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

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

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/second^{2}Force = Mass * Acceleration Newtons = kg meters/second^{2}Energy = Force * Distance Joules = kg meters^{2}/second^{2}Power = Energy / Time Watts = kg meters^{2}/second^{3}Area = Length$2$ meters$2$ Volume = Length$3$ meters$3$ Density = Mass / Volume kg / meters^{2}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

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/inch^{2}(pounds interpreted as force) Pound/inch^{2}= 6895 Pascals Bar = 101325 Pascals (Atmosphere pressure at sea level) = 14.50 pounds/inch^{2}(pounds interpreted as force) Earth gravity= 9.807 meters/second^{2}= 32.2 feet/second^{2}Standard sheet of paper = 11 x 8.5 inches = 27.94 x 21.59 cm

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/s^{2}= 4.448 Newtons EarthGravity = 9.8 m/s^{2}Force = Mass * Acceleration PoundAsForce = PoundAsMass * EarthGravity

Metric prefixes

Scientific notation

Precision and error

Unit conversion

Meters Earth Earth Light travel radii orbits time (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 radius .273 Mars radius .532 Earth radius 6371000 1.0 Jupiter radius 10.9 Moon distance 60.3 .00257 1.5 seconds Sun radius 109 .00474 Earth orbit 1.496⋅10^{11}1.0 8 minutes Jupiter orbit 5.2 40 minutes Neptune orbit 30.1 3 days Light year 9.461⋅10^{15}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

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 10^{7}Fusion rocket 10^{7}Light 3⋅10^{8}880000

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 Bradley tank 27000 Argentinosaurus 70000 Largest dinosaur Blue whale 200000 Moon 7.35⋅10^{22}.0123 Mars 6.42⋅10^{23}.107 Earth 5.92⋅10^{24}1 Jupiter 1.90⋅10^{27}318 .00096 Sun 1.99⋅10^{30}330000 1.0 White dwarf max 2.9⋅10^{30}1.44 Milky Way black hole7.4⋅10^{36}4.2 million Milky Way 2.5⋅10^{42}1.2 trillion Andromeda 2.5⋅10^{42}1.2 trillion M87 galaxy 10 trillion Virgo galaxy cluster 1200 trillion

Meters/second^{2}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

Mass Diameter Height Density Density Copper Nickel Zinc Manganese (g) (mm) (mm) (g/cmThe above objects are all to scale. The dimensions of a dollar bill are 155.956 mm * 66.294 mm * .11 mm^{3}) ($/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

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/cmThe 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.^{2}) 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 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. Broad: Broad jump in inches Vertical: Vertical leap in inches

grams/cm^{2}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 Lead 11.3 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

Energies in MJoules = 10^{6} 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⋅10^{7}= 16 kilotons of TNT Plutonium fission bomb (Trinity) = 8⋅10^{7}= 20 kilotons of TNT Uranium fission bomb (Fat man) = 9⋅10^{7}= 22 kilotons of TNT Fusion bomb = 4⋅10^{10}= 10 megatons of TNT 1 kg of antimatter = 9⋅10^{10}= 20 megatons of TNT Krakatoa volcano, 1883 = 8⋅10^{11}World energy production in 1 year = 6⋅10^{14}Dinosaur-extinction asteroid = 5⋅10^{17}Typical energy of a supernova = 1⋅10^{38}Intense gamma ray burst = 1⋅10^{41}

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⋅10^{6}Blue whale 2.5⋅10^{6}Boeing 747 1.4⋅10^{8}Hoover Dam 2.1⋅10^{9}U.S. power consumption 3.4⋅10^{12}World power consumption 1.5⋅10^{13}Earth geologic heat 4.4⋅10^{13}World photosynthesis 7.5⋅10^{13}Hurricane 1.0⋅10^{14}Earth solar power 1.7⋅10^{17}Total solar power falling on the Earth

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 = V_{alc}= F V Volume of one beer = V_{beer}= 12 ounces Fraction of alcohol in beer = F_{beer}= .05 Volume of alcohol in one beer = V_{Bond}= .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

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 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 ocean1 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 .00036The 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 (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

Speed of light 2.9979e8 m/s Gravitational constant 6.6738e-11 m^{3}/kg/s^{2}Planck constant 6.6261e-34 J s Earth surface gravity 9.8067 m/s Electric force constant 8.9876e9 N m^{2}/ C^{2}Magnetic constant 4 Pi e-7 N/A^{2}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 = hbar^{2}/ (ElectronMass*ElectronCharge^{2}*Ke) Boltzmann constant 1.3806e-23 J/K Avogadro number 6.0221e23 particles/mole Gas constant 8.3145 J/K/mole Stefan-Boltzmann constant 5.6704e-8 Watts/m^{2}/K^{4}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/meter^{2}/Kelvin^{4}= (2π^{5}/15) Boltzmann^{4}/ SpeedOfLight^{2}/ PlanckConstant^{3}Wein 2.898e-3 Kelvin meters Electron spin 5.2729e-35 Joule seconds = PlanckConstant / (4 Pi) Pi 3.14159 Euler number 2.71828

