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What are electric units

  1. Jul 24, 2014 #1
    Definition/Summary

    Electric and magnetic units have symbols which are (or begin with) a capital letter, but have names which begin with a small letter.

    The units below (except for eV) are SI units.

    dim. = dimension; M = mass; L = length; T = time; Q = charge.

    Units such as [itex]A.s^{-1}[/itex] have been written as fractions, to make easier comparison between different units, but this is generally bad practice, and is not to be copied.

    Equations

    Charge (dim. [itex]Q[/itex]):

    [tex]\text{C}\ \equiv\ \text{coulomb}[/tex]

    Current = charge/time = energy/magnetic flux (dim. [itex]Q/T[/itex]):

    [tex]\text{A}\ \equiv\ \text{amp (or ampere)}\ \equiv\ \frac{\text{C}}{\text{s}}\ \equiv\ \frac{\text{coulomb}}{\text{second}}\ \equiv\ \frac{\text{J}}{\text{Wb}}\ \equiv\ \frac{\text{joule}}{\text{weber}}[/tex]

    Magnetic flux = voltage.time = energy/current (dim. [itex]ML^2/QT[/itex]):

    [tex]\text{Wb}\ \equiv\ \text{weber}\ \equiv\ \text{V.s}\ \equiv\ \text{volt.second}\ \equiv\ \frac{\text{J.s}}{\text{C}}\ \equiv\ \frac{\text{joule.second}}{\text{coulomb}}[/tex]

    Magnetic pole-strength:

    [tex]\text{A-m}\ \equiv\ \text{amp-metre}[/tex]

    Magnetic dipole moment = pole-strength.distance = current.area:

    [tex]\text{A-m.m}\ \equiv\ \text{A.m}^2\ \equiv\ \text{amp-square metre}\ \equiv\ \frac{\text{J}}{\text{T}}\ \equiv\ \frac{\text{joule}}{\text{tesla}}[/tex]

    Magnetic intensity ([itex]\boldsymbol{H}[/itex]) and magnetisation density ([itex]\boldsymbol{M}[/itex]) = current/distance (dim. [itex]Q/LT[/itex]):

    [tex]\frac{\text{A}}{\text{m}}\ \equiv\ \frac{\text{amp-turns}}{\text{metre}}\ \equiv\ \frac{\text{amp}}{\text{metre}}\ \equiv\ \frac{\text{A-m.m}}{\text{m}^3}\ \equiv\ \frac{\text{magnetic dipole moment}}{\text{volume}}[/tex]

    Electric potential = voltage = energy/charge = emf (dim. [itex]ML^2/QT^2[/itex]):

    [tex]\text{V}\ \equiv\ \text{volt}\ \equiv\ \frac{\text{J}}{\text{C}}\ \equiv\ \frac{\text{joule}}{\text{coulomb}}\ \equiv\ \frac{\text{W.s}}{\text{C}}\ \equiv\ \frac{\text{watt.second}}{\text{coulomb}}\ \equiv\ \frac{\text{W}}{\text{A}}\ \equiv\ \frac{\text{watt}}{\text{amp}}[/tex]

    Power = voltage.current = energy/time (dim. [itex]ML^2/T^3[/itex]):

    [tex]\text{W}\ \equiv\ \text{watt}\ \equiv\ \frac{\text{J}}{\text{s}}\ \equiv\ \frac{\text{joule}}{\text{second}}\ \equiv\ \frac{\text{N.m}}{\text{s}}\ \equiv\ \frac{\text{newton.metre}}{\text{second}}\ \equiv\ \text{V.A}\ \equiv\ \text{volt.amp}\ \equiv\ \Omega\text{.A}^2\ \equiv\ \text{ohm.amp}^2[/tex]

    Energy = voltage.charge (dim. [itex]ML^2/T^2[/itex]):

    [tex]\text{J}\ \equiv\ \text{joule}\ \equiv\ \text{CV}\ \equiv\ \text{coulomb.volt}\ \equiv\ \frac{\text{eV}}{1.602\ 10^{-19}}\ \equiv\ \frac{\text{electron.volt}}{1.602\ 10^{-19}}[/tex]

