
Water (20 C)    80
Water (100 C)   55
TiO2         ~ 150
SrTiO3         310
BaSrTiO3       500
Ba TiO3     ~ 5000
CaCuTiO3    250000


Capacitance
A   =  Plate area               meter^2
Z   =  Plate spacing            meter
Ke  =  Electric force constant
    =  8.9876e9 N m^2 / C^2
Q   =  Max charge on the plate  Coulomb
Emax=  Max electric field
    =  4 Pi Ke Q / A
V   =  Voltage between plates   Volt
    =  E Z
    =  4 Pi Ke Q Z / A
En  =  Energy                   Joule
    =  .5 Q V
    =  .5 A Z E^2 / (4 Pi Ke)
e   =  Energy density           Joule/m^3
    =  E / A Z
    =  .5 E^2 / (4 Pi Ke)
q   =  Charge density           Coulomb/m^3
    =  Q / A / Z
C   =  Capacitance              Farad
    =  Q/V
    =  (4 Pi Ke)-1 A/Z
c   =  Capacitance density      Farad/kg
    =  C / A / Z
    =  (4 Pi Ke)-1 Emax^2 / V^2
Eair=  Max electric field in air
    =  3 MVolt/meter
k   =  Dielectric factor
    =  Emax / Eair


Continuum                                                 Macroscopic

Energy/Volume  =  .5 E^2 / (4 Pi Ke)           <->        Energy = .5 C V^2
               =  .5 q V                                         =  .5 Q V
c              =  (4 Pi Ke)-1 Emax^2 / V^2     <->        C      = (4 Pi Ke)^-1 A / Z


A capacitor can be specified by two parameters:


*)    Maximum energy density or maximum electric field


*)    Voltage between the plates

The maximum electric field is equal to the max field for air times
a dimensionless number characterizing the dielectric
Eair =  Maximum electric field for air before electical breakdown
Emax =  Maximum electric field in the capacitor
Rbohr=  Bohr radius
     =  Characteristic size of atoms
     =  5.2918e-11 m
     =  hbar^2 / (ElectronMass*ElectronCharge^2*Ke)
Ebohr=  Bohr electric field
     =  Field generated by a proton at a distance of 1 Bohr radius
     =  5.142e11 Volt/m
Maximum energy density  =  .5 * 8.854e-12 Emax^2


                         Emax (MVolt/m)   Energy density
                                            (Joule/kg)
Al electrolyte capacitor     15.0            1000
Supercapacitor               90.2           36000
Bohr limit               510000            1.2e12            Capacitor with a Bohr electric field



Inductance
N  =  Number of wire loops   Dimensionless
Z  =  Length                 meter
A  =  Area                   meter^2
Mu =  Magnetic constant      Henry/meter
   =  4 Pi 10^-7
I  =  Current                Ampere
It =  Current change/time    Ampere/second
F  =  Magnetic flux          Tesla meter^2
   =  N B A
Ft =  Flux change/time       Tesla meter^2 / second
B  =  Magnetic field         Tesla
   =  Mu N I / Z
V  =  Voltage                Volt
   =  Ft
   =  L It
   =  N A Bt
   =  Mu N^2 A It / Z
L  =  Inductance             Henry
   =  Ft / It
   =  Mu N^2 A / Z
E  =  Energy                 Joule
   =  .5 L I^2

Hyperphysics:  Inductor

Viscosity
                Pascal second

Hydrogen            .00000876
Nitrogen            .0000178
Air                 .0000183
Helium              .000019
Oxygen              .0000202
Xenon               .0000212
Acetone             .00031
Benzine             .00061
H2O at  10 C        .00131
H2O at  20 C        .00100
H2O at  30 C        .000798
H2O at 100 C        .000282
Mercury             .00153
Blood               .0035
Motor oil           .065
Olive oil           .081
Honey              6
Peanut butter    250
Asthenosphere   7e19         Weak layer between the curst and mantle
Upper mantle   .8e21
Lower mantle  1.5e21


Fluid mechanics
Schmidt number           = Momentum diffusivity / Mass diffusivity
Prandtl number           = Momentum diffusivity / Thermal diffusivity
Magnetic Prandtl number  = Momentum diffusivity / Magnetic diffusivity

                Prandtl   Schmidt
Air                .7       .7
Water             7
Liquid metals  << 1
Oils           >> 1


Refractive index
             Index
Vacuum         1
Air            1.000293
Water          1.333
Olive oil      1.47
Ice            1.309
Glass          1.5
Plexiglass     1.5
Cubic zirconia 2.15
Diamond        2.42

The refractive index is dimensionless.

Plasma physics
n              =  Electron density
Q              =  Proton charge
DebyeLength^2  =  Boltzmann Temperature / n / Q^2 / (4 Pi Ke)


              Electron  Temp  Debye   Mag
              density   (K)    (m)    (T)
              (m^-3)
Solar core      e32     e7    e-11    -
Tokamak         e20     e8    e-4    e1
Gas discharge   e16     e4    e-4     -
Ionosphere      e12     e3    e-3    e-5
Magnetosphere   e7      e7    e2     e-8
Solar wind      e6      e5    e1     e-9
Interstellar    e5      e4    e1     e-10
Intergalactic   e0      e6    e5      -


Art of Order of Magnitude Physics

Order of magnitude physics is a style for generating numerical estimates with a
minimum of calculation, and using units arguments to obtain formulae.

