Units
One-second ticks

The fundamental units are the meter, second, kilogram, and Coulomb. They were defined in 1793 as the "Standard International" (SI) units, or "MKS" units.

```
Quantity   Unit       Definition

Length     Meter      The Earth's circumference is 40 million meters
Time       Second     There are 86400 seconds in one Earth day
Mass       Kilogram   The mass of a cube of water 10 cm on a side is 1 kilogram
Charge     Coulomb    The force between two charges of one Coulomb each and
separated by 1 meter is 9 billion Newtons

```
1 kg of water
1 kg of air
```Density of water  =1000   kg/meter$3$  = 1      g/cm$3$
Density of air    =   1.2 kg/meter$3$  =  .0012 g/cm$3$
```

Derived units

The fundamental units are length, mass, time, and charge, and all other units are derived from these.

```
Quanity                        Composition            Units

Length           = X                                  meters
Mass             = M                                  kg
Time             = T                                  seconds
Charge           = C                                  Coulomb
Speed            = V = X/T     Length / Time          meters/second
Acceleration     = A = V/T     Speed  / Time          meters/second2
Momentum         = Q = M V     Mass   * Speed         kg meters/second
Force            = F = M A     Mass   * Acceleration  Newtons =  kg meters/second2
Energy           = E = F X     Force  * Distance      Joules  =  kg meters2/second2
Power            = P = E/T     Energy / Time          Watts   =  kg meters2/second3
Area             = S = X2      Length$2$                meters$2$
Volume           = Υ = X3      Length$3$                meters$3$
Density          = ρ = M/Υ     Mass   / Volume        kg / meters2
Pressure         = Φ = F/S     Force  / Area          Pascals =  Newtons/meter$2$  =  Joules/meter$3$
Angular momentum = L = M V X   Momentum*Length        kg meters$2$/second
Torque           = Γ = F X     Force  * Length        kg meters$2$/second$2$
Frequency        = f = 1/T     1      / Time          Hertz   =  1/second
```

Unit conversions
```Meter        =    3.281 feet
=   39.37  inches
Mile         = 5280     feet       (exact)
= 1609     meters
Foot         =   12     inches     (exact)
Inch         =   25.4   mm         (exact)

Minute       =   60     seconds    (exact)
Hour         =   60     minutes    (exact)
Day          =   24     hours      (exact)
Year         =  365.25  days

Ton          = 1000     kg         (exact)
Kilogram     = 1000     grams      (exact)
=    2.205 pounds     (pounds interpreted as mass)
Newton       =     .225 pounds     (pounds interpreted as force)
Pound        =   16     ounces     (exact)  (interpreted as mass)
=     .454 kg
4.448 Newtons    (Newtons interpreted as force)
Ounce        =   28.35  grams      (ounces interpreted as mass)
Meter/second =    2.24  miles/hour
Km/hour      =     .621 miles/hour
Miles/hour   =    1.609 km/hour
Pascal       = .0001450 pounds/inch2   (pounds interpreted as force)
Pound/inch2  = 6895     Pascals
Bar          =101325    Pascals    (Atmosphere pressure at sea level)
=   14.50  pounds/inch2     (pounds interpreted as force)
Earth gravity=    9.807 meters/second2
=   32.2   feet/second2
Standard sheet of paper  =  11 x 8.5 inches  =  27.94 x 21.59 cm
```

Distance

Larger Picture
```                    Meters       Earth      Earth        Light travel
(AU)
Nucleus                2⋅10$-15$
Atom                   2⋅10$-10$
Green light            5.5⋅10$-7$
Neuron                    .00002
Dime thickness            .00135
Dime diameter             .0178
Quarter diameter          .024
Tennis ball diameter      .067
Soccer ball diameter      .22
Average person           1.78
Central Park width     800
Mount Everest         8848
Moon distance     3.844⋅108         60.3         .00257      1.5 seconds
Sun radius        6.957⋅108        109           .00474      2.3 seconds
Earth orbit       1.496⋅1011     23481          1.0          8   minutes
Jupiter orbit                                  5.2         40   minutes
Neptune orbit                                 30.1          3   days
Light year        9.461⋅1015                63241            1   year
Alpha Centauri                                              4.4 years      Nearest star
Galaxy thickness                                         1000   years
Galaxy center                                           27200   years
Galaxy diameter                                        100000   years
Andromeda distance                                          2.5 million years
Virgo cluster distance                                     54   million years
Size of universe                                           14   billion years
```
A photon traveling from the Earth to the moon

Speed

Cheetah
Needletail
Concorde
F-16 Falcon

F-15 Eagle
F-35 Lightning
F-22 Raptor
SR-71 Blackbird

```            meters/second   Mach

Walk                 1.5
Running sprint      10
Cycling sprint      20
Cheetah             30             Fastest land animal
70 miles/hour       31
Baseball pitch      45             100 miles/hour
Human neuron       100
Sound at altitude  295             Speed of sound at altitude 10 km to 20 km
747 airplane       266      .9
Sound at sea level 340     1.0     At sea level and 15 degrees Celsius
F-35 Lightning     475     1.6     Stealth fighter
F-16 Falcon        590     2.0
Concorde           606     2.05
F-22 Raptor        670     2.3     Stealth fighter
F-15 Eagle         740     2.5
SR-71 Blackbird    980     3.3
Orbit speed       7800    26.4     Minimum speed to orbit the Earth
Escape speed     11200    38.0     Minimum speed to escape the Earth's gravity
Ion rocket      100000             Fastest spacecraft we can build
Fission rocket     107
Fusion rocket      107
Light            3⋅108  1020000
```
Aircraft typical fly at altitude 10 km to 20 km, where the speed of sound is 295 m/s. Mach 1 for aircraft is defined using this speed.