    Energy density = energy/volume = work done/volume = force/area = pressure (dim. [itex]M/LT^2[/itex]):

    [tex]\text{Pa}\ \equiv\ \text{pascal}\ \equiv\ \frac{\text{J}}{\text{m}^3}\ \equiv\ \frac{\text{joule}}{\text{metre}^3}\ \equiv\ \frac{\text{N}}{\text{m}^2}\ \equiv\ \frac{\text{N}}{\text{C}}\ \frac{\text{C}}{\text{m}^2}\ \equiv\ \frac{\text{N}}{\text{A.m}}\ \frac{\text{A}}{\text{m}}\ \equiv\ \frac{\text{newton}}{\text{metre}^2}[/tex]

    Impedance ([itex]Z\ =\ R\ +\ jX[/itex]) (resistance plus [itex]j[/itex]reactance) = voltage/current = electric field per magnetic intensity ([itex]\boldsymbol{E}/\boldsymbol{H}[/itex]) = power/current-squared = inductance/time = inductance.frequency (dim. [itex]ML^2/Q^2T[/itex]):

    [tex]\Omega\ \equiv\ \text{ohm}\ \equiv\ \frac{\text{V}}{\text{A}}\ \equiv\ \frac{\text{volt}}{\text{amp}}\ \equiv\ \frac{\text{W}}{\text {A}^2}\ \equiv\ \frac{\text{watt}}{\text{amp}^2}\ \equiv\ \frac{\text{H}}{\text {s}}\ \equiv\ \frac{\text{henry}}{\text{second}}[/tex]

    Conductance = current/voltage = capacitance/time = capacitance.frequency (dim. [itex]Q^2T/ML^2[/itex]):

    [tex]S\text{ or }\mho\ \equiv\ \text{siemens}\ \equiv\ \frac{\text{A}}{\text{V}}\ \equiv\ \frac{\text{amp}}{\text{volt}}\ \equiv\ \frac{\text{F}}{\text {s}}\ \equiv\ \frac{\text{farad}}{\text{second}}[/tex]

    Inductance = magnetic flux/current = voltage.time/current = energy.time-squared/charge-squared (dim. [itex]ML^2/Q^2[/itex]):

    [tex]
    \begin{eqnarray*}
    \text{H} & \equiv & \text{henry}\ \equiv\ \frac{\text{Wb}}{\text{A}}\ \equiv\ \frac{\text{weber}}{\text{amp}}\ \equiv\ \frac{\text{V.s}}{\text{A}}\ \equiv\ \frac{\text{volt.second}}{\text{amp}}\ \equiv\ \Omega\text{.s}\ \equiv\ \text{ohm.second}\\
    & \equiv & \frac{\text{J.s}^2}{\text{C}^2}\ \equiv\ \frac{\text{joule.second}^{\,2}}{\text{coulomb}^{\,2}}\ \equiv\ \frac{\text{s}^2}{\text{F}}\ \equiv\ \frac{\text{second}^{\,2}}{\text{farad}}
    \end{eqnarray*}[/tex]

    Capacitance = charge/voltage = current.time/voltage = charge-squared/energy (dim. [itex]Q^2T^2/ML^2[/itex]):

    [tex]
    \begin{eqnarray*}
    \text{F} & \equiv & \text{farad}\ \equiv\ \frac{\text{C}}{\text{V}}\ \equiv\ \frac{\text{coulomb}}{\text{volt}}\ \equiv\ \frac{\text{C}^2}{\text{J}}\ \equiv\ \frac{\text{coulomb}^{\,2}}{\text{joule}}\ \equiv\ \frac{\text{C}^{\,2}}{\text{N.m}}\ \equiv\ \frac{\text{coulomb}^2}{\text{newton.metre}}\\
    & \equiv & \frac{\text{A.s}}{\text{V}}\ \equiv\ \frac{\text{amp.second}}{\text{volt}}\ \equiv\ \frac{\text{s}}{\Omega}\ \equiv\ \frac{\text{second}}{\text{ohm}}
    \end{eqnarray*}[/tex]

