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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
```

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
```

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.
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
```

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
```

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