Mass

```                       kg         Earth      Solar
masses     masses
Electron          9.109⋅10-31
Proton            1.673⋅10-27
Neutron           1.675⋅10-27
1 ounce                    .0283
Tennis ball                .058
Soccer ball                .44
1 pound                    .454
Typical human            67
Sumo wrestler           230
Ton                    1000
Honda Civic            1200
Elephant               5000
Argentinosaurus       70000                                   Largest dinosaur
Blue whale           200000
Moon               7.35⋅1022         .0123
Mars               6.42⋅1023         .107
Earth              5.92⋅1024        1
Jupiter            1.90⋅1027      318           .00096
Sun                1.99⋅1030   330000          1.0
White dwarf max     2.9⋅1030                   1.44
Milky Way black hole7.4⋅1036                   4.2 million
Milky Way           2.5⋅1042                   1.2 trillion
Andromeda           2.5⋅1042                   1.2 trillion
M87 galaxy                                   10   trillion
Virgo galaxy cluster                       1200   trillion
```

Balls

Ball sizes are in scale with each other and court sizes are in scale with each other.
Ball sizes are magnified by 10 with respect to court sizes.
The distance from the back of the court to the ball is the characteristic distance the ball travels before losing half its speed to air drag.

```             Ball    Ball   Court   Court    Ball
diameter  Mass   length  width   density
(mm)    (g)     (m)     (m)    (g/cm3)

Ping pong      40      2.7    2.74    1.525   .081
Squash         40     24      9.75    6.4     .716
Golf           43     46                     1.10
Badminton      54      5.1   13.4     5.18    .062
Racquetball    57     40     12.22    6.10    .413
Billiards      59    163      2.84    1.42   1.52
Tennis         67     58     23.77    8.23    .368
Baseball       74.5  146                      .675   Pitcher-batter distance = 19.4 m
Hockey puck    76    163     61      26      1.44    25 mm thick
Whiffle        76     45                      .196
Football      178    420     91.44   48.76    .142
Rugby         191    435    100      70       .119
Bowling       217   7260     18.29    1.05   1.36
Soccer        220    432    105      68       .078
Basketball    239    624     28      15       .087
Cannonball    220  14000                     7.9     For an iron cannonball
```

Density

```               grams/cm2     \$/kg   Year of discovery

Magnesium         1.74         2.8     1808
Aluminum          2.70         1.7     1827
Titanium          4.51        10       1910
Zinc              7.14         2.0     1300
Manganese         7.21         2.3     1774
Iron              7.9           .3    -1200
Nickel            8.91        15       1751
Copper            8.96         6      -5000
Silver           10.49       640    Ancient
Tungsten         19.25        50       1783
Gold             19.30     43000    Ancient
Platinum         21.45     37000       1735
Osmium           22.59     12000       1803    Densest element

Air at Everest     .0004                       10 km altitude
Air at Denver      .001                        1 Mile altitude
Air at sea level   .00127
Ice                .92
Water             1.0
Rock              2.8

Earth             5.52
Moon              3.35
Mars              3.95
Europa            3.10
Ganymede          1.94
Callisto          1.83
Titan             1.88

Balsa              .12
Corkwood           .21
Cedar              .32
Pine               .37
Spruce, red        .41
Oak, red           .66
Hickory            .81
Bamboo             .85
Oak, live          .98
Ironwood          1.1
Lignum Vitae      1.26
```

Coins

```            Mass  Diameter  Height  Density  Price/kg  Copper    Nickel     Zinc    Manganese
g       mm       mm     g/cm3     \$/kg   fraction  fraction  fraction  fraction

Penny        2.5     19.05   1.52    5.77       4.0     .025               .975
Nickel       5.000   21.21   1.95    7.26      10.0     .75      .25
Dime         2.268   17.91   1.35    4.62      44.1     .9167    .0833
Quarter      5.670   24.26   1.75    6.29      44.1     .9167    .0833
Half dollar 11.340   30.61   2.15    7.90      44.1     .9167    .0833
Dollar       8.100   26.5    2.00    7.53     123.5     .885     .02       .06      .035
Dollar bill  1.0              .11     .88    1000
```
The above objects are all to scale. The dimensions of a dollar bill are 155.956 mm * 66.294 mm * .11 mm.
For a coin,
```Mass     =  M
Diameter =  D
Height   =  H
Volume   =  Vol  =  π H D2 / 4
Density  =  M / Vol
```
Gold was the densest element known until the discovery of tungsten in 1783 and was hence valuable as an uncounterfeitable currency. Silver can be counterfeited with lead because lead is more dense and cheaper than silver.

The price of the metal in a penny is

```Metal price  =  Penny mass * (Copper fraction * Copper price/kg + Zinc fraction * Zinc price/kg)
=  .0025 kg   * (     .025              6 \$/kg           .975           2 \$/kg    )
=  .0052 \$
```
For a penny made of pure copper the price of the metal is 1.5 cents. A penny made of gold, silver, or zinc has a value of:
```       Price/Mass   Price
\$/kg        \$

Zinc        2        .005
Copper      6        .015
Silver    640       1.6
Gold    43000     108
```

Frequency

```                        Frequency   Note
(Hertz)

Whale songs                 10
Human ear lower limit       20
Bass lowest note            41       E
Bass guitar lowest note     41       E
Cello lowest note           65       C
Bass singer lowest note     82       E
Viola & tenor lowest note  131       C
Violin & alto lowest note  196       G
Soprano lowest note        262       C
Violin D string            293       D
Violin A string            440       A
Violin E string            660       E
Human ear upper limit    20000
```

Temperature

```                        Kelvin   Celsius   Fahrenheit

Absolute zero             0      -273.2     -459.7
Water melting point     273.2       0         32
Room temperature        294        21         70
Human body temperature  310        37         98.6
Water boiling point     373.2     100        212

Kelvin

Absolute zero             0
Helium boiling point      4.2
Hydrogen boiling point   20.3
Triton                   38
Pluto                    44
Titania                  70
Nitrogen boiling point   77.4
Oxygen boiling point     90.2
Titan                    94
Europa                  102
Hottest superconductor  135          HgBaCaCuO
Ceres                   168
Mars                    210
Water melting point     273.15
Earth average           288
Room temperature        293
Water boiling point     373.15
Venus                   740
Wood fire              1170
Copper melting point   1358
Iron melting point     1811
Bunsen burner          1830
Tungsten melting point 3683          Highest melting point among metals
Earth's core           5650          Inner-core boundary
Sun's surface          5780
Solar core             13.6 million
Helium-4 fusion         200 million
Carbon-12 fusion        230 million
```