    Electric field ([itex]\boldsymbol{E}[/itex]) = force/charge = voltage/distance (dim. [itex]ML/QT^2[/itex]):

    [tex]\frac{\text{N}}{\text{C}}\ \equiv\ \frac{\text{newton}}{\text{coulomb}} \equiv\ \frac{\text{V}}{\text{m}}\ \equiv\ \frac{\text{volt}}{\text{metre}}[/tex]

    Electric displacement field ([itex]\boldsymbol{D}[/itex]) and polarisation density ([itex]\boldsymbol{P}[/itex]) = charge/area (dim. [itex]Q/L^2[/itex]):

    [tex]\frac{\text{C}}{\text{m}^2}\ \equiv\ \frac{\text{coulomb}}{\text{metre}^2}[/tex]

    Magnetic field ([itex]\boldsymbol{B}[/itex]) = force/charge.speed = magnetic flux/area = voltage.time/area = force/current.distance = mass/charge.time = mass/current.time-squared = energy.time/charge.area (dim. [itex]M/QT[/itex]):

    [tex]\begin{eqnarray*}
    \text{T} & \equiv & \text{tesla}\ \equiv\ \frac{\text{Wb}}{\text{m}^2}\ \equiv\ \frac{\text{weber}}{\text{metre}^2}\ \equiv\ \frac{\text{V.s}}{\text{m}^2}\ \equiv\ \frac{\text{volt.second}}{\text{metre}^2}\\
    & \equiv & \frac{\text{N}}{\text{A.m}}\ \equiv\ \frac{\text{newton}}{\text{amp.metre}}\ \equiv\ \frac{\text{kg}}{\text{C.s}}\ \equiv\ \frac{\text{kilogram}}{\text{coulomb.second}}\ \equiv\ \frac{\text{kg}}{\text{A.s}^2}\ \equiv\ \frac{\text{kilogram}}{\text{amp.second}^{\,2}}
    \end{eqnarray*}[/tex]

    Time (dim. [itex]T[/itex]):

    [tex]\text{s}\ \equiv\ \text{second}\ \equiv\ \frac{\text{H}}{\Omega}\ \equiv\ \frac{\text{henry}}{\text{ohm}}\ \equiv\ \Omega\text{.F}\ \equiv\ \text{ohm.farad}\ \equiv\ \text{H}^{1/2}\text{.F}^{1/2}\ \equiv\ \text{henry}^{1/2}\text{.farad}^{1/2}[/tex]

    Extended explanation

    Two ways of defining voltage:

    voltage = energy/charge = work/charge = force"dot"distance/charge = (from the Lorentz force) electric field"dot"distance, or dV = E.dr

    but also voltage = energy/charge = (energy/time)/(charge/time) = power/current, or V = W/I

    Velocity:

    Note that, dimensionally, the relationship between the electric and magnetic fields [itex]\mathbf{E}[/itex] and [itex]\mathbf{B}[/itex] is the inverse of the analogous relationship between [itex]\mathbf{D}[/itex] and [itex]\mathbf{H}[/itex] or between [itex]\mathbf{P}[/itex] and [itex]\mathbf{M}[/itex]:

    [tex]\text{velocity}\ =\ \frac{\text{electric field (E)}}{magnetic\text{ field (B)}}\ =\ \frac{magnetic\text{ intensity (H)}}{\text{electric displacement field (D)}}\ =\ \frac{magnetic\text{ density (M)}}{\text{polarisation density (P)}}[/tex]

    and so, for example, we expect to find [itex](1/c)\mathbf{E}[/itex] and [itex]\mathbf{B}[/itex] together, but [itex]c\mathbf{D}[/itex] and [itex]\mathbf{H}[/itex] together, and [itex]c\mathbf{P}[/itex] and [itex]\mathbf{M}[/itex] together.

    Time constants:

    In "RLC" AC circuits (with resistance R, inductance L and/or capacitance C), combinations with dimensions of time, such as RC, or L/R, occur as "time constants", and combinations with dimensions of 1/time, such as [itex]\sqrt{(1/LC - R^2/4L^2)}[/itex], occur as frequencies.

