Electric and Magnetic Units: Definition and Summary

[Greg Bernhardt]

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 $A.s^{-1}$ 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. $Q$):

$$\text{C}\ \equiv\ \text{coulomb}$$

Current = charge/time = energy/magnetic flux (dim. $Q/T$):

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

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

$$\text{Wb}\ \equiv\ \text{weber}\ \equiv\ \text{V.s}\ \equiv\ \text{volt.second}\ \equiv\ \frac{\text{J.s}}{\text{C}}\ \equiv\ \frac{\text{joule.second}}{\text{coulomb}}$$

Magnetic pole-strength:

$$\text{A-m}\ \equiv\ \text{amp-metre}$$

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

$$\text{A-m.m}\ \equiv\ \text{A.m}^2\ \equiv\ \text{amp-square metre}\ \equiv\ \frac{\text{J}}{\text{T}}\ \equiv\ \frac{\text{joule}}{\text{tesla}}$$

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

$$\frac{\text{A}}{\text{m}}\ \equiv\ \frac{\text{amp-turns}}{\text{metre}}\ \equiv\ \frac{\text{amp}}{\text{metre}}\ \equiv\ \frac{\text{A-m.m}}{\text{m}^3}\ \equiv\ \frac{\text{magnetic dipole moment}}{\text{volume}}$$

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

$$\text{V}\ \equiv\ \text{volt}\ \equiv\ \frac{\text{J}}{\text{C}}\ \equiv\ \frac{\text{joule}}{\text{coulomb}}\ \equiv\ \frac{\text{W.s}}{\text{C}}\ \equiv\ \frac{\text{watt.second}}{\text{coulomb}}\ \equiv\ \frac{\text{W}}{\text{A}}\ \equiv\ \frac{\text{watt}}{\text{amp}}$$

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

$$\text{W}\ \equiv\ \text{watt}\ \equiv\ \frac{\text{J}}{\text{s}}\ \equiv\ \frac{\text{joule}}{\text{second}}\ \equiv\ \frac{\text{N.m}}{\text{s}}\ \equiv\ \frac{\text{Newton.metre}}{\text{second}}\ \equiv\ \text{V.A}\ \equiv\ \text{volt.amp}\ \equiv\ \Omega\text{.A}^2\ \equiv\ \text{ohm.amp}^2$$

Energy = voltage.charge (dim. $ML^2/T^2$):

$$\text{J}\ \equiv\ \text{joule}\ \equiv\ \text{CV}\ \equiv\ \text{coulomb.volt}\ \equiv\ \frac{\text{eV}}{1.602\ 10^{-19}}\ \equiv\ \frac{\text{electron.volt}}{1.602\ 10^{-19}}$$

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

$$\text{Pa}\ \equiv\ \text{pascal}\ \equiv\ \frac{\text{J}}{\text{m}^3}\ \equiv\ \frac{\text{joule}}{\text{metre}^3}\ \equiv\ \frac{\text{N}}{\text{m}^2}\ \equiv\ \frac{\text{N}}{\text{C}}\ \frac{\text{C}}{\text{m}^2}\ \equiv\ \frac{\text{N}}{\text{A.m}}\ \frac{\text{A}}{\text{m}}\ \equiv\ \frac{\text{Newton}}{\text{metre}^2}$$

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

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

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

$$S\text{ or }\mho\ \equiv\ \text{siemens}\ \equiv\ \frac{\text{A}}{\text{V}}\ \equiv\ \frac{\text{amp}}{\text{volt}}\ \equiv\ \frac{\text{F}}{\text {s}}\ \equiv\ \frac{\text{farad}}{\text{second}}$$

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

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

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

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

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

$$\frac{\text{N}}{\text{C}}\ \equiv\ \frac{\text{Newton}}{\text{coulomb}} \equiv\ \frac{\text{V}}{\text{m}}\ \equiv\ \frac{\text{volt}}{\text{metre}}$$

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

$$\frac{\text{C}}{\text{m}^2}\ \equiv\ \frac{\text{coulomb}}{\text{metre}^2}$$

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

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

Time (dim. $T$):

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

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 $\mathbf{E}$ and $\mathbf{B}$ is the inverse of the analogous relationship between $\mathbf{D}$ and $\mathbf{H}$ or between $\mathbf{P}$ and $\mathbf{M}$:

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

and so, for example, we expect to find $(1/c)\mathbf{E}$ and $\mathbf{B}$ together, but $c\mathbf{D}$ and $\mathbf{H}$ together, and $c\mathbf{P}$ and $\mathbf{M}$ 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 $\sqrt{(1/LC - R^2/4L^2)}$, occur as frequencies.

Electric displacement field:

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

Permittivity and permeability:

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

$$\frac{\text{F}}{\text{m}}\ =\ \frac{\text{farad}}{\text{metre}}$$

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

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

$$\frac{\text{H}}{\text{m}}\ =\ \frac{\text{henry}}{\text{metre}}\ =\ \frac{\text{T.m}}{\text{A}}\ =\ \frac{\text{tesla.metre}}{\text{amp}}\ =\ \frac{\text{N}}{\text{A}^2}\ =\ \frac{\text{Newton}}{\text{amp}^2}$$

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

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

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

$$\widetilde{\mathbf{\varepsilon}}\widetilde{\mathbf{\mu}}\ =\ \frac{1}{v^2}$$

Vacuum constants:

Vacuum permeability is defined as exactly:

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

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

Vacuum permittivity is defined as exactly:

$$\varepsilon_0\ \equiv\ \frac{1}{\mu_0\,c^2}$$

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

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

Characteristic impedance of vacuum ($Z_0=\mu_0c$) is defined as exactly:

$$Z_0\ =\ 119.9169832\pi\ \Omega$$

which is approximately: $376.73\ \Omega$

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, $\mathbf{H}$)
gauss (magnetic field, $\mathbf{B}$)

* 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!