Pressure

```Surface area =  A
Force        =  F
Pressure     =  P  =  F / A    (Pascals or Newtons/meter2 or Joules/meter3)
```

Atmospheric pressure

```Mass of the Earth's atmosphere  =  M              =  5.15e18 kg
Surface area of the Earth       =  A              =  5.10e14 m2
Gravitational constant          =  g              =  9.8 m/s2
Pressure on Earth's surface     =  P  =  M g / A  =  101000 Pascals
=  15 pounds/inch2
=  1 Bar
```
One bar is defined as the Earth's mean atmospheric pressure at sea level
```              Height   Pressure   Density
(km)     (Bar)     (kg/m3)

Sea level         0      1.00     1.225
Denver            1.6     .82     1.05        One mile
Everest           8.8     .31      .48
Airbus A380      13.1     .16      .26
F-22 Raptor      19.8     .056      .091
SR-71 Blackbird  25.9     .022      .034
Space station   400       .000009   .000016
```

Energy and power

```Energy          =  E          Joules
Time            =  T          seconds
Power           =  P  =  E/T  Watts
Mass            =  M          kilograms
Energy/Mass     =  e  =  E/M  Joules/kilogram
Power/Mass      =  p  =  P/M  Watts/kilogram
```

Energy

Energies in MJoules = 106 Joules

```1 Watt hour                             .0036      1 Watt * 3600 seconds
1 food calorie                          .0042
Sprinting person                        .004       (80 kg moving at 10 m/s)
Battery, lithium, AAA                   .0047
Battery, lithium, AA                    .0107
Battery, lithium, A                     .046
Battery, lithium, C                     .060
Battery, iPhone                         .018       ( 5 Watt hours)
Battery, laptop                         .180       (50 Watt hours)
1 kg of Lithium battery                1.0
1 kg of TNT                            4.2
1 kg of sugar                         20     =  5000 Food Calories
1 kg of protein                       20     =  5000 Food Calories
1 kg of alcohol                       25     =  7000 Food Calories
1 kg of fat                           38     =  9000 Food Calories
1 kg of gasoline                      48     = 13000 Food Calories
Uranium fission bomb (Little boy)    7⋅107    = 16 kilotons of TNT
Plutonium fission bomb (Trinity)     8⋅107    = 20 kilotons of TNT
Uranium fission bomb (Fat man)       9⋅107    = 22 kilotons of TNT
Fusion bomb                          4⋅1010   = 10 megatons of TNT
World energy production in 1 year    6⋅1014

Energy    Mass   Velocity
(Joule)   (kg)    (m/s)

Ping pong ball        2.2    .0027     40
Squash ball          43      .024      60
Golf ball           230      .046     100
Tennis ball         104      .058      60
Baseball ball       116      .146      40
Soccer ball         778      .432      60
Bullet, 9 mm        338      .0065    323
Cannonball      1900000    14         518     220 mm diameter
Human sprint       4000    80          10
Car, freeway     540000  1200          30
```
Forms of energy:
```Distance         =  X          meters
Force            =  F          Newtons
Mass             =  M          kg
Velocity         =  V          meters/second
Gravity constant =  g  =  9.8  meters/second2
Pressure         =  P          Pascals
Volume           =  U          meters3
Mechanical energy=  Ew =  F X  Joules
Gravity energy   =  Eg =  MgX  Joules    (X = height above ground)
Kinetic energy   =  Ek =  ½MV2 Joules
Pressure energy  =  Ep =  P U  Joules
```

Power
```                         Watts

Human cell                 10-12
Laptop computer            10
Human brain                20
Incandescent Light bulb    60
Human at rest             100
1 horsepower              746
Strenuous exercise       1000
Maximum human power      1600
World power per person   2500
Tesla S Ludicrous      397000       532 horsepower
Wind turbine               1⋅106
Blue whale               2.5⋅106
Boeing 747               1.4⋅108
Hoover Dam               2.1⋅109
U.S. power consumption   3.4⋅1012
World power consumption  1.5⋅1013
Earth geologic heat      4.4⋅1013
World photosynthesis     7.5⋅1013
Earth solar power        1.7⋅1017    Total solar power falling on the Earth
```

Fuel

Methane (Natural gas)
Ethane
Propane
Butane (Lighter fluid)
Octane (gasoline)
Dodecane (Kerosene)

Palmitic acid (fat)
Ethanol (alcohol)

Glucose (sugar)
Fructose (sugar)
Galactose (sugar)
Lactose = Glucose + Galactose
Starch (sugar chain)
Leucine (amino acid)

Phosphocreatine
Nitrocellulose (smokeless powder)
TNT
HMX (plastic explosive)

Lignin (wood)
Coal

Medival-style black powder
Modern smokeless powder
Capacitor
Lithium-ion battery
Nuclear fission
Nuclear fusion
Antimatter

Black: Carbon        White: Hydrogen        Red: Oxygen

Energy/Mass and Power/Mass

The energy sources that can be used by vehicles are:

```              Energy/Mass   Power/mass   Energy/\$   Rechargeable   Charge   Maximum charging
MJoule/kg     Watt/kg     MJoule/\$                  time          cycles

Gasoline            45                   60
Battery, aluminum    4.6       130                      No
Battery, lithium      .8      1600         .010         Yes        1 hour      1000
Supercapacitor        .016    8000         .00005       Yes        Instant     Infinite
Aluminum capacitor    .010   10000         .0001        Yes        Instant     Infinite
```