Reference material:


*  Order of Magnitude Physics at Caltech


*  Order of Magnitude Physics at Berkeley


*
"Order-of-Magnitude Physics: Understanding the Wo\ rld with Dimensional Analysis,
Educated Guesswork, and White Lies." - Peter Goldreich, Sanjoy Mahajan, and Sterl Phinney



*
"Street-Fighting Mathematics: The Art of Educated Guessing and Opportunistic Problem Solving." - Sanjoy Mahajan

Using units to derive formulae

Equations can often be derived using units.  For example, what is the
aerodynamic drag force on a moving object?  Such a formula will depend on:
V  =  The object's velocity
A  =  The object's cross sectional area
D  =  The density of the medium the objects is moving through.

Assume the formula has the form
Aerodynamic drag force  =  Dimensionless_Constant * Density^x * Cross_Section^y * Velocity^z

for some value of {x,y,z}. The values that give units of force are
Aerodynamic drag force  =  Dimensionless_Constant * Density * Cross_Section * Velocity^2

Units arguments often give the right formula up to a dimensionless constant and a more
involved derivation usually required to produce the constant.  For the aerodynamic drag
formula, the constant is 1/2.  The formula with the dimensionless constant included
can always be found on Wikipedia.

Examples of equations that can be obtained with units arguments
Aerodynamic drag force    =  1/2  Density  CrossSection  Velocity^2

Aerodynamic drag power    =  1/2  Density  CrossSection  Velocity^3

Gravitational energy      =  G  Mass1  Mass2  /  Distance

Gravitational self-energy =  3/5  G  Mass^2  /  Radius                 For a sphere of uniform density

Kinetic energy            =  1/2 Mass Velocity^2

Gas pressure              =  2/3  KineticEnergyDensity
                          =  1/3  GasDensity  ThermalSpeed^2

Sound Speed               =  [Gamma  Pressure  /  Density]^1/2         Gamma=7/5 for air
                          =  [1/3  Gamma]^1/2  ThermalSpeed

Wave speed for a string   =  [Tension  /  MassPerLength]^1/2


Unit scaling

Suppose you are estimating the maximum speed of a car.
v = Maximum speed           V = v / 55.6 meters per second         55.6 m/s = 200 kilometers per hour
a = Area                    A = a / 3 meters^2
d = Air density             D = d / 1.2 kg/m^3                     Atmospheric density = 1.2 kg/m^3
p = Engine power            P = p / 149200 Watts                   149200 Watts = 200 Horsepower

Lower case variables are in S.I. units.
Upper case variables are scaled so that they have a magnitude of ~ 1 for a typical car.

Drag formula:
p  =  1/2   d a v^3
P  =  2.07  D A V^3

With the scaled variables, values can be estimated at a glance.  For example,
a car with a 200 horsepower engine and a cross sectional area of 3 meters^2 has a maximum speed of
V = (2.07)^(-1/3) = .78                       ->  v = V * 200 kph = 157 kph

For a Formula-1 car,
P  ~   4
A  ~  2/3
V  ~  1.43           ->  v = 286 kilometers per hourm


Examples of order of magnitude analysis

*  Telescopes


*  Gravity, Pluto, and the definition of a planet


*  Gases


*  Blackbody radiation, stars, and the habitable zone


*  Hubble's Law


*  Mountains and the roundness of solar system objects


*  Ancient Greek astronomy


*  Human powered flight on Titan


*  Heating of the Earth by radioactivity


*  Solar energy


*  Tides


*  Viola strings


*  Asteroid deflection


*  Tables of numbers for order of magnitude estimation

Appendix


Prefixes
               Meters
terameter   =  10$12$
gigameter   =  10$9$
megameter   =  10$6$
kilometer   =  10$3$
cm          =  10$-2$
mm          =  10$-3$
micrometer  =  10$-6$
nanometer   =  10$-9$
picometer   =  10$-12$
femtometer  =  10$-15$



Scientific notation

Examples of scientific notation.
    1   = $100$  =  e0
   10   = $101$  =  e1
  100   = $102$  =  e2
  123   = 1.23*$102$  =  1.23e2
0.123   = 1.23*$10-1$  = 1.23e-1

The abbreviation "e" for "10^" comes from Fortran
and is standard in all programming languages.


Precision

A measurement consists of a quantity and an estimated error.  For example, you might
measure the length of a room to be
Length  =  6.35 +- .02 meters
"6.35" is the quantity and ".02" is the estimated error>

Care should be taken to use an appropriate number of digits.  For example,
Length  =  6.3     +- .02 meters                   Not enough digits in the measured quantity
Length  =  6.34    +- .02 meters                   Minimum number of digits to state the measured quantity
Length  =  6.342   +- .02 meters                   It is wise to to include an extra digit
Length  =  6.3421  +- .02 meters                   Too many digits.  The last digit is unnecessary.

The fractional error is defined as
Fractional error  =  Error  /  Measured quanitity
                  =   .02   /       6.34
                  =       .0032

Rounding:
6.3424  ->  6.342
6.3425  ->  6.342
6.3426  ->  6.343

If the last digit is even then round down, and if odd then round up.  This helps to
prevent bias in rounding.  For example:
6.3405  ->  6.340
6.3415  ->  6.342
6.3425  ->  6.342
6.3435  ->  6.344
6.3445  ->  6.344



Unit conversion
                                 1609 meters        1 hour
1 mile/hour  =  1 miles/hour  *  -----------  *  ------------  =  .447 meters/second
                                   1 mile        3600 seconds







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