Main site of science textbooks
Crowdfunding site for the free
online science textbooks project

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
Capacitance                      C   Farad                       cm          1.11e-12
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
Capacitance      =  C             Farads
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

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
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                       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
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

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

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
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^-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

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
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       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.

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

Main page

Support the free online science textbooks project






© Jason Maron, all rights reserved.