Energy and power sources

```                    Energy/Mass   Power/mass  Density   Energy/\$
MJoule/kg     Watt/kg              MJoule/\$

Antimatter           90000000000
Fusion bomb            250000000                                  Max for d+t fusion
Fission bomb            83000000                                  Max for a uranium bomb
Nuclear battery, Pu-238  2265000          10             7.6      88 year half life
Nuclear battery, Sr-90    589000          10            59        29 year half life
Hydrogen ( 0 carbons)        141.8              .07
Methane  ( 1 carbon )         55.5              .42               Natural gas
Ethane   ( 2 carbons)         51.9              .54
Propane  ( 3 carbons)         50.4              .60
Butane   ( 4 carbons)         49.5              .60
Octane   ( 8 carbons)         47.8              .70
Kerosene (12 carbons)         46                .75
Diesel   (16 carbons)         46                .77
Oil      (36 carbons)         46                .8
Fat      (20 carbons)         37                .9       1.0      9 Calories/gram
Pure carbon                   32.8             2.0
Coal                          32                .8                Similar to pure carbon
Ethanol                       29                .79       .2      7 Calories/gram
Wood                          22                .6
Sugar                         17               1.54      6        4 Calories/gram
Protein                       17               1.06      2        4 Calories/gram
Plastic explosive              8.0    600000   1.91               HMX
Smokeless powder               5.2             1.23               Modern gunpowder
TNT                            4.7             1.65
Black powder                   2.6             1.7                Medieval gunpowder
Phosphocreatine                 .137                              Recharges ATP
Battery, aluminum-air          4.68      130
Battery, Li-S                  1.44      670
Battery, Li-ion                 .8      1600              .007
Battery, Li-polymer             .6      4000
Battery, Alkaline               .4
Lithium supercapacitor          .054   15000
Supercapacitor                  .016    8000              .00005
Aluminum capacitor              .010   10000              .0001
Spring                          .0003
Human                                     20
Solar cell                                77
Gasoline engine                         8000
Electric motor                          8000
Jet engine                             10000
Rocket engine                        3200000
```
The energy cost to convert water to hydrogen and oxygen is 13.16 MJ/kg. If hydrogen and oxygen are reacted to produce one kg of water, the energy produced is equivalent to a 1 kg mass moving at 5.13 km/s.

Acceleration
```                    Meters/second2

Ceres gravity            .27
Europa gravity          1.31
Titan gravity           1.35
Moon gravity            1.62
Mars gravity            3.8
Venus gravity           8.87
Earth gravity           9.8
Bugatti Veyron         15.2  0 to 100 km/h in 2.4 seconds
Red out                30    Max long-term acceleration in the direction of blood rushing to your head
Blackout               50    Max long-term acceleration while sitting
Formula-1 car          50    High-speed breaking and cornering with a downforce wing
Blackout with g suit   90    Max long-term acceleration while sitting with a g-suit
Max long-term (front) 120    Max long-term acceleration while lying on one's front
Max long-term (back)  170    Max long-term acceleration while lying on one's back
Max short-term        500    Max short-term acceleration
Bullet             310000    9x19 Parabellum handgun, average acceleration along the barrel
```

Physical constants
```Speed of light               2.9979e8   m/s
Gravitational constant       6.6738e-11 m3/kg/s2
Planck constant              6.6261e-34 J s
Earth surface gravity        9.8067     m/s
Electric force constant      8.9876e9   N m2 / C2
Magnetic constant            4 Pi e-7   N/A2
Proton mass                  1.6726e-27 kg  =  938.272 GeV
Neutron mass                 1.6749e-27 kg  =  939.565 GeV
Electron mass                9.1094e-31 kg
Electron charge              1.6022e-19 C
Atomic mass unit             1.6605e-27 kg
Bohr radius                  5.2918e-11 m           = hbar2 / (ElectronMass*ElectronCharge2*Ke)
Boltzmann constant           1.3806e-23 J/K
Gas constant                 8.3145     J/K/mole
Stefan-Boltzmann constant    5.6704e-8  Watts/m2/K4
Wein constant                2.8978e-3  m K
Mole of Carbon-12            .012       kg          Exact
Planck length                1.6162e-35 m
Planck mass                  2.1765e-8  kg
Planck time                  5.3911e-44 s
Planck charge                1.8755e-18 C
Planck temperature           1.4168e32  K
Water heat capacity          4200       J/kg/K
Steam heat capacity          2080       J/kg/K      At 100 Celsius
Ice heat capacity            2110       J/kg/K      At -10 Celsius
Air heat capacity            1004       J/kg/K
Stefan-Boltzmann             5.67e-8    Watts/meter2/Kelvin4
= (2π5/15) Boltzmann4 / SpeedOfLight2 / PlanckConstant3
Wein                         2.898e-3   Kelvin meters
Electron spin                5.2729e-35 Joule seconds  =  PlanckConstant / (4 Pi)

Pi                           3.14159
Euler number                 2.71828
```

Systems of units
```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
```

Particle energies

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

Index of variables and equations
```
Quantity                         MKS units                       CGS units     Conversion factor

Mass                             M   kg                          gram            .001
Wire length                      Z   meter                       cm               .01
Radial distance from wire        R   meter                       cm               .01
Time                             T   second                      second             1
Force                            F   Newton                      dyne          100000
Charge                           Q   Coulomb                     Franklin   3.336e-10
Velocity of a charge             V   meter/second                cm/s             .01
Speed of light                   C   2.999e8 meter/second        cm/s             100
Energy                           E   Joule                       erg              e-7
Electric current                 I   Ampere = Coulomb/s          Franklin/s 3.336e-10
Electric potential               V   Volt                        Statvolt      299.79
Electric field                   E   Volt/meter                  StatVolt/cm    29979
Magnetic field                   B   Tesla                       Gauss          10000
Inductance                       L   Henry                       s2/cm  9e-11
Electric force constant          Ke  = 8.988e9 N m2/C2            Ke = 1 dyne cm2 / Franklin2
Magnetic force constant          Km  = 2e-7 = Ke/C2               Km = 1/C2
Vacuum permittivity              ε   = 8.854e-12 F/m =1/4/π/Ke
Vacuum permeability              μ   = 4 π e-7 Vs/A/m =2 π Km
Proton charge                    Qpro = 1.602e-19 Coulomb         Qpro= 4.803e-10 Franklin
Electric field from a charge     E   = Ke Q / R2                  E  = Q / R2
Electric force on a charge       F   = Q E                        F  = Q E
Electric force between charges   F   = Ke Q Q / R2                F  = Q Q / R2
Magnetic field of moving charge  B   = Km V Q / R2                B  = (V/C) Q / R2
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 B  Z                    F  = I B z
Magnetic force between charges   F   = Km V2 Q1 Q2 / R2            F  = (V/C)2 Q Q / R2
Magnetic force between wires     F   = Km I1 I2 Z / R              F  = I1 I2 Z / R
Energy of a capacitor            E   = .5 C V2
Field energy per volume          Z   = (8 π Ke)-1 (E2 + B2/C2)      Z = .5 (E2 + B2/C2)
```