    Electric displacement field:

    The electric displacement field was designed specifically for parallel-plate capacitors: it is always [itex]Q/A[/itex], the charge (on either plate) divided by the area, in coulombs per square metre ([itex]C/m^2[/itex]).

    Permittivity and permeability:

    Permittivity (a tensor) = capacitance/distance = electric displacement field/electric field (dim. [itex]Q^2T^2/ML^3[/itex]):

    [tex]\frac{\text{F}}{\text{m}}\ =\ \frac{\text{farad}}{\text{metre}}[/tex]

    [tex]\mathbf{D}\ =\ \widetilde{\mathbf{\varepsilon}}\mathbf{E}\ =\ \varepsilon_0\,\mathbf{E}\ +\ \mathbf{P}\ \,\text{(or }\mathbf{E}\ =\ \mu_0\,c^2\,(\mathbf{D}\ -\ \mathbf{P})\ \text{)}[/tex]

    Permeability (a tensor) = inductance/distance = magnetic field/auxiliary magnetic field (dim. [itex]ML/Q^2[/itex]):

    [tex]\frac{\text{H}}{\text{m}}\ =\ \frac{\text{henry}}{\text{metre}}\ =\ \frac{\text{T.m}}{\text{A}}\ =\ \frac{\text{tesla.metre}}{\text{amp}}\ =\ \frac{\text{N}}{\text{A}^2}\ =\ \frac{\text{newton}}{\text{amp}^2}[/tex]

    [tex]\mathbf{H} = \widetilde{\mathbf{\mu}}^{-1}\mathbf{B}\ =\ \frac{1}{\mu_0}\,\mathbf{B}\ \,-\ \,\mathbf{M}\ \,\text{(or }\mathbf{B}\ =\ \mu_0(\mathbf{H}\ +\ \mathbf{M})\ \text{)}[/tex]

    Note that, since the magnetic analogies of [itex]{\mathbf{E}}[/itex] and [itex]{\mathbf{D}}[/itex] are [itex]{\mathbf{B}}[/itex] and [itex]{\mathbf{H}}[/itex], respectively, the magnetic analogy of permittivity is the inverse of permeability, and the magnetic analogy of [itex]\mathbf{P}[/itex] is minus [itex]\mathbf{M}[/itex].
    This is purely for historical reasons.


    Permeability times permittivity = 1/velocity-squared (dim. [itex]T^2/L^2[/itex]):

    [tex]\widetilde{\mathbf{\varepsilon}}\widetilde{\mathbf{\mu}}\ =\ \frac{1}{v^2}[/tex]

    Vacuum constants:

    Vacuum permeability is defined as exactly:

    [tex]\mu_0\ \equiv\ 4\pi\,10^{-7}\ \text{H/m}[/tex]

    which is approximately: [itex]1.26\,10^{-6}\ \text{H/m}[/itex]

    Vacuum permittivity is defined as exactly:

    [tex]\varepsilon_0\ \equiv\ \frac{1}{\mu_0\,c^2}[/tex]

    which is approximately: [itex]8.85 \, 10^{-12}\ \text{F/m}[/itex]

    (If it wasn't for that arbitrary [itex]10^{-7}[/itex] in the definition of [itex]\mu_0[/itex], then [itex]\varepsilon_0[/itex] would simply be [itex]1/4\pi c^2 F/m[/itex])​

    Characteristic impedance of vacuum ([itex]Z_0=\mu_0c[/itex]) is defined as exactly:

    [tex]Z_0\ =\ 119.9169832\pi\ \Omega[/tex]

    which is approximately: [itex]376.73\ \Omega[/itex]

    cgs units:

    The following are cgs units, and more details may be found at http://en.wikipedia.org/wiki/CGS and http://www.qsl.net/g4cnn/units/units.htm:

    esu (charge)
    biot (current)
    statvolt (electric potential)
    maxwell (magnetic flux)
    oersted (magnetic intensity, [itex]\mathbf{H}[/itex])
    gauss (magnetic field, [itex]\mathbf{B}[/itex])

    * This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!
     
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