Units

One-second ticks

The fundamental units are the meter, second, kilogram, and Coulomb. They were originally 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                                  meters
Mass                                    kg
Time                                    seconds
Charge                                  Coulomb
Speed            = Length / Time        meters/second
Momentum         = Mass * Speed         kg meters/second
Acceleration     = Velocity / Time      meters/second2
Force            = Mass * Acceleration  Newtons =  kg meters/second2
Energy           = Force * Distance     Joules  =  kg meters2/second2
Power            = Energy / Time        Watts   =  kg meters2/second3
Area             = Length$2$              meters$2$
Volume           = Length$3$              meters$3$
Density          = Mass / Volume        kg / meters2
Pressure         = Force / Area         Pascals =  Newtons/meter$2$  =  Joules/meter$3$
Angular momentum = Momentum * Length    kg meters$2$/second
Torque           = Force * Length       kg meters$2$/second$2$
Frequency        = 1 / Time             Hertz   =  1/second
```

Unit Conversions
```Meter        =   39.37  inches
=   1.0936 yards
=   3.281  feet
=   1/1609 miles
Mile         =   1609   meters
=   1760   yards   (exact)
Yard         =   3      feet    (exact)
=   .9144  meters
Foot         =   12     inches  (exact)
=   .3048  meters
Inch         =   25.4   mm      (exact)
Minute       =   60     seconds
Hour         =   60     minutes
Day          =   24     hours
Year         =   365.25 days
Ton          =   1000   kg         (exact)
Kilogram     =   1000   grams      (exact)
=   2.205  pounds     (pounds interpreted as mass)
Newton       =   .2248  pounds     (For Earth gravity at the surface)
(pounds interpreted as force)
Pound        =   16     ounces     (exact)  (interpreted as mass)
=   .4535  kg
4.448  Newtons    (interpreted as force)
Ounce        =   28.35  grams      (interpreted as mass)
Meter/second =   2.24   miles/hour
Km/hour      =   .6214  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
```

Pounds
```PoundAsMass  =  Pound interpreted as mass, with units of kg
=  .4535 kg
PoundAsForce =  Pound interpreted as force, with units of Newtons
=  The force exerted by .4535 kg in Earth's gravity
=  .4535 kg  *  9.8 m/s2
=  4.448 Newtons
EarthGravity =  9.8 m/s2

Force        =  Mass        * Acceleration
PoundAsForce =  PoundAsMass * EarthGravity
```

Tools

Metric prefixes
Scientific notation
Precision and error
Unit conversion
Magnitudes

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                     60.3       .00257     1.5 seconds
Earth orbit       1.496⋅1011                 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                                   2.54   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-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
747 airplane      300        .9
Sound             340       1.0
F-16 Falcon       590       2.0
Concorde          605       1.7
F-22 Raptor       670       2.0
F-15 Eagle        740       2.2
SR-71 Blackbird   980       2.9
Orbit speed      7800      22.9      Minimum speed to orbit the Earth   (Mach 23)
Ion rocket     100000                Fastest spacecraft we can build
Fission rocket    107
Fusion rocket     107
Light           3⋅108  880000
```

Mass

```                       kg         Earth    Solar
masses   masses
Electron          9.109e-31
Proton            1.673e-27
Neutron           1.675e-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
```

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
```

Coins
```            Mass  Diameter  Height  Density  Density  Copper  Nickel  Zinc  Manganese
(g)     (mm)     (mm)   (g/cm3)  (\$/kg)    frac    frac   frac    frac

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
Silver                              10.49     640
Gold                                19.30   43000
Platinum                            21.45   37000
```
The above objects are all to scale. The dimensions of a dollar bill are 155.956 mm * 66.294 mm * .11 mm
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.
```             Ball    Ball   Court   Court    Ball
diameter  Mass   length  width   density
(mm)    (g)     (m)     (m)    (g/cm2)

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 dist. = 19.4 m
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
```
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.
Football

The following data is a five-year average of results from the NFL Combine, from 2008-2013.