Maxwell's equations
```Speed of light                      C
Electric field                      E
Electric field, time derivative     Et
Magnetic field                      B
Magnetic field, time derivative     Bt
Charge                              Q
Charge density                      q
Current density                     J

MKS                           CGS

Ke=8.988e9                    Ke=1
Km=2e-7                       Km=2/C

∇˙E = 4 π Ke q                ∇˙E = 4 π q
∇˙B = 0                       ∇˙B = 0
∇×E = -Bt                     ∇×E = -Bt / C
∇×B = 2 π Km J + Et / C2       ∇×B = 4 π J / C + Et / C
```

Electromagnetism

Electric force

The fundamental unit of charge is the "Coulomb", and the electric force follows the same equations as the gravitational force.

Charges of the same sign repel and charges of opposite sign attract.

```Charge 1    Charge 2     Electric Force

+           +         Repel
-           -         Repel
+           -         Attract
-           +         Attract

Charge                   =  Q  (Coulombs)       1 Proton = 1.602e-19 Coulombs
Distance between charges =  R
Mass of the charges      =  M

Gravity constant         =  G  = 6.67e-11 Newton m2 / kg2
Electric constant        =  K  = 8.99e9 Newton m2 / Coulomb2

Gravity force            =  F  =  -G M1 M2 / R2  =  M2 g
Electric force           =  F  =  -K Q1 Q2 / R2  =  Q2 E

Gravity field from M1    =  g  =  G M1 / R2
Electric field from Q1   =  E  =  K Q1 / R2

Gravity voltage          =  H g               (H = Height, g = Gravitational acceleration)
Electric voltage         =  H E               (H = Distance parallel to the electric field)

Gravity energy           =  -G M1 M2 / R
Electric energy          =  -K Q1 Q2 / R
```

A charge generates an electric field. The electric field points away from positive charges and toward negative charges.

Electric current

A moving charge is an "electric current". In an electric circuit, a battery moves electrons through a wire.

```Charge            =  Q
Time              =  T
Electric current  =  I  =  Q / T   (Coulombs/second)
```
The current from a positive charge moving to the right is equivalent to that from a negative charge moving to the left.
Magnetic force

Parallel currents attract

Moving charges and currents exert forces on each other. Parallel currents attract and antiparallel currents repel.

```Charge                          =  Q
Velocity of the charges         =  V
Current                         =  I
Length of a wire                =  L
Distance between the charges    =  R
Electric force constant         =  Ke  =  8.988e9 N m2/C2
Magnetic force constant         =  Km  =  2e-7 = Ke/C2
Electric force between charges  =  Fe  =  Ke Q1 Q2 / R2
Magnetic force between charges  =  Fm  =  Km V2 Q1 Q2 / R2  =  (V2/C2) Fe
Magnetic force between currents =  Fm  =  Km I1 I2 Z / R
Magnetic force / Electric force =  V2 / C2
```
The magnetic force is always less than the electric force.
Magnetic field

A current generates a magnetic field
A magnetic field exerts a force on a current

The electric force can be interpreted as an electric field, and the magnetic force can be interpreted as a magnetic field. Both interpretations produce the same force.

```Radial distance                =  R                 (Distance perpendicular to the velocity of the charge)
Magnetic field from charge Q1  =  B  =  Km V Q1 / R2
Magnetic field from current I1 =  B  =  Km I1 / R
Magnetic force on charge Q2    =  Fm =  Q2 V B  =  Km V2 Q1 Q2 / R2
Magnetic force on current I2   =  Fm =  I2 Z B  =  Km I1 I2 Z / R
```

Right hand rule

The direction of the magnetic force on a positive charge is given by the right hand rule. The force on a negative charge is in the opposite direction (the left hand rule).

Positive and negative charge

A vertical magnetic field deflects positive charges rightward and negative charges leftward
A vertical field causes positive charges to circle clockwise and negative charges to circle counterclockwise.

We use the above symbols to depict vectors in the Z direction. The vector on the left points into the plane and the vector on the right points out of the plane.

Magnetic field generated by a magnet
Iron filings align with a magnetic field

Cross product

The direction of the force is the cross product "×" of V and B. The direction is given by the "right hand rule".

```Magnetic field              =  B
Magnetic force on a charge  =  F  =  Q V × B
Magnetic force on a current =  F  =  2e-7 I × B
```

Capacitors
```Voltage          =  V             Volts
Total energy     =  E  =  ½ C V2  Joules
Effective        =  Ee =  ¼ C V2  Joules
```
Not all of the energy in a capacitor is harnessable because the voltage diminishes as the charge diminishes, hence the effective energy is less than the total energy.
Conductivity

```White: High conductivity
Red:   Low conductivity
```

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/g/K) (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   147      22.56       .094     144       .131   2.96     2917  13000   528  1.32      .26     1.67
Rhodium     2.0    150      12.41       .161     133       .243   3.02     2237  13000   275   .95      .26     1.1
Tungsten    1.89   173      19.25       .098     137       .132   2.54     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       7.14                          .388             693      2   108   .2       .25      .41
Nickel      1.4     90.9     8.91                          .444            1728     15
Ruthenium   1.25   117      12.45                                          2607   5600
Cadmium     1.25    96.6     8.65                                           594      2    50   .078     .30      .20
Osmium      1.23    87.6    22.59                          .130            3306  12000
Indium      1.19    81.8     7.31                                           430    750    11   .004     .45      .009
Iron        1.0     80.4     7.87                          .449            1811          211   .35      .29      .49
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
Thorium      .68
Niobium      .55    53.7                                                   2750
Rhenium      .52                                           .137            3459
Uranium      .35
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 (usually 10% chromium)
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
```
For most metals,
```Electric conductivity / Thermal conductivity  ~  140  J/g/K
```

Magnetic field magnitudes
```                                     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
Field for frog levitation          16
Strongest electromagnet            32.2         without using superconductors
Strongest electromagnet            45           using superconductors
Neutron star                       1010
Magnetar neutron star              1014
```

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
```

