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Circuits

Battery

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

```Charge          =  Q           Coulombs
Voltage         =  V           Volts
Energy          =  E  =  VQ    Joules
Time            =  T           seconds
Current         =  I  =  Q/T   Amperes
Resistance      =  R  =  V/I   Ohms
Power           =  P  =  QV/T  Watts
=  IV
=  V2/R
=  I2R

Ohm's Law:  V = IR
```

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
```

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

Conductivity

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

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

Silver      6.30   429      10.49       .60      147       .235   2.47     1235    590    83   .17      .37      .024
Copper      5.96   401       8.96       .67      147       .385   3.21     1358      6   130   .21      .34      .87
Gold        4.52   318      19.30       .234     142       .129   2.49     1337  24000    78   .124     .44      .24
Aluminum    3.50   237       2.70      1.30      148       .897   2.42      933      2    70   .05      .35      .245
Beryllium   2.5    200       1.85      1.35      125      1.825   3.38     1560    850   287   .448     .032     .6
Magnesium   2.3    156       1.74      1.32      147      1.023   1.78      923      3    45   .22      .29      .26
Iridium     2.12   147      22.56       .094     144       .131   2.96     2917  13000   528  1.32      .26     1.67
Rhodium     2.0    150      12.41       .161     133       .243   3.02     2237  13000   275   .95      .26     1.1
Tungsten    1.89   173      19.25       .098     137       .132   2.54     3695     50   441  1.51      .28     2.57
Molybdenum  1.87   138      10.28       .182     136       .251            2896     24   330   .55      .31     1.5
Cobalt      1.7    100       8.90       .170               .421            1768     30   209   .76      .31      .7
Zinc        1.69   116       7.14                          .388             693      2   108   .2       .25      .41
Nickel      1.4     90.9     8.91                          .444            1728     15
Ruthenium   1.25   117      12.45                                          2607   5600
Cadmium     1.25    96.6     8.65                                           594      2    50   .078     .30      .20
Osmium      1.23    87.6    22.59                          .130            3306  12000
Indium      1.19    81.8     7.31                                           430    750    11   .004     .45      .009
Iron        1.0     80.4     7.87                          .449            1811          211   .35      .29      .49
Tin          .83    66.8                                                    505     22    47   .20      .36      .005
Chromium     .79    93.9                                   .449            2180
Platinum     .95                                           .133            2041
Tantalum     .76                                           .140            3290
Gallium      .74                                                            303
Thorium      .68
Niobium      .55    53.7                                                   2750
Rhenium      .52                                           .137            3459
Uranium      .35
Titanium     .25    21.9                                   .523            1941
Scandium     .18    15.8                                                   1814
Neodymium    .156                                                          1297
Mercury      .10     8.30                                  .140             234
Manganese    .062    7.81                                                  1519
Germanium    .00019                                                        1211

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

Siemens:    Amperes^2 Seconds^3 / kg / meters^2     =   1 Ohm^-1
```
For most metals,
```Electric conductivity / Thermal conductivity  ~  140  J/g/K
```

Magnetic field magnitudes
```                                     Teslas

Field generated by brain             10-12
Wire carrying 1 Amp                  .00002     1 cm from the wire
Earth magnetic field                 .0000305   at the equator
Neodymium magnet                    1.4
Magnetic resonance imaging machine  8
Field for frog levitation          16
Strongest electromagnet            32.2         without using superconductors
Strongest electromagnet            45           using superconductors
Neutron star                       1010
Magnetar neutron star              1014
```

Dielectric strength

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

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

Relative permittivity

Relative permittivity is the factor by which the electric field between charges is decreased relative to vacuum. Relative permittivity is dimensionless. Large permittivity is desirable for capacitors.

```             Relative permittivity
Vacuum            1                   (Exact)
Air               1.00059
Polyethylene      2.5
Sapphire         10
Concrete         4.5
Glass          ~ 6
Rubber           7
Diamond        ~ 8
Graphite       ~12
Silicon         11.7
Water (0 C)     88
Water (20 C)    80
Water (100 C)   55
TiO2         ~ 150
SrTiO3         310
BaSrTiO3       500
Ba TiO3     ~ 5000
CaCuTiO3    250000
```

Magnetic permeability

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

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

Effect of temperature on conductivity

Resistivity in 10^-9 Ohm Meters

```              293 K   300 K   500 K

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

Current density

Current density
Resistor

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

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

Continuum
```Continuum quantity       Macroscopic quantity

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

Wire gauges
```Gauge  Diameter  Continuous  10 second  1 second  32 ms    Resistance
mm      current    current    current   current
Ampere     Ampere     Ampere    Ampere   mOhm/meter

0        8.3      125        1900      16000     91000       .32
2        6.5       95        1300      10200     57000       .51
4        5.2       70         946       6400     36000       .82
6        4.1       55         668       4000     23000      1.30
12        2.0       20         235       1000      5600      5.2
18        1.02      10          83        250      1400     21.0
24         .51       3.5        29         62       348     84
30         .255       .86       10         15        86    339
36         .127       .18        4         10        22   1361
40         .080                  1          1.5       8   3441
```

Superconductors

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

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

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

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

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

```         Boiling point (Kelvin)

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

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

Superconducting critical current

The maximum current density decreases with temperature and magentic field.

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

```
Teslas               16    12     8      4    2

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

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

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

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

Commercial supercapacitors
```                Mass     Energy      E/M    Power   P/M   Price  Energy/\$    C     Voltage
kg      kJoule     kJ/kg   kWatt  kW/kg    \$    kJoule/\$  Farads   Volts

PM-5R0V105-R      .000454   .0062   13.8                    5.7   .0011       1      5.0
Maxwell BCAP0350  .060      .638    10.6     .459  7.65    16     .040      350      2.7
Adafruit          .135      .984     7.3                   20     .049      630      2.5
BMOD0006E160B02  5.2      37.1       7.1    2.08    .40  1170     .032        5.8  160
XLM-62R1137-R   15       125.3       8.4  124.2    8.3   1396     .090      130     62.1
```

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