```                  Pounds  Reps   40    20  Broad Vertical

Wide receiver      202.3  15.4  4.55  4.25  120  35
Cornerback         193.2  15.5  4.55  4.17  121  35
Running back       213.3  20.5  4.59  4.28  117  34.5
Safety             208.9  18.1  4.62  4.24  114  34.5
Outside linebacker 238.1  22.7  4.74  4.34  117  33.5
Tight end          251.6  21.5  4.77  4.37  116  33.5
Fullback           242.6  24.1  4.80  4.39  120  33.5
Inside linebacker  241.5  22.7  4.80  4.31  115  33
Quarterback        223.1  17.8  4.87  4.34  110  31
Defensive end      266.3  25.6  4.88  4.46  113  32.5
Defensive tackle   304.8  28.3  5.13  4.66  105  29
Offensive center   303.1  27.3  5.30  4.66  100  27
Offensive tackle   314.7  25.3  5.32  4.80  101  27
Offensive guard    314    26.2  5.36  4.85   99  27

Pounds:    Weight of the player
Reps:      Bench press repetitions at 225 pounds.
40:        Time for the 40 yard dash.  Reaction time is not counted.
20:        20 yard shuttle.  5 yards to the right, 10 yards to the left, and 5 yards to the right.
Vertical:  Vertical leap in inches
```

Density
```                  grams/cm2

Air on Mars         .00002
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               2.5

Magnesium          1.7
Aluminum           2.7
Titanium           4.5
Iron               7.9
Silver            10.5
Gold              19.3
Tungsten          19.3
Osmium            22.6           Densest element

Earth              5.52
Moon               3.35
Mars               3.95
Europa             3.103
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
```

Energy

Energies in MJoules = 106 Joules

```Raise 1 kg of water by 1 Kelvin   =      .00419 =  1 Food Calorie  =  1000 calories
Sprinting person                  =      .004       (80 kg moving at 10 m/s)
Phone battery                     =      .018       (5 Watt hours)
Laptop battery                    =      .180       (50 Watt hours)
1 kg of supercapacitors           =      .04
1 kg of Lithium battery           =     1.3
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     =  6000 Food Calories
1 kg of fat                       =    38     = 10000 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
1 kg of antimatter                =   9⋅1010   = 20 megatons of TNT
Krakatoa volcano, 1883            =   8⋅1011
World energy production in 1 year =   6⋅1014
Dinosaur-extinction asteroid      =   5⋅1017
Typical energy of a supernova     =   1⋅1038
Intense gamma ray burst           =   1⋅1041
```

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
Hurricane                1.0⋅1014
Earth solar power        1.7⋅1017    Total solar power falling on the Earth
```

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
Fraction of alcohol in beer   =  Fbeer  =  .05
Volume of alcohol in one beer =  VBond  =  .6 ounces
One ounce                     =  29.6 mL
One "Bond" of alcohol         =    .6 ounces
One wine or Scotch bottle     =  25.4 ounces  =  750 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
Distilled ethanol .95     25.4   24.1      40.2   15      .37   Everclear
```

Wine regions

Temperature

```                       Kelvin   Celsius   Fahrenheit
Absolute zero            0      -273.2     -459.7
Water freezing point   273.2       0         32
Room temperature       294        21         70
Water boiling point    373.2     100        212