Magnetic permeability

A ferromagnetic material amplifies a magnetic field by a factor called the "relative permeability".

```                Relative    Magnetic   Maximum    Critical
permeability  moment     frequency  temperature
(kHz)      (K)
Metglas 2714A    1000000                100               Rapidly-cooled metal
Iron              200000      2.2                 1043
Iron + nickel     100000                                  Mu-metal or permalloy
Cobalt + iron      18000
Nickel               600       .606                627
Cobalt               250      1.72                1388
Carbon steel         100
Neodymium magnet       1.05
Manganese              1.001
Air                    1.000
Superconductor         0
Dysprosium                   10.2                   88
EuO                           6.8                   69
Y3Fe5O12                      5.0                  560
MnBi                          3.52                 630
MnAs                          3.4                  318
NiO + Fe                      2.4                  858
CrO2                          2.03                 386
```

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
```

Current density

Current density
Resistor

```                  Electric quantities             |                Thermal quantities
|
Q  =  Charge                 Coulomb              |   Etherm=  Thermal energy          Joule
I  =  Current                Amperes              |   Itherm=  Thermal current         Watts
E  =  Electric field         Volts/meter          |   Etherm=  Thermal field           Kelvins/meter
C  =  Electric conductivity  Amperes/Volt/meter   |   Ctherm=  Thermal conductivity    Watts/meter/Kelvin
A  =  Area                   meter^2              |   A     =  Area                    meter^2
Z  =  Distance               meter                |   Z     =  Distance                meter^2
J  =  Current flux           Amperes/meter^2      |   Jtherm=  Thermal flux            Watts/meter^2
=  I / A                                       |         =  Ittherm / A
=  C * E                                       |         =  Ctherm * Etherm
V  =  Voltage                Volts                |   Temp  =  Temperature difference  Kelvin
=  E Z                                         |         =  Etherm Z
=  I R                                         |         =  Itherm Rtherm
R  =  Resistance             Volts/Ampere = Ohms  |   Rtherm=  Thermal resistance      Kelvins/Watt
=  Z / (A C)                                   |         =  Z / (A Ct)
H  =  Current heating        Watts/meter^3        |
=  E J                                         |
P  =  Current heating power  Watts                |
=  E J Z A                                     |
=  V I                                         |
```

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              |   Etherm=  Thermal energy          Joule
I  =  Current                Amperes              |   Itherm=  Thermal current         Watts
E  =  Electric field         Volts/meter          |   Etherm=  Thermal field           Kelvins/meter
C  =  Electric conductivity  Amperes/Volt/meter   |   Ctherm=  Thermal conductivity    Watts/meter/Kelvin
A  =  Area                   meter^2              |   A     =  Area                    meter^2
Z  =  Distance               meter                |   Z     =  Distance                meter^2
J  =  Current flux           Amperes/meter^2      |   Jtherm=  Thermal flux            Watts/meter^2
=  I / A                                       |         =  Ittherm / A
=  C * E                                       |         =  Ctherm * Etherm
V  =  Voltage                Volts                |   Temp  =  Temperature difference  Kelvin
=  E Z                                         |         =  Etherm Z
=  I R                                         |         =  Itherm Rtherm
R  =  Resistance             Volts/Ampere = Ohms  |   Rtherm=  Thermal resistance      Kelvins/Watt
=  Z / (A C)                                   |         =  Z / (A Ct)
H  =  Current heating        Watts/meter^3        |
=  E J                                         |
P  =  Current heating power  Watts                |
=  E J Z 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
```

Viscosity

Viscosity is analogous to electrical conductivity and thermal conductivity.

```Quantity                    Electricity            Thermal               Viscosity

Stuff                       Coulomb                Joule                 Momentum
Stuff/volume                Coulomb/m^3            Joule/m^3             Momentum/m^3
Flow = Stuff/time           Coulomb/second         Joule/s               Momentum/s
Potential                   Volts                  Kelvin                Momentum/m^3
Field                       Volts/meter            Kelvins/meter         Momentum/m^3/m
Flow density = Flow/m^2     Amperes/meter^2        Watts/meter^2         Momentum/s/m^2
Conductivity                Amperes/Volt/meter     Watts/meter/Kelvin    m^2/s
Resistance                  Volts/Ampere           Kelvins/Watt          s/m^3

Flow density  =  Conductivity  *  Field

Flow          =  Potential  /  Resistance

```

Kinematic and dynamic viscosity
```Fluid density          =  ρ              (kg/meter3)
Fluid velocity         =  V
Fluid momentum density =  U  =  D V
Kinematic viscosity    =  νk             (meters2 / second)
Dynamic viscosity      =  νd  =  ρ νk    (Pascal seconds)
Lagrangian time deriv. =  Dt

Dt U =  ∇⋅(νd∇U)
Dt V =  ∇⋅(νk∇V)
```

Viscosity
```               Dynamic       Kinematic  Density
viscosity     viscosity  (kg/m3)
(Pa s)      (m2/s)
Hydrogen            .00000876
Nitrogen            .0000178
Air                 .0000183  .0000150     1.22
Helium              .000019
Oxygen              .0000202
Xenon               .0000212
Acetone             .00031
Benzine             .00061
Water at   2 C      .00167
Water at  10 C      .00131    .0000010  1000
Water at  20 C      .00100              1000
Water at  30 C      .000798             1000
Water at 100 C      .000282             1000
Mercury             .00153    .00000012
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
```
1 Stokes = 1 cm2/s = 10-4 m2/s
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
```

Superconductors

```                 Critical    Critical  Type
temperature  field
(Kelvin)    (Teslas)