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

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

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 currentAmpere = Coulomb/s Franklin/s 3.336e-10 Electric potentialIVolt Statvolt 299.79 Electric fieldVVolt/meter StatVolt/cm 29979 Magnetic fieldETesla Gauss 10000 CapacitanceBFarad cm 1.11e-12 InductanceCHenry sL^{2}/cm 9e-11 Electric force constant K_{e}= 8.988e9 N m^{2}/C^{2}K_{e}= 1 dyne cm^{2}/ Franklin^{2}Magnetic force constant K_{m}= 2e-7 = K_{e}/C^{2}K_{m}= 1/C^{2}Vacuum permittivity ε = 8.854e-12 F/m =1/4/π/K_{e}Vacuum permeability μ = 4 π e-7 Vs/A/m =2 π K_{m}Proton charge Q_{pro}= 1.602e-19 Coulomb Q_{pro}= 4.803e-10 Franklin Electric field from a charge= KE_{e}Q / R^{2}= Q / RE^{2}Electric force on a charge F = QF = QEElectric force between charges F = KE_{e}Q Q / R^{2}F = Q Q / R^{2}Magnetic field of moving charge= KB_{m}V Q / R^{2}= (V/C) Q / RB^{2}Magnetic field around a wire= KB_{m}/ RI= (V/C)B/ R Magnetic force on a charge F = Q VIF = (V/C) QBMagnetic force on a wire F = KB_{m}Z F =BIz Magnetic force between charges F = KB_{m}V^{2}Q_{1}Q_{2}/ R^{2}F = (V/C)^{2}Q Q / R^{2}Magnetic force between wires F = K_{m}I_{1}I_{2}Z / R F =I_{1}I_{2}Z / R Energy of a capacitor E = .5 C V^{2}Field energy per volume Z = (8 π K_{e})^{-1}(E^{2}+B^{2}/C^{2}) Z = .5 (E^{2}+B^{2}/C^{2})

Speed of light C Electric field E Electric field, time derivative E_{t}Magnetic field B Magnetic field, time derivative B_{t}Charge Q Charge density q Current density J MKS CGS K_{e}=8.988e9 K_{e}=1 K_{m}=2e-7 K_{m}=2/C ∇˙E = 4 π K_{e}q ∇˙E = 4 π q ∇˙B = 0 ∇˙B = 0 ∇×E = -B_{t}∇×E = -B_{t}/ C ∇×B = 2 π K_{m}J + E_{t}/ C^{2}∇×B = 4 π J / C + E_{t}/ C

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 m^{2}/ kg^{2}Electric constant = K = 8.99e9 Newton m^{2}/ Coulomb^{2}Gravity force = F = -G M_{1}M_{2}/ R^{2}= M_{2}g Electric force = F = -K Q_{1}Q_{2}/ R^{2}= Q_{2}E Gravity field from M_{1}= g = G M_{1}/ R^{2}Electric field from Q_{1}= E = K Q_{1}/ R^{2}Gravity voltage = H g (H = Height, g = Gravitational acceleration) Electric voltage = H E (H = Distance parallel to the electric field) Gravity energy = -G M_{1}M_{2}/ R Electric energy = -K Q_{1}Q_{2}/ R

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

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

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

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 = KThe magnetic force is always less than the electric force._{e}= 8.988e9 N m^{2}/C^{2}Magnetic force constant = K_{m}= 2e-7 = K_{e}/C^{2}Electric force between charges = F_{e}= K_{e}Q_{1}Q_{2}/ R^{2}Magnetic force between charges = F_{m}= K_{m}V^{2}Q_{1}Q_{2}/ R^{2}= (V^{2}/C^{2}) F_{e}Magnetic force between currents = F_{m}= K_{m}I_{1}I_{2}Z / R Magnetic force / Electric force = V^{2}/ C^{2}

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 Q_{1}== KB_{m}V Q_{1}/ R^{2}Magnetic field from currentI_{1}== KB_{m}I_{1}/ R Magnetic force on charge Q_{2}= F_{m}= Q_{2}V= KB_{m}V^{2}Q_{1}Q_{2}/ R^{2}Magnetic force on currentI_{2}= F_{m}=I_{2}Z= KB_{m}I_{1}I_{2}Z / R

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).

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.

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

Voltage == I R Charge = Q Energy of a charge at voltage "V" = E = QVCurrent = I Power = P = QV/ T =VIResistance =VOhm's law:R=VIPower: P =RI=VI^{2}=RV^{2}/R

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

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 WattsBattery energy is often quoted in WattHours.