Kelvin  Celsius  Fahrenheit
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
H2O melting point       273.15      0       32
Earth average           288        15       59
Room temperature        293        20       68
H2O boiling point       373.15    100      212
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

```                 Pressure in bars

Air on Mars              .0063
Air at Everest           .30       10  km altitude
Air at Denver            .8        1.6 km altitude
Air at sea level        1.0        15 pounds per square inch
Air on Titan            1.46
Inside a football       1.9        15 + 13 pounds per square inch
10 meters underwater    2.0
Air on Venus           92.1
Seawolf nuclear sub    50          Maximum depth of 490 meters
11 km underwater     1100          Mariana trench, deepest part of the ocean
```
1 bar = 101300 Pascals
```              Energy density
(MegaJoules/kg)

Antimatter     90 billion
Hydrogen bomb  25,000,000    theoretical maximum yield
Hydrogen bomb  21,700,000    highest achieved yield
Uranium        20,000,000    nuclear fuel
Hydrogen              143
Natural gas            53.6
Gasoline               47
Jet fuel               43
Fat                    37
Coal                   24
Carbohydrates & sugar  17
Protein                16.8
Wood                   16
Lithium-air battery     9
TNT                     4.6
Gunpowder               3
Lithium battery         1.3
Lithium-ion battery      .72
Alkaline battery         .59
Compressed air           .5        300 atmospheres
Supercapacitor           .1
Capacitor                .00036
```
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.

Frequency

```                       Frequency
(Hertz)

Whale songs                10
Human ear lower limit      20
Bass lowest note           41
Bass guitar lowest note    41
Cello lowest note          65
Bass singer lowest note    82
Viola lowest note         131
Tenor lowest note         131
Alto lowest note          196
Soprano lowest note       262
Violin A string           440
Human ear upper limit   20000
```

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
```

Circuits

Battery

A battery moves charge upwards in voltage
A resistor dissipates energy as charges fall downwards in voltage

```Voltage                           =  V  =  I R
Charge                            =  Q
Energy of a charge at voltage "V" =  E  =  Q V
Current                           =  I
Power                             =  P  =  Q V / T  =  I V
Resistance                        =  R

Ohm's law:  V =  I R

Power:      P  =  I V  =  I2 R  =  V2 / R
```

Resistance

Superconductor
Resistor

In a superconductor, electrons move without interference.
In a resistor, electrons collide with atoms and lose energy.

```                 Resistance (Ohms)

Copper wire            .02          1 meter long and 1 mm in diameter
1 km power line        .03
AA battery             .1           Internal resistance
Light bulb          200
Human             10000
```

Batteries

Typical values for battery energies are:

```                      Energy    Energy     Time     Power
(kJoule)  (WattHour)  (hour)   (Watt)

Smartphone               28.7     8         10        .80
Tablet                   57.6    16         10       1.60
Macbook air             129      36          5       7.2
Small external battery   42      11          -        -
Large external battery  160      44          -        -
```
All lithium batteries have a voltage of 3.7 Volts. If a lithium battery delivers 1 Amp of current for 1 hour,
```Voltage of a lithium battery  =  V                    =  3.7 Volts
Energy of a lithium battery   =  E                    =  13320 Joules
Current                       =  I                    =  1 Amp
Time the battery lasts        =  T                    =  3600 seconds
Power                         =  P  =  E / T  =  V I  =  3.7 Watts
```
Battery energy is often quoted in WattHours.
```1 WattHour  =  1 Watt * 3600 seconds  =  3600 Joules
```

Capacitance
```A   =  Plate area
Z   =  Plate spacing
Ke  =  Electric force constant  =  8.9876e9 N m2 / C2
Q   =  Max charge on the plate     (Coulombs)
Emax=  Max electric field       =  4 Pi Ke Q / A
V   =  Voltage between plates   =  E Z     =  4 Pi Ke Q Z / A
En  =  Energy                   =  .5 Q V  =  .5 A Z E2 / (4 π Ke)
e   =  Energy/Volume            =  E / A Z =  .5 E2 / (4 π Ke)
q   =  Charge/Volume            =  Q / A / Z
C   =  Capacitance              =  Q/V     =  (4 Pi Ke)-1 A/Z   (Farads)
c   =  Capacitance/Volume       =  C / A / Z =  (4 Pi Ke)-1 Emax2 / V2
Eair=  Max electric field in air=  3 MVolt/meter
k   =  Dielectric factor        =  Emax / Eair