Magnesium-Boron2     39        55       2   MRI machines
Niobium3-Germanium   23.2      37       2   Field for thin films.  Not widely used
Magnesium-Boron2-C   34        36           Doped with 5% carbon
Niobium3-Tin         18.3      30       2   High-performance magnets.  Brittle
Niobium-Titanium     10        15       2   Cheaper than Niobium3-Tin.  Ductile
Niobium3-Aluminum

Technetium           11.2               2
Niobium               9.26       .82    2
Tantalum              4.48       .09    1
Lanthanum             6.3               1
Mercury               4.15       .04    1
Tungsten              4                 1    Not BCS
Tin                   3.72       .03    1
Indium                3.4        .028
Rhenium               2.4        .03    1
Thallium              2.4        .018
Thallium              2.39       .02    1
Aluminum              1.2        .01    1
Gallium               1.1
Protactinium          1.4
Thorium               1.4
Thallium              2.4
Molybdenum             .92
Zinc                   .85       .0054
Osmium                 .7
Zirconium              .55
Ruthenium              .5
Titanium               .4        .0056
Iridium                .1
Lutetium               .1
Hafnium                .1
Uranium                .2
Beryllium              .026
Tungsten               .015

HgBa2Ca2Cu3O8       134                 2
HgBa2Ca Cu2O6       128                 2
YBa2Cu3O7            92                 2
C60Cs2Rb             33                 2
C60Rb                28         2       2
C60K3                19.8        .013   2
C6Ca                 11.5        .95    2    Not BCS
Diamond:B            11.4       4       2    Diamond doped with boron
In2O3                 3.3       3       2
```
The critical fields for Niobium-Titanium, Niobium3-Tin, and Vanadium3-Gallium are for 4.2 Kelvin.

All superconductors are described by the BCS theory unless stated otherwise.

```         Boiling point (Kelvin)

Water      273
Ammonia    248
Freon R12  243
Freon R22  231
Propane    230
Acetylene  189
Ethane     185
Xenon      165.1
Krypton    119.7
Oxygen      90.2
Argon       87.3
Nitrogen    77.4     Threshold for cheap superconductivity
Neon        27.1
Hydrogen    20.3     Cheap MRI machines
Helium-4     4.23    High-performance magnets
Helium-3     3.19
```
The record for Niobium3-Tin is 2643 Amps/mm^2 at 12 T and 4.2 K.

Titan has a temperature of 94 Kelvin, allowing for superconducting equipment. The temperature of Mars is too high at 210 Kelvin.

Superconducting critical current

The maximum current density decreases with temperature and magentic field.

Maximum current density in kAmps/mm2 for 4.2 Kelvin (liquid helium):

```
Teslas               16    12     8      4    2

Niobium3-Tin         1.05  3
Niobium3-Aluminum           .6   1.7
Niobium-Titanium            -    1.0    2.4   3
Magnesium-Boron2-C          .06   .6    2.5   4
Magnesium-Boron2            .007  .1    1.5   3

```
Maximum current density in Amps/mm2 for 20 Kelvin (liquid hydrogen):
```
Teslas               4     2

Magnesium-Boron2-C   .4   1.5
Magnesium-Boron2     .12  1.5
```

History of superconductivity
```1898  Dewar liquefies hydrogen (20 Kelvin) using regenerative cooling and
his invention, the vacuum flask, which is now known as a "Dewar".
1908  Helium liquified by Onnes. His device reached a temperature of 1.5 K
1911  Superconductivity discovered by Onnes.  Mercury was the first superconductor
found
1935  Type 2 superconductivity discovered by Shubnikov
1953  Vanadium3-Silicon found to be superconducting, the first example of a
superconducting alloy with a 3:1 chemical ratio.  More were soon found
1954  Niobium3-Tin superconductivity discovered
1955  Yntema builds the first superconducting magnet using niobium wire, reaching
a field of .7 T at 4.2 K
1961  Niobium3-Tin found to be able to support a high current density and
magnetic field (Berlincourt & Hake). This was the first material capable of
producing a high-field superconducting magnet and paved the way for MRIs.
1962  Niobium-Titanium found to be able to support a high current density and
magnetic field.  (Berlincourt & Hake)
1965  Superconducting material found that could support a large
current density (1000 Amps/mm^2 at 8.8 Tesla)
(Kunzler, Buehler, Hsu, and Wernick)
1986  Superconductor with a high critical temperature discovered in a ceramic
(35 K) (Lanthanum Barium Copper Oxide) (Bednorz & Muller).
More ceramics are soon found to be superconducting at even higher temperatures.
1987  Nobel prize awarded to Bednorz & Muller, one year after the discovery of
high-temperature superconductivity.  Nobel prizes are rarely this fast.
```

Plasma physics
```n       =  Electron density
M       =  Electron mass
V       =  Electron thermal velocity
Q       =  Proton charge
k       =  Boltzmann constant
Temp    =  Temperature
Xdebye  =  Debye length                   (k*Temp/n/Q^2/(4 Pi Ke))^.5
Xgyro   =  Electron gyro radius           M V / Q B
Fgyro   =  Electron gyrofrequency

Electron  Temp  Debye   Magnetic
density   (K)    (m)    field (T)
(m^-3)
Solar core       e32     e7    e-11    -
ITER          1.0e20     e8    e-4     5.3
Laser fusion  6.0e32     e8            -    National Ignition Facility.  density=1000 g/cm^3
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      -

ITER ion temperature      = 8.0 keV
ITER electron temperature = 8.8 keV
ITER confinement time     = 400 seconds
```

Inertial confinement fusion
```                                 Compression  Heating     Fusion   Heating  Density    Year
laser (MJ)   laser (MJ)  energy   time     (kg/m^3)
(MJ)     (s)
NOVA                                                                          .3    1984.  LLNL
National Ignition Facility (NIF)   330           -          20                .9       2010
HiPER                                 .2        .07         30      e-11      .3       Future
```

Science fundamentals

Prefixes
```
terameter   =  Tm  =  10$12$  meters
gigameter   =  Bm  =  10$9$   meters
megameter   =  Mm  =  10$6$   meters
kilometer   =  km  =  10$3$   meters
meter       =   m  =  10$0$   meters
centimeter  =  cm  =  10$-2$  meters
millimeter  =  mm  =  10$-3$  meters
micrometer  =  μm  =  10$-6$  meters
nanometer   =  nm  =  10$-9$  meters
picometer   =  pm  =  10$-12$ meters
femtometer  =  fm  =  10$-15$ meters

1 million kg       =  1 Mkg
1 million dollars  =  1 M\$
```

Scientific notation

Examples of scientific notation.