1 WattHour = 1 Watt * 3600 seconds = 3600 Joules

A = Plate area Z = Plate spacing Ke = Electric force constant = 8.9876e9 N mA capacitor can be specified by two parameters:^{2}/ C^{2}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 E^{2}/ (4 π Ke) e = Energy/Volume = E / A Z = .5 E^{2}/ (4 π Ke) q = Charge/Volume = Q / A / Z C = Capacitance = Q/V = (4 Pi Ke)

*) Maximum energy density or maximum electric field

*) Voltage between the plates

The maximum electric field is equal to the max field for air times a dimensionless number characterizing the dielectric

Eair = Maximum electric field for air before electical breakdown Emax = Maximum electric field in the capacitor Rbohr= Bohr radius = Characteristic size of atoms = 5.2918e-11 m = hbar^{2}/ (ElectronMass*ElectronCharge^{2}*Ke) Ebohr= Bohr electric field = Field generated by a proton at a distance of 1 Bohr radius = 5.142e11 Volt/m Maximum energy density = .5 * 8.854e-12 Emax^{2}Emax (MVolt/m) Energy density (Joule/kg) Al electrolyte capacitor 15.0 1000 Supercapacitor 90.2 36000 Bohr limit 510000 1.2e12 Capacitor with a Bohr electric field

A solenoid is a wire wound into a coil.

N = Number of wire loops Z = Length A = Area Mu = Magnetic constant = 4 π 10Hyperphysics: Inductor^{-7}I = Current It = Current change/time F = Magnetic flux = N B A (Tesla meter^{2}) Ft = Flux change/time (Tesla meter^{2}/ second) B = Magnetic field = Mu N I / Z V = Voltage = Ft = L It = N A Bt = Mu N^{2}A It / Z L = Inductance = Ft / It = Mu N^{2}A / Z (Henrys) E = Energy = .5 L I^{2}

White: High conductivity Red: Low conductivity

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

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 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

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 Gadolinium 7.63 292 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

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 |

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

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 Field = -Gradient(Potential)

Fluid density = ρ (kg/meter^{3}) Fluid velocity = V Fluid momentum density = U = D V Kinematic viscosity = ν_{k}(meters^{2}/ second) Dynamic viscosity = ν_{d}= ρ ν_{k}(Pascal seconds) Lagrangian time deriv. = D_{t}D_{t}U = ∇⋅(ν_{d}∇U) D_{t}V = ∇⋅(ν_{k}∇V)

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 Palladium .95 71.8 1828 Tin .83 66.8 505 22 47 .20 .36 .005 Chromium .79 93.9 .449 2180 Platinum .95 .133 2041 Tantalum .76 .140 3290 Gallium .74 303 Thorium .68 Niobium .55 53.7 2750 Rhenium .52 .137 3459 Vanadium .5 30.7 2183 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^-1For most metals,

Electric conductivity / Thermal conductivity ~ 140 J/g/K

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

Dynamic Kinematic Density viscosity viscosity (kg/m1 Stokes = 1 cm^{3}) (Pa s) (m^{2}/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

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

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

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 Vanadium3-Gallium 14.2 19 2 Niobium-Titanium 10 15 2 Cheaper than Niobium3-Tin. Ductile Niobium3-Aluminum Technetium 11.2 2 Niobium 9.26 .82 2 Vanadium 5.03 1 2 Tantalum 4.48 .09 1 Lead 7.19 .08 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 Gadolinium 1.1 Protactinium 1.4 Thorium 1.4 Thallium 2.4 Molybdenum .92 Zinc .85 .0054 Osmium .7 Zirconium .55 Cadmium .52 .0028 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 2The 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 3.19The 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.

The maximum current density decreases with temperature and magentic field.

Maximum current density in kAmps/mm^{2} 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 3Maximum current density in Amps/mm

Teslas 4 2 Magnesium-Boron2-C .4 1.5 Magnesium-Boron2 .12 1.5

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

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.

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

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$

Examples of scientific notation.

1 = 10The abbreviation "e" for "10^" comes from Fortran and is standard in all programming languages.^{0}= e0 10 = 10^{1}= e1 100 = 10^{2}= e2 123 = 1.23⋅10^{2}= 1.23e2 0.123 = 1.23⋅10^{-1}= 1.23e-1 11000 ⋅ .012 = 1.1⋅10^{4}⋅ 1.2⋅10^{-2}= 1.32⋅10^{2}= 132

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 = .0032Rounding:

6.3424 -> 6.342 6.3425 -> 6.342 6.3426 -> 6.343If 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

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

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 metersOne 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.

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 = V_{alc}= F V Volume of one beer = V_{beer}= 12 ounces Fraction of alcohol in beer = F_{beer}= .05 Volume of alcohol in one beer = V_{Bond}= .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

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/secondThere 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:^{2}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

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

Exact Approximation log_{10}1 0 0 log_{10}2 .301 .3 log_{10}3 .477 .5 log_{10}4 .602 .6 log_{10}5 .700 .7 log_{10}6 .778 .8 log_{10}7 .845 .85 log_{10}8 .903 .9 log_{10}9 .954 .95 log_{10}10 1 1 log_{10}1.585 .2 log_{10}2.512 .4 log_{10}3.162 .5 log_{10}5.012 .7

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

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^zfor some value of {x,y,z}. The values that give units of force are

Aerodynamic drag force = Dimensionless_Constant * Density * Cross_Section * Velocity^2Units 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.

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

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 HorsepowerLower 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^3With 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

***** 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

© Jason Maron, all rights reserved.