Continuum                                                 Macroscopic

Energy/Volume  =  .5 E2  / (4 Pi Ke)           <->        Energy = .5 C V2
=  .5 q V                                         =  .5 Q V
c              =  (4 Pi Ke)-1 Emax2  / V2      <->        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
=  Characteristic size of atoms
=  5.2918e-11 m
=  hbar2 / (ElectronMass*ElectronCharge2*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 Emax2

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

A solenoid is a wire wound into a coil.

```N  =  Number of wire loops
Z  =  Length
A  =  Area
Mu =  Magnetic constant  =  4 π 10-7
I  =  Current
It =  Current change/time
F  =  Magnetic flux      =  N B A        (Tesla meter2)
Ft =  Flux change/time                   (Tesla meter2 / second)
B  =  Magnetic field     =  Mu N I / Z
V  =  Voltage            =  Ft =  L It  =  N A Bt  =  Mu N2 A It / Z
L  =  Inductance         =  Ft / It  =  Mu N2 A / Z      (Henrys)
E  =  Energy             =  .5 L I2
```
Hyperphysics: Inductor
Conductivity

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

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
```

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)
```

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
```

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
```

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
```

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
```

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
```

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
```

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
```

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
0.123   = 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:
```6.3405  ->  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
```

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.

2D tables of numbers

Many phenomena are most properly 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.

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
Fraction of alcohol in beer   =  Fbeer  =  .05
Volume of alcohol in one beer =  VBond  =  .6 ounces
One ounce                     = 29.57 mL
One "Bond" of alcohol         =      .6 ounces     (Volume of alcohol in a 12 ounce beer)
One pint                      =    16   ounces
One wine or Scotch bottle     =    25.4 ounces  =  750 ml
One pitcher                   =    64   ounces
One gallon                    =   128   ounces
One keg                       =  1984   ounces  =  15.5 gallons

Alcohol   Volume  Alcohol   Alcohol
fraction   (oz)   vol (oz)  vol (Bonds)

Beer (12 oz)      .05     12       .6      1
Wine glass        .13      4.6     .6      1
Scotch shot       .40      1.5     .6      1
Beer pitcher      .05     64      3.2      5.3
Beer keg          .05   1984     99.2    165.3
Wine bottle       .13     25.4    3.3      5.5
Scotch bottle     .40     25.4   10.1     16.9
Distilled ethanol .95     25.4   24.1     40.2

\$   Bonds   \$/Bond

Bottle of Everclear       15    40.2    .37
Charles Shaw wine          3     5.5    .55
Keg of Budweiser         100   165.3    .60
24 pack of Budweiser      16    24      .67
Bottle of Smirnoff Vodka  15    16.9    .89
```

Lion Slash problem-solving style

Suppose you measure the power exerted in climbing a set of stairs.

```
Time to climb stairs    =  T          =    10  seconds
Mass of climber         =  M          =   100  kg
Gravity constant        =  g          =    10  meters/second2
Height of stairs        =  H          =    20  meters
Vertical speed          =  V  =  H/T  =     2  meters/second
Gravity energy          =  E  =  MgH  = 20000  Joules
Power                   =  P  =  E/T  =  2000  Watts
```
There is a row for each variable and there are 5 columns showing the properties of each variable. One column is a concrete numerical example. The columns are:
Description of the variable
Symbol for the variable
Units-style equation
Numerical example
Units

More examples of the Lion Slash style:

```Battery energy     =  E  =       =  20000  Joules    (Typical smartphone battery)
Battery life       =  T  =       =  14400  seconds   (While playing League of Legends)
Battery power      =  P  =  E/T  =   1.39  Watts
```

Logarithms
```            Exact     Approximation

log10 1      0          0
log10 2       .301       .3
log10 3       .477       .5
log10 4       .602       .6
log10 5       .700       .7
log10 6       .778       .8
log10 7       .845       .85
log10 8       .903       .9
log10 9       .954       .95
log1010      1          1

log10 1.585   .2
log10 2.512   .4
log10 3.162   .5
log10 5.012   .7
```

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

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