```    1  =     100  =  e0
10  =     101  =  e1
100  =     102  =  e2
123  = 1.23⋅102  = 1.23e2
.123  = 1.23⋅10-1 = 1.23e-1

11000 * .012  =  1.1⋅104 * 1.2⋅10-2  =  1.32⋅102  =  132
```
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 measurement 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 prevents bias in rounding. For example:
```6.05  →  6.0
6.15  →  6.2
6.25  →  6.2
6.35  →  6.4
6.45  →  6.4
6.55  →  6.6
6.65  →  6.6
6.75  →  6.8
6.85  →  6.8
6.95  →  7.0
```

Unit conversion
```1 mile  =  1609 meters
1 hour  =  3600 seconds

1609 meters        1 hour
1 mile/hour  =  1 mile/hour  *  -----------  *  ------------  =  .447 meters/second
1 mile        3600 seconds
```

SI units

If you have data that is not in SI units, then the safest procedure is to convert everything to SI units do the calculation. You can't go wrong with this. For example, if a car moving at 70 mph travels for 2 hours, how far does it go?

```Speed of a car     =  V          =   70 mph    =    31.3 meters/second
Time traveled      =  T          =    2 hours  =    7200 seconds
Distance traveled  =  X  =  V T  =  140 miles  =  225360 meters
```
One first converts 70 mph and 2 hours to SI units, then apply X=VT to arrive at X=225360 meters, and then convert this to mph.

Alternatively, you can do the calculation in non-SI units but care must be taken to make sure the units are consistent.

Logarithm

The logarithm is the inverse of the exponential function.

```10-2  =   .01               log10 .01 = -2
10-1  =   .1                log10 .1  = -1
100   =  1                  log10  1  =  0
101   = 10                  log10 10  =  1
102   =100                  log10100  =  2

10log10x  =  log10 10x  =  x
```

2D tables of numbers

Many phenomena are besty understood by constructing a 2D table of numbers. For example, suppose you're wondering how to compare the alcohol content of a 6 pack of beer, a bottle of wine, and a bottle of Scotch. The following section constructs a table to show the alcohol content.

Alcohol

A typical bottle of beer has a volume of 12 ounces, is 5% alcohol, and contains
.6 ounces of alcohol. We use this amount as a reference unit and define
.6 ounces of alcohol to be one "Bond".

```Volume of the drink        =  V
Fraction of alcohol        =  F
Volume of alcohol          =  Valc  = F V
Volume of one beer         =  Vbeer = 12    ounces
Alcohol fraction of beer   =  Fbeer =   .05
Alcohol volume in one beer =  VBond =   .6  ounces
One "Bond" of alcohol              =   .6  ounces
One wine or Scotch bottle          = 25.4  ounces  =  750 ml
One ounce                          = 29.6  mL

Alcohol   Volume  Alcohol  Alcohol   \$    \$/Bond
fraction   (oz)    (oz)    (Bonds)

Beer (12 oz)      .05     12       .6       1       .67   .67   Budweiser
Wine glass        .13      4.6     .6       1      8     8.0    Napa Valley
Scotch shot       .40      1.5     .6       1      8     8.0    Laphroaig
Beer pitcher      .05     64      3.2       5.3   16     3.0    Budweiser
Beer keg          .05   1984     99.2     165.3  100      .60   Budweiser
Wine bottle       .13     25.4    3.3       5.5    3      .55   Charles Shaw
Vodka bottle      .40     25.4   10.1      16.9   15      .89   Smirnoff
Scotch bottle     .40     25.4   10.1      16.9   50     3.0    Laphroaig
Distilled ethanol .95     25.4   24.1      40.2   15      .37   Everclear
```

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.
Using units to derive formulae

Equations can often be derived using units. For example, what is the formula for kinetic energy? The variables that will be present in the formula are:

```Mass                   =  M  (kg)
Velocity               =  V  (meters/second)
Kinetic energy         =  E  (Joules  =  kg meters2/second2)
Dimensionless constant =  K  (Unitless)
```
Assume the formula as the form
```E = K Mm Vv
```
For some value of m and v. The values that gives units of energy are:
```E = K M1 V2
```
Units arguments often give the right formula up to a dimensionless constant and a more involved derivation is usually required to produce the constant. The formula with the dimensionless constant included can always be found on Wikipedia. For the kinetic energy, K=½ and E = ½ M V2.

Another example of using units to derive formulae is the aerodynamic drag force. The variables that will be present in the formula are:

```Velocity              =  V        meters/second
Cross sectional area  =  A        meters2
Density of air        =  D = 1.22 kg/meter3
Drag force            =  F        Newtons = kg meters/second2
Dimensionless constant=  K        Unitless
```
Assume the formula has the form
```F  =  K Dd Aa Vv
```
for some value of {d,a,v}. The values that give units of force are
```F  =  K D1 A1 V2
```

Examples of equations that can be obtained with units arguments
```Aerodynamic drag force    =  1/2  Density  CrossSection  Velocity2

Aerodynamic drag power    =  1/2  Density  CrossSection  Velocity3

Gravitational force       = -G  Mass1  Mass2  /  Distance2

Gravitational energy      = -G  Mass1  Mass2  /  Distance

Gravitational self-energy =  3/5  G  Mass2  /  Radius                 For a sphere of uniform density

Kinetic energy            =  1/2  Mass  Velocity2

Sound Speed               =  [Γ Pressure / Density]1/2         Γ=7/5 for air

Wave speed for a string   =  [Tension Length / Mass]1/2
```

Main page

Support the free online